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␈↓ α,␈↓∧␈↓ β␈↓ βm␈↓&AM:␈↓)αβ  An Artificial Intelligence Approach to

␈↓ α,␈↓∧␈↓ β␈↓ β]Discovery in Mathematics as Heuristic Search










␈↓ α,␈↓¬␈↓ β␈↓ ε
A DISSERTATION
␈↓ α,␈↓¬␈↓ β␈↓ ∧#SUBMITTED TO THE COMPUTER SCIENCE DEPARTMENT
␈↓ α,␈↓¬␈↓ β␈↓ ∧YAND THE COMMITTEE ON GRADUATE STUDIES
␈↓ α,␈↓¬␈↓ β␈↓ ¬POF STANFORD UNIVERSITY
␈↓ α,␈↓¬␈↓ β␈↓ ∧CIN PARTIAL FULFILLMENT OF THE REQUIREMENTS
␈↓ α,␈↓¬␈↓ β␈↓ ¬vFOR THE DEGREE OF
␈↓ α,␈↓¬␈↓ β␈↓ ¬\DOCTOR OF PHILOSOPHY
















␈↓ α,␈↓↓␈↓ β␈↓ εb␈↓By␈↓↓

␈↓ α,␈↓↓␈↓ β␈↓ ¬↑Douglas  Bruce  Lenat

␈↓ α,␈↓↓␈↓ β␈↓ ε1July,  l976




























␈↓ α,␈↓␈↓ ε∂␈↓λc␈↓ Copyright 1976

␈↓ α,␈↓␈↓ εtby

␈↓ α,␈↓␈↓ ε	Douglas  B.  Lenat


















␈↓ α,␈↓␈↓ εr-␈↓εii␈↓-␈↓ \




␈↓ α,␈↓βI␈αcertify␈αthat␈αI␈αhave␈αread␈αthis␈αthesis␈αand␈αthat␈αin␈αmy␈αopinion␈αit␈αis␈αfully␈αadequate,␈αin␈αscope␈αand
␈↓ α,␈↓β␈↓ β≤quality, as a dissertation for the degree of Doctor of Philosophy.

␈↓ α,␈↓β␈↓ ¬|␈↓&                                                                        ␈↓ R ␈↓)αβ 
␈↓ α,␈↓β␈↓ ¬|␈↓ π≤␈↓εC. Cordell Green (Principal Adviser)␈↓β



␈↓ α,␈↓βI␈αcertify␈αthat␈αI␈αhave␈αread␈αthis␈αthesis␈αand␈αthat␈αin␈αmy␈αopinion␈αit␈αis␈αfully␈αadequate,␈αin␈αscope␈αand
␈↓ α,␈↓β␈↓ β≤quality, as a dissertation for the degree of Doctor of Philosophy.

␈↓ α,␈↓β␈↓ ¬|␈↓&                                                                        ␈↓ R ␈↓)αβ 
␈↓ α,␈↓β␈↓ ¬|␈↓ π≤␈↓εBruce G. Buchanan␈↓β



␈↓ α,␈↓βI␈αcertify␈αthat␈αI␈αhave␈αread␈αthis␈αthesis␈αand␈αthat␈αin␈αmy␈αopinion␈αit␈αis␈αfully␈αadequate,␈αin␈αscope␈αand
␈↓ α,␈↓β␈↓ β≤quality, as a dissertation for the degree of Doctor of Philosophy.

␈↓ α,␈↓β␈↓ ¬|␈↓&                                                                        ␈↓ R ␈↓)αβ 
␈↓ α,␈↓β␈↓ ¬|␈↓ π≤␈↓εEdward A. Feigenbaum␈↓β



␈↓ α,␈↓βI␈αcertify␈αthat␈αI␈αhave␈αread␈αthis␈αthesis␈αand␈αthat␈αin␈αmy␈αopinion␈αit␈αis␈αfully␈αadequate,␈αin␈αscope␈αand
␈↓ α,␈↓β␈↓ β≤quality, as a dissertation for the degree of Doctor of Philosophy.

␈↓ α,␈↓β␈↓ ¬|␈↓&                                                                        ␈↓ R ␈↓)αβ 
␈↓ α,␈↓β␈↓ ¬|␈↓ π≤␈↓εDonald E. Knuth␈↓β



␈↓ α,␈↓βI␈αcertify␈αthat␈αI␈αhave␈αread␈αthis␈αthesis␈αand␈αthat␈αin␈αmy␈αopinion␈αit␈αis␈αfully␈αadequate,␈αin␈αscope␈αand
␈↓ α,␈↓β␈↓ β≤quality, as a dissertation for the degree of Doctor of Philosophy.

␈↓ α,␈↓β␈↓ ¬|␈↓&                                                                        ␈↓ R ␈↓)αβ 
␈↓ α,␈↓β␈↓ ¬|␈↓ π≤␈↓εAllen Newell␈↓β




␈↓ α,␈↓Approved for the University Committee on Graduate Studies:

␈↓ α,␈↓␈↓ ¬|␈↓&                                                                        ␈↓ R ␈↓)αβ 
␈↓ α,␈↓␈↓ ¬|␈↓ π≤␈↓ε(Dean of Graduate Studies)␈↓


␈↓ α,␈↓␈↓ εm-␈↓εiii␈↓-␈↓ \











␈↓ α,␈↓∧␈↓ ¬g␈↓&Acknowledgments␈↓)αβ




␈↓ α,␈↓I␈αowe␈α
a␈αgreat␈αdebt␈α
of␈αthanks␈αto␈α
many␈αpeople,␈α
both␈αfor␈αthe␈α
input␈αof␈αnew␈α
ideas␈αand␈α
for␈αthe
␈↓ α,␈↓evaluation, channelling, and pruning of my own.

␈↓ α,␈↓Let␈αme␈α
begin␈αby␈α
alphabetically␈αthanking␈α
my␈αcommittee:␈α
Bruce␈αBuchanan,␈α
Ed␈αFeigenbaum,
␈↓ α,␈↓Cordell␈αGreen,␈αDon␈αKnuth,␈α
and␈αAllen␈αNewell.␈α Interacting␈αwith␈α
each␈αof␈αthem␈αhas␈α
been␈αan
␈↓ α,␈↓exciting experience, and my thesis has greatly bene≡ted from their guidance.

␈↓ α,␈↓The␈α⊃following␈α⊃individuals␈α⊃have␈α⊃each␈α⊃informally␈α⊃supplied␈α⊃some␈α⊃ideas␈α⊃or␈α⊃comments␈α⊃that
␈↓ α,␈↓appear␈α∪within␈α∀this␈α∪thesis.␈α∪They␈α∀all␈α∪have␈α∪earned␈α∀my␈α∪gratitude,␈α∪and␈α∀have␈α∪signi≡cantly
␈↓ α,␈↓improved␈αthe␈αexperence␈αyou␈αare␈αabout␈αto␈αhave,␈αthat␈αof␈αreading␈αthis␈αthesis:␈αDanny␈αBobrow,
␈↓ α,␈↓Don␈α
Cohen,␈α
Paul␈α
Coehn,␈α
Avra␈α
Cohn,␈α
Randy␈α
Davis,␈α
Bob␈α
Floyd,␈α
Carl␈α
Hewitt,␈α
Earl␈α
Sacerdoti,
␈↓ α,␈↓Richard␈αWeyrauch,␈αand␈αTerry␈αWinograd.␈α Let␈αme␈αalso␈αthank␈αSAIL,␈αSRI,␈αand␈αSUMEX␈αfor
␈↓ α,␈↓providing a sophisticated computing environment in which to work.


␈↓ α,␈↓Around␈α∞this␈α∞point␈α∞in␈α∞the␈α∞␈↓βAcknowledgements␈↓,␈α∞most␈α∞theses␈α∞have␈α∞some␈α∞sort␈α∞of␈α∞tribute␈α∞to␈α
the
␈↓ α,␈↓candidate's␈α∞wife.␈α∂Until␈α∞I␈α∞was␈α∂in␈α∞the␈α∞throes␈α∂of␈α∞this␈α∞research,␈α∂I␈α∞never␈α∞fully␈α∂appreciated␈α∞the
␈↓ α,␈↓importance␈α∪of␈α∩such␈α∪support.␈α∩ So␈α∪let␈α∪me␈α∩sincerely␈α∪acknowledge␈α∩the␈α∪indispensable␈α∪aid␈α∩I
␈↓ α,␈↓received␈α
from␈α
Merle,␈αmy␈α
wonderful␈α
wife,␈αwho␈α
put␈α
up␈α
with␈αinverted␈α
schedules␈α
and␈αwho␈α
gave
␈↓ α,␈↓me the con≡dence to tackle this problem and the enthusiasm to keep going.














␈↓ α,␈↓␈↓ εn-␈↓εiv␈↓-␈↓ \

␈↓ α,␈↓␈↓ β␈↓ ¬S␈↓∧␈↓&Table of Contents␈↓)αβ␈↓



␈↓ α,␈↓␈↓ β,␈↓↓1. ␈↓&Overview␈↓)αβ␈↓
␈↓ α,␈↓␈↓ β,      1.1. Abstract .............................................................................................................␈↓ 
# 1
␈↓ α,␈↓␈↓ β,      1.2. Five-page Summary of the Project .....................................................␈↓ 
# 2
␈↓ α,␈↓␈↓ β,           ␈↓εDetour: Analysis of a discovery␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εWhat AM does: Syntheses of discoveries␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εResults␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εMotivation␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εConclusions␈↓
␈↓ α,␈↓␈↓ β,      1.3. Ways of viewing AM as some common process ..........................␈↓ 
# 6
␈↓ α,␈↓␈↓ β,           ␈↓εAM as Hill-climbing␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAM as Heuristic Search␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAM as a Mathematician␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAM as a Book␈↓

␈↓ α,␈↓␈↓ β,␈↓↓2. ␈↓&An Example: Discovering Prime Numbers␈↓)αβ␈↓
␈↓ α,␈↓␈↓ β,      2.1. Discussion of the AM Program .........................................................␈↓ 
∀ 13
␈↓ α,␈↓␈↓ β,           ␈↓εRepresentation␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAgenda and Heuristics␈↓
␈↓ α,␈↓␈↓ β,      2.2. What to get out of ¬ and NOT get out of ¬ this example ␈↓ 
∀ 16
␈↓ α,␈↓␈↓ β,      2.3. Deciphering the Example .....................................................................␈↓ 
∀ 17
␈↓ α,␈↓␈↓ β,      2.4. The Example Itself ...................................................................................␈↓ 
∀ 19
␈↓ α,␈↓␈↓ β,      2.5. Recapping the Example .........................................................................␈↓ 
∀ 26

␈↓ α,␈↓␈↓ β,␈↓↓3. ␈↓&Control Structure␈↓)αβ␈↓
␈↓ α,␈↓␈↓ β,      3.1. AM's Search .................................................................................................␈↓ 
∀ 27
␈↓ α,␈↓␈↓ β,      3.2. Constraining AM's Search ...................................................................␈↓ 
∀ 29
␈↓ α,␈↓␈↓ β,      3.3. The Agenda .................................................................................................␈↓ 
∀ 31
␈↓ α,␈↓␈↓ β,           ␈↓εWhy an Agenda?␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εDetails of the Agenda scheme␈↓

␈↓ α,␈↓␈↓ β,␈↓↓4. ␈↓&Heuristic Rules␈↓)αβ␈↓
␈↓ α,␈↓␈↓ β,      4.1. Syntax of the Heuristics ........................................................................␈↓ 
∀ 34
␈↓ α,␈↓␈↓ β,           ␈↓εSyntax of the Left-hand Side␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εSyntax of the Right-hand Side␈↓
␈↓ α,␈↓␈↓ β,      4.2. Heuristics Suggest New Tasks ............................................................␈↓ 
∀ 37
␈↓ α,␈↓␈↓ β,           ␈↓εAn Illustration: "Fill in Generalizations of Equality"␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εThe Ratings Game␈↓
␈↓ α,␈↓␈↓ β,      4.3. Heuristics Create New Concepts .......................................................␈↓ 
∀ 41
␈↓ α,␈↓␈↓ β,           ␈↓εAn Illustration: Discovering Primes␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εThe Theory of Creating New Concepts␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAnother Illustration: Squaring a number␈↓






␈↓ α,␈↓␈↓ εr-␈↓εv␈↓-␈↓ \

␈↓ α,␈↓␈↓ β,      4.4. Heuristics Fill in Entries for a Speci≡c Facet .............................␈↓ 
∀ 46
␈↓ α,␈↓␈↓ β,           ␈↓εAn Illustration: "Fill in Examples of Set-union"␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics Propose New Conjectures␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAn Illustration: "All primes except 2 are odd"␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAnother illustration: Discovering Unique Factorization␈↓
␈↓ α,␈↓␈↓ β,      4.5. Gathering Relevant Heuristics ...........................................................␈↓ 
∀ 52
␈↓ α,␈↓␈↓ β,           ␈↓εDomain of Applicability␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εRippling␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εOrdering the Relevant Heuristics␈↓
␈↓ α,␈↓␈↓ β,      4.6. AM's Starting Heuristics .......................................................................␈↓ 
∀ 56
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics Grouped by the Knowledge They Embody␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics Grouped by How Specific They Are␈↓

␈↓ α,␈↓␈↓ β,␈↓↓5. ␈↓&AM's Concepts␈↓)αβ␈↓
␈↓ α,␈↓␈↓ β,      5.1. Motivation and Overview ....................................................................␈↓ 
∀ 59
␈↓ α,␈↓␈↓ β,           ␈↓εA Glimpse of a Typical Concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εThe main constraint: Fixed set of facets␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εBEINGs Representation of Knowledge␈↓
␈↓ α,␈↓␈↓ β,      5.2. Facets ................................................................................................................␈↓ 
∀ 63
␈↓ α,␈↓␈↓ β,           ␈↓εGeneralizations/Specializations␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εExamples/Isa's␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εIn-Domain-of/In-Range-of␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εViews␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εIntuitions␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAnalogies␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εConjec's␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εDefinitions␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAlgorithms␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εDomain/Range␈↓

␈↓ α,␈↓␈↓ β,␈↓↓6. ␈↓&Results␈↓)αβ␈↓
␈↓ α,␈↓␈↓ β,      6.1. What AM Did ............................................................................................␈↓ 
∀ 83
␈↓ α,␈↓␈↓ β,           ␈↓εLinear Task-by-task Summary of a Good Run␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εTwo-Dimensional Behavior Graph␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAM as a Computer Program␈↓
␈↓ α,␈↓␈↓ β,      6.2. Experiments with AM ............................................................................␈↓ 
∀ 94
␈↓ α,␈↓␈↓ β,           ␈↓εMust the Worth numbers be finely tuned?␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHow finely tuned is the Agenda?␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHow valuable is tacking reasons onto each task?␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εWhat if certain concepts are eliminated/added?␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εCan AM work in a new domain: Plane Geometry?␈↓

␈↓ α,␈↓␈↓ β,␈↓↓7. ␈↓&Evaluating AM␈↓)αβ␈↓








␈↓ α,␈↓␈↓ εn-␈↓εvi␈↓-␈↓ \

␈↓ α,␈↓␈↓ β,      7.1. Judging Performance ............................................................................␈↓ 
¬ 103
␈↓ α,␈↓␈↓ β,           ␈↓εAM's Ultimate Discoveries␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εThe Magnitude of AM's Progress␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εThe Quality of AM's Route␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εThe Character of the User-System Interactions␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εAM's Intuitive Powers␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εExperiments on AM␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHow to Perform Experiments on AM␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εFuture Implications of this Project␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εOpen Problems: Suggestions for Future Research␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εComparison to Other Systems␈↓
␈↓ α,␈↓␈↓ β,      7.2. Capabilities and Limitations of AM ............................................␈↓ 
¬ 118
␈↓ α,␈↓␈↓ β,           ␈↓εCurrent Abilities␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εCurrent Limitations␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εLimiting Assumptions␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εChoice of Domain␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εLimitations of the Model of Math Research␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εUltimate powers and weaknesses␈↓
␈↓ α,␈↓␈↓ β,      7.3. Final Conclusions ....................................................................................␈↓ 
¬ 126

␈↓ α,␈↓␈↓ β,␈↓↓Appendix 1. ␈↓&AM's Concepts␈↓)αβ␈↓ ...............................................................................␈↓ 
¬ 128
␈↓ α,␈↓␈↓ β,      Concepts and Heuristics as coded in LISP
␈↓ α,␈↓␈↓ β,           ␈↓εThe `Compose' Concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εThe `Osets' Concept␈↓
␈↓ α,␈↓␈↓ β,      Concepts created by AM

␈↓ α,␈↓␈↓ β,␈↓↓Appendix 2. ␈↓&AM's Heuristics␈↓)αβ␈↓ ............................................................................␈↓ 
¬ 138
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Anything
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Any-concept
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for any facet of Any-concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for the Examples facets of Any-concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for the Conjecs facet of Any-concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for the Analogies facet of Any-concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for the Genl/Spec facets of Any-concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for the View facet of Any-concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for the In-dom/ran-of facets of Any-concept␈↓
␈↓ α,␈↓␈↓ β,           ␈↓εHeuristics for the Definition facet of Any-concept␈↓
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with any Active concept
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with any Predicate
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with any Operation
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with any Composition
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with any Insertions
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with the operation Coalesce
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with the operation Canonize
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with the operation Substitute
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with the operation Restrict
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with the operation Invert
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Logical combinations
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Structures


␈↓ α,␈↓␈↓ εi-␈↓εvii␈↓-␈↓ \

␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Ordered-structures
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Unordered-structures
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Multiple-eles-structures
␈↓ α,␈↓␈↓ β,      Heuristics for dealing with Sets

␈↓ α,␈↓␈↓ β,␈↓↓Appendix 3. ␈↓&Traces of AM in Action␈↓)αβ␈↓ ..........................................................␈↓ 
¬ 187

␈↓ α,␈↓␈↓ β,␈↓↓Appendix 4. ␈↓&Bibliography␈↓)αβ␈↓ ..................................................................................␈↓ 
¬ 205











































␈↓ α,␈↓␈↓ εe-␈↓εviii␈↓-␈↓ \
␈↓ α,␈↓␈↓ ;-␈↓ε1␈↓-




␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬>␈↓∧Chapter 1.   Overview␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓








␈↓ α,␈↓β␈↓ α|Indeed,␈αyou␈αcan␈αbuild␈αa␈αmachine␈αto␈αdraw␈αdemonstrative␈αconclusions␈αfor␈αyou,␈αbut␈αI
␈↓ α,␈↓β␈↓ α|think you can never build a machine that will draw plausible inferences.

␈↓ α,␈↓¬␈↓ ε\-- Polya








␈↓ α,␈↓␈↓ ε↔␈↓∧␈↓&1.1. Abstract␈↓)αβ␈↓

␈↓ α,␈↓A␈α
program,␈α
called␈α
"AM",␈α
is␈α
described␈αwhich␈α
models␈α
one␈α
aspect␈α
of␈α
elementary␈αmathematics
␈↓ α,␈↓research:␈α∂developing␈α∞new␈α∂concepts␈α∞under␈α∂the␈α∞guidance␈α∂of␈α∞a␈α∂large␈α∞body␈α∂of␈α∂heuristic␈α∞rules.
␈↓ α,␈↓"Mathematics"␈α⊃is␈α⊃considered␈α⊃as␈α⊃a␈α⊃type␈α∩of␈α⊃intelligent␈α⊃behavior,␈α⊃not␈α⊃merely␈α⊃as␈α∩a␈α⊃≡nished
␈↓ α,␈↓product.

␈↓ α,␈↓The␈α∞local␈α
heuristics␈α∞communicate␈α
via␈α∞an␈α∞agenda␈α
mechanism,␈α∞a␈α
global␈α∞list␈α
of␈α∞tasks␈α∞for␈α
the
␈↓ α,␈↓system␈α
to␈α
perform␈α
and␈α
reasons␈α
why␈α
each␈αtask␈α
is␈α
plausible.␈α
 A␈α
single␈α
task␈α
might␈α
direct␈αAM␈α
to
␈↓ α,␈↓de≡ne␈α
a␈α
new␈α
concept,␈α
or␈α
to␈α
explore␈α
some␈α
facet␈α
of␈α
an␈α
existing␈α
concept,␈α
or␈α
to␈α
examine␈α
some
␈↓ α,␈↓empirical␈α∂data␈α∂for␈α∂regularities,␈α∂etc.␈α∂ Repeatedly,␈α∂the␈α∂program␈α∂selects␈α∂from␈α∂the␈α∂agenda␈α∂the
␈↓ α,␈↓task having the best supporting reasons, and then executes it.

␈↓ α,␈↓Each␈α∪concept␈α∩is␈α∪an␈α∪active,␈α∩structured␈α∪knowledge␈α∪module.␈α∩ A␈α∪hundred␈α∪very␈α∩incomplete
␈↓ α,␈↓modules␈α∀are␈α∀initially␈α∃provided,␈α∀each␈α∀one␈α∃corresponding␈α∀to␈α∀an␈α∃elementary␈α∀set-theoretic
␈↓ α,␈↓concept␈α∂(e.g.,␈α∞union).␈α∂ This␈α∞provides␈α∂a␈α∞de≡nite␈α∂but␈α∞immense␈α∂"space"␈α∞which␈α∂AM␈α∂begins␈α∞to
␈↓ α,␈↓explore,␈α∪guided␈α∪by␈α∪a␈α∪corpus␈α∪of␈α∪250␈α∪heuristic␈α∪rules.␈α∪ AM␈α∪extends␈α∪its␈α∪knowledge␈α∪base,
␈↓ α,␈↓ultimately␈α∂rediscovering␈α∂hundreds␈α∞of␈α∂common␈α∂concepts␈α∞(e.g.,␈α∂numbers)␈α∂and␈α∂theorems␈α∞(e.g.,
␈↓ α,␈↓unique factorization).
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε2␈↓-

␈↓ α,␈↓␈↓ ∧.␈↓∧␈↓&1.2. Five-page Summary of the Project␈↓)αβ␈↓

␈↓ α,␈↓Scientists␈αoften␈αface␈αthe␈αdi≠cult␈αtask␈αof␈αformulating␈αnontrivial␈αresearch␈αproblems␈αwhich␈αare
␈↓ α,␈↓solvable.␈α⊃ In␈α⊃any␈α⊃given␈α⊃branch␈α⊂of␈α⊃science,␈α⊃it␈α⊃is␈α⊃usually␈α⊂easier␈α⊃to␈α⊃tackle␈α⊃a␈α⊃speci≡c␈α⊂given
␈↓ α,␈↓problem␈α∪than␈α∪to␈α∪propose␈α∪interesting␈α∪yet␈α∪managable␈α∪new␈α∪questions␈α∪to␈α∀investigate.␈α∪ For
␈↓ α,␈↓example,␈αcontrast␈α␈↓βsolving␈↓␈αthe␈αMissionaries␈αand␈αCannibals␈αproblem␈αwith␈αthe␈αmore␈αill-de≡ned
␈↓ α,␈↓reasoning which led to ␈↓βinventing␈↓ it.

␈↓ α,␈↓This␈α
half␈αof␈α
the␈αbook␈α
is␈α
concerned␈αwith␈α
creative␈αtheory␈α
formation␈αin␈α
mathematics:␈α
how␈αto
␈↓ α,␈↓propose␈α~interesting␈α~new␈α~concepts␈α≠and␈α~plausible␈α~hypotheses␈α~connecting␈α≠them.␈α~ The
␈↓ α,␈↓experimental␈α∂vehicle␈α∂of␈α∂my␈α∂research␈α∂is␈α∂a␈α∂computer␈α∂program␈α∂called␈α∂␈↓↓AM␈↓␈↓	1␈↓␈α∂Initially,␈α∂AM␈α∂is
␈↓ α,␈↓given␈α∂the␈α∂de≡nitions␈α∂of␈α∂115␈α∞simple␈α∂set-theoretic␈α∂concepts␈α∂(like␈α∂"Delete",␈α∂"Equality").␈α∞ Each
␈↓ α,␈↓concept␈αis␈αrepresented␈αinternally␈αas␈αa␈αdata␈αstructure␈αwith␈αa␈αcouple␈αdozen␈αslots␈αor␈αfacets␈α(like
␈↓ α,␈↓"De≡nition",␈α∂"Examples",␈α∂"Worth").␈α∂ Initially,␈α∂most␈α∂facets␈α∂of␈α∂most␈α∂concepts␈α∂are␈α∂blank,␈α∂and
␈↓ α,␈↓AM␈αuses␈αa␈αcollection␈αof␈α
250␈αheuristics␈α¬␈αplausible␈αrules␈αof␈α
thumb␈α¬␈αfor␈αguidance,␈αas␈αit␈α
tries
␈↓ α,␈↓to␈α
≡ll␈α
in␈αthose␈α
blanks.␈α
Some␈αheuristics␈α
are␈α
used␈αto␈α
select␈α
which␈αspeci≡c␈α
facet␈α
of␈αwhich␈α
speci≡c
␈↓ α,␈↓concept␈αto␈αexplore␈αnext,␈αwhile␈αothers␈α
are␈αused␈αto␈αactually␈α≡nd␈αsome␈αappropriate␈α
information
␈↓ α,␈↓about␈α⊂the␈α⊂chosen␈α⊂facet.␈α⊂ Other␈α⊂rules␈α⊂prompt␈α⊂AM␈α⊂to␈α⊂notice␈α⊂simple␈α⊂relationships␈α∂between
␈↓ α,␈↓known␈α∩concepts,␈α∪to␈α∩de≡ne␈α∩promising␈α∪new␈α∩concepts␈α∩to␈α∪investigate,␈α∩and␈α∩to␈α∪estimate␈α∩how
␈↓ α,␈↓interesting each concept is.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.2.1. Detour: Analysis of a discovery␈↓)αβ␈↓

␈↓ α,␈↓Before␈αdiscussing␈αhow␈αto␈α␈↓βsynthesize␈↓␈αa␈αnew␈αtheory,␈αconsider␈αbrie∨y␈αhow␈αto␈α␈↓βanalyze␈↓␈αone,␈αhow
␈↓ α,␈↓to␈αconstruct␈αa␈αplausible␈αchain␈αof␈αreasoning␈αwhich␈αterminates␈αin␈αa␈αgiven␈αdiscovery.␈α One␈α
can
␈↓ α,␈↓do␈αthis␈αby␈αworking␈αbackwards,␈αby␈αreducing␈αthe␈αcreative␈αact␈αto␈αsimpler␈αand␈αsimpler␈αcreative
␈↓ α,␈↓acts.␈α For␈αexample,␈αconsider␈αthe␈α
concept␈αof␈αprime␈αnumbers.␈α How␈α
might␈αone␈αbe␈αled␈αto␈α
de≡ne
␈↓ α,␈↓such a notion?  Notice the following plausible strategy:

␈↓ α,␈↓¬␈↓ β<"If␈αf␈αis␈αa␈αfunction␈αwhich␈αtransforms␈αelements␈αof␈αA␈αinto␈αelements␈αof␈αB,␈αand
␈↓ α,␈↓¬␈↓ β<B␈α↔is␈α↔ordered,␈α↔then␈α↔consider␈α↔just␈α↔those␈α↔members␈α↔of␈α↔A␈α_which␈α↔are
␈↓ α,␈↓¬␈↓ β<transformed␈αinto␈α
␈↓βextremal␈↓¬␈αelements␈αof␈α
B.␈α This␈αset␈α
is␈αan␈αinteresting␈α
subset
␈↓ α,␈↓¬␈↓ β<of A."

␈↓ α,␈↓When␈α⊂f(x)␈α∂means␈α⊂"divisors␈α∂of␈α⊂x",␈α∂and␈α⊂the␈α∂ordering␈α⊂is␈α∂"by␈α⊂length",␈α∂this␈α⊂heuristic␈α⊂says␈α∂to
␈↓ α,␈↓consider␈αthose␈α
numbers␈αwhich␈αhave␈α
a␈αminimal␈↓	2␈↓␈α
number␈αof␈αfactors␈α
¬␈αthat␈α
is,␈αthe␈αprimes.␈α
 So
␈↓ α,␈↓this␈α∞rule␈α∞actually␈α∞␈↓βreduces␈↓␈α∞our␈α∞task␈α∞from␈α∞"proposing␈α∞the␈α∞concept␈α∞of␈α∞prime␈α∞numbers"␈α∞to␈α
the
␈↓ α,␈↓more elementary problems of "discovering ordering-by-length" and "inventing divisors-of".

␈↓ α,␈↓But␈αsuppose␈αwe␈αknow␈αthis␈αgeneral␈αrule:␈α␈↓¬"If␈αf␈αis␈αan␈αinteresting␈αfunction,␈αconsider␈αits␈αinverse."␈↓␈αIt



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε The original meaning of this mnemonic has been abandoned.  As Exodus states: I ␈↓&AM␈↓)αβ that I ␈↓&AM␈↓)αβ.
␈↓ α,␈↓ε␈↓	2␈↓ε␈αλThe␈αλother␈αλextreme,␈αλnumbers␈αλwith␈αλa␈αλMAXIMAL␈αλnumber␈αλof␈αλfactors,␈αλwas␈αλalso␈αλproposed␈αλby␈αλAM␈αλas␈αλworth␈αλinvestigating.␈α	 This␈αλled
␈↓ α,␈↓ε␈↓ βLAM to many interesting questions.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε3␈↓-

␈↓ α,␈↓reduces␈αthe␈α
task␈αof␈α
discovering␈αdivisors-of␈α
to␈αthe␈α
simpler␈αtask␈α
of␈αdiscovering␈α
multiplication␈↓	3␈↓.
␈↓ α,␈↓Eventually,␈α∂this␈α∞task␈α∂reduces␈α∂to␈α∞the␈α∂discovery␈α∞of␈α∂very␈α∂basic␈α∞notions,␈α∂like␈α∂substitution,␈α∞set-
␈↓ α,␈↓union,␈α∀and␈α∪equality.␈α∀ To␈α∪explain␈α∀how␈α∪a␈α∀given␈α∪researcher␈α∀might␈α∪have␈α∀made␈α∀a␈α∪given
␈↓ α,␈↓discovery,␈αsuch␈αan␈αanalysis␈αis␈αcontinued␈αuntil␈αthat␈αinductive␈αtask␈αis␈αreduced␈αto␈α
"discovering"
␈↓ α,␈↓notions which the researcher already knew, which were his conceptual primitives.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.2.2. What AM does: Syntheses of discoveries␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|This␈α
leads␈α
to␈α
the␈α∞paradox␈α
that␈α
the␈α
more␈α
original␈α∞a␈α
discovery␈α
the␈α
more␈α∞obvious␈α
it
␈↓ α,␈↓β␈↓ α|seems␈α
afterwards.␈α∞ The␈α
creative␈α
act␈α∞is␈α
not␈α
an␈α∞act␈α
of␈α
creation␈α∞in␈α
the␈α
sense␈α∞of␈α
the
␈↓ α,␈↓β␈↓ α|Old␈α
Testament.␈α
 It␈α
does␈α
not␈α
create␈αsomething␈α
out␈α
of␈α
nothing;␈α
it␈α
uncovers,␈αselects,
␈↓ α,␈↓β␈↓ α|re-shu∞es,␈αcombines,␈αsynthesizes␈αalready␈αexisting␈αfacts,␈αfaculties,␈αskills.␈α The␈αmore
␈↓ α,␈↓β␈↓ α|familiar the parts, the more striking the new whole.

␈↓ α,␈↓¬␈↓ ε\-- Koestler



␈↓ α,␈↓Suppose␈αa␈α
large␈αcollection␈αof␈α
these␈αheuristic␈α
strategies␈αhas␈αbeen␈α
assembled␈α(e.g.,␈αby␈α
analyzing
␈↓ α,␈↓a␈α∀great␈α∪many␈α∀discoveries,␈α∀and␈α∪writing␈α∀down␈α∀new␈α∪heuristic␈α∀rules␈α∀whenever␈α∪necessary).
␈↓ α,␈↓Instead␈α
of␈α∞using␈α
them␈α
to␈α∞␈↓βexplain␈↓␈α
how␈α
a␈α∞given␈α
idea␈α
might␈α∞have␈α
evolved,␈α
one␈α∞can␈α
imagine
␈↓ α,␈↓starting␈α∩from␈α∩a␈α∩basic␈α∩core␈α∩of␈α⊃knowledge␈α∩and␈α∩"running"␈α∩the␈α∩heuristics␈α∩to␈α∩␈↓βgenerate␈↓␈α⊃new
␈↓ α,␈↓concepts.␈α We're␈αtalking␈α
about␈αreversing␈αthe␈αprocess␈α
described␈αin␈αthe␈α
last␈αsection:␈αnot␈αhow␈α
to
␈↓ α,␈↓␈↓βexplain␈↓ discoveries, but how to ␈↓βmake␈↓ them.

␈↓ α,␈↓Such␈α∂syntheses␈α∂are␈α∂precisely␈α∂what␈α∂AM␈α∂does.␈α∂ The␈α∂program␈α∂consists␈α∂of␈α∂a␈α∂large␈α∂corpus␈α∞of
␈↓ α,␈↓primitive␈αmathematical␈αconcepts,␈αeach␈αwith␈αa␈αfew␈αassociated␈αheuristics␈↓	4␈↓.␈α AM's␈αactivities␈αall
␈↓ α,␈↓serve␈α
to␈αexpand␈α
AM␈αitself,␈α
to␈α
enlarge␈αupon␈α
a␈αgiven␈α
body␈αof␈α
mathematical␈α
knowledge.␈α To
␈↓ α,␈↓cope␈α
with␈α
the␈α
enormity␈α
of␈α
the␈α∞potential␈α
"search␈α
space"␈α
involved,␈α
AM␈α
uses␈α
its␈α∞heuristics␈α
as
␈↓ α,␈↓judgmental␈α
criteria␈αto␈α
guide␈αdevelopment␈α
in␈α
the␈αmost␈α
promising␈αdirection.␈α
 It␈α
appears␈αthat
␈↓ α,␈↓the␈α∩process␈α∩of␈α∩inventing␈α∩worthwhile␈α∩new␈↓	5␈↓␈α∩concepts␈α∩can␈α∩be␈α∩guided␈α∩successfully␈α∩using␈α⊃a
␈↓ α,␈↓collection of a few hundred such heuristics.

␈↓ α,␈↓Each␈α
concept␈α
is␈α
represented␈α
as␈α
a␈α
frame-like␈α
data␈α
structure␈α
with␈α
25␈α
di≥erent␈α
facets␈α
or␈αslots.
␈↓ α,␈↓The␈α⊂types␈α⊂of␈α⊂facets␈α⊂include:␈α⊂␈↓¬Examples,␈α⊂Definitions,␈α⊂Generalizations,␈α⊃Domain/Range,␈α⊂Analogies,
␈↓ α,␈↓¬Interestingness,␈↓␈αand␈α
many␈αothers.␈α Modular␈α
representation␈αof␈α
concepts␈αprovides␈αa␈α
convenient
␈↓ α,␈↓scheme␈α∀for␈α∪organizing␈α∀the␈α∪heuristics;␈α∀for␈α∪example,␈α∀the␈α∪following␈α∀strategy␈α∪≡ts␈α∀into␈α∪the
␈↓ α,␈↓␈↓βExamples␈↓␈αfacet␈αof␈αthe␈α␈↓βPredicate␈↓␈αconcept:␈α␈↓¬"If,␈αempirically,␈α10␈αtimes␈αas␈αmany␈αelements␈α␈↓βfail␈↓¬␈αsome
␈↓ α,␈↓¬predicate␈α⊂P,␈α⊂as␈α⊂␈↓βsatisfy␈↓¬␈α∂it,␈α⊂then␈α⊂some␈α⊂␈↓βgeneralization␈↓¬␈α⊂(weakened␈α∂version)␈α⊂of␈α⊂P␈α⊂might␈α⊂be␈α∂more
␈↓ α,␈↓¬interesting␈α∂than␈α∂P"␈↓.␈α∂ AM␈α∂considers␈α∂this␈α∂suggestion␈α∞after␈α∂trying␈α∂to␈α∂≡ll␈α∂in␈α∂examples␈α∂of␈α∞each

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	3␈↓ε Plus noticing that multiplication is associative and commutative.
␈↓ α,␈↓ε␈↓	4␈↓ε␈α
Situation/action␈α
rules␈α
which␈α
function␈α
as␈α
local␈α
"plausible␈α
move␈α
generators".␈α
 Some␈α
suggest␈α
tasks␈α
for␈α
the␈α
system␈αto␈α
carry
␈↓ α,␈↓ε␈↓ βLout, some suggest ways of satisfying a given task, etc.
␈↓ α,␈↓ε␈↓	5␈↓ε Typically, "new" means new to AM, not to Mankind; and "worthwhile" can only be judged in hindsight.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε4␈↓-

␈↓ α,␈↓predicate␈↓	6␈↓.

␈↓ α,␈↓AM␈α
is␈α
initially␈α
given␈α
a␈α
collection␈α
of␈α
115␈α∞core␈α
concepts,␈α
with␈α
only␈α
a␈α
few␈α
facets␈α
≡lled␈α∞in␈α
for
␈↓ α,␈↓each.␈α Its␈α
sole␈αactivity␈αis␈α
to␈αchoose␈αsome␈α
facet␈αof␈αsome␈α
concept,␈αand␈α≡ll␈α
in␈αthat␈αparticular␈α
slot.
␈↓ α,␈↓In␈α∪so␈α∀doing,␈α∪new␈α∀notions␈α∪will␈α∪often␈α∀emerge.␈α∪ Uninteresting␈α∀ones␈α∪are␈α∀forgotten,␈α∪mildly
␈↓ α,␈↓interesting␈αones␈αare␈αkept␈αas␈αparts␈αof␈αone␈αfacet␈αof␈αone␈αconcept,␈αand␈αvery␈αinteresting␈αones␈αare
␈↓ α,␈↓granted␈α∞full␈α∞concept-module␈α∞status.␈α∞Each␈α∞of␈α∞these␈α∞new␈α∞modules␈α∞has␈α∞dozens␈α∞of␈α∞blank␈α∞slots,
␈↓ α,␈↓hence␈α∪the␈α∪space␈α∩of␈α∪possible␈α∪actions␈α∪(blank␈α∩facets␈α∪to␈α∪≡ll␈α∩in)␈α∪grows␈α∪rapidly.␈α∪ The␈α∩same
␈↓ α,␈↓heuristics␈α
are␈α
used␈α
both␈α∞to␈α
suggest␈α
new␈α
directions␈α∞for␈α
investigation,␈α
and␈α
to␈α∞limit␈α
attention:
␈↓ α,␈↓both to sprout and to prune.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.2.3. Results␈↓)αβ␈↓

␈↓ α,␈↓The␈α∂particular␈α∞mathematical␈α∂domains␈α∞in␈α∂which␈α∞AM␈α∂operates␈α∞depend␈α∂upon␈α∞the␈α∂choice␈α∞of
␈↓ α,␈↓initial␈α∂concepts.␈α∂ Currently,␈α∂AM␈α∞begins␈α∂with␈α∂nothing␈α∂but␈α∞a␈α∂scanty␈α∂knowledge␈α∂of␈α∞concepts
␈↓ α,␈↓which␈αPiaget␈αmight␈αdescribe␈αas␈α␈↓βprenumerical␈↓:␈αSets,␈αsubstitution,␈αoperations,␈αequality,␈αand␈αso
␈↓ α,␈↓on.␈α∃ In␈α∀particular,␈α∃AM␈α∀is␈α∃not␈α∀told␈α∃anything␈α∀about␈α∃proof,␈α∀single-valued␈α∃functions,␈α∀or
␈↓ α,␈↓numbers.

␈↓ α,␈↓From␈α~this␈α→primitive␈α~basis,␈α→AM␈α~quickly␈α→discovered␈↓	7␈↓␈α~elementary␈α~numerical␈α→concepts
␈↓ α,␈↓(corresponding␈αto␈αthose␈αwe␈αrefer␈αto␈αas␈αnatural␈αnumbers,␈αmultiplication,␈αfactors,␈α
and␈αprimes)
␈↓ α,␈↓and␈αwandered␈αaround␈αin␈αthe␈αdomain␈αof␈αelementary␈αnumber␈αtheory.␈α AM␈αwas␈αnot␈αdesigned
␈↓ α,␈↓to␈α⊃␈↓βprove␈↓␈α⊂anything,␈α⊃but␈α⊂it␈α⊃did␈α⊂␈↓βconjecture␈↓␈α⊃many␈α⊂well-known␈α⊃relationships␈α⊂(e.g.,␈α⊃the␈α⊂unique
␈↓ α,␈↓factorization theorem).

␈↓ α,␈↓AM␈αwas␈αnot␈α
able␈αto␈αdiscover␈αany␈α
"new-to-Mankind"␈αmathematics␈αpurely␈α
on␈αits␈αown,␈αbut␈α
␈↓βhas␈↓
␈↓ α,␈↓discovered␈αseveral␈αinteresting␈αnotions␈α
hitherto␈αunknown␈αto␈αthe␈α
author.␈αA␈αcouple␈αbits␈αof␈α
new
␈↓ α,␈↓mathematics␈α∪have␈α∪been␈α∀␈↓βinspired␈↓␈α∪by␈α∪AM.␈↓	2␈↓␈α∀A␈α∪synergetic␈α∪AM¬human␈α∀combination␈α∪can
␈↓ α,␈↓sometimes␈αproduce␈αbetter␈αresearch␈αthan␈αeither␈αcould␈αalone.␈↓	8␈↓␈αAlthough␈αmost␈αof␈α
the␈αconcepts
␈↓ α,␈↓AM␈α
proposed␈αand␈α
developed␈αwere␈α
already␈α
very␈αwell␈α
known,␈αAM␈α
de≡ned␈αsome␈α
of␈α
them␈αin
␈↓ α,␈↓novel␈α⊂ways␈α⊂(e.g.,␈α⊂prime␈α⊂pairs␈α⊂were␈α⊃de≡ned␈α⊂by␈α⊂restricting␈α⊂addition␈α⊂to␈α⊂primes;␈α⊂that␈α⊃is,␈α⊂for
␈↓ α,␈↓which primes p,q,r is it possible that p+q=r?␈↓	9␈↓).

␈↓ α,␈↓Everything␈α
that␈α
AM␈αdoes␈α
can␈α
be␈α
viewed␈αas␈α
testing␈α
the␈αunderlying␈α
body␈α
of␈α
heuristic␈αrules.
␈↓ α,␈↓Gradually,␈αthis␈αknowledge␈αbecomes␈α
better␈αorganized,␈αits␈αimplications␈αclearer.␈α
 The␈αresultant
␈↓ α,␈↓body␈α∞of␈α
detailed␈α∞heuristics␈α
may␈α∞be␈α∞the␈α
germ␈α∞of␈α
a␈α∞more␈α
e≠cient␈α∞programme␈α∞for␈α
educating
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	6␈↓ε␈αIn␈αfact,␈αafter␈αAM␈αattempts␈αto␈αfind␈αexamples␈αof␈αSET-EQUALITY,␈αso␈αfew␈αare␈αfound␈αthat␈αAM␈αdecides␈αto␈αgeneralize␈αthat
␈↓ α,␈↓ε␈↓ βLpredicate.␈α The␈αresult␈α
is␈αthe␈αcreation␈α
of␈αa␈αnew␈α
predicate␈αwhich␈αmeans␈α
"Has-the-same-length-as"␈α--␈αi.e.,␈α
a
␈↓ α,␈↓ε␈↓ βLrudimentary precursor to natural numbers.
␈↓ α,␈↓ε␈↓	7␈↓ε␈α	"Discovering"␈α	a␈α	concept␈α
means␈α	that␈α	(1)␈α	AM␈α	recognized␈α
it␈α	as␈α	a␈α	distinguished␈α	entity␈α
(e.g.,␈α	by␈α	formulating␈α	its␈α
definition)␈α	and
␈↓ α,␈↓ε␈↓ βLalso␈α(2)␈αAM␈αdecided␈αit␈αwas␈αworth␈αinvestigating␈α
(either␈αbecause␈αof␈αthe␈αinteresting␈αway␈αit␈αwas␈αformed,␈α
or
␈↓ α,␈↓ε␈↓ βLbecause of surprising preliminary empirical results).
␈↓ α,␈↓ε␈↓	8␈↓ε␈αThis␈αis␈αsupported␈αby␈αGelernter's␈αexperiences␈αwith␈αhis␈αgeometry␈αprogram:␈αWhile␈αlecturing␈αabout␈αhow␈αit␈αmight␈αprove␈αa
␈↓ α,␈↓ε␈↓ βLcertain␈α	theorem␈α
about␈α	isosceles␈α
triangles,␈α	he␈α
came␈α	up␈α	with␈α
a␈α	new,␈α
cute␈α	proof.␈α
Similarly,␈α	Guard␈α
and␈α	Eastman
␈↓ α,␈↓ε␈↓ βLnoticed␈α
an␈α
intermediate␈α∞result␈α
of␈α
their␈α
SAM␈α∞resolution␈α
theorem␈α
prover,␈α
and␈α∞wisely␈α
interpreted␈α
it␈α∞as␈α
a
␈↓ α,␈↓ε␈↓ βLnontrivial result in lattice theory (now known as SAM's lemma).
␈↓ α,␈↓ε␈↓	9␈↓ε The answer is that either p or q must be 2, and that the other two primes are a prime pair -- i.e., they differ by two.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε5␈↓-

␈↓ α,␈↓math students than the current dogma␈↓	10␈↓.

␈↓ α,␈↓Another␈α∞bene≡t␈α∞of␈α∞actually␈α∞constructing␈α∞AM␈α∂is␈α∞that␈α∞of␈α∞␈↓βexperimentation␈↓:␈α∞one␈α∞can␈α∂vary␈α∞the
␈↓ α,␈↓concepts␈α
AM␈α∞starts␈α
with,␈α∞vary␈α
the␈α∞heuristics␈α
available,␈α
etc.,␈α∞and␈α
study␈α∞the␈α
e≥ects␈α∞on␈α
AM's
␈↓ α,␈↓behavior.␈α Several␈αsuch␈αexperiments␈αwere␈αperformed.␈α One␈αinvolved␈αadding␈αa␈αcouple␈α
dozen
␈↓ α,␈↓new␈α∂concepts␈α∂from␈α∂an␈α∂entirely␈α∂new␈α∞domain:␈α∂plane␈α∂geometry.␈α∂ AM␈α∂busied␈α∂itself␈α∞exploring
␈↓ α,␈↓elementary␈αgeometric␈α
concepts,␈αand␈α
was␈αalmost␈α
as␈αproductive␈α
there␈αas␈α
in␈αits␈αoriginal␈α
domain.
␈↓ α,␈↓New␈α
concepts␈α
were␈α
de≡ned,␈α
and␈α
new␈α
conjectures␈α
formulated.␈α
 Other␈α
experiments␈αindicated
␈↓ α,␈↓that␈αAM␈αwas␈α
more␈αrobust␈αthan␈α
anticipated;␈αit␈αwithstood␈α
many␈αkinds␈αof␈α"de-tuning".␈α
 Others
␈↓ α,␈↓demonstrated␈α
the␈α
tremendous␈α
impact␈α
that␈α
a␈α
few␈α
key␈α
concepts␈α
(e.g.,␈α
Equality)␈α
had␈α
on␈α
AM's
␈↓ α,␈↓behavior.  Several more experiments and extensions have been planned for the future.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.2.4. Motivation␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|We␈α∂need␈α⊂a␈α∂super-mathematics␈α⊂in␈α∂which␈α∂the␈α⊂operations␈α∂are␈α⊂as␈α∂unknown␈α⊂as␈α∂the
␈↓ α,␈↓β␈↓ α|quantities␈α
they␈α
operate␈α∞on,␈α
and␈α
a␈α
super-mathematician,␈α∞who␈α
does␈α
not␈α∞know␈α
what
␈↓ α,␈↓β␈↓ α|he is doing when he performs these operations.

␈↓ α,␈↓¬␈↓ ε\-- Eddington



␈↓ α,␈↓Although␈αthe␈αmotivation␈αfor␈αcarrying␈αout␈αthis␈α
research␈αof␈αcourse␈αpreceded␈αthe␈αe≥ort,␈αI␈α
have
␈↓ α,␈↓delayed until this section a discussion of why this is worthwhile, why it was attempted.

␈↓ α,␈↓First␈α
there␈α
was␈α
the␈α
inherent␈α
interest␈α
of␈α
getting␈α
a␈α
handle␈α
on␈α
scienti≡c␈α
creativity.␈α
 AM␈α
is␈α
partly
␈↓ α,␈↓a␈α
demonstration␈α
that␈α
some␈α
aspects␈α
of␈α
creative␈α
theory␈α
formation␈α
can␈α
be␈α
demysti≡ed,␈α∞can␈α
be
␈↓ α,␈↓modelled as simple rule-governed behavior.

␈↓ α,␈↓Related␈α
to␈αthis␈α
is␈αthe␈α
potential␈αfor␈α
learning␈αfrom␈α
AM␈αmore␈α
about␈αthe␈α
processes␈α
of␈αconcept
␈↓ α,␈↓formation.␈αThis␈αwas␈αtouched␈αon␈α
previously,␈αand␈αseveral␈αexperiments␈αalready␈α
performed␈αon
␈↓ α,␈↓AM will be detailed later.

␈↓ α,␈↓Third,␈α∞AM␈α∞itself␈α∞may␈α∞grow␈α∞into␈α∞something␈α∞of␈α∞pragmatic␈α∞value.␈α∞Perhaps␈α∞it␈α∞will␈α∞become␈α
a
␈↓ α,␈↓useful␈α∂tool␈α∂for␈α∂mathematicians,␈α⊂for␈α∂educators,␈α∂or␈α∂as␈α∂a␈α⊂model␈α∂for␈α∂similar␈α∂systems␈α⊂in␈α∂more
␈↓ α,␈↓"practical"␈α⊂≡elds.␈α⊂ Perhaps␈α⊂in␈α⊃the␈α⊂future␈α⊂we␈α⊂scientists␈α⊂will␈α⊃be␈α⊂able␈α⊂to␈α⊂rely␈α⊃on␈α⊂automated
␈↓ α,␈↓assistants␈α⊂to␈α⊂carry␈α∂out␈α⊂the␈α⊂"hack"␈α∂phases␈α⊂of␈α⊂research,␈α∂the␈α⊂tiresome␈α⊂legwork␈α⊂necessary␈α∂for
␈↓ α,␈↓"secondary" creativity.

␈↓ α,␈↓Historically,␈α
the␈αdomain␈α
of␈α
AM␈αcame␈α
from␈α
a␈αsearch␈α
for␈α
a␈αscienti≡c␈α
≡eld␈α
whose␈αactivities␈α
had
␈↓ α,␈↓no␈αspeci≡c␈αgoal,␈αand␈αin␈αwhich␈αnatural␈αlanguage␈αabilities␈αwere␈αunnecessary.␈α This␈αwas␈αto␈αtest
␈↓ α,␈↓out the BEINGs [Lenat 75b] ideas for a modular representation of knowledge.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	10␈↓ε␈αCurrently,␈αan␈αeducator␈αtakes␈αthe␈α
very␈αbest␈αwork␈αany␈αmathematician␈αhas␈α
ever␈αdone,␈αpolishes␈αit␈αuntil␈αits␈α
brilliance␈αis
␈↓ α,␈↓ε␈↓ βLblinding,␈α
then␈αpresents␈α
it␈α
to␈αthe␈α
student␈α
to␈αinduce␈α
upon.␈αMany␈α
individuals␈α
(e.g.,␈αKnuth␈α
and␈α
Polya)␈αhave
␈↓ α,␈↓ε␈↓ βLpointed␈α
out␈αthis␈α
blunder.␈α A␈α
few␈α(e.g.,␈α
Papert␈αat␈α
MIT,␈αAdams␈α
at␈αStanford)␈α
are␈αexperimenting␈α
with␈αmore
␈↓ α,␈↓ε␈↓ βLrealistic strategies for "teaching" creativity.  See the references by these authors in the bibliography.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε6␈↓-

␈↓ α,␈↓It␈αwould␈α
be␈αunfair␈α
not␈αto␈αmention␈α
the␈αusual␈α
bad␈αreasons␈αfor␈α
this␈αresearch:␈α
the␈α"Look␈αma,␈α
no
␈↓ α,␈↓hands" syndrome, the AI researcher's classic maternal urges, ego, the usual thesis drives, etc.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.2.5. Conclusions␈↓)αβ␈↓

␈↓ α,␈↓AM␈α∞is␈α∞forced␈α∞to␈α
judge␈α∞␈↓βa␈α∞priori␈↓␈α∞the␈α
value␈α∞of␈α∞each␈α∞new␈α
concept,␈α∞to␈α∞lose␈α∞interest␈α∞quickly␈α
in
␈↓ α,␈↓concepts␈α
which␈α
aren't␈α
going␈α
to␈α
develop␈αinto␈α
anything.␈α
 Often,␈α
such␈α
judgments␈α
can␈α
only␈αbe
␈↓ α,␈↓based␈α⊃on␈α∩hindsight.␈α⊃ For␈α⊃similar␈α∩reasons,␈α⊃AM␈α⊃has␈α∩di≠culty␈α⊃formulating␈α∩new␈α⊃heuristics
␈↓ α,␈↓which␈α⊂are␈α⊂relevant␈α⊂to␈α⊂the␈α⊂new␈α∂concepts␈α⊂it␈α⊂creates.␈α⊂ Heuristics␈α⊂are␈α⊂often␈α⊂merely␈α∂compiled
␈↓ α,␈↓hindsight.␈α∂ While␈α∂AM's␈α∂"approach"␈α∂to␈α∂empirical␈α∂research␈α∂may␈α∂be␈α∂used␈α∂in␈α∂other␈α∞scienti≡c
␈↓ α,␈↓domains,␈α⊂the␈α∂main␈α⊂limitation␈α⊂(reliance␈α∂on␈α⊂hindsight)␈α⊂will␈α∂probably␈α⊂recur.␈α⊂ This␈α∂prevents
␈↓ α,␈↓AM from progressing inde≡nitely far on its own.

␈↓ α,␈↓This␈α⊂ultimate␈α∂limitation␈α⊂was␈α∂reached.␈α⊂AM's␈α⊂performance␈α∂degraded␈α⊂more␈α∂and␈α⊂more␈α⊂as␈α∂it
␈↓ α,␈↓progressed␈α
further␈α
away␈α
from␈α
its␈α
initial␈α
base␈α
of␈α
concepts.␈α
 Nevertheless,␈α
AM␈α
demonstrated
␈↓ α,␈↓that␈α∞selected␈α∂aspects␈α∞of␈α∂creative␈α∞discovery␈α∂in␈α∞elementary␈α∂mathematics␈α∞could␈α∂be␈α∞adequately
␈↓ α,␈↓represented␈α∞as␈α∞a␈α∞heuristic␈α∞search␈α∞process.␈α∞ Actually␈α∞constructing␈α∞a␈α∞computer␈α∞model␈α∞of␈α∞this
␈↓ α,␈↓activity␈α∪has␈α∪provided␈α∪an␈α∪experimental␈α∪vehicle␈α∪for␈α∪studying␈α∪the␈α∪dynamics␈α∪of␈α∪plausible
␈↓ α,␈↓empirical inference.


␈↓ α,␈↓␈↓ βE␈↓∧␈↓&1.3. Ways of viewing AM as some common process␈↓)αβ␈↓

␈↓ α,␈↓This␈α∞section␈α∞will␈α∞provide␈α
a␈α∞few␈α∞metaphors:␈α∞some␈α
hints␈α∞for␈α∞squeezing␈α∞AM␈α∞into␈α
paradigms
␈↓ α,␈↓with␈αwhich␈αthe␈α
reader␈αmight␈αbe␈αfamiliar.␈α
 For␈αexample,␈αthe␈αexistence␈α
of␈αheuristics␈αin␈αAM␈α
is
␈↓ α,␈↓functionally␈α⊂the␈α⊃same␈α⊂as␈α⊃the␈α⊂presence␈α⊃of␈α⊂domain-speci≡c␈α⊃information␈α⊂in␈α⊃any␈α⊂knowledge-
␈↓ α,␈↓based system.

␈↓ α,␈↓Consider␈α∀assumptions,␈α∀axioms,␈α∀de≡nitions,␈α∀and␈α∃theorems␈α∀to␈α∀be␈α∀syntactic␈α∀rules␈α∃for␈α∀the
␈↓ α,␈↓language␈α⊃that␈α⊃we␈α⊃call␈α⊃Mathematics.␈α⊃ Thus␈α⊃theorem-proving,␈α⊃and␈α⊃the␈α⊃whole␈α⊃of␈α⊂textbook
␈↓ α,␈↓mathematics,␈α≠is␈α~a␈α≠purely␈α~syntactic␈α≠process.␈α~ Then␈α≠the␈α~heuristic␈α≠rules␈α~used␈α≠by␈α~a
␈↓ α,␈↓mathematician␈α
(and␈α∞by␈α
AM)␈α∞would␈α
correspond␈α∞to␈α
the␈α∞semantic␈α
knowledge␈α∞associated␈α
with
␈↓ α,␈↓these more formal methods.

␈↓ α,␈↓Just␈α→as␈α→one␈α→can␈α→upgrade␈α→natural-language-understanding␈α→by␈α→incorporating␈α→semantic
␈↓ α,␈↓knowledge, so AM is only as successful as the heuristics it knows.

␈↓ α,␈↓Four␈α∂more␈α∂ways␈α∂of␈α∂"viewing"␈α∞AM␈α∂as␈α∂something␈α∂else␈α∂will␈α∞be␈α∂provided:␈α∂(i)␈α∂AM␈α∂as␈α∂a␈α∞hill-
␈↓ α,␈↓climber,␈α(ii)␈αAM␈αas␈αa␈αheuristic␈αsearch␈α
program,␈α(iii)␈αAM␈αas␈αa␈αmathematician,␈αand␈α
(iv)␈αAM
␈↓ α,␈↓as half a book.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.3.1. AM as Hill-climbing␈↓)αβ␈↓

␈↓ α,␈↓Let's␈αdraw␈αan␈αanalogy␈αbetween␈αthe␈αprocess␈αof␈αdeveloping␈αnew␈αmathematics␈αand␈αthe␈α
familiar
␈↓ α,␈↓process␈αof␈αhill-climbing.␈α We␈αmay␈αvisualize␈αAM␈αas␈αexploring␈αa␈αspace␈αusing␈αa␈αmeasuring␈αor
␈↓ α,␈↓"evaluation" function which imparts to it a topography.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε7␈↓-

␈↓ α,␈↓Consider␈α∞AM's␈α∞core␈α∞of␈α∞very␈α∞simple␈α
knowledge.␈α∞ By␈α∞compounding␈α∞its␈α∞known␈α∞concepts␈α
and
␈↓ α,␈↓methods,␈αAM␈αcan␈αexplore␈αbeyond␈αthe␈αfrontier␈α
of␈αthis␈αfoundation␈αa␈αlittle␈αwherever␈αit␈α
wishes.
␈↓ α,␈↓The␈α
incredible␈αvariety␈α
of␈αalternatives␈α
to␈αinvestigate␈α
includes␈αall␈α
known␈α
mathematics,␈αmuch
␈↓ α,␈↓trivia,␈α∂countless␈α∂deadends,␈α∂and␈α∂so␈α∂on.␈α∂ The␈α∂only␈α∂"successful"␈α∂paths␈α∂near␈α∂the␈α∂core␈α∂are␈α∞the
␈↓ α,␈↓narrow␈α∪ridges␈α∪of␈α∪known␈α∪mathematics␈α∩(plus␈α∪perhaps␈α∪a␈α∪few␈α∪as-yet-undiscovered␈α∩isolated
␈↓ α,␈↓peaks).

␈↓ α,␈↓How␈α
can␈α
AM␈α
walk␈αthrough␈α
this␈α
immense␈α
space,␈α
with␈αany␈α
hope␈α
of␈α
following␈α
the␈αfew,␈α
slender
␈↓ α,␈↓trails␈αof␈αalready-established␈αmathematics␈α(or␈α
some␈αequally␈αsuccessful␈αnew␈α≡elds)?␈α
 AM␈αmust
␈↓ α,␈↓do␈α
hill-climbing:␈α
As␈α
new␈α∞concepts␈α
are␈α
formed,␈α
decide␈α∞how␈α
promising␈α
they␈α
are,␈α∞and␈α
always
␈↓ α,␈↓explore␈α∃the␈α∃currently␈α∃most-promising␈α∃new␈α∃concept.␈α∃ The␈α∃evaluation␈α∃function␈α⊗is␈α∃quite
␈↓ α,␈↓nontrivial,␈αand␈αthe␈αwork␈αreported␈αin␈αthis␈αhalf␈αof␈αthe␈αbook␈αmay␈αbe␈αviewed␈αas␈αan␈αattempt␈αto
␈↓ α,␈↓study␈α∀and␈α∀explain␈α∀and␈α∀duplicate␈α∀the␈α∀judgmental␈α∀criteria␈α∀people␈α∃employ.␈α∀ Preliminary
␈↓ α,␈↓attempts␈↓	11␈↓␈α∪at␈α∩codifying␈α∪such␈α∩"mysterious"␈α∪emotive␈α∩forces␈α∪as␈α∩intuition,␈α∪aesthetics,␈α∩utility,
␈↓ α,␈↓richness,␈αinterestingness,␈αrelevance...␈α indicated␈αthat␈αa␈αlarge␈αbut␈αnot␈αunmanageable␈αcollection
␈↓ α,␈↓of heuristic rules should su≠ce.

␈↓ α,␈↓The␈α∞important␈α
visualization␈α∞to␈α
make␈α∞is␈α∞that␈α
with␈α∞proper␈α
evaluation␈α∞criteria,␈α∞AM's␈α
planar
␈↓ α,␈↓mass␈αof␈αinterrelated␈αconcepts␈αis␈αtransformed␈αinto␈αa␈αthree-dimensional␈αrelief␈αmap:␈αthe␈αknown
␈↓ α,␈↓lines␈α
of␈α
development␈α
become␈α
mountain␈α
ranges,␈α∞soaring␈α
above␈α
the␈α
vast␈α
∨at␈α
plains␈α∞of␈α
trivia
␈↓ α,␈↓and inconsistency below.

␈↓ α,␈↓Occasionally␈α∪an␈α∀isolated␈α∪hill␈α∪is␈α∀discovered␈α∪near␈α∪the␈α∀core;␈↓	12␈↓␈α∪certainly␈α∪whole␈α∀ranges␈α∪lie
␈↓ α,␈↓undiscovered␈αfor␈αlong␈αperiods␈αof␈αtime␈↓	13␈↓,␈αand␈αthe␈αterrrain␈αfar␈αfrom␈αthe␈αinitial␈αcore␈αis␈αnot␈αyet
␈↓ α,␈↓explored at all.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.3.2. AM as Heuristic Search␈↓)αβ␈↓

␈↓ α,␈↓We␈αearlier␈αreferred␈αto␈α
AM␈αas␈αa␈αkind␈αof␈α
"heuristic␈αsearch"␈αprogram.␈α That␈αmust␈α
mean␈αthat
␈↓ α,␈↓AM is exploring a particular "space," using some informal evaluation criteria to guide it.

␈↓ α,␈↓The␈α∞∨avor␈α
of␈α∞search␈α
which␈α∞is␈α
used␈α∞here␈α
is␈α∞that␈α
of␈α∞progressively␈α
enlarging␈α∞a␈α∞tree.␈α
Certain
␈↓ α,␈↓"evaluation-function"␈α
heuristics␈α
are␈α
used␈α
to␈α
decide␈α
which␈α
node␈α
of␈α
the␈α
tree␈α
to␈α
expand␈αnext,
␈↓ α,␈↓and␈αother␈α
guiding␈αrules␈αare␈α
then␈αused␈α
to␈αproduce␈αfrom␈α
that␈αnode␈α
a␈αfew␈αinteresting␈α
successor
␈↓ α,␈↓nodes.␈α
To␈α
do␈α
mathematical␈α
research␈αwell,␈α
I␈α
claim␈α
that␈α
it␈αis␈α
necessary␈α
and␈α
su≠cent␈α
to␈αhave
␈↓ α,␈↓good␈α∀methods␈α∪for␈α∀proposing␈α∪new␈α∀concepts␈α∪from␈α∀existing␈α∪ones,␈α∀and␈α∪for␈α∀deciding␈α∪how
␈↓ α,␈↓interesting each "node" (partially-studied concept) is.

␈↓ α,␈↓AM␈α⊂is␈α⊃initially␈α⊂supplied␈α⊂with␈α⊃a␈α⊂few␈α⊂facts␈α⊃about␈α⊂some␈α⊂simple␈α⊃math␈α⊂concepts.␈α⊃ AM␈α⊂then
␈↓ α,␈↓explores␈α
mathematics␈αby␈α
selectively␈αenlarging␈α
that␈αbasis.␈α
 One␈αcould␈α
say␈αthat␈α
AM␈αconsists␈α
of

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	11␈↓ε␈α
These␈α
took␈α	the␈α
form␈α
of␈α	informal␈α
simulations.␈α
Although␈α
far␈α	from␈α
controlled␈α
experiments,␈α	they␈α
indicated␈α
the␈α
feasability␈α	of
␈↓ α,␈↓ε␈↓ βLattempting␈αto␈αcreate␈α
AM,␈αby␈αyielding␈α
an␈αapproximate␈αfigure␈α
for␈αthe␈αamount␈α
of␈αinformal␈αknowledge␈αsuch␈α
a
␈↓ α,␈↓ε␈↓ βLsystem would need.
␈↓ α,␈↓ε␈↓	12␈↓ε E.g., Conway's numbers, as described in [Knuth 74].
␈↓ α,␈↓ε␈↓	13␈↓ε E.g., non-Euclidean geometries weren't thought of until 1848.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε8␈↓-

␈↓ α,␈↓an␈α∞active␈α
body␈α∞of␈α
mathematical␈α∞concepts,␈α
plus␈α∞enough␈α
"wisdom"␈α∞to␈α
use␈α∞and␈α∞develop␈α
them
␈↓ α,␈↓e≥ectively.␈αFor␈α"wisdom",␈αread␈α"heuristics".␈αLoosely␈αspeaking,␈αthen,␈αAM␈αis␈αa␈αheuristic␈αsearch
␈↓ α,␈↓program.␈α
 To␈αsee␈α
this␈αmore␈α
clearly,␈αwe␈α
must␈αexplain␈α
what␈αthe␈α
nodes␈αof␈α
AM's␈α
search␈αspace
␈↓ α,␈↓are, what the successor operators or links are, and what the evaluation function is.

␈↓ α,␈↓AM's␈αspace␈αcan␈αbe␈αconsidered␈αto␈αconsist␈αof␈αall␈αnodes␈αwhich␈αare␈αconsistent,␈αpartially-≡lled-in
␈↓ α,␈↓concepts.␈α Then␈α
a␈αprimitive␈α
"legal␈αmove"␈α
for␈αAM␈α
would␈αbe␈α
to␈α(i)␈α
enlarge␈αsome␈α
facet␈αof␈α
some
␈↓ α,␈↓concept,␈α∂or␈α∞(ii)␈α∂create␈α∞a␈α∂new,␈α∞partially-complete␈α∂concept.␈α∞Consider␈α∂momentarily␈α∞the␈α∂size␈α∞of
␈↓ α,␈↓this␈α
space.␈α If␈α
there␈αwere␈α
no␈αconstraint␈α
on␈α
what␈αthe␈α
new␈αconcepts␈α
can␈αbe,␈α
and␈α
no␈αinformal
␈↓ α,␈↓knowledge␈α⊂for␈α⊃quickly␈α⊂≡nding␈α⊃entries␈α⊂for␈α⊂a␈α⊃desired␈α⊂facet,␈α⊃a␈α⊂blind␈α⊃"legal-move"␈α⊂program
␈↓ α,␈↓would␈αgo␈αnowhere␈α¬␈αslowly!␈α One␈αshouldn't␈αeven␈αcall␈αthe␈αactivity␈αsuch␈αa␈αprogram␈αwould␈αbe
␈↓ α,␈↓doing "math research."

␈↓ α,␈↓The␈αheuristic␈αrules␈αare␈αused␈αas␈αlittle␈α"plausible␈αmove␈αgenerators".␈α They␈αsuggest␈αwhich␈αfacet
␈↓ α,␈↓of␈αwhich␈αconcept␈αto␈αenlarge␈αnext,␈αand␈αthey␈αsuggest␈αspeci≡c␈αnew␈αconcepts␈αto␈αcreate.␈αThe␈αonly
␈↓ α,␈↓activities␈α∂which␈α∂AM␈α∂will␈α∂consider␈α∂doing␈α∂are␈α∂those␈α∂which␈α∂have␈α∂been␈α∂motivated␈α∂for␈α∂some
␈↓ α,␈↓speci≡c␈α⊃good␈↓	14␈↓␈α⊃reason.␈α⊃A␈α⊃global␈α⊃␈↓βagenda␈α⊂of␈α⊃tasks␈↓␈α⊃is␈α⊃maintained,␈α⊃listing␈α⊃all␈α⊃the␈α⊂activities
␈↓ α,␈↓suggested but not yet worked on.

␈↓ α,␈↓AM␈αhas␈αa␈αde≡nite␈αalgorithm␈αfor␈αrating␈αthe␈αnodes␈αof␈αits␈αspace.␈α Many␈αheuristics␈αexist␈α
merely
␈↓ α,␈↓to␈α∂estimate␈α∂the␈α∂worth␈α⊂of␈α∂any␈α∂given␈α∂concept.␈α∂ Other␈α⊂heuristics␈α∂use␈α∂these␈α∂worth␈α⊂ratings␈α∂to
␈↓ α,␈↓order␈αthe␈αtasks␈αon␈αthe␈αglobal␈αagenda␈αlist.␈α Yet␈αAM␈αhas␈αno␈αspeci≡c␈αgoal␈αcriteria:␈αit␈αcan␈αnever
␈↓ α,␈↓"halt", never succeed or fail in any absolute sense. AM goes on forever␈↓	15␈↓.

␈↓ α,␈↓Consider␈αNilsson's␈αdescriptions␈αof␈αdepth-≡rst␈αsearching␈αand␈αbreadth-≡rst␈αsearching␈α([Nilsson
␈↓ α,␈↓71]).␈α∞ He␈α∞has␈α∞us␈α∞maintain␈α∞a␈α∞list␈α∞of␈α∞"open"␈α∞nodes.␈α∞ Repeatedly,␈α∞he␈α∞plucks␈α∞the␈α∞top␈α∂one␈α∞and
␈↓ α,␈↓expands␈αit.␈α In␈αthe␈αprocess,␈αsome␈αnew␈αnodes␈αmay␈αbe␈αadded␈αto␈αthe␈αOpen␈αlist.␈α In␈αthe␈αcase␈αof
␈↓ α,␈↓depth-≡rst␈α∞searching,␈α
they␈α∞are␈α
added␈α∞at␈α
the␈α∞top;␈α
the␈α∞next␈α
node␈α∞to␈α
expand␈α∞is␈α
the␈α∞one␈α
most
␈↓ α,␈↓recently␈αcreated;␈αthe␈αOpen-list␈αis␈αbeing␈αused␈αas␈αa␈αpush-down␈αstack.␈α For␈αbreadth-≡rst␈αsearch,
␈↓ α,␈↓new␈α∞nodes␈α∞are␈α∂added␈α∞at␈α∞the␈α∞bottom;␈α∂they␈α∞aren't␈α∞expanded␈α∞until␈α∂all␈α∞the␈α∞older␈α∂nodes␈α∞have
␈↓ α,␈↓been;␈α
the␈αOpen-list␈α
is␈α
used␈αas␈α
a␈α
queue.␈α For␈α
heuristic␈α
search,␈αor␈α
"best-≡rst"␈α
search,␈αnew␈α
nodes
␈↓ α,␈↓are␈αevaluated␈αin␈αsome␈αnumeric␈αway,␈αand␈αthen␈α"merged"␈αinto␈αthe␈αalready-sorted␈αlist␈αof␈αOpen
␈↓ α,␈↓nodes.

␈↓ α,␈↓This␈αprocess␈αis␈αvery␈αsimilar␈αto␈αthe␈αagenda␈αmechanism␈αAM␈αuses␈αto␈αmanage␈αits␈αsearch.␈α This
␈↓ α,␈↓will␈αbe␈αdiscussed␈α
in␈αdetail␈αin␈αChapter␈α
3.␈α Each␈αentry␈αon␈α
the␈αagenda␈αconsists␈αof␈α
three␈αparts:
␈↓ α,␈↓(i)␈α∂a␈α∂plausible␈α⊂task␈α∂for␈α∂AM␈α∂to␈α⊂do,␈α∂(ii)␈α∂a␈α⊂list␈α∂of␈α∂reasons␈α∂supporting␈α⊂that␈α∂task,␈α∂and␈α⊂(iii)␈α∂a
␈↓ α,␈↓numeric␈α
estimate␈αof␈α
the␈α
overall␈αpriority␈α
this␈α
task␈αshould␈α
have.␈α
 When␈αa␈α
task␈α
is␈αsuggested␈α
for
␈↓ α,␈↓some␈αreason,␈αit␈αis␈αadded␈α
to␈αthe␈αagenda.␈α A␈αtask␈α
may␈αbe␈αsuggested␈αseveral␈αtimes,␈αfor␈α
di≥erent
␈↓ α,␈↓reasons.␈α The␈αglobal␈αpriority␈αvalue␈αassigned␈αto␈αeach␈αtask␈αis␈αbased␈αon␈αthe␈αcombined␈αvalue␈αof
␈↓ α,␈↓its␈α⊃reasons.␈α⊃ The␈α⊃control␈α⊃structure␈α⊃of␈α⊃AM␈α⊃is␈α⊃simply␈α⊃to␈α⊃select␈α⊃the␈α⊃task␈α⊃with␈α⊃the␈α⊂highest
␈↓ α,␈↓priority,␈α
execute␈α
it,␈αand␈α
select␈α
a␈α
new␈αone.␈α
 The␈α
agenda␈α
mechanism␈αappears␈α
to␈α
be␈α
a␈αvery␈α
well-
␈↓ α,␈↓suited data structure for managing a "best-≡rst" search process.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	14␈↓ε␈αλOf␈αλcourse,␈αλAM␈αλthinks␈αλa␈α	reason␈αλis␈αλ"good"␈αλif␈αλ--␈αλand␈αλonly␈α	if␈αλ--␈αλit␈αλwas␈αλtold␈αλthat␈αλby␈α	a␈αλheuristic␈αλrule;␈αλso␈αλthose␈αλrules␈α	had␈αλbetter
␈↓ α,␈↓ε␈↓ βLbe plausible, preferably the ones actually used by the experts.
␈↓ α,␈↓ε␈↓	15␈↓ε Technically, forever is about 100,000 list cells and a couple cpu hours.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε9␈↓-

␈↓ α,␈↓Similar␈αcontrol␈αstructures␈αwere␈α
used␈αin␈αLT␈α[Newell,␈αShaw,␈α
&␈αSimon␈α57],␈αthe␈α
predictor␈αpart
␈↓ α,␈↓of␈α
Dendral␈α
[Buchanan␈α
et␈α
al␈α
69],␈α∞SIMULA-67␈α
[Dahl␈α
68],␈α
and␈α
KRL␈α
[Bobrow␈α∞&␈α
Winograd
␈↓ α,␈↓77].␈α The␈αmain␈αdi≥erence␈αis␈αthat␈αin␈αAM,␈αsymbolic␈αreasons␈αare␈αused␈α(albeit␈αin␈αtrivial␈αtoken-
␈↓ α,␈↓like␈αways)␈αto␈αdecide␈α
whether␈α¬␈αand␈αhow␈αmuch␈α
¬␈αto␈αboost␈αthe␈αpriority␈α
of␈αa␈αtask␈αwhen␈α
it␈αis
␈↓ α,␈↓suggested again.

␈↓ α,␈↓There␈α∀are␈α∪several␈α∀di≠culties␈α∪and␈α∀anomalies␈α∀in␈α∪forcing␈α∀AM␈α∪into␈α∀the␈α∀heuristic␈α∪search
␈↓ α,␈↓paradigm.␈α In␈αa␈αtypical␈αheuristic␈αsearch␈α(e.g.,␈αDendral␈α[Feigenbaum␈αet␈αal␈α71],␈αMeta-Dendral
␈↓ α,␈↓[Buchanan␈αet␈αal␈α72],␈αmost␈αgame-playing␈α
programs␈α[Samuel␈α67]),␈αa␈α"search␈αspace"␈α
is␈αde≡ned
␈↓ α,␈↓implicitly␈αby␈αa␈α"legal␈α
move␈αgenerator".␈αHeuristics␈αare␈α
present␈αto␈αconstrain␈αthat␈α
generator␈αso
␈↓ α,␈↓that␈αonly␈αplausible␈αnodes␈αare␈αproduced.␈αThe␈αsecond␈αkind␈αof␈αheuristic␈αsearch,␈αof␈αwhich␈αAM
␈↓ α,␈↓is␈α∂an␈α∂example,␈α∂contains␈α∂no␈α∂"legal␈α⊂move␈α∂generator".␈α∂ Instead,␈α∂AM's␈α∂heuristics␈α∂are␈α⊂used␈α∂as
␈↓ α,␈↓plausible␈α
move␈α
generators.␈α∞ Those␈α
heuristics␈α
themselves␈α∞implicitly␈α
de≡ne␈α
the␈α∞possible␈α
tasks
␈↓ α,␈↓AM␈αmight␈αconsider,␈αand␈α␈↓βall␈↓␈αsuch␈αtasks␈αshould␈αbe␈αplausible␈αone.␈αIn␈αthe␈α≡rst␈αkind␈αof␈αsearch,
␈↓ α,␈↓removing␈αa␈αheuristic␈αwidens␈αthe␈αsearch␈αspace;␈αin␈αAM's␈αkind␈αof␈αsearch,␈αremoving␈αa␈αheuristic
␈↓ α,␈↓␈↓βreduces␈↓ it.

␈↓ α,␈↓Another␈αanomaly␈αis␈αthat␈αthe␈αoperators␈αwhich␈αAM␈αuses␈αto␈αenlarge␈αand␈αexplore␈αthe␈αspace␈αof
␈↓ α,␈↓concepts␈α∀are␈α∪themselves␈α∀mathematical␈α∪concepts␈α∀(e.g.,␈α∪some␈α∀heuristic␈α∪rules␈α∀result␈α∀in␈α∪the
␈↓ α,␈↓creation␈αof␈αnew␈αheuristic␈αrules;␈α"Compose"␈αis␈αboth␈αa␈αconcept␈αand␈αan␈αoperation␈αwhich␈αresults
␈↓ α,␈↓in␈α
new␈αconcepts).␈α
 Thus␈α
AM␈αshould␈α
be␈α
viewed␈αas␈α
a␈α
mass␈αof␈α
knowledge␈α
which␈αenlarges␈α
␈↓βitself␈↓
␈↓ α,␈↓repeatedly.␈α∃ Typically,␈α∃computer␈α∃programs␈α∃keep␈α∃the␈α∃information␈α∃they␈α∃"discover"␈α∀quite
␈↓ α,␈↓separate from the knowledge they use to make discoveries␈↓	16␈↓

␈↓ α,␈↓Perhaps␈αthe␈αgreatest␈αdi≥erence␈αbetween␈αAM␈αand␈αtypical␈αheuristic␈αsearch␈αprocedures␈αis␈αthat
␈↓ α,␈↓AM␈α
has␈α
no␈α
well-de≡ned␈αtarget␈α
concepts␈α
or␈α
target␈αrelationships.␈α
 Rather,␈α
its␈α
"goal␈αcriterion"␈α
¬
␈↓ α,␈↓its␈α⊂sole␈α⊂aim␈α⊃¬␈α⊂is␈α⊂to␈α⊂maximize␈α⊃the␈α⊂interestingness␈α⊂level␈α⊂of␈α⊃the␈α⊂activities␈α⊂it␈α⊃performs,␈α⊂the
␈↓ α,␈↓priority␈αratings␈αof␈αthe␈αtop␈αtasks␈αon␈αthe␈αagenda.␈α It␈αdoesn't␈αmatter␈αprecisely␈αwhich␈αde≡nitions
␈↓ α,␈↓or␈αconjectures␈αAM␈αdiscovers␈α¬␈αor␈αmisses␈α¬␈αso␈αlong␈αas␈αit␈αspends␈αits␈αtime␈αon␈αplausible␈αtasks.
␈↓ α,␈↓There␈αis␈αno␈α≡xed␈αset␈αof␈αtheorems␈αthat␈αAM␈αshould␈αdiscover,␈αso␈αAM␈αis␈αnot␈αa␈αtypical␈α
␈↓βproblem-
␈↓ α,␈↓βsolver␈↓.␈αThere␈αis␈αno␈α≡xed␈αset␈αof␈αtraps␈αAM␈αshould␈αavoid,␈αno␈αsmall␈αset␈αof␈αlegal␈αmoves,␈αand␈αno
␈↓ α,␈↓winning/losing behavior, so AM is not a typical ␈↓βgame-player␈↓.

␈↓ α,␈↓For␈αexample,␈αno␈αstigma␈αis␈αattached␈αto␈αthe␈αfact␈αthat␈αAM␈αnever␈αdiscovered␈αreal␈αnumbers␈↓	17␈↓;␈αit
␈↓ α,␈↓was␈α∞rather␈α∞surprising␈α∞that␈α∂AM␈α∞managed␈α∞to␈α∞discover␈α∂␈↓βnatural␈↓␈α∞numbers!␈α∞ Even␈α∞if␈α∂it␈α∞hadn't
␈↓ α,␈↓done␈αthat,␈αit␈αwould␈αhave␈α
been␈αacceptable␈↓	18␈↓␈αif␈αAM␈αhad␈α
simply␈αgone␈αo≥␈αand␈αdeveloped␈α
ideas
␈↓ α,␈↓in set theory.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	16␈↓ε␈αOf␈αcourse␈αthis␈α
is␈αoften␈αbecause␈αthe␈α
two␈αkinds␈αof␈αknowledge␈α
are␈αvery␈αdifferent:␈αFor␈α
a␈αchess-player,␈αthe␈αfirst␈αkind␈α
is
␈↓ α,␈↓ε␈↓ βL"good␈αboard␈αpositions,"␈αand␈αthe␈αsecond␈αis␈α"strategies␈αfor␈αmaking␈αa␈αgood␈αmove."␈αTheorem-provers␈αare␈α
an
␈↓ α,␈↓ε␈↓ βLexception.␈α
They␈αproduce␈α
a␈αnew␈α
theorem,␈αand␈α
then␈αuse␈α
it␈α
(almost␈αlike␈α
a␈αnew␈α
operator)␈αin␈α
future␈αproofs.␈α
A
␈↓ α,␈↓ε␈↓ βLprogram␈α	to␈α	learn␈α	to␈α	play␈α
checkers␈α	[Samuel␈α	67]␈α	has␈α	this␈α
same␈α	flavor,␈α	thereby␈α	indicating␈α	that␈α
this␈α	`self-help'
␈↓ α,␈↓ε␈↓ βLproperty is not a function of the task domain, not simply a characteristic of mathematics.
␈↓ α,␈↓ε␈↓	17␈↓ε␈α	There␈α	are␈α	many␈α	"nice"␈α	things␈α	which␈α	AM␈α	didn't␈α	--␈α	and␈α	can't␈α	--␈α	do:␈α	e.g.,␈α	devising␈α	␈↓&geometric␈↓)αβ␈α	concepts␈α	from␈α	its␈α	initial␈α	simple
␈↓ α,␈↓ε␈↓ βLset-theoretic knowledge.  See the discussion of the limitations of AM, Section ..
␈↓ α,␈↓ε␈↓	18␈↓ε␈αAcceptable␈α
to␈αwhom?␈α Is␈α
there␈αreally␈α
a␈αdomain-invariant␈αcriterion␈α
for␈αjudging␈α
the␈αquality␈αof␈α
AM's␈αactions?␈α
 See␈αthe
␈↓ α,␈↓ε␈↓ βLdiscussions in Section 7.1.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε10␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.3.3. AM as a Mathematician␈↓)αβ␈↓

␈↓ α,␈↓Before␈α∂diving␈α∞into␈α∂the␈α∂innards␈α∞of␈α∂AM,␈α∂let's␈α∞take␈α∂a␈α∞moment␈α∂to␈α∂discuss␈α∞the␈α∂totality␈α∂of␈α∞the
␈↓ α,␈↓mathematics␈αwhich␈αAM␈αcarried␈αout.␈α Like␈αa␈αcontemporary␈αhistorian␈αsummarizing␈αthe␈αwork
␈↓ α,␈↓of␈α
the␈α
Babylonian␈α
mathematicians,␈α
we␈α
shan't␈α
hesitate␈α
to␈α
use␈α
current␈α
terms␈α
and␈α
criticize␈αby
␈↓ α,␈↓current standards.

␈↓ α,␈↓AM␈α⊂began␈α⊂its␈α⊂investigations␈α⊂with␈α⊂scanty␈α⊂knowledge␈α⊂of␈α⊂a␈α⊂few␈α⊂set-theoretic␈α⊂concepts␈α⊂(sets,
␈↓ α,␈↓equality␈α
of␈αsets,␈α
set␈αoperations).␈α
 Most␈αof␈α
the␈αobvious␈α
set-theory␈αrelations␈α
(e.g.,␈α
de␈αMorgan's
␈↓ α,␈↓laws)␈α⊂were␈α⊂eventually␈α⊂uncovered;␈α⊂since␈α⊃AM␈α⊂never␈α⊂fully␈α⊂understood␈α⊂abstract␈α⊃algebra,␈α⊂the
␈↓ α,␈↓statement␈αand␈αveri≡cation␈αof␈αeach␈αof␈αthese␈αwas␈αquite␈αobscure.␈α AM␈αnever␈αderived␈αa␈αformal
␈↓ α,␈↓notion␈α
of␈α
in≡nity,␈αbut␈α
it␈α
naively␈α
established␈αconjectures␈α
like␈α
"a␈α
set␈αcan␈α
never␈α
be␈α
a␈αmember␈α
of
␈↓ α,␈↓itself",␈α∪and␈α∀procedures␈α∪for␈α∀making␈α∪chains␈α∀of␈α∪new␈α∪sets␈α∀("insert␈α∪a␈α∀set␈α∪into␈α∀itself").␈α∪ No
␈↓ α,␈↓sophisticated set theory (e.g., diagonalization) was ever done.

␈↓ α,␈↓After␈αthis␈αinitial␈αperiod␈αof␈αexploration,␈αAM␈αdecided␈αthat␈α"equality"␈αwas␈αworth␈αgeneralizing,
␈↓ α,␈↓and␈α
thereby␈α
discovered␈αthe␈α
relation␈α
"same-size-as".␈α
 "Natural␈αnumbers"␈α
were␈α
based␈α
on␈αthis,
␈↓ α,␈↓and soon most simple arithmetic operations were de≡ned.

␈↓ α,␈↓Since␈α∞addition␈α∞arose␈α∞as␈α∞an␈α∞analog␈α∞to␈α∞union,␈α∞and␈α∞multiplication␈α∞as␈α∞a␈α∂repeated␈α∞substitution
␈↓ α,␈↓followed␈αby␈αa␈αgeneralized␈αkind␈αof␈αunioning␈↓	19␈↓␈αit␈αcame␈αas␈αquite␈αa␈αsurprise␈αwhen␈αAM␈αnoticed
␈↓ α,␈↓that␈α
they␈α
were␈α
related␈α
(namely,␈α
N+N=2␈↓εx␈↓N).␈α
 AM␈α
later␈α
re-discovered␈α
multiplication␈α
in␈αthree
␈↓ α,␈↓other␈α∞ways:␈α∞as␈α∞repeated␈α∞addition,␈α∞as␈α∂the␈α∞numeric␈α∞analog␈α∞of␈α∞the␈α∞Cartesian␈α∞product␈α∂of␈α∞sets,
␈↓ α,␈↓and␈αby␈αstudying␈αthe␈αcardinality␈αof␈αpower␈αsets␈↓	20␈↓.␈α These␈αoperations␈αwere␈αde≡ned␈αin␈αdi≥erent
␈↓ α,␈↓ways,␈αso␈αit␈αwas␈αan␈αunexpected␈α(to␈αAM)␈αdiscovery␈αwhen␈αthey␈αall␈αturned␈αout␈αto␈αbe␈α
equivalent.
␈↓ α,␈↓These surprises caused AM to give the concept `Times' quite a high Worth rating.

␈↓ α,␈↓Exponentiation␈αwas␈αde≡ned␈αas␈αrepeated␈α
multiplication.␈αUnfortunately,␈αAM␈αnever␈αfound␈α
any
␈↓ α,␈↓obvious properties of exponentiation, hence lost all interest in it.

␈↓ α,␈↓Soon␈αafter␈αde≡ning␈αmultiplication,␈αAM␈αinvestigated␈αthe␈αprocess␈αof␈αmultiplying␈αa␈αnumber␈αby
␈↓ α,␈↓itself:␈α
squaring.␈α The␈α
inverse␈αof␈α
this␈αturned␈α
out␈α
to␈αbe␈α
interesting,␈αand␈α
led␈αto␈α
the␈αde≡nition␈α
of
␈↓ α,␈↓square-root.␈α AM␈α
remained␈αcontent␈α
to␈αplay␈α
around␈αwith␈α
the␈αconcept␈α
of␈α␈↓βinteger␈↓-square-root.
␈↓ α,␈↓Although␈αit␈αisolated␈αthe␈αset␈αof␈αnumbers␈α
which␈αhad␈αno␈αsquare␈αroot,␈αAM␈αwas␈αnever␈α
close␈αto
␈↓ α,␈↓discovering rationals, let alone irrationals.

␈↓ α,␈↓Raising␈αto␈αfourth-powers,␈αand␈αfourth-rooting,␈αwere␈αdiscovered␈αat␈αthis␈αtime.␈α Perfect␈αsquares
␈↓ α,␈↓and␈α⊃perfect␈α⊃fourth-powers␈α⊃were␈α⊃isolated.␈α⊃ Many␈α⊃other␈α⊃numeric␈α⊃operations␈α⊃and␈α∩kinds␈α⊃of
␈↓ α,␈↓numbers␈αwere␈αisolated:␈αOdds,␈αEvens,␈αDoubling,␈αHalving,␈αetc.␈α Primitive␈αnotions␈αof␈αnumeric
␈↓ α,␈↓inequality were de≡ned but AM never even discovered Trichotomy.

␈↓ α,␈↓The␈αassociativity␈αand␈αcommutativity␈αof␈αmultiplication␈αindicated␈αthat␈αit␈αcould␈αaccept␈αa␈αBAG

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	19␈↓ε␈αλTake␈αλtwo␈αλbags␈α	A␈αλand␈αλB.␈αλReplace␈αλeach␈α	element␈αλof␈αλA␈αλby␈αλthe␈α	bag␈αλB.␈αλRemove␈αλone␈αλlevel␈α	of␈αλparentheses␈αλby␈αλtaking␈αλthe␈α	union␈αλof
␈↓ α,␈↓ε␈↓ βLall␈αλelements␈αλof␈αλthe␈αλtransfigured␈αλbag␈αλA.␈αλ Then␈αλthat␈α	new␈αλbag␈αλwill␈αλhave␈αλas␈αλmany␈αλelements␈αλas␈αλthe␈αλproduct␈α	of␈αλthe
␈↓ α,␈↓ε␈↓ βLlengths of the two original bags.
␈↓ α,␈↓ε␈↓	20␈↓ε␈αThe␈αsize␈αof␈αthe␈α
set␈αof␈αall␈αsubsets␈αof␈α
S␈αis␈α2␈↓#
|␈↓#␈↓#
S␈↓#␈↓#
|␈↓#.␈α Thus␈αthe␈α
power␈αset␈αof␈αA∪B␈αhas␈α
length␈αequal␈αto␈αthe␈α␈↓&product␈↓)αβ␈αof␈α
the
␈↓ α,␈↓ε␈↓ βLlengths of the power sets of A and B individually (assuming A and B are disjoint).
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε11␈↓-

␈↓ α,␈↓of␈α⊂numbers␈α∂as␈α⊂its␈α∂argument.␈α⊂ When␈α∂AM␈α⊂de≡ned␈α∂the␈α⊂inverse␈α∂operation␈α⊂corresponding␈α∂to
␈↓ α,␈↓Times,␈α
this␈αproperty␈α
allowed␈αthe␈α
de≡nition␈αto␈α
be:␈α"any␈α
␈↓βbag␈↓␈αof␈α
numbers␈α(>1)␈α
whose␈αproduct␈α
is
␈↓ α,␈↓x".␈α⊃ This␈α∩was␈α⊃just␈α∩the␈α⊃notion␈α⊃of␈α∩factoring␈α⊃a␈α∩number␈α⊃x.␈α∩ Minimally-factorable␈α⊃numbers
␈↓ α,␈↓turned␈αout␈αto␈αbe␈α
what␈αwe␈αcall␈αprimes.␈α
 Maximally-factorable␈αnumbers␈αwere␈αalso␈α
thought␈αto
␈↓ α,␈↓be interesting.

␈↓ α,␈↓Prime␈αpairs␈αwere␈αdiscovered␈αin␈αa␈αbizarre␈αway:␈αby␈αrestricting␈αaddition␈α(its␈αarguments␈αand␈αits
␈↓ α,␈↓values)␈α∀to␈α∀Primes.␈↓	21␈↓␈α∀AM␈α∃conjectured␈α∀the␈α∀fundamental␈α∀theorem␈α∀of␈α∃arithmetic␈α∀(unique
␈↓ α,␈↓factorization␈α
into␈αprimes)␈α
and␈α
Goldbach's␈αconjecture␈α
(every␈αeven␈α
number␈α
>2␈αis␈α
the␈α
sum␈αof
␈↓ α,␈↓two␈α∞primes)␈α∞in␈α∞a␈α
surprisingly␈α∞symmetric␈α∞way.␈α∞ The␈α
unary␈α∞representation␈α∞of␈α∞numbers␈α
gave
␈↓ α,␈↓way␈αto␈αa␈αrepresentation␈αas␈αa␈αbag␈αof␈αprimes␈α(based␈αon␈αunique␈αfactorization),␈αbut␈α
AM␈αnever
␈↓ α,␈↓thought␈αof␈αexponential␈αnotation.␈α ␈↓	22␈↓␈αSince␈αthe␈αkey␈αconcepts␈αof␈αremainder,␈αgreater-than,␈αgcd,
␈↓ α,␈↓and exponentiation were never mastered, progress in number theory was arrested.

␈↓ α,␈↓When␈α
a␈α
new␈α
base␈αof␈α
␈↓βgeometric␈↓␈α
concepts␈α
was␈α
added,␈αAM␈α
began␈α
≡nding␈α
some␈α
more␈αgeneral
␈↓ α,␈↓associations.␈α In␈αplace␈αof␈αthe␈αstrict␈αde≡nitions␈αfor␈αthe␈αequality␈αof␈αlines,␈αangles,␈αand␈αtriangles,
␈↓ α,␈↓came␈αnew␈αde≡nitions␈αof␈αconcepts␈αwe␈αrefer␈αto␈αas␈αParallel,␈αEqual-measure,␈αSimilar,␈α
Congruent,
␈↓ α,␈↓Translation,␈αRotation,␈αplus␈αmany␈αwhich␈αhave␈α
no␈αcommon␈αname␈α(e.g.␈αthe␈αrelationship␈αof␈α
two
␈↓ α,␈↓triangles␈α_sharing␈α_a␈α↔common␈α_angle).␈α_ A␈α↔cute␈α_geometric␈α_interpretation␈α_of␈α↔Goldbach's
␈↓ α,␈↓conjecture␈α∞was␈α∞found␈↓	23␈↓.␈α∞ Lacking␈α∞a␈α∞geometry␈α∞"model"␈α∞(an␈α∞analogic␈α∞representation␈α∂like␈α∞the
␈↓ α,␈↓one␈α∀Gelernter␈α∀employed),␈α∀AM␈α∀was␈α∀doomed␈α∪to␈α∀failure␈α∀with␈α∀respect␈α∀to␈α∀proposing␈α∪only
␈↓ α,␈↓plausible geometric conjectures.

␈↓ α,␈↓Similar␈α∂restrictions␈α∂due␈α∂to␈α∂poor␈α∂"visualization"␈α∂abilities␈α∂would␈α∂crop␈α∂up␈α∂in␈α∂topology.␈α∞ The
␈↓ α,␈↓concepts␈α∞of␈α
continuity,␈α∞in≡nity,␈α
and␈α∞measure␈α
would␈α∞have␈α
to␈α∞be␈α
fed␈α∞to␈α
AM␈α∞before␈α∞it␈α
could
␈↓ α,␈↓enter␈α
the␈α
domains␈α
of␈α
analysis.␈α
More␈α
and␈αmore␈α
drastic␈α
changes␈α
in␈α
its␈α
initial␈α
base␈α
would␈αbe
␈↓ α,␈↓required,␈α
as␈α
the␈αdesired␈α
domain␈α
gets␈α
further␈αand␈α
further␈α
from␈αsimple␈α
≡nite␈α
set␈α
theory␈αand
␈↓ α,␈↓elementary number theory.












␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	21␈↓ε␈α	That␈α
is,␈α	consider␈α
the␈α	set␈α	of␈α
triples␈α	p,q,r,␈α
all␈α	primes,␈α
for␈α	which␈α	p+q=r.␈α
Then␈α	one␈α
of␈α	them␈α	must␈α
be␈α	"2",␈α
and␈α	the␈α
other␈α	two
␈↓ α,␈↓ε␈↓ βLmust therefore form a prime pair.
␈↓ α,␈↓ε␈↓	22␈↓ε␈αA␈αtangential␈α
note:␈αAll␈αof␈αthe␈α
discoveries␈αmentioned␈αabove␈αwere␈α
made␈αby␈αAM␈αworking␈α
by␈αitself,␈αwith␈αa␈α
human␈αbeing
␈↓ α,␈↓ε␈↓ βLobserving␈α∞its␈α∂behavior.␈α∞If␈α∞the␈α∂level␈α∞of␈α∂sophistication␈α∞of␈α∞AM's␈α∂concepts␈α∞were␈α∞higher␈α∂(or␈α∞the␈α∂level␈α∞of
␈↓ α,␈↓ε␈↓ βLsophistication␈α∞of␈α∞its␈α∞users␈α∞were␈α∞lower),␈α∞then␈α
it␈α∞might␈α∞be␈α∞worthwhile␈α∞to␈α∞develop␈α∞a␈α∞nice␈α
user--system
␈↓ α,␈↓ε␈↓ βLinterface. The user in that case could -- and ought to -- work right along with AM as a co-researcher.
␈↓ α,␈↓ε␈↓	23␈↓ε␈αGiven␈αall␈αangles␈αof␈αa␈αprime␈αnumber␈αof␈αdegrees,␈α(0,1,2,3,5,7,11,...,179␈αdegrees),␈αthen␈αany␈αangle␈αbetween␈α0␈αand␈α180
␈↓ α,␈↓ε␈↓ βLdegrees can be approximated (to within 1 degree) as the sum of two of those angles.
␈↓ α,␈↓␈↓εChapter 1␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε12␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&1.3.4. AM as a Book␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|Walking␈α⊃home␈α⊃along␈α⊃a␈α⊃deserted␈α⊃street␈α⊃late␈α⊃at␈α⊃night,␈α⊃the␈α⊃reader␈α⊃may␈α⊂imagine
␈↓ α,␈↓β␈↓ α|himself␈αto␈αfeel␈αin␈αthe␈αsmall␈αof␈αhis␈αback␈αa␈αcold,␈αhard␈αobject;␈αand␈αto␈αhear␈αthe␈αwords
␈↓ α,␈↓β␈↓ α|spoken␈αbehind␈αhim,␈α`Easy␈αnow.␈αThis␈αis␈αa␈αstick-up.␈α Hand␈αover␈αyour␈αmoney.'␈αWhat
␈↓ α,␈↓β␈↓ α|does␈α∞the␈α∞reader␈α
do?␈α∞ He␈α∞attempts␈α
to␈α∞generate␈α∞the␈α
utterance.␈α∞He␈α∞says␈α∞to␈α
himself,
␈↓ α,␈↓β␈↓ α|now␈α∂if␈α∂I␈α∂were␈α∂standing␈α∂behind␈α∂someone␈α∂holding␈α∂a␈α∂cold,␈α∂hard␈α∂object␈α∂against␈α∞his
␈↓ α,␈↓β␈↓ α|back,␈α∂what␈α∂would␈α∞make␈α∂me␈α∂say␈α∂that?␈α∞What␈α∂would␈α∂I␈α∂mean␈α∞by␈α∂it?␈α∂The␈α∂reader␈α∞is
␈↓ α,␈↓β␈↓ α|advised␈αthat␈α
he␈αcan␈α
only␈αarrive␈α
at␈αthe␈α
deep␈αstructure␈α
of␈αthis␈α
book,␈αand␈αthrough␈α
the
␈↓ α,␈↓β␈↓ α|deep␈αstructure␈αthe␈αsemantics,␈αif␈αhe␈α
attempts␈αto␈αgenerate␈αthe␈αbook␈αfor␈αhimself.␈α
 The
␈↓ α,␈↓β␈↓ α|author wishes him luck.

␈↓ α,␈↓¬␈↓ ε\-- Linderholm



␈↓ α,␈↓Each␈αchapter␈αis␈αof␈αroughly␈αequal␈αimportance,␈αwhich␈αexplains␈αthe␈αhuge␈αvariation␈α
in␈αlength.
␈↓ α,␈↓Start␈αlooking␈αover␈αChapter␈α2␈αright␈αaway:␈αit␈αcontains␈αa␈αdetailed␈αexample␈αof␈αwhat␈α
AM␈αdoes.
␈↓ α,␈↓Since␈α
you're␈α
reading␈α
this␈α
sentence␈α
now,␈α
we'll␈αassume␈α
that␈α
you␈α
want␈α
a␈α
preview␈α
of␈α
what's␈αto
␈↓ α,␈↓come in the rest of this document.

␈↓ α,␈↓Chapter␈α∞3␈α∞covers␈α
the␈α∞top-level␈α∞control␈α∞structure␈α
of␈α∞the␈α∞system,␈α
which␈α∞is␈α∞based␈α∞around␈α
the
␈↓ α,␈↓notion␈α∞of␈α
an␈α∞`agenda'␈α∞of␈α
tasks␈α∞to␈α
perform.␈α∞ In␈α∞Chapter␈α
4␈α∞the␈α
low-level␈α∞control␈α∞structure␈α
is
␈↓ α,␈↓revealed:␈αAM␈αis␈αreally␈αguided␈αby␈αa␈αmass␈αof␈αheuristic␈αrules␈αof␈αvarying␈αgenerality.␈α Chapter␈α
5
␈↓ α,␈↓contains␈α
more␈αthan␈α
you␈αwant␈α
to␈α
know␈αabout␈α
the␈αrepresentation␈α
of␈αknowledge␈α
in␈α
AM.␈αThe
␈↓ α,␈↓diagram␈α⊃showing␈α⊃some␈α⊂of␈α⊃AM's␈α⊃starting␈α⊂concepts␈α⊃(page␈α⊃)␈α⊂is␈α⊃worth␈α⊃a␈α⊂look,␈α⊃even␈α⊃out␈α⊂of
␈↓ α,␈↓context.

␈↓ α,␈↓Most␈α
of␈α
the␈α
results␈α
of␈α
the␈α
project␈α
are␈α
presented␈α
in␈α
Chapter␈α
6.␈α
In␈α
addition␈α
to␈αsimply␈α
`running'
␈↓ α,␈↓AM,␈αseveral␈α
experiments␈αhave␈αbeen␈α
conducted␈αwith␈α
it.␈α It's␈αawkward␈α
to␈αevaluate␈α
AM,␈αand
␈↓ α,␈↓therefore Chapter 7 is quite long and detailed.

␈↓ α,␈↓The␈α∪appendices␈α∩provide␈α∪material␈α∪which␈α∩supplements␈α∪the␈α∪text.␈α∩Appendix␈α∪1␈α∪contains␈α∩a
␈↓ α,␈↓description␈α∂of␈α∂a␈α∞few␈α∂of␈α∂the␈α∂initial␈α∞concepts,␈α∂some␈α∂examples␈α∂of␈α∞how␈α∂they␈α∂were␈α∂coded␈α∞into
␈↓ α,␈↓Lisp,␈α∪and␈α∪a␈α∪partial␈α∪list␈α∪of␈α∪the␈α∪concepts␈α∪AM␈α∪de≡ned␈α∪and␈α∪investigated␈α∪along␈α∪the␈α∩way.
␈↓ α,␈↓Appendix␈α∞2␈α∞exhibits␈α
all␈α∞242␈α∞heuristics␈α
that␈α∞AM␈α∞is␈α
explicitly␈α∞provided␈α∞with.␈α∞ Appendix␈α
3
␈↓ α,␈↓contains traces of AM in action.

␈↓ α,␈↓This␈α
book␈α∞¬␈α
and␈α
its␈α∞readers␈α
¬␈α
must␈α∞come␈α
to␈α
grips␈α∞with␈α
a␈α
very␈α∞interdisciplinary␈α
problem.
␈↓ α,␈↓For␈αthe␈αreader␈αwhose␈αbackground␈αis␈αin␈αArti≡cial␈αIntelligence,␈αmost␈αof␈αthe␈αsystem's␈αactions␈α¬
␈↓ α,␈↓the␈α"mathematics"␈αit␈αdoes␈α¬␈α
may␈αseem␈αinherently␈αuninteresting.␈αFor␈αthe␈α
mathematician,␈αthe
␈↓ α,␈↓word␈α"LISP"␈αsigni≡es␈α
nothing␈αbeyond␈αa␈α
speech␈αimpediment␈α(to␈α
Arti≡cial␈αIntelligence␈αtypes␈α
it
␈↓ α,␈↓also␈αconnotes␈αa␈αprogramming␈αimpediment).␈αAs␈αa␈αresult,␈αthere␈αis␈αa␈αgood␈αbit␈αof␈αde≡nition␈αand
␈↓ α,␈↓explanation, and the reader's indulgence is entreated.
␈↓ α,␈↓␈↓ ,-␈↓ε13␈↓-




␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ β!␈↓∧Chapter 2.   An Example: Discovering Prime Numbers␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓



␈↓ α,␈↓This␈αchapter␈αwill␈αpresent␈αan␈αexample␈αof␈αAM␈αin␈αaction,␈αan␈αexcerpt␈αfrom␈αthe␈αoutput␈αof␈αAM,
␈↓ α,␈↓as it investigates some concepts.

␈↓ α,␈↓After␈α∂a␈α∂brief␈α∂discussion␈α∂of␈α∞AM's␈α∂control␈α∂structure␈α∂in␈α∂Section␈α∞2.1,␈α∂the␈α∂reader␈α∂will␈α∂be␈α∞told
␈↓ α,␈↓what␈α
the␈α
point␈α
of␈α
this␈α
example␈α
is␈α
¬␈α
and␈α
is␈α
␈↓βnot␈↓.␈α
 Section␈α
2.3␈α
provides␈α
a␈α
few␈αeleventh-hour
␈↓ α,␈↓hints at decoding the example.

␈↓ α,␈↓The␈αexcerpt␈αitself␈αfollows␈αin␈αSection␈α2.4.␈α It␈αskips␈αthe␈α≡rst␈αhalf␈αof␈αthe␈αsession,␈αand␈αpicks␈αup
␈↓ α,␈↓at␈αa␈α
point␈αjust␈α
after␈αAM␈αhas␈α
de≡ned␈αthe␈α
concept␈α"Divisors-of".␈α
 Soon␈αafterward,␈αAM␈α
de≡nes
␈↓ α,␈↓Primes,␈α
and␈αbegins␈α
to␈α
≡nd␈αinteresting␈α
conjectures␈α
related␈αto␈α
them.␈α
 The␈αexcerpt␈α
goes␈α
on␈αto
␈↓ α,␈↓show␈α↔how␈α⊗AM␈α↔conjectured␈α⊗the␈α↔fundamental␈α⊗theorem␈α↔of␈α⊗arithmetic␈α↔and␈α⊗Goldbach's
␈↓ α,␈↓conjecture.␈α∂ AM␈α∂derived␈α∂the␈α∂notion␈α∂of␈α∂partitioning␈α∂a␈α∂collection␈α∂of␈α∂n␈α∂objects␈α⊂into␈α∂smaller
␈↓ α,␈↓bundles,␈αbut␈αfailed␈αto␈α≡nd␈αany␈αinteresting␈αconjectures␈αabout␈αthat␈αprocess.␈α Instead,␈αAM␈αwas
␈↓ α,␈↓side-tracked␈αinto␈αthe␈α(probably)␈αfruitless␈αinvestigation␈αof␈αnumbers␈αwhich␈αcan␈αbe␈αrepresented
␈↓ α,␈↓as the sum of two primes in one unique way.

␈↓ α,␈↓The␈α∞≡nal␈α∞section␈α
of␈α∞this␈α∞chapter␈α∞will␈α
recap␈α∞this␈α∞example␈α∞the␈α
way␈α∞a␈α∞math␈α∞historian␈α
might
␈↓ α,␈↓report it.


␈↓ α,␈↓␈↓ ∧O␈↓∧␈↓&2.1. Discussion of the AM Program␈↓)αβ␈↓


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&2.1.1. Representation␈↓)αβ␈↓

␈↓ α,␈↓AM␈α
is␈α
a␈α
program␈α
which␈α
expands␈αa␈α
knowledge␈α
base␈α
of␈α
mathematical␈α
concepts.␈α Each␈α
concept
␈↓ α,␈↓is␈α⊂stored␈α⊃as␈α⊂a␈α⊃particular␈α⊂kind␈α⊂of␈α⊃data␈α⊂structure,␈α⊃namely␈α⊂as␈α⊂a␈α⊃collection␈α⊂of␈α⊃properties␈α⊂or
␈↓ α,␈↓"facets" of the concept.  For example, here is a miniature example of a concept␈↓	1␈↓:








␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε␈αλThe␈αλright␈αλarrow␈αλ("→")␈α	in␈αλthe␈αλbox␈αλon␈αλthe␈αλnext␈α	page␈αλis␈αλthe␈αλsymbol␈αλfor␈αλ"implies".␈α	"Nos."␈αλis␈αλan␈αλabbreviation␈αλfor␈α	"Numbers".␈αλThe
␈↓ α,␈↓ε␈↓ βLvertical␈αλbar␈αλ"|"␈αλis␈αλa␈αλsymbol␈αλfor␈αλthe␈αλpredicate␈αλ"divides␈αλevenly␈αλinto";␈αλthe␈αλhook␈αλ"¬"␈αλis␈αλa␈αλsymbol␈αλfor␈αλthe␈αλpredicate
␈↓ α,␈↓ε␈↓ βL"the negation of". "⊗" indicates exclusive or, and the symbol "∀" is read "for all".
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε14␈↓-

␈↓"␈↓ α,␈↓π⊂ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π}                                                                         ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ NAME: Prime Numbers ␈↓π                                                     ␈↓ L}
␈↓"␈↓ α,␈↓π}                                                                         ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ DEFINITIONS: ␈↓π                                                            ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            ORIGIN: Number-of-divisors-of(x) = 2 ␈↓π                                   ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            PREDICATE-CALCULUS: Prime(x) ≡ (∀z)(z|x →  z=1 ⊗ z=x) ␈↓π                   ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            ITERATIVE: (for x>1): For i from 2 to Sqrt(x), ¬(i|x) ␈↓π                         ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ EXAMPLES: 2, 3, 5, 7, 11, 13, 17 ␈↓π                                            ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            BOUNDARY: 2, 3 ␈↓π                                                   ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            BOUNDARY-FAILURES: 0, 1 ␈↓π                                          ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            FAILURES: 12 ␈↓π                                                     ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ GENERALIZATIONS: Nos., Nos. with an even no. of divisors, Nos. with a prime no. of divisors ␈↓π ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ SPECIALIZATIONS: Odd Primes, Prime Pairs, Prime Uniquely-addables ␈↓π                 ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ CONJECS: Unique factorization, Goldbach's conjecture, Extremes of Number-of-divisors-of ␈↓π  ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ INTU'S: ␈↓βA metaphor to the e≥ect that Primes are the building blocks of all numbers␈↓¬ ␈↓π     ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ANALOGIES: ␈↓π                                                             ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            Maximally-divisible numbers are converse extremes of Number-of-divisors-of ␈↓π     ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬            Factor a non-simple group into simple groups ␈↓π                             ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ INTEREST: Conjectures tying Primes to TIMES, to Divisors-of, to closely related operations ␈↓π  ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ WORTH: 800 ␈↓π                                                             ␈↓ L}
␈↓"␈↓ α,␈↓π} ␈↓¬ ␈↓π                                                                       ␈↓ L}
␈↓"␈↓ α,␈↓π%ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$

␈↓ α,␈↓"Creating␈α∩a␈α∩new␈α∩concept"␈α∪is␈α∩a␈α∩well-de≡ned␈α∩activity:␈α∩it␈α∪involves␈α∩setting␈α∩up␈α∩a␈α∪new␈α∩data
␈↓ α,␈↓structure␈αlike␈αthe␈αone␈α
above,␈αand␈α≡lling␈αin␈αentries␈α
for␈αsome␈αof␈αits␈α
facets␈αor␈αslots.␈α Filling␈αin␈α
a
␈↓ α,␈↓particular␈α∂facet␈α∞of␈α∂a␈α∂particular␈α∞concept␈α∂is␈α∞also␈α∂quite␈α∂well-de≡ned,␈α∞and␈α∂is␈α∂accomplished␈α∞by
␈↓ α,␈↓executing␈α∂a␈α∞collection␈α∂of␈α∞relevant␈α∂heuristic␈α∞rules.␈α∂ This␈α∞process␈α∂will␈α∞be␈α∂described␈α∂in␈α∞great
␈↓ α,␈↓detail in later chapters.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&2.1.2. Agenda and Heuristics␈↓)αβ␈↓

␈↓ α,␈↓An␈α∞agenda␈α
of␈α∞plausible␈α∞tasks␈α
is␈α∞maintained␈α
by␈α∞AM.␈α∞ A␈α
typical␈α∞task␈α
is␈α∞␈↓¬"Fill-in␈α∞examples␈α
of
␈↓ α,␈↓¬Primes"␈↓.␈α∂The␈α∂agenda␈α∂may␈α⊂contain␈α∂hundreds␈α∂of␈α∂entries␈α⊂such␈α∂as␈α∂this␈α∂one.␈α⊂ AM␈α∂repeatedly
␈↓ α,␈↓selects␈α∞the␈α∂top␈α∞task␈α∂from␈α∞the␈α∂agenda␈α∞and␈α∂tries␈α∞to␈α∂carry␈α∞it␈α∂out.␈α∞ This␈α∂is␈α∞the␈α∂whole␈α∞control
␈↓ α,␈↓structure!␈αOf␈αcourse,␈αwe␈αmust␈αstill␈αexplain␈α
how␈αAM␈αcreates␈αplausible␈αnew␈αtasks␈αto␈α
place␈αon
␈↓ α,␈↓the␈α
agenda,␈α
how␈α
AM␈α
decides␈α
which␈α
task␈α
will␈α
be␈α
the␈α
best␈α
one␈α
to␈α
execute␈α
next,␈α
and␈α∞how␈α
it
␈↓ α,␈↓carries out a task.
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε15␈↓-

␈↓ α,␈↓If␈α∞the␈α
task␈α∞is␈α∞␈↓¬"Fill␈α
in␈α∞new␈α
Algorithms␈α∞for␈α∞Set-union"␈↓,␈α
then␈α∞␈↓βsatisfying␈↓␈α
it␈α∞would␈α∞mean␈α
actually
␈↓ α,␈↓synthesizing␈α∞some␈α∞new␈α∞procedures,␈α
some␈α∞new␈α∞LISP␈α∞code␈α
capable␈α∞of␈α∞forming␈α∞the␈α∞union␈α
of
␈↓ α,␈↓any␈αtwo␈αsets.␈α A␈αheuristic␈αrule␈αis␈α␈↓βrelevant␈↓␈αto␈αa␈αtask␈αi≥␈αexecuting␈αthat␈αrule␈αbrings␈αAM␈αcloser
␈↓ α,␈↓to␈αsatisfying␈αthat␈αtask.␈αRelevance␈αis␈αdetermined␈αa␈αpriori␈αby␈αwhere␈αthe␈αrule␈αis␈αstored.␈αA␈αrule
␈↓ α,␈↓tacked␈αonto␈αthe␈α
Domain/range␈αfacet␈αof␈α
the␈αCompose␈αconcept␈α
would␈αbe␈αpresumed␈αrelevant␈α
to
␈↓ α,␈↓the task ␈↓¬"Check the Domain/range of Insert␈↓εo␈↓¬Delete"␈↓.

␈↓ α,␈↓Once␈α
a␈α
task␈α
is␈α
chosen␈α
from␈α
the␈α∞agenda,␈α
AM␈α
gathers␈α
some␈α
heuristic␈α
rules␈α
which␈α∞might␈α
be
␈↓ α,␈↓relevant␈αto␈αsatisfying␈αthat␈αtask.␈α They␈αare␈αexecuted,␈αand␈αthen␈αAM␈αpicks␈αa␈αnew␈αtask.␈α While
␈↓ α,␈↓a rule is executing, three kinds of actions or e≥ects can occur:

␈↓ α,␈↓(i)␈αFacets␈αof␈αsome␈αconcepts␈αcan␈αget␈α≡lled␈αin␈α(e.g.,␈αexamples␈αof␈αprimes␈αmay␈αactually␈αbe␈αfound
␈↓ α,␈↓␈↓ αland␈αtacked␈α
onto␈αthe␈α
"Examples"␈αfacet␈α
of␈αthe␈α
"Primes"␈αconcept).␈α
 A␈αtypical␈αheuristic␈α
rule
␈↓ α,␈↓␈↓ αlwhich might have this e≥ect is:

␈↓ α,␈↓¬␈↓ β,To fill in examples of X, where X is a kind of Y (for some more general concept Y),
␈↓ α,␈↓¬␈↓ β,Check the examples of Y; some of them may be examples of X as well.

␈↓ α,␈↓␈↓ αlFor␈α∂the␈α∂task␈α∞of␈α∂≡lling␈α∂in␈α∞examples␈α∂of␈α∂Primes,␈α∞this␈α∂rule␈α∂would␈α∞have␈α∂AM␈α∂notice␈α∞that
␈↓ α,␈↓␈↓ αlPrimes␈α⊃is␈α⊃a␈α⊃kind␈α⊃of␈α⊃Number,␈α⊃and␈α⊃therefore␈α⊃look␈α⊃over␈α⊃all␈α⊃the␈α⊃known␈α⊃examples␈α⊂of
␈↓ α,␈↓␈↓ αlNumber.␈α
Some␈α∞of␈α
those␈α
would␈α∞be␈α
primes,␈α∞and␈α
would␈α
be␈α∞transferred␈α
to␈α∞the␈α
Examples
␈↓ α,␈↓␈↓ αlfacet of Primes.

␈↓ α,␈↓(ii)␈αNew␈αconcepts␈α
may␈αbe␈αcreated␈α
(e.g.,␈αthe␈αconcept␈α
"primes␈αwhich␈αare␈αuniquely␈α
representable
␈↓ α,␈↓␈↓ αlas␈α⊂the␈α⊂sum␈α⊂of␈α⊂two␈α⊂other␈α⊂primes"␈α⊂may␈α⊂be␈α⊂somehow␈α⊂be␈α⊂deemed␈α⊂worth␈α⊂studying).␈α∂ A
␈↓ α,␈↓␈↓ αltypical heuristic rule which might result in this new concept is:

␈↓ α,␈↓¬␈↓ β,If some (but not most) examples of X are also examples of Y (for some concept Y),
␈↓ α,␈↓¬␈↓ β,Create a new concept defined as the intersection of those 2 concepts (X and Y).

␈↓ α,␈↓␈↓ αlSuppose␈αAM␈αhas␈α
already␈αisolated␈αthe␈αconcept␈α
of␈αbeing␈αrepresentable␈α
as␈αthe␈αsum␈αof␈α
two
␈↓ α,␈↓␈↓ αlprimes␈α⊃in␈α⊃only␈α⊂one␈α⊃way␈α⊃(AM␈α⊂actually␈α⊃calls␈α⊃such␈α⊃numbers␈α⊂"Uniquely-prime-addable
␈↓ α,␈↓␈↓ αlnumbers").␈α∂ When␈α∂AM␈α∞notices␈α∂that␈α∂some␈α∂primes␈α∞are␈α∂in␈α∂this␈α∞set,␈α∂the␈α∂above␈α∂rule␈α∞will
␈↓ α,␈↓␈↓ αlcreate␈α
a␈αbrand␈α
new␈αconcept,␈α
de≡ned␈αas␈α
the␈αset␈α
of␈αnumbers␈α
which␈αare␈α
both␈α
prime␈αand
␈↓ α,␈↓␈↓ αluniquely prime addable.

␈↓ α,␈↓(iii)␈α
New␈αtasks␈α
may␈αbe␈α
added␈αto␈α
the␈α
agenda␈α(e.g.,␈α
the␈αcurrent␈α
activity␈αmay␈α
suggest␈α
that␈αthe
␈↓ α,␈↓␈↓ αlfollowing␈α∂task␈α∂is␈α∂worth␈α∂considering:␈α∞"Generalize␈α∂the␈α∂concept␈α∂of␈α∂prime␈α∂numbers").␈α∞ A
␈↓ α,␈↓␈↓ αltypical heuristic rule which might have this e≥ect is:

␈↓ α,␈↓¬␈↓ β,If very few examples of X are found,
␈↓ α,␈↓¬␈↓ β,Then add the following task to the agenda: "Generalize the concept X".

␈↓ α,␈↓␈↓ αlOf␈α
course,␈α
AM␈α
contains␈αa␈α
precise␈α
meaning␈α
for␈α
the␈αphrase␈α
"very␈α
few".␈α
 When␈αAM␈α
looks
␈↓ α,␈↓␈↓ αlfor␈αprimes␈αamong␈αexamples␈αof␈αalready-known␈αkinds␈αof␈αnumbers,␈αit␈αwill␈α≡nd␈αdozens␈αof
␈↓ α,␈↓␈↓ αlnon-examples␈αfor␈α
every␈αexample␈αof␈α
a␈αprime␈α
it␈αuncovers.␈α "Very␈α
few"␈αis␈α
thus␈αnaturally
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε16␈↓-

␈↓ α,␈↓␈↓ αlimplemented as a statistical con≡dence level␈↓	2␈↓.

␈↓ α,␈↓The␈α∂concept␈α∞of␈α∂an␈α∞agenda␈α∂is␈α∞certainly␈α∂not␈α∞new:␈α∂schedulers␈α∞have␈α∂been␈α∞around␈α∂for␈α∂a␈α∞long
␈↓ α,␈↓time.␈α∞ But␈α
one␈α∞important␈α∞feature␈α
of␈α∞AM's␈α
agenda␈α∞scheme␈α∞␈↓βis␈↓␈α
a␈α∞new␈α
idea:␈α∞attaching␈α∞¬␈α
and
␈↓ α,␈↓using␈α¬␈αa␈αlist␈αof␈αquasi-symbolic␈↓	3␈↓␈αreasons␈αto␈αeach␈αtask␈αwhich␈αexplain␈αwhy␈αthe␈αtask␈αis␈αworth
␈↓ α,␈↓considering,␈αwhy␈α
it's␈αplausible.␈α
 ␈↓βIt␈αis␈α
the␈αresponsibility␈α
of␈αthe␈α
heuristic␈αrules␈α
to␈αinclude␈α
reasons
␈↓ α,␈↓βfor␈αany␈α
tasks␈αthey␈α
propose␈↓.␈↓	4␈↓␈αFor␈αexample,␈α
let's␈αreconsider␈α
the␈αheuristic␈α
rule␈αmentioned␈αin␈α
(iii)
␈↓ α,␈↓above.  It really looks more like the following:

␈↓ α,␈↓¬␈↓ β,If very few examples of X are found,
␈↓ α,␈↓¬␈↓ β,Then␈α
add␈α
the␈α
following␈α
task␈α
to␈α
the␈α
agenda:␈α
"Generalize␈α
the␈α
concept␈α
X",␈α
for␈α
the␈α
following
␈↓ α,␈↓¬␈↓ ∧,reason:␈α"X's␈αare␈αquite␈αrare;␈αa␈αslightly␈αless␈αrestrictive␈αconcept␈αmight␈αbe␈α
more
␈↓ α,␈↓¬␈↓ ∧,interesting".


␈↓ α,␈↓If␈α
the␈α
same␈α∞task␈α
is␈α
proposed␈α∞by␈α
several␈α
rules,␈α∞then␈α
several␈α
di≥erent␈α∞reasons␈α
for␈α
it␈α∞may␈α
be
␈↓ α,␈↓present.␈α⊃ In␈α⊃addition,␈α∩one␈α⊃ephemeral␈α⊃reason␈α⊃also␈α∩exists:␈α⊃"Focus␈α⊃of␈α⊃attention".␈α∩Any␈α⊃tasks
␈↓ α,␈↓which␈αare␈αsimilar␈αto␈αthe␈αone␈αlast␈αexecuted␈αget␈α"Focus␈αof␈αattention"␈αas␈αa␈αbonus␈αreason.␈α AM
␈↓ α,␈↓uses␈α⊗all␈α∃these␈α⊗reasons,␈α⊗e.g.␈α∃ to␈α⊗decide␈α∃how␈α⊗to␈α⊗rank␈α∃the␈α⊗tasks␈α∃on␈α⊗the␈α⊗agenda.␈α∃ The
␈↓ α,␈↓"intelligence"␈α
AM␈α
exhibits␈α
is␈α
not␈α
so␈α
much␈α
"what␈α
it␈α
does",␈α
but␈α
rather␈α
the␈α
␈↓βorder␈↓␈α
in␈α
which␈α
it
␈↓ α,␈↓arranges␈α∞its␈α∞agenda␈↓	5␈↓.␈α
 AM␈α∞uses␈α∞the␈α
list␈α∞of␈α∞reasons␈α
in␈α∞another␈α∞way:␈α
Once␈α∞a␈α∞task␈α∞has␈α
been
␈↓ α,␈↓selected,␈αthe␈αquality␈αof␈αthe␈αreasons␈αis␈αused␈αto␈αdecide␈αhow␈αmuch␈αtime␈αand␈αspace␈αthe␈αtask␈αwill
␈↓ α,␈↓be␈α∪permitted␈α∩to␈α∪absorb,␈α∪before␈α∩AM␈α∪quits␈α∩and␈α∪moves␈α∪on␈α∩to␈α∪a␈α∩new␈α∪task.␈α∪ This␈α∩whole
␈↓ α,␈↓mechanism will be detailed in Section 3.3.2, on Page 31.


␈↓ α,␈↓␈↓ αT␈↓∧␈↓&2.2. What to get out of -- and NOT get out of -- this example␈↓)αβ␈↓

␈↓ α,␈↓The␈α∞purpose␈α
of␈α∞the␈α
example␈α∞which␈α∞begins␈α
on␈α∞page␈α
19␈α∞is␈α∞to␈α
convey␈α∞a␈α
bit␈α∞of␈α∞AM's␈α
∨avor.
␈↓ α,␈↓After␈αreading␈αthrough␈αit,␈αthe␈αreader␈αshould␈αbe␈αconvinced␈αthat␈αAM␈αis␈α␈↓βnot␈↓␈αa␈αtheorem-prover,
␈↓ α,␈↓nor␈α∃is␈α∃it␈α∃␈↓βrandomly␈↓␈α∃manipulating␈α∃entries␈α∃in␈α∃a␈α∃knowledge␈α∃base,␈α∃nor␈α∃is␈α∃it␈α∃␈↓βexhaustively␈↓
␈↓ α,␈↓manipulating␈α↔or␈α↔searching.␈α↔ AM␈α↔is␈α↔carefully␈α↔growing␈α↔a␈α↔network␈α↔of␈α↔data␈α↔structures
␈↓ α,␈↓representing␈αmathematical␈α
concepts,␈αby␈αrepeatedly␈α
using␈αheuristics␈αboth␈α
(a)␈αfor␈α
guidance␈αin
␈↓ α,␈↓choosing a task to work on next, and (b) to provide methods to satisfy the chosen task.



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	2␈↓ε␈αλThe␈α	ratio␈αλof␈α	examples␈αλfound␈αλto␈α	non-examples␈αλstumbled␈α	over␈αλlies␈α	between␈αλ.001␈αλand␈α	.05.␈αλPhilosophers␈α	outraged␈αλby␈α	this␈αλmay
␈↓ α,␈↓ε␈↓ βLbe␈αsomewhat␈αappeased␈αby␈αknowledge␈αthat␈αlarge␈αchanges␈αin␈αthe␈αprecise␈αnumbers␈αvery␈αrarely␈αalter␈αAM's
␈↓ α,␈↓ε␈↓ βLbehavior.
␈↓ α,␈↓ε␈↓	3␈↓ε␈α	Each␈α	reason␈α	is␈αλan␈α	English␈α	sentence.␈α	While␈αλAM␈α	can␈α	tell␈α	whether␈αλtwo␈α	given␈α	reasons␈α	coincide,␈αλit␈α	can't␈α	actually␈α	do␈α	any␈αλinternal
␈↓ α,␈↓ε␈↓ βLprocessing␈α
on␈α
them.␈α
If␈α
this␈α
lack␈α
of␈α
intelligence␈α
had␈α
proved␈α
to␈α
be␈α
a␈α
limiting␈α
problem,␈α
then␈α
more␈α
work␈α	would
␈↓ α,␈↓ε␈↓ βLhave been expended on giving AM some such abilities.
␈↓ α,␈↓ε␈↓	4␈↓ε␈αAn␈αalternative␈αscheme,␈αperhaps␈αeven␈αa␈αbit␈αmore␈αhuman-like,␈αwould␈αbe␈αto␈α(perhaps␈αonly␈αoccasionally)␈αallow␈αa␈αburst␈αof
␈↓ α,␈↓ε␈↓ βLpoorly-motivated␈α	tasks␈α	to␈α	be␈α	proposed,␈α	and␈α	then␈α	use␈α	some␈α	pruning␈α	criteria␈α	to␈α	weed␈α	out␈α	the␈α	obvious␈α	losers.
␈↓ α,␈↓ε␈↓ βLDuring␈α	this␈α	time,␈αλAM␈α	could␈α	type␈α	out␈αλto␈α	the␈α	user␈αλ(who␈α	otherwise␈α	would␈α	be␈αλclosely␈α	monitoring␈α	its␈α	activities)␈αλa
␈↓ α,␈↓ε␈↓ βLcute anthropomorphic phrase like "I'm now sitting back and puffing on my pipe, lost in contemplation."
␈↓ α,␈↓ε␈↓	5␈↓ε␈αFor␈αexample,␈α
alternating␈αa␈αrandomly-chosen␈α
task␈αand␈αthe␈α"best"␈α
task␈α(the␈αone␈α
AM␈αchose␈αto␈α
do)␈αonly␈αslows␈αthe␈α
system
␈↓ α,␈↓ε␈↓ βLdown␈αby␈αa␈αfactor␈αof␈α2,␈αyet␈αit␈αtotally␈αdestroys␈αits␈αcredibility␈αas␈αa␈αrational␈αresearcher␈α(as␈αjudged␈αby␈αthe
␈↓ α,␈↓ε␈↓ βLhuman user of AM).  This is one conclusion of experiment 2 (see Section 6.2.2, page 98).
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε17␈↓-


␈↓ α,␈↓The following points are important but can't be conveyed by any lone example:

␈↓ α,␈↓(i)␈αAlthough␈αAM␈αappears␈αto␈αhave␈αreasonable␈αnatural␈αlanguage␈αabilities,␈αthis␈αis␈αa␈αtypical␈αAI
␈↓ α,␈↓␈↓ αlillusion:␈αmost␈αof␈αthe␈αphrases␈αAM␈αtypes␈αare␈αmere␈αtokens,␈αand␈αthe␈αsyntax␈αwhich␈αthe␈αuser
␈↓ α,␈↓␈↓ αlmust␈αobey␈α
is␈αunnaturally␈αconstrained.␈α
For␈αthe␈αsake␈α
of␈αclarity,␈αI␈α
have␈α"touched␈αup"␈α
some
␈↓ α,␈↓␈↓ αlof␈α
the␈α
wording,␈αindentation,␈α
syntax,␈α
etc.␈αof␈α
what␈α
AM␈α
actually␈αoutputs,␈α
but␈α
left␈αthe␈α
spirit
␈↓ α,␈↓␈↓ αlof␈αeach␈α
phrase␈αintact.␈α
 As␈αthe␈α
reader␈αbecomes␈α
more␈αfamiliar␈α
with␈αAM,␈αfuture␈α
examples
␈↓ α,␈↓␈↓ αlcan␈α⊂be␈α⊂"unretouched".␈α⊂ If␈α⊂he␈α⊂wishes,␈α⊂he␈α⊂may␈α⊂glance␈α⊂at␈α⊂Appendix␈α⊂3.0,␈α⊂which␈α∂shows
␈↓ α,␈↓␈↓ αlsome actual listings of AM in action.

␈↓ α,␈↓(ii)␈αThe␈αreader␈αshould␈αbe␈αskeptical␈αof␈αthe␈αgenerality␈αof␈αthe␈αprogram;␈αis␈αthe␈αknowledge␈αbase
␈↓ α,␈↓␈↓ αl"just␈α⊂right"␈α⊂(i.e.,␈α⊂≡nely␈α∂tuned␈α⊂to␈α⊂elicit␈α⊂this␈α⊂one␈α∂chain␈α⊂of␈α⊂behaviors)?␈α⊂ The␈α⊂answer␈α∂is
␈↓ α,␈↓␈↓ αl"␈↓βNo␈↓"␈↓	6␈↓.␈αThe␈αwhole␈αpoint␈αof␈αthis␈αproject␈αis␈αto␈αshow␈αthat␈αa␈αrelatively␈αsmall␈αset␈αof␈αgeneral
␈↓ α,␈↓␈↓ αlheuristics␈α∩can␈α∪guide␈α∩a␈α∪nontrivial␈α∩discovery␈α∪process.␈α∩ Each␈α∪activity,␈α∩each␈α∪task,␈α∩was
␈↓ α,␈↓␈↓ αlproposed␈α∞by␈α
some␈α∞heuristic␈α
rule␈α∞(like␈α∞"look␈α
for␈α∞extreme␈α
cases␈α∞of␈α
X")␈α∞which␈α∞was␈α
used
␈↓ α,␈↓␈↓ αltime␈αand␈αtime␈αagain,␈αin␈αmany␈α
situations.␈α It␈αwas␈αnot␈αconsidered␈αfair␈αto␈α
insert␈αheuristic
␈↓ α,␈↓␈↓ αlguidance which could only "guide" in a single situation.

␈↓ α,␈↓␈↓ αlThis␈α∂kind␈α⊂of␈α∂generality␈α⊂can't␈α∂be␈α⊂shown␈α∂convincingly␈α⊂in␈α∂one␈α⊂example.␈α∂ Nevertheless,
␈↓ α,␈↓␈↓ αleven␈α↔within␈α↔this␈α↔small␈α↔excerpt,␈α↔the␈α↔same␈α↔line␈α↔of␈α↔development␈α↔which␈α↔leads␈α↔to
␈↓ α,␈↓␈↓ αldecomposing␈α
numbers␈α(using␈α
TIMES␈↓	-1␈↓)␈α
and␈αthereby␈α
discovering␈α
unique␈αfactorization,
␈↓ α,␈↓␈↓ αlalso␈αleads␈αto␈αdecomposing␈αnumbers␈α
(using␈αADD␈↓	-1␈↓)␈αand␈αthereby␈αdiscovering␈α
Goldbach's
␈↓ α,␈↓␈↓ αlconjecture.␈α
 The␈αsame␈α
heuristic␈α
which␈αcaused␈α
AM␈α
to␈αexpect␈α
that␈α
unique␈αfactorization
␈↓ α,␈↓␈↓ αlwill be useful, also caused AM to suspect that Goldbach's conjecture will be useless.

␈↓ α,␈↓Let␈α∪me␈α∪reemphasize␈α∪that␈α∩the␈α∪"point"␈α∪of␈α∪this␈α∩example␈α∪is␈α∪␈↓βnot␈↓␈α∪the␈α∪speci≡c␈α∩mathematical
␈↓ α,␈↓concepts,␈αnor␈αthe␈αparticular␈αchains␈αof␈αplausible␈αreasoning␈αAM␈αproduces,␈αnor␈αthe␈αfew␈α∨ashy
␈↓ α,␈↓conjectures AM spouts, but rather an illustration of the ␈↓βkinds␈↓ of things AM does.


␈↓ α,␈↓␈↓ ¬↓␈↓∧␈↓&2.3. Deciphering the Example␈↓)αβ␈↓

␈↓ α,␈↓Recall␈αthat␈αin␈αgeneral,␈αeach␈αtask␈αon␈αthe␈αagenda␈αwill␈αhave␈αseveral␈αreasons␈αattached␈αto␈αit.␈α In
␈↓ α,␈↓the␈αexample␈αexcerpt,␈αthe␈αreasons␈αfor␈αeach␈αtask␈αare␈αprinted␈αjust␈αafter␈αthe␈αtask␈αis␈αchosen,␈αand
␈↓ α,␈↓before it's executed.

␈↓ α,␈↓AM␈α∩numbers␈α∪its␈α∩activities␈α∪sequentially.␈α∩Each␈α∪time␈α∩a␈α∩new␈α∪task␈α∩is␈α∪chosen,␈α∩a␈α∪counter␈α∩is
␈↓ α,␈↓incremented.␈αThe␈α≡rst␈αtask␈αin␈αthe␈αexample␈αexcerpt␈αis␈αlabelled␈α␈↓¬**␈α␈↓&TASK␈α65␈↓)αβ␈α**␈↓,␈αmeaning␈αthat
␈↓ α,␈↓the␈αexample␈αskips␈αthe␈α
≡rst␈α64␈αtasks␈αwhich␈α
AM␈αselects␈αand␈αcarries␈α
out.␈α The␈αreason␈αsimply␈α
is
␈↓ α,␈↓that␈α∪the␈α∪development␈α∪of␈α∪simple␈α∪concepts␈α∪related␈α∪to␈α∪divisibility␈α∪will␈α∪probably␈α∪be␈α∪more
␈↓ α,␈↓intelligible and palatable to the reader, than AM's early ramblings in ≡nite set theory.




␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	6␈↓ε The ␈↓&design␈↓)αβ of AM was finely tuned so that the answer to this question would be "No". Ponder that one!
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε18␈↓-

␈↓ α,␈↓In␈α⊂the␈α⊂example␈α⊃itself,␈α⊂several␈α⊂irrelevant␈α⊃tasks␈α⊂have␈α⊂been␈α⊃excised␈↓	7␈↓.␈α⊂ About␈α⊂half␈α⊃of␈α⊂those
␈↓ α,␈↓omitted␈αtasks␈αwere␈αinteresting␈αin␈αthemselves,␈αbut␈αall␈αof␈αthem␈αwere␈αtangential␈αor␈αunrelated␈αto
␈↓ α,␈↓the␈αdevelopment␈αshown.␈α The␈αreader␈αcan␈αtell␈αby␈αthe␈αglobal␈αtask␈αnumbering␈αhow␈αmany␈αwere
␈↓ α,␈↓skipped.  For example, notice that the excerpt jumps from Task 67 to Task 79.

␈↓ α,␈↓To␈α⊃help␈α⊃gauge␈α⊃AM's␈α⊃abilities,␈α⊃the␈α⊃reader␈α⊃may␈α⊃be␈α⊃interested␈α⊃to␈α⊃know␈α⊃that␈α⊃AM␈α⊃de≡ned
␈↓ α,␈↓"Natural␈α∩Numbers"␈α⊃during␈α∩Task␈α⊃44,␈α∩and␈α∩"TIMES"␈α⊃was␈α∩de≡ned␈α⊃during␈α∩Task␈α∩57.␈α⊃ AM
␈↓ α,␈↓started␈α
with␈α
no␈α
knowledge␈α
of␈α
numbers,␈α
and␈α
only␈α
scanty␈α
knowledge␈α
of␈α
sets␈αand␈α
set-operations.
␈↓ α,␈↓Task 3, e.g., was to ≡ll in examples of Sets.

␈↓ α,␈↓The␈α
concepts␈α
that␈α
AM␈αtalks␈α
about␈α
are␈α
self-explanatory␈α
¬␈αby␈α
and␈α
large.␈α
 Below␈αare␈α
discussed
␈↓ α,␈↓some nonstandard ones.

␈↓ α,␈↓␈↓β␈↓&BAG␈↓)αβ␈↓␈αis␈αa␈αkind␈αof␈αlist␈αstructure,␈αa␈α
bunch␈αof␈αelements␈αwhich␈αare␈αunordered,␈αbut␈αone␈αin␈α
which
␈↓ α,␈↓multiple␈α∞copies␈α∞of␈α∞the␈α∞same␈α∞element␈α∞are␈α∞permitted.␈α∞ One␈α∞may␈α∞visualize␈α∞a␈α∞paper␈α∞bag␈α
≡lled
␈↓ α,␈↓with␈αcardboard␈αletters.␈αTechnically,␈αwe␈αshall␈αsay␈αthat␈αa␈αset␈αis␈α␈↓βnot␈↓␈αconsidered␈αto␈αbe␈αa␈αbag.␈α A
␈↓ α,␈↓bag␈α
is␈α
denoted␈α∞by␈α
enclosure␈α
within␈α
parentheses,␈α∞just␈α
as␈α
sets␈α
are␈α∞within␈α
braces.␈α
So␈α∞the␈α
bag
␈↓ α,␈↓containing␈α⊃X␈α⊃and␈α⊃four␈α⊃Y's␈α⊃might␈α⊃be␈α⊃written␈α⊃(X␈α⊃Y␈α⊃Y␈α⊃Y␈α⊃Y),␈α⊃and␈α⊃would␈α⊃be␈α⊃considered
␈↓ α,␈↓indistinguishable from the bag (Y Y Y X Y).

␈↓ α,␈↓␈↓&␈↓βNumber␈↓␈↓)αβ will mean (typically) a positive integer.

␈↓ α,␈↓␈↓&␈↓βTIMES␈↓␈↓	-1␈↓␈↓)αβ␈α∞is␈α∞a␈α
particular␈α∞relation.␈α∞ For␈α
any␈α∞number␈α∞x,␈α
TIMES␈↓	-1␈↓(x)␈α∞is␈α∞a␈α
set␈α∞of␈α∞bags.␈α
Each
␈↓ α,␈↓bag contains some numbers which, when multiplied together, equal x.  For example,
␈↓ α,␈↓TIMES␈↓	-1␈↓(18)␈α⊂=␈α∂{␈α⊂(18)␈α∂(2␈α⊂9)␈α∂(2␈α⊂3␈α⊂3)␈α∂(3␈α⊂6)␈α∂}.␈α⊂ Checking,␈α∂we␈α⊂see␈α∂that␈α⊂multiplying,␈α⊂e.g.,␈α∂the
␈↓ α,␈↓numbers␈αin␈αthe␈αbag␈α(2␈α3␈α3)␈αtogether,␈αwe␈αdo␈αget␈α2x2x3=18.␈α TIMES␈↓	-1␈↓(x)␈αcontains␈αall␈α
possible
␈↓ α,␈↓such bags (containing natural numbers >1).

␈↓ α,␈↓␈↓β␈↓&ADD␈↓	-1␈↓β␈↓)αβ␈↓␈α
is␈α
a␈α∞relation␈α
analogous␈α
to␈α
TIMES␈↓	-1␈↓.␈α∞For␈α
any␈α
number␈α
x,␈α∞ADD␈↓	-1␈↓(x)␈α
is␈α
also␈α
a␈α∞set␈α
of
␈↓ α,␈↓bags.␈α
 Each␈α
bag␈α∞contains␈α
a␈α
bunch␈α∞of␈α
numbers␈α
which,␈α
when␈α∞added␈α
together,␈α
equal␈α∞x.␈α
 For
␈↓ α,␈↓example,␈α
ADD␈↓	-1␈↓(4)␈α
=␈α
{␈α(4)␈α
(1␈α
1␈α
1␈α1)␈α
(1␈α
1␈α
2)␈α(1␈α
3)␈α
(2␈α
2)␈α}.␈α
 ADD␈↓	-1␈↓(x)␈α
contains␈α
all␈αpossible␈α
such
␈↓ α,␈↓bags (containing numbers >0); it ≡nds all possible ␈↓βpartitions␈↓ of x.

␈↓ α,␈↓␈↓β␈↓&Divisors-of␈↓)αβ␈↓␈α
is␈α
a␈α
more␈α
standard␈α
relation.␈α
 For␈α∞any␈α
number␈α
x,␈α
Divisors-of(x)␈α
is␈α
the␈α
set␈α∞of␈α
all
␈↓ α,␈↓positive numbers which divide evenly into x.  For example, Divisors-of(18) = {1 2 3 6 9 18}.

␈↓ α,␈↓Whenever␈αthere␈αis␈αa␈α
con∨ict␈αbetween␈α"computer␈αscience␈α
jargon"␈αand␈α"math␈αjargon",␈α
I␈αhave
␈↓ α,␈↓opted␈αfor␈αthe␈αlatter.␈αSo,␈αe.g.,␈αall␈α"functions"␈αare␈αnecessarily␈αsingle-valued␈αfor␈αeach␈αmember␈αof
␈↓ α,␈↓their domain.

␈↓ α,␈↓AM␈αis␈α
an␈α␈↓βinteractive␈↓␈αcomputer␈α
program.␈α It␈α
prints␈αout␈αphrases␈α
indicating␈αwhat␈α
it␈αis␈αdoing␈α
at
␈↓ α,␈↓each␈αmoment,␈αand␈αa␈αhuman␈αbeing,␈αreferred␈αto␈αas␈αthe␈α␈↓βUser␈↓,␈αwatches␈αAM's␈αactivities.␈α At␈αany
␈↓ α,␈↓moment,␈α
the␈α
user␈αcan␈α
interrupt␈α
AM␈α
to␈αask␈α
questions␈α
or␈α
re-direct␈αits␈α
energies.␈α
 In␈αthe␈α
excerpt,
␈↓ α,␈↓the␈α∩user␈α∩never␈α⊃re-directs␈α∩AM,␈α∩although␈α⊃he␈α∩does␈α∩occasionally␈α⊃ask␈α∩AM␈α∩to␈α∩rename␈α⊃some
␈↓ α,␈↓concept␈α
and␈αto␈α
explain␈α
its␈αbehavior␈α
(e.g.,␈α
just␈αafter␈α
␈↓¬**␈α
␈↓&␈αTASK␈α
106␈↓)αβ␈α
**␈↓).␈α For␈α
the␈α
bene≡t␈αof␈α
the
␈↓ α,␈↓reader, I have bold-faced all the user's comments.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	7␈↓ε␈αThis␈αis␈αfair,␈αdespite␈αthe␈αresults␈αof␈αExperiment␈α2␈α(see␈αSection␈α6.2.2)␈αbecause␈αthe␈αremaining␈αtasks␈αclump␈αtogether␈αin
␈↓ α,␈↓ε␈↓ βLtwos,␈αthrees,␈αetc;␈αthey␈αare␈αuninterrupted␈αlines␈αof␈αresearch␈α(e.g.,␈αTasks␈α65-67),␈αseparated␈αby␈αvery␈αlarge
␈↓ α,␈↓ε␈↓ βLgaps (e.g., the jump from Task 67 to 79).
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε19␈↓-

␈↓ α,␈↓My personal notes and comments occur right-justi≡ed, in italics.


␈↓ α,␈↓␈↓ ¬4␈↓∧␈↓&2.4. The Example Itself␈↓)αβ␈↓

␈↓ α,␈↓α␈↓¬** ␈↓&Task 65:␈↓)αβ **␈↓α
␈↓ α,␈↓αFilling in examples of the following concept: "Divisors-of".

␈↓ α,␈↓α        3 Reasons:␈↓ ∧<(1) No known examples for Divisors-of so far.
␈↓ α,␈↓α␈↓ ∧<(2) TIMES, which is related to Divisors-of, is now very interesting.
␈↓ α,␈↓α␈↓ ∧<(3) Focus of attention: AM recently defined Divisors-of.

␈↓ α,␈↓α  26 examples found, in 9.2 seconds. e.g.: Divisors-of(6)={1 2 3 6}.
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε20␈↓-

␈↓ α,␈↓α␈↓¬** ␈↓&Task 66:␈↓)αβ **␈↓α
␈↓ α,␈↓αConsidering numbers which have very small sets of Divisors-of (i.e., very few divisors).

␈↓ α,␈↓α        2 Reasons:␈↓ ∧<(1) Worthwhile to look for extreme cases.
␈↓ α,␈↓α␈↓ ∧<(2) Focus of attention: AM recently worked on Divisors-of.

␈↓ α,␈↓α  Filling in examples of numbers with 0 divisors.
␈↓ α,␈↓α      0 examples found, in 4.0 seconds.
␈↓ α,␈↓α      Conjecture: no numbers have 0 divisors.

␈↓ α,␈↓α  Filling in examples of numbers with 1 divisors.
␈↓ α,␈↓α      1 examples found, in 4.0 seconds. e.g.: Divisors-of(1)={1}.
␈↓ α,␈↓α      Conjecture: 1 is the only number with 1 divisor.

␈↓ α,␈↓α  Filling in examples of numbers with 2 divisors.
␈↓ α,␈↓α      24 examples found, in 4.0 seconds. e.g.: Divisors-of(13)={1 13}.
␈↓ α,␈↓α      No obvious conjecture.  This kind of number merits more study.
␈↓ α,␈↓α      Creating a new concept: "Numbers-with-2-divisors".

␈↓ α,␈↓β␈↓ εAM␈α∃had␈α∃previously␈α∃derived␈α∃the␈α∃notion␈α∃of␈α∃singleton,
␈↓ α,␈↓β␈↓ εdoubleton,␈α
etc.␈α Above,␈α
AM␈α
was␈αactually␈α
de≡ning,␈αe.g.,␈α
the
␈↓ α,␈↓β␈↓ εset␈α∂of␈α∂numbers␈α∂which␈α∞gets␈α∂mapped␈α∂into␈α∂a␈α∂doubleton␈α∞of
␈↓ α,␈↓β␈↓ εnumbers,␈α
by␈α
the␈α
operation␈α
Divisors-of.␈α
 I␈α
took␈α
the␈αliberty
␈↓ α,␈↓β␈↓ εof␈α
translating␈α
this␈α
into␈α∞the␈α
above␈α
form,␈α
which␈α∞uses,␈α
e.g.,
␈↓ α,␈↓β␈↓ εthe number "2".

␈↓ α,␈↓α  Filling in examples of numbers with 3 divisors.
␈↓ α,␈↓α      11 examples found, in 4.0 seconds. e.g.: Divisors-of(49)={1 7 49}.
␈↓ α,␈↓α      All numbers with 3 divisors are also Squares.  This kind of number merits more study.
␈↓ α,␈↓α      Creating a new concept: "Numbers-with-3-divisors".

␈↓ α,␈↓β␈↓ εHow␈α∂did␈α∂AM␈α∂notice␈α∂that␈α∂conjecture?␈α∂It␈α∂took␈α∂a␈α∂random
␈↓ α,␈↓β␈↓ εexample␈α∩of␈α∩Numbers-with-3-divisiors,␈α∩say␈α∩49.␈α∩Then␈α∩it
␈↓ α,␈↓β␈↓ εasked␈α
what␈α
other␈α∞known␈α
concepts␈α
"49"␈α
was␈α∞an␈α
example
␈↓ α,␈↓β␈↓ εof.␈α∩The␈α∩two␈α⊃answers␈α∩were:␈α∩Odd-numbers␈α∩and␈α⊃Perfect-
␈↓ α,␈↓β␈↓ εsquares.␈α∂AM␈α⊂then␈α∂tested␈α∂these␈α⊂conjectures␈α∂on␈α⊂the␈α∂other
␈↓ α,␈↓β␈↓ εten␈α∞examples␈α∞just␈α∞found.␈α∞The␈α∞only␈α∞surviving␈α∞conjecture
␈↓ α,␈↓β␈↓ εwas␈α∩that␈α∩all␈α∩numbers-with-3-divisors␈α∩are␈α∩also␈α∩perfect-
␈↓ α,␈↓β␈↓ εsquares.
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε21␈↓-

␈↓ α,␈↓α␈↓¬** ␈↓&Task 67:␈↓)αβ **␈↓α
␈↓ α,␈↓αConsidering the square-roots of Numbers-with-3-divisors.

␈↓ α,␈↓α        2 Reasons:␈↓ ∧<(1) Numbers-with-3-divisors are unexpectedly also perfect Squares.
␈↓ α,␈↓α␈↓ ∧<(2) Focus of attention: AM recently worked on Numbers-with-3-divisors.

␈↓ α,␈↓α  All square-roots of Numbers-with-3-divisors seem to be Numbers-with-2-divisors.
␈↓ α,␈↓α        e.g., Divisors-of( Square-root(169) ) = Divisors-of(13) = {1 13}.

␈↓ α,␈↓α  Formulating the converse to this statement. Empirically, it seems to be true.
␈↓ α,␈↓α        The square of each Number-with-2-divisors seems to be a Number-with-3-divisors.
␈↓ α,␈↓α        This is very unusual. It is not plausibly a coincidence. (Chance of coincidence is < .001)

␈↓ α,␈↓α  Boosting interestingness factor of the concepts involved:
␈↓ α,␈↓α        Interestingness factor of "Divisors-of" raised from 300 to 400.
␈↓ α,␈↓α        Interestingness factor of "Numbers-with-2-divisors" raised from 100 to 600.
␈↓ α,␈↓α        Interestingness factor of "Numbers-with-3-divisors" raised from 200 to 700.


␈↓ α,␈↓α␈↓↓USER: Call the set of numbers with 2 divisors "Primes".␈↓α



␈↓ α,␈↓α␈↓¬** ␈↓&Task 68:␈↓)αβ **␈↓α
␈↓ α,␈↓αConsidering the squares of Numbers-with-3-divisors.

␈↓ α,␈↓α        2 Reasons:␈↓ ∧<(1) Squares of Numbers-with-2-divisors were interesting.
␈↓ α,␈↓α␈↓ ∧<(2) Focus of attention: AM recently worked on Numbers-with-3-divisors.
␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓β␈↓ εThis␈α
gap␈α
in␈α
the␈α
sequencing␈α¬␈α
from␈α
task␈α
67␈α
to␈α
task␈α79␈α
¬
␈↓ α,␈↓β␈↓ εeliminates␈α∂some␈α∂tangential␈α∂and␈α∂boring␈α∂tasks.␈α∂See␈α∂page
␈↓ α,␈↓β␈↓ ε18 for an explanation.
␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε22␈↓-

␈↓ α,␈↓α␈↓¬** ␈↓&Task 79:␈↓)αβ **␈↓α
␈↓ α,␈↓αExamining TIMES␈↓	-1␈↓α(x), looking for patterns involving its values.

␈↓ α,␈↓α        2 Reasons:␈↓ ∧<(1) TIMES␈↓	-1␈↓α is related to the newly-interesting concept "Divisors-of".
␈↓ α,␈↓α␈↓ ∧<(2) Many examples of TIMES␈↓	-1␈↓α are known, to induce from.

␈↓ α,␈↓α  Looking specifically at TIMES␈↓	-1␈↓α(12), which is { (12) (2 6) (2 2 3) (3 4) }.
␈↓ α,␈↓α    13 conjectures proposed, after 2.0 seconds.
␈↓ α,␈↓α    e.g., "TIMES␈↓	-1␈↓α(x) always contains a bag containing only even numbers".
␈↓ α,␈↓α    Testing the conjectures on other examples of TIMES␈↓	-1␈↓α.

␈↓ α,␈↓α    5 false conjectures deal with even numbers.
␈↓ α,␈↓α    AM will sometime consider the restriction of TIMES␈↓	-1␈↓α to even numbers.

␈↓ α,␈↓α    Only 2 out of the 13 conjectures are verified for all 26 known examples of TIMES␈↓	-1␈↓α:

␈↓ α,␈↓α  Conjecture 1: TIMES␈↓	-1␈↓α(x) always contains a singleton bag.
␈↓ α,␈↓α    e.g., TIMES␈↓	-1␈↓α(12), which is { (12) (2 6) (2 2 3) (3 4) }, contains (12).
␈↓ α,␈↓α    e.g., TIMES␈↓	-1␈↓α(13), which is { (13) }, contains (13).

␈↓ α,␈↓α  Creating a new concept, "Single-times".
␈↓ α,␈↓α    Single-times is a relation from Numbers to Bags-of-numbers.
␈↓ α,␈↓α    Single-times(x) is all bags in TIMES␈↓	-1␈↓α(x) which are singletons.
␈↓ α,␈↓α    e.g., Single-times(12)={ (12) }.
␈↓ α,␈↓α    e.g., Single-times(13)={ (13) }.

␈↓ α,␈↓α  Conjecture 2: TIMES␈↓	-1␈↓α(x) always contains a bag containing only primes.
␈↓ α,␈↓α    e.g., TIMES␈↓	-1␈↓α(12), which is { (12) (2 6) (2 2 3) (3 4) }, contains (2 2 3).
␈↓ α,␈↓α    e.g., TIMES␈↓	-1␈↓α(13), which is { (13) }, contains (13).

␈↓ α,␈↓α  Creating a new concept, "Prime-times".
␈↓ α,␈↓α    Prime-times is a relation from Numbers to Bags-of-numbers.
␈↓ α,␈↓α    Prime-times(x) is all bags in TIMES␈↓	-1␈↓α(x) which contain only primes.
␈↓ α,␈↓α    e.g., Prime-times(12)={ (2 3 3) }.
␈↓ α,␈↓α    e.g., Prime-times(13)={ (13) }.
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε23␈↓-

␈↓ α,␈↓α␈↓¬** ␈↓&Task 80:␈↓)αβ **␈↓α
␈↓ α,␈↓αConsidering the concept "Prime-times".

␈↓ α,␈↓α        2 Reasons:␈↓ ∧<(1) Conjecs about Prime-times will tell much about Primes and TIMES␈↓	-1␈↓α.
␈↓ α,␈↓α␈↓ ∧<(2) Focus of attention: AM recently defined Prime-times.

␈↓ α,␈↓α  Looking specifically at Prime-times(48), which is { (2 2 2 2 3) }.
␈↓ α,␈↓α    4 conjectures proposed, after .6 seconds.
␈↓ α,␈↓α    e.g., "x is never inside any member of Prime-times(x)".
␈↓ α,␈↓α    Testing them on other examples of Prime-times.

␈↓ α,␈↓α    Only 1 out of the 4 conjectures are verified for all 23 known examples of Prime-times:

␈↓ α,␈↓α  Conjecture 1: Prime-times(x) is always a singleton set.
␈↓ α,␈↓α    That is, Prime-times is a function, not just a relation.
␈↓ α,␈↓α    e.g., Prime-times(48), which is { (2 2 2 2 3) }, is a singleton set.
␈↓ α,␈↓α    e.g., Prime-times(47), which is { (47) }, is a singleton set.
␈↓ α,␈↓α    This holds for all 17 known examples of Prime-times. (Chance of coincidence is .0001)
␈↓ α,␈↓α    This fails for 2 of the boundary cases (extreme numbers): 0 and 1.
␈↓ α,␈↓α    Conjecture is amended: Each number >1 is the product of a unique bag of primes.
␈↓ α,␈↓α    I suspect that this conjecture may be very useful.␈↓	8␈↓α

␈↓ α,␈↓α␈↓↓USER: Call this conjecture  "Unique factorization conjecture".␈↓α

␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓α  ␈↓π#␈↓α
␈↓ α,␈↓α  ␈↓π#␈↓α



␈↓ α,␈↓β␈↓ εTo␈αshow␈αthat␈αAM␈αisn't␈αreally␈αalways␈αright␈αon␈αthe␈αmark,
␈↓ α,␈↓β␈↓ εthe␈α
next␈α∞sequence␈α
of␈α∞tasks␈α
includes␈α∞a␈α
crime␈α∞of␈α
omission
␈↓ α,␈↓β␈↓ ε(ignoring␈α⊂the␈α⊂concept␈α⊂of␈α⊂Partitions)␈α⊂and␈α⊂a␈α⊂false␈α⊂start
␈↓ α,␈↓β␈↓ ε(worrying␈α
about␈α
numbers␈αwhich␈α
can␈α
be␈α
represented␈αas␈α
the
␈↓ α,␈↓β␈↓ εsum␈α⊃of␈α⊃two␈α⊂primes␈α⊃in␈α⊃precisely␈α⊂one␈α⊃way).␈α⊃ Notice␈α⊂the
␈↓ α,␈↓β␈↓ εskip here; 3 tasks have been omitted.







␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	8␈↓ε␈α	How␈α
did␈α	AM␈α
know␈α	this?␈α
One␈α	of␈α
the␈α	(unfortunately␈α	few!)␈α
meta-heuristics␈α	in␈α
AM␈α	said␈α
the␈α	following:␈α
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the␈α	␈↓¬`look
␈↓ α,␈↓¬␈↓ βLat␈α
the␈α
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the␈α
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Tack␈α
the␈α
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note␈αonto␈α
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␈↓ α,␈↓ε␈↓ βLInterest␈α	facet␈α
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concept␈α	which␈α	is␈α
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␈↓ α,␈↓¬␈↓ βLf␈↓	-1␈↓¬)␈α
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␈↓ α,␈↓	␈↓ βLusing␈αλthe␈α	`extrema'␈αλheuristic␈α	rule,␈αλwith␈α	f=Divisors-of.␈αλWhen␈α	PRIMES␈αλwas␈α	first␈αλcreated,␈α	the␈αλmeta-rule␈α	we␈αλjust
␈↓ α,␈↓	␈↓ βLpresented␈α∞tacked␈α∞the␈α∞following␈α∞note␈α∞onto␈α∞Primes.Interest:␈α∞␈↓β`Conjectures␈α∞involving␈α∞Primes␈α∞and
␈↓ α,␈↓β␈↓ βLdivision␈α(or␈αmultiplication)␈αare␈αnatural,␈αinteresting,␈αand␈αprobably␈αuseful.'␈↓	␈αThus␈αthe
␈↓ α,␈↓	␈↓ βLunique factorization conjecture triggers this feature, whereas Goldbach's conjecture wouldn't.
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε24␈↓-

␈↓ α,␈↓α␈↓¬** ␈↓&Task 84:␈↓)αβ **␈↓α
␈↓ α,␈↓αExamining ADD␈↓	-1␈↓α(x), looking for patterns involving its values.

␈↓ α,␈↓α        2 Reasons:␈↓ ∧<(1) ADD␈↓	-1␈↓α is analogous to the newly-interesting concept "TIMES␈↓	-1␈↓α".
␈↓ α,␈↓α␈↓ ∧<(2) Many examples of ADD␈↓	-1␈↓α are known, to induce from.

␈↓ α,␈↓α  Looking specifically at ADD␈↓	-1␈↓α(6), which is { (1 1 1 1 1 1) (1 1 1 1 2) (1 1 1 3) (1 1 2 2)
␈↓ α,␈↓α                 (1 1 4) (1 2 3) (1 5) (2 2 2) (2 4) (3 3) (6)}.
␈↓ α,␈↓α    17 conjectures proposed, after 3.9 seconds.
␈↓ α,␈↓α    e.g., "ADD␈↓	-1␈↓α(x) always contains a bag of primes".
␈↓ α,␈↓α    Testing them on other examples of ADD␈↓	-1␈↓α.
␈↓ α,␈↓α    Only 11 out of the 17 conjectures are verified for all 19 known examples of ADD␈↓	-1␈↓α:
␈↓ α,␈↓α    3 out of the 11 conjectures were false until amended.

␈↓ α,␈↓α  Conjecture 1: ADD␈↓	-1␈↓α(x) never contains a singleton bag.

␈↓ α,␈↓α  Conjecture 2: ADD␈↓	-1␈↓α(x) always contains a bag of size 2 (also called a "pair" or a "doubleton").
␈↓ α,␈↓α    e.g., ADD␈↓	-1␈↓α(6) contains (1 5), (2 4), and (3 3).
␈↓ α,␈↓α    e.g., ADD␈↓	-1␈↓α(4) contains (1 3), and (2 2).

␈↓ α,␈↓α  Creating a new concept, "Pair-add".
␈↓ α,␈↓α    Pair-add is a relation from Numbers to Pairs-of-numbers.
␈↓ α,␈↓α    Pair-add(x) is all bags in ADD␈↓	-1␈↓α(x) which are doubletons (i.e., of size 2).
␈↓ α,␈↓α    e.g., Pair-add(12)={ (1 11) (2 10) (3 9) (4 8) (5 7) (6 6) }.
␈↓ α,␈↓α    e.g., Pair-add(4)={ (1 3) (2 2) }.

␈↓ α,␈↓α  Conjecture 3: ADD␈↓	-1␈↓α(x) always contains a bag containing only 1's.
␈↓ α,␈↓α    ␈↓π#␈↓α
␈↓ α,␈↓α    ␈↓π#␈↓α
␈↓ α,␈↓α    ␈↓π#␈↓α
␈↓ α,␈↓α  Conjecture 10: ADD␈↓	-1␈↓α(x) always contains a pair of primes.
␈↓ α,␈↓α    This conjecture is false. Conjecture is amended:
␈↓ α,␈↓α    "ADD␈↓	-1␈↓α(x) usually (but not always) contains a pair of primes."
␈↓ α,␈↓α    e.g., ADD␈↓	-1␈↓α(10) contains (3 7), and (5 5).
␈↓ α,␈↓α    e.g., ADD␈↓	-1␈↓α(4) contains (2 2).
␈↓ α,␈↓α    e.g., ADD␈↓	-1␈↓α(11) does not contain a pair of primes.

␈↓ α,␈↓α  Creating a new concept, "Prime-add".
␈↓ α,␈↓α    Prime-add is a relation from Numbers to Pairs-of-numbers.
␈↓ α,␈↓α    Prime-add(x) is all bags in ADD␈↓	-1␈↓α(x) which are pairs of primes.
␈↓ α,␈↓α    e.g., Prime-add(12)={ (5 7) }.
␈↓ α,␈↓α    e.g., Prime-add(10)={ (3 7) (5 5) }.
␈↓ α,␈↓α    e.g., Prime-add(11) = {  }
␈↓ α,␈↓α    ␈↓π#␈↓α
␈↓ α,␈↓α    ␈↓π#␈↓α
␈↓ α,␈↓α    ␈↓π#␈↓α
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε25␈↓-

␈↓ α,␈↓α␈↓¬** ␈↓&Task 106:␈↓)αβ **␈↓α
␈↓ α,␈↓αConsidering the set of numbers for which Prime-add is defined (has non-empty value).

␈↓ α,␈↓α        1 Reason:␈↓ ∧<(1) Prime-add often has non-empty value. Worth isolating that case.

␈↓ α,␈↓α  Warning: no task on the agenda has an interestingness value above 200!!!

␈↓ α,␈↓α  Creating a new concept "Prime-addable".
␈↓ α,␈↓α    Prime-addable is a kind of Number.  x is Prime-addable if Prime-add(x) is non-empty.
␈↓ α,␈↓α    Will spend 5.0 seconds filling in examples of Prime-addable.
␈↓ α,␈↓α    18 examples found.   Here are some of them:  4 5 6 7 8 9 10 12 13 14 17 16 18.
␈↓ α,␈↓α    Empirically, all even numbers are also in this set (ignoring boundary cases: 0 2 4 6).
␈↓ α,␈↓α    So conjectured.  Danger: must examine boundary cases: the numbers 0, 2, 4, and 6.
␈↓ α,␈↓α    Two exceptions noticed. The only exceptions are the smallest boundary cases: 0, 2.
␈↓ α,␈↓α    Conjecture is amended: All even numbers >2 are the sum of two primes.

␈↓ α,␈↓α  Warning: I expect this conjecture will be cute but useless.


␈↓ α,␈↓α␈↓↓USER: Why?␈↓α


␈↓ α,␈↓αBecause␈α
the␈α
concept␈αof␈α
"Primes"␈α
originally␈αarose␈α
as␈α
extreme␈αcases␈α
of␈α
Divisors-of.␈α
 The␈αmore
␈↓ α,␈↓α␈↓ αLclosely␈α
an␈α
operation␈α
X␈α∞is␈α
related␈α
to␈α
the␈α
concept␈α∞Divisors-of,␈α
the␈α
more␈α
natural␈α
will␈α∞be␈α
any
␈↓ α,␈↓α␈↓ αLconjecture␈α⊃involving␈α⊂both␈α⊃that␈α⊂operation␈α⊃X␈α⊂and␈α⊃Primes.␈α⊂ E.g.,␈α⊃conjectures␈α⊃involving␈α⊂both
␈↓ α,␈↓α␈↓ αLPrimes␈αand␈αTimes␈α
will␈αbe␈αnatural␈αand␈α
useful.␈α But␈αthis␈α
conjecture,␈αwhich␈αinvolves␈αPrimes␈α
and
␈↓ α,␈↓α␈↓ αLADD␈↓	-1␈↓α,␈α⊂will␈α⊃be␈α⊂cute␈α⊃but␈α⊂useless,␈α⊃since␈α⊂the␈α⊂relation␈α⊃"ADD␈↓	-1␈↓α"␈α⊂is␈α⊃unrelated␈α⊂to␈α⊃the␈α⊂relation
␈↓ α,␈↓α␈↓ αL"Divisors-of".
␈↓ α,␈↓αEnd of explanation.


␈↓ α,␈↓α␈↓↓USER: Call this conjecture "Goldbach's conjecture".␈↓α


␈↓ α,␈↓α␈↓¬** ␈↓&Task 107:␈↓)αβ **␈↓α
␈↓ α,␈↓αConsidering the set of numbers for which the relation Prime-add is single-valued.

␈↓ α,␈↓α        3 Reasons:␈↓ ∧<(1) Prime-add often has singleton value. Worth isolating that case.
␈↓ α,␈↓α␈↓ ∧<(2) Restricted to this set, Prime-add would be a function.
␈↓ α,␈↓α␈↓ ∧<(3) Focus of attention: AM recently worked on Prime-add.

␈↓ α,␈↓α  Creating a new concept "Uniquely-prime-addable".
␈↓ α,␈↓α    "Uniquely-prime-addable" is a kind of Number.
␈↓ α,␈↓α    x is Uniquely-prime-addable if Prime-add(x) is a singleton.
␈↓ α,␈↓α    Will spend 10.0 seconds filling in examples of Uniquely-prime-addable.
␈↓ α,␈↓α    11 examples found. Here are some of them:  4 5 7 8 9 12 13.
␈↓ α,␈↓α    No obvious conjecture derived empirically.

␈↓ α,␈↓α  Will forget "Uniquely-prime-addable numbers", if no Ties found in near future.
␈↓ α,␈↓␈↓εChapter 2␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε26␈↓-

␈↓ α,␈↓␈↓ ¬⊃␈↓∧␈↓&2.5. Recapping the Example␈↓)αβ␈↓

␈↓ α,␈↓Let's␈α∞once␈α∞again␈α∞eavesdrop␈α∞on␈α∞a␈α∂mathematician,␈α∞as␈α∞he␈α∞describes␈α∞to␈α∞a␈α∞colleague␈α∂what␈α∞AM
␈↓ α,␈↓did.

␈↓ α,␈↓This␈α∞example␈α∞was␈α∞preceded␈α∞by␈α∞the␈α∞momentous␈α∞discoveries␈α∞of␈α∞multiplication␈α∞and␈α
division.
␈↓ α,␈↓Several␈α∂interesting␈α∂properites␈α∞of␈α∂these␈α∂operations␈α∞were␈α∂noticed.␈α∂ The␈α∞≡rst␈α∂task␈α∂which␈α∞was
␈↓ α,␈↓illustrated␈α∪(␈↓¬**␈α∪␈↓&Task␈α∪65␈↓)αβ␈α∪**␈↓)␈α∪involves␈α∪exploring␈α∪the␈α∪concept␈α∪of␈α∪"divisors␈α∪of␈α∪a␈α∩number"
␈↓ α,␈↓(meaning␈α
all␈α
positive␈α
integers␈α
which␈α
divide␈αevenly␈α
into␈α
the␈α
given␈α
number).␈α
After␈α
tiring␈αof
␈↓ α,␈↓≡nding␈α
examples␈α
of␈α
this␈αrelation,␈α
AM␈α
investigates␈α
extreme␈αcases:␈α
that␈α
is,␈α
it␈α
wonders␈αwhich
␈↓ α,␈↓numbers have very few or very many divisors.

␈↓ α,␈↓AM␈α
thus␈α
discovers␈α
Primes␈α
in␈α
a␈α
curious␈α
way.␈α
 Numbers␈α
with␈α
0␈α
or␈α
1␈α
divisor␈α∞are␈α
essentially
␈↓ α,␈↓nonexistent,␈αso␈αthey're␈α
not␈αfound␈αto␈αbe␈α
interesting.␈αAM␈αnotices␈α
that␈αnumbers␈αwith␈α3␈α
divisors
␈↓ α,␈↓always␈α↔seem␈α↔to␈α_be␈α↔squares␈α↔of␈α_numbers␈α↔with␈α↔2␈α_divisors␈α↔(primes).␈α↔ This␈α_raises␈α↔the
␈↓ α,␈↓interestingness␈α∪of␈α∪several␈α∪concepts,␈α∪including␈α∀primes.␈α∪ Soon␈α∪(␈↓¬**␈α∪␈↓&TASK␈α∪79␈↓)αβ␈α∀**␈↓),␈α∪another
␈↓ α,␈↓conjecture␈α⊂involving␈α⊂primes␈α⊂is␈α⊂noticed:␈α⊂Many␈α⊂numbers␈α⊂seem␈α⊂to␈α⊂factor␈α⊂into␈α⊃primes.␈α⊂This
␈↓ α,␈↓causes␈αa␈αnew␈αrelation␈αto␈αbe␈αde≡ned,␈αwhich␈αassociates␈αto␈αa␈αnumber␈αx,␈αall␈αprime␈αfactorizations
␈↓ α,␈↓of␈αx.␈α The␈α≡rst␈αquestion␈αAM␈αasks␈αabout␈αthis␈αrelation␈αis␈α"is␈αit␈αa␈αfunction?".␈α This␈αquestion␈αis
␈↓ α,␈↓the␈α∪full␈α∪statement␈α∪of␈α∪the␈α∪unique␈α∪factorization␈α∪conjecture:␈α∪the␈α∪fundamental␈α∪theorem␈α∪of
␈↓ α,␈↓arithmetic.␈α↔ AM␈α↔recognized␈α↔the␈α⊗value␈α↔of␈α↔this␈α↔relationship,␈α⊗and␈α↔assigned␈α↔it␈α↔a␈α⊗high
␈↓ α,␈↓interestingness rating.

␈↓ α,␈↓In␈αa␈αsimilar␈αmanner,␈αthough␈αwith␈αlower␈αhopes,␈αit␈αnoticed␈αsome␈αmore␈αrelationships␈α
involving
␈↓ α,␈↓primes,␈αincluding␈αGoldbach's␈αconjecture.␈α AM␈αquite␈αcorrectly␈αpredicted␈αthat␈αthis␈αwould␈αturn
␈↓ α,␈↓out to be cute but of no future use mathematically.

␈↓ α,␈↓The␈αlast␈αactivity␈αmentioned␈α(␈↓¬**␈α␈↓&TASK␈α107␈↓)αβ␈α**␈↓)␈αshows␈αAM␈αexamining␈αa␈αrather␈αnonstandard
␈↓ α,␈↓concept:␈α
"numbers␈αwhich␈α
can␈αbe␈α
written␈αas␈α
the␈αsum␈α
of␈αa␈α
pair␈αof␈α
primes,␈αin␈α
only␈α
one␈αway".
␈↓ α,␈↓These␈α∞are␈α
termed␈α∞"uniquely-prime-addable"␈α
numbers.␈α∞ It␈α
was␈α∞mildly␈α
unfortunate␈α∞that␈α
AM
␈↓ α,␈↓gave␈αup␈αon␈α
this␈αconcept␈αbefore␈α
noticing␈αthat␈αp+2␈α
is␈αuniquely-prime-addable,␈αfor␈α
any␈αprime
␈↓ α,␈↓number␈α∞p,␈α∞and␈α∞that␈α
in␈α∞fact␈α∞these␈α∞are␈α
the␈α∞only␈α∞odd␈α∞uniquely-prime-addable␈α∞numbers.␈α
The
␈↓ α,␈↓session␈α
was␈α
repeated␈α
once,␈α
with␈α
a␈α
human␈α
user␈α
telling␈α
AM␈α
explicitly␈α
to␈α
continue␈αstudying␈α
this
␈↓ α,␈↓concept.␈α AM␈αdid␈αin␈αfact␈αconstruct␈α"Uniquely-prime-addable-odd-numbers",␈αand␈αthen␈αnotice
␈↓ α,␈↓this␈α∞relationship.␈α∞Here␈α
we␈α∞see␈α∞an␈α
example␈α∞of␈α∞unstable␈α
equilibrium:␈α∞if␈α∞pushed␈α∞slightly␈α
this
␈↓ α,␈↓way,␈αAM␈αwill␈αget␈αvery␈αinterested␈αand␈αspend␈αa␈αlot␈αof␈αtime␈αworking␈αon␈αthis␈αkind␈αof␈αnumber.
␈↓ α,␈↓Since␈α∞it␈α
doesn't␈α∞have␈α
all␈α∞the␈α
sophistication␈α∞(i.e.,␈α
compiled␈α∞hindsight)␈α
that␈α∞we␈α
have,␈α∞it␈α
can't
␈↓ α,␈↓know instantly whether what it's doing will be fruitless.
␈↓ α,␈↓␈↓ ,-␈↓ε27␈↓-




␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ∧t␈↓∧Chapter 3.   Control Structure␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓




␈↓ α,␈↓β␈↓ α|`Objectively'␈αgiven,␈α`important'␈α
problems␈αmay␈αarise␈α[in␈α
math].␈α But␈αeven␈α
then␈αthe
␈↓ α,␈↓β␈↓ α|mathematician␈αis␈αessentially␈αfree␈αto␈αtake␈αit␈αor␈αleave␈αit␈αand␈αturn␈αto␈αsomething␈αelse,
␈↓ α,␈↓β␈↓ α|while␈α∩an␈α∪`important'␈α∩problem␈α∪in␈α∩[any␈α∩other␈α∪science]␈α∩is␈α∪usually␈α∩a␈α∪con∨ict,␈α∩a
␈↓ α,␈↓β␈↓ α|contradiction,␈α∂which␈α∞`must'␈α∂be␈α∂resolved.␈α∞The␈α∂mathematician␈α∂has␈α∞a␈α∂wide␈α∂choice␈α∞of
␈↓ α,␈↓β␈↓ α|which way to turn, and he enjoys a very considerable freedom in what he does.

␈↓ α,␈↓¬␈↓ ε\-- von Neumann



␈↓ α,␈↓Section␈α
3.1␈α
will␈α
give␈α
the␈α
reader␈α
a␈α
feeling␈α
for␈α
the␈α
immensity␈α
of␈α
AM's␈α
search␈α
space.␈α∞This␈α
is
␈↓ α,␈↓the␈α⊂"problem".␈α∂ The␈α⊂next␈α∂section␈α⊂will␈α∂give␈α⊂the␈α∂top-level␈α⊂"solution":␈α∂the␈α⊂∨ow␈α∂of␈α⊂control␈α∂is
␈↓ α,␈↓governed␈αby␈αa␈α
job-list,␈αan␈αagenda␈α
of␈αplausible␈αtasks.␈α
 Section␈α3.3␈αwill␈α
present␈αsome␈αdetails␈α
of
␈↓ α,␈↓this␈α⊂global␈α⊂control␈α⊂scheme.␈α⊂ Chapter␈α⊂4␈α⊃deals␈α⊂with␈α⊂the␈α⊂way␈α⊂AM's␈α⊂heuristics␈α⊃operate;␈α⊂this
␈↓ α,␈↓could be viewed as the "low-level" or ␈↓βlocal␈↓ control structure of AM.


␈↓ α,␈↓␈↓ ¬u␈↓∧␈↓&3.1. AM's Search␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|To develop mathematics, one must always labor to substitute ideas for calculations.

␈↓ α,␈↓¬␈↓ ε\-- Dirichlet



␈↓ α,␈↓Let's␈α≡rst␈αspend␈αa␈αparagraph␈αreviewing␈α
how␈αconcepts␈αare␈αstored.␈α AM␈αcontains␈α
a␈αcollection
␈↓ α,␈↓of␈α∞data␈α∞structures,␈α
called␈α∞␈↓βconcepts␈↓.␈α∞ Each␈α
concept␈α∞is␈α∞meant␈α
to␈α∞coincide␈α∞intuitively␈α∞with␈α
one
␈↓ α,␈↓mathematical␈αidea␈α(e.g.,␈αSets,␈αUnion,␈αTrichotomy).␈α As␈αsuch,␈αa␈αconcept␈αhas␈αseveral␈αaspects␈αor
␈↓ α,␈↓parts,␈αcalled␈α␈↓βfacets␈↓␈α(e.g.,␈αExamples,␈αDe≡nitions,␈αDomain/range,␈αWorth).␈α If␈αyou␈αwish␈αto␈αthink
␈↓ α,␈↓of␈αa␈αconcept␈αas␈αa␈α"frame",␈αthen␈αits␈αfacets␈αare␈α"slots"␈αto␈αbe␈α≡lled␈αin.␈α Each␈αfacet␈αof␈α
a␈αconcept
␈↓ α,␈↓will␈α⊃either␈α∩be␈α⊃totally␈α∩blank,␈α⊃or␈α⊃else␈α∩will␈α⊃contain␈α∩a␈α⊃bunch␈α⊃of␈α∩␈↓βentries␈↓.␈α⊃ For␈α∩example,␈α⊃the
␈↓ α,␈↓Algorithms␈α
facet␈α
of␈αthe␈α
concept␈α
Union␈αmay␈α
point␈α
to␈αseveral␈α
equivalent␈α
LISP␈αfunctions,␈α
each
␈↓ α,␈↓of␈αwhich␈αcan␈αbe␈αused␈αto␈αform␈αthe␈αunion␈αof␈αtwo␈αsets␈↓	1␈↓.␈αEven␈αthe␈α"heuristic␈αrules"␈α
are␈αmerely
␈↓ α,␈↓entries␈α∩on␈α∪the␈α∩appropriate␈α∪kind␈α∩of␈α∩facet␈α∪(e.g.,␈α∩the␈α∪entries␈α∩on␈α∩the␈α∪Interest␈α∩facet␈α∪of␈α∩the
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε␈α
The␈α
reasons␈α
for␈α
having␈α
multiple␈αalgorithms␈α
is␈α
that␈α
sometimes␈α
AM␈α
will␈α
want␈αone␈α
that␈α
is␈α
fast,␈α
sometimes␈α
AM␈α
will␈αbe␈α
more
␈↓ α,␈↓ε␈↓ βLconcerned␈αwith␈αeconomizing␈αon␈αstorage,␈αsometimes␈αAM␈αwill␈αwant␈αto␈α"analyze"␈αan␈αalgorithm,␈αand␈αfor␈αthat
␈↓ α,␈↓ε␈↓ βLpurpose it must be a very un-optimized function, etc.
␈↓ α,␈↓␈↓εChapter 3␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε28␈↓-

␈↓ α,␈↓Structure concept are rules for judging the interestingness of Structures␈↓	2␈↓).

␈↓ α,␈↓At␈α
any␈α
moment,␈α
AM␈α
contains␈α
a␈α
couple␈αhundred␈α
concepts,␈α
each␈α
of␈α
which␈α
has␈α
only␈α
␈↓βsome␈↓␈αof␈α
its
␈↓ α,␈↓facets␈α⊃≡lled␈α⊃in.␈α⊃ AM␈α⊃starts␈α⊃with␈α∩115␈α⊃concepts,␈α⊃and␈α⊃grows␈α⊃to␈α⊃about␈α⊃300␈α∩concepts␈α⊃before
␈↓ α,␈↓running␈α⊂out␈α⊃of␈α⊂time/space.␈α⊂ Most␈α⊃facets␈α⊂of␈α⊃most␈α⊂concepts␈α⊂are␈α⊃totally␈α⊂blank.␈α⊃ AM's␈α⊂basic
␈↓ α,␈↓activity␈α
is␈αto␈α
select␈αsome␈α
facet␈αof␈α
some␈αconcept,␈α
and␈αthen␈α
try␈αto␈α
≡ll␈αin␈α
some␈αentries␈α
for␈αthat
␈↓ α,␈↓slot␈↓	3␈↓.␈α
Thus␈α
the␈α∞primitive␈α
kind␈α
of␈α
"␈↓βtask␈↓"␈α∞for␈α
AM␈α
is␈α
to␈α∞deal␈α
with␈α
a␈α∞particular␈α
facet/concept
␈↓ α,␈↓pair.  A typical task looks like this:

␈↓ α,␈↓¬␈↓ β,Check the entries on the "Domain/range" facet of the "Bag-Insert" concept

␈↓ α,␈↓If␈α⊂the␈α⊂average␈α⊂concept␈α⊂has␈α⊂ten␈α⊂or␈α∂twenty␈α⊂blank␈α⊂facets,␈α⊂and␈α⊂there␈α⊂are␈α⊂a␈α⊂couple␈α∂hundred
␈↓ α,␈↓concepts,␈α
then␈α
clearly␈α
there␈α∞will␈α
be␈α
about␈α
20x200=4000␈α
"≡ll-in"␈α∞type␈α
tasks␈α
for␈α
AM␈α∞to␈α
work
␈↓ α,␈↓on,␈αat␈α
any␈αgiven␈αmoment.␈α
 If␈αseveral␈αhundred␈α
facets␈αhave␈αrecently␈α
been␈α≡lled␈αin,␈α
there␈αwill
␈↓ α,␈↓be␈αthat␈αmany␈α"check-entries"␈αtype␈αtasks␈αavailable.␈α Executing␈αa␈αtask␈αhappens␈αto␈αtake␈α
around
␈↓ α,␈↓ten␈α
or␈α
twenty␈α
cpu␈α
seconds,␈α
so␈α
over␈α
the␈α
course␈α
of␈α
a␈α
few␈α
hours␈α
only␈α
a␈α
small␈α
percentage␈α
of␈α
these
␈↓ α,␈↓tasks can ever be executed.␈↓	4␈↓

␈↓ α,␈↓Since␈α∞most␈α
of␈α∞these␈α
tasks␈α∞will␈α
never␈α∞be␈α∞explored,␈α
what␈α∞will␈α
make␈α∞AM␈α
appear␈α∞smart␈α∞¬␈α
or
␈↓ α,␈↓stupid␈α
¬␈αare␈α
its␈α
choices␈αof␈α
which␈α
task␈αto␈α
pick␈α
at␈αeach␈α
moment.␈↓	5␈↓␈α
So␈αit's␈α
worth␈αAM's␈α
spending
␈↓ α,␈↓a␈αnontrivial␈α
amount␈αof␈αtime␈α
deciding␈αwhich␈αtask␈α
to␈αexecute␈α
next.␈αOn␈αthe␈α
other␈αhand,␈αit␈α
had
␈↓ α,␈↓better not be ␈↓βtoo␈↓ much time, since a task does take only a dozen seconds.␈↓	6␈↓

␈↓ α,␈↓One␈α⊂question␈α⊂that␈α⊂must␈α⊂be␈α⊂answered␈α⊃is:␈α⊂What␈α⊂percentage␈α⊂of␈α⊂AM's␈α⊂legal␈α⊂moves␈α⊃(at␈α⊂any
␈↓ α,␈↓typical␈α⊃moment)␈α⊂would␈α⊃be␈α⊂considered␈α⊃intelligent␈α⊂choices,␈α⊃and␈α⊂what␈α⊃percentage␈α⊃would␈α⊂be
␈↓ α,␈↓irrational?␈α↔ The␈α⊗answer␈α↔comes␈α↔from␈α⊗empirical␈α↔results.␈α⊗ The␈α↔percentages␈α↔vary␈α⊗wildly
␈↓ α,␈↓depending␈αon␈αthe␈αprevious␈αfew␈αtasks.␈α Sometimes,␈αAM␈αwill␈αbe␈αobviously␈α"in␈αthe␈αmiddle"␈αof
␈↓ α,␈↓a␈α∞sequence␈α∞of␈α∞tasks,␈α∞and␈α∞only␈α∞one␈α∞or␈α∞two␈α∞of␈α∞the␈α∞legal␈α∞tasks␈α∞would␈α∞seem␈α∞plausible.␈α
 Other
␈↓ α,␈↓times,␈α
AM␈α
has␈α
just␈α
completed␈α
an␈α
investigation␈α
by␈α
running␈α
into␈α
dead-ends,␈α
and␈α
there␈αmay␈α
be
␈↓ α,␈↓hundreds␈α
of␈α∞tasks␈α
it␈α
could␈α∞choose␈α
and␈α
not␈α∞be␈α
criticized.␈α
 The␈α∞median␈α
case␈α∞would␈α
perhaps
␈↓ α,␈↓permit about 6 of the legal tasks to be judged reasonable.

␈↓ α,␈↓It␈α∞is␈α∞important␈α∞for␈α∞AM␈α∞to␈α∞locate␈α∞one␈α∞of␈α∞these␈α∞currently-plausible␈α∞tasks,␈α∞but␈α∞it's␈α∂not␈α∞worth
␈↓ α,␈↓spending␈αmuch␈α
time␈αdeciding␈α
which␈αof␈α
␈↓βthem␈↓␈αto␈αwork␈α
on␈αnext.␈α
 AM␈αstill␈α
faces␈αa␈αhuge␈α
search:
␈↓ α,␈↓≡nd one of the 6 winners out of a few thousand candidates.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	2␈↓ε A typical such rule is: "A structure is very interesting if all its elements are mildly interesting in precisely the same way."
␈↓ α,␈↓ε␈↓	3␈↓ε␈α	This␈α	is␈α	not␈α	quite␈α	complete.␈α	In␈α	addition␈α	to␈α	filling␈α	in␈α	entries␈α	for␈α	a␈α	given␈α	facet/concept␈α	pair,␈α	AM␈α	may␈α	wish␈α	to␈α	check␈α	it,␈α	split␈αλit
␈↓ α,␈↓ε␈↓ βLup, reorganize it, etc.
␈↓ α,␈↓ε␈↓	4␈↓ε␈αλThe␈αλprecise␈αλ"18␈α	seconds␈αλaverage"␈αλfigure␈αλis␈α	not␈αλimportant.␈αλAll␈αλheuristic-search␈αλprograms␈α	suffer␈αλthis␈αλsame␈αλhandicap:␈α	As␈αλthe
␈↓ α,␈↓ε␈↓ βLdepth␈αto␈αwhich␈αthey've␈αsearched␈αincreases,␈αthe␈αpercentage␈αof␈αnodes␈α(at␈αor␈αabove␈αthat␈αlevel)␈αwhich␈αhave
␈↓ α,␈↓ε␈↓ βLbeen examined ␈↓&decreases␈↓)αβ exponentially (assuming the branching factor b is strictly larger than unity).
␈↓ α,␈↓ε␈↓	5␈↓ε This is true of all heuristic search programs. The branchier the search, the more it applies.
␈↓ α,␈↓ε␈↓	6␈↓ε␈αλThe␈α	answer␈αλis␈αλthat␈α	AM␈αλspends␈α	this␈αλ"deciding"␈αλtime␈α	not␈αλjust␈αλbefore␈α	a␈αλtask␈α	is␈αλ␈↓βpicked␈↓ε,␈αλbut␈α	rather␈αλeach␈αλtime␈α	a␈αλtask␈α	is␈αλadded
␈↓ α,␈↓ε␈↓ βLto␈αthe␈αagenda.␈αA␈αlittle␈αunder␈α1␈αcpu␈αsecond␈αis␈αspent,␈αon␈αthe␈αaverage,␈αto␈αplace␈αthe␈αtask␈αproperly␈αon␈αthe
␈↓ α,␈↓ε␈↓ βLagenda,␈α	to␈αλassign␈α	it␈αλa␈α	meaningful␈αλnumeric␈α	priority␈αλvalue.␈α	 So␈αλ"action␈α	time"␈αλis␈α	roughly␈αλone␈α	order␈α	of␈αλmagnitude
␈↓ α,␈↓ε␈↓ βLlarger than "deciding time".
␈↓ α,␈↓␈↓εChapter 3␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε29␈↓-

␈↓ α,␈↓Its␈α
choice␈α
of␈αtasks␈α
is␈α
made␈αeven␈α
more␈α
important␈α
due␈αto␈α
the␈α
10-second␈α"cycle␈α
time"␈α
¬␈αthe␈α
time
␈↓ α,␈↓to␈αinvestigate/execute␈αone␈αtask.␈α A␈αhuman␈αuser␈αis␈αwatching,␈αand␈αten␈αseconds␈αis␈αa␈αnontrivial
␈↓ α,␈↓amount␈α∞of␈α∞time␈α∞to␈α∞him.␈α∞ He␈α∞can␈α∞therefore␈α∞observe,␈α∞perceive,␈α∞and␈α∞analyze␈α∞each␈α∂and␈α∞every
␈↓ α,␈↓task␈α
that␈α
AM␈α
selects.␈α
 Even␈α
just␈α
a␈αfew␈α
bizarre␈α
choices␈α
will␈α
greatly␈α
lower␈α
his␈α
opinion␈αof␈α
AM's
␈↓ α,␈↓intelligence.␈α∞ The␈α∞trace␈α
of␈α∞AM's␈α∞actions␈α∞is␈α
what␈α∞counts,␈α∞not␈α∞its␈α
≡nal␈α∞results.␈α∞ So␈α∞AM␈α
can't
␈↓ α,␈↓draw much of its apparent intelligence from the ␈↓βspeed␈↓ of the computer.

␈↓ α,␈↓Chess-playing␈α
programs␈α
have␈α
had␈α
to␈α
face␈α
the␈α
dilemma␈α
of␈α
the␈α
trade-o≥␈α
between␈α
"intelligence"
␈↓ α,␈↓(foresight,␈α
inference,␈αprocessing,...)␈α
and␈αtotal␈α
number␈αof␈α
board␈αsituations␈α
examined.␈α In␈α
chess,
␈↓ α,␈↓the␈αcharacteristics␈α
of␈αcurrent-day␈α
machines,␈αlanguage␈αpower␈α
␈↓βvs.␈↓␈αspeed,␈α
and␈α(to␈α
some␈αextent)
␈↓ α,␈↓the␈αlimitations␈αof␈αour␈αunderstanding␈αof␈αhow␈αto␈αbe␈αsophisticated,␈αhave␈αto␈αdate␈αunfortunately
␈↓ α,␈↓still␈α∞favored␈α∞fast,␈α∂nearly-blind␈↓	7␈↓␈α∞search.␈α∞ Although␈α∞machine␈α∂speed␈α∞and␈α∞LISP␈α∂slowness␈α∞may
␈↓ α,␈↓allow␈αblind␈αsearch␈αto␈αwin␈αover␈αsymbolic␈αinference␈αfor␈α␈↓βshallow␈↓␈αsearches,␈αit␈αcan't␈αprovide␈αany
␈↓ α,␈↓more␈αthan␈αa␈αconstant␈αspeed-up␈αfactor␈αfor␈αan␈αexponential␈αsearch.␈αInference␈αis␈αslowly␈αgaining
␈↓ α,␈↓on brute force,␈↓	8␈↓ and must someday triumph.

␈↓ α,␈↓Since␈α∩the␈α⊃number␈α∩of␈α∩"legal␈α⊃moves"␈α∩for␈α∩AM␈α⊃at␈α∩any␈α⊃moment␈α∩is␈α∩in␈α⊃the␈α∩thousands,␈α∩it␈α⊃is
␈↓ α,␈↓unrealistic␈α
to␈α
consider␈α
"systematically"␈↓	9␈↓␈α
walking␈α
through␈α
the␈α
entire␈α
space␈α
that␈α
AM␈αcan␈α
reach.
␈↓ α,␈↓In␈α∞AM's␈α∞problem␈α∂domain,␈α∞there␈α∞is␈α∂so␈α∞much␈α∞"freedom"␈α∂that␈α∞symbolic␈α∞inference␈α∂≡nally␈α∞␈↓βcan␈↓
␈↓ α,␈↓win over the "simple but fast" exploration strategy␈↓	10␈↓.


␈↓ α,␈↓␈↓ ∧⎇␈↓∧␈↓&3.2. Constraining AM's Search␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|There␈α
exist␈α
too␈αmany␈α
combinations␈α
to␈α
consider␈αall␈α
combinations␈α
of␈αexisting␈α
entities;
␈↓ α,␈↓β␈↓ α|the creative mind must only ␈↓&propose␈↓)αβ those of potential interest.

␈↓ α,␈↓¬␈↓ ε\-- Poincare'



␈↓ α,␈↓A␈α∪great␈α∪deal␈α∪of␈α∪heuristic␈α∪knowledge␈α∪is␈α∪required␈α∪to␈α∪constrain␈α∪the␈α∪necessary␈α∩processing
␈↓ α,␈↓e≥ectively, to zero in on a good task to tackle next.  This is done in two stages.
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α
A␈α
list␈α
of␈α
plausible␈α
facet/concept␈α∞pairs␈α
is␈α
maintained.␈α
Nothing␈α
can␈α
get␈α
onto␈α∞this␈α
list
␈↓ α,␈↓␈↓ β≤unless␈α∞there␈α
is␈α∞some␈α
reason␈α∞why␈α
≡lling␈α∞in␈α
(or␈α∞checking)␈α
that␈α∞facet␈α
of␈α∞that␈α
concept
␈↓ α,␈↓␈↓ β≤would be worthwhile.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈αAll␈αthe␈α
plausible␈αtasks␈αon␈αthis␈α
"job␈αlist"␈αare␈αranked␈α
by␈αthe␈αnumber␈αand␈α
strength␈αof
␈↓ α,␈↓␈↓ β≤the␈α
di≥erent␈αreasons␈α
supporting␈αthem.␈α
 Thus␈α
the␈αfacet/concept␈α
pairs␈αnear␈α
the␈αtop␈α
of
␈↓ α,␈↓␈↓ β≤the list will all be very promising tasks to work on.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	7␈↓ε i.e., using a very simple static evaluation function.
␈↓ α,␈↓ε␈↓	8␈↓ε␈α	E.g.,␈α	see␈α	[Berliner␈α	74].␈α	There,␈α
searching␈α	is␈α	used␈α	mainly␈α	to␈α	verify␈α
plausible␈α	moves␈α	(a␈α	convergent␈α	process),␈α	not␈α
to␈α	discover
␈↓ α,␈↓ε␈↓ βLthem (a bushier search).
␈↓ α,␈↓ε␈↓	9␈↓ε e.g., exhaustively, or using αβ minimaxing, etc.
␈↓ α,␈↓ε␈↓	10␈↓ε␈α	This␈α	is␈α	the␈αλauthor's␈α	opinion,␈α	partially␈α	supported␈α	by␈αλthe␈α	results␈α	of␈α	AM.␈α	 Paul␈αλCohen␈α	disagrees,␈α	feeling␈α	that␈α	machine␈αλspeed
␈↓ α,␈↓ε␈↓ βLshould be the key to an automated mathematician's success.
␈↓ α,␈↓␈↓εChapter 3␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε30␈↓-

␈↓ α,␈↓The␈α∂≡rst␈α∂of␈α∂these␈α⊂constraints␈α∂is␈α∂akin␈α∂to␈α∂replacing␈α⊂a␈α∂␈↓βlegal␈↓␈α∂move␈α∂generator␈α∂by␈α⊂a␈α∂␈↓βplausible␈↓
␈↓ α,␈↓move␈α⊂generator.␈α⊃ The␈α⊂second␈α⊃kind␈α⊂of␈α⊃constraint␈α⊂is␈α⊃akin␈α⊂to␈α⊃using␈α⊂a␈α⊃heuristic␈α⊂evaluation
␈↓ α,␈↓function to select the best move from among the plausible ones.␈↓	11␈↓

␈↓ α,␈↓The␈α
job-list␈α
or␈α␈↓βagenda␈↓␈α
is␈α
a␈α
data␈αstructure␈α
which␈α
is␈αa␈α
natural␈α
way␈α
to␈αstore␈α
the␈α
results␈αof␈α
these
␈↓ α,␈↓procedures.␈α It␈αis␈α(1)␈αa␈αlist␈α
of␈αall␈αthe␈αplausible␈αtasks␈αwhich␈α
have␈αbeen␈αgenerated,␈αand␈α(2)␈αit␈α
is
␈↓ α,␈↓kept␈α
ordered␈α
by␈α
the␈α
numeric␈α
estimate␈α
of␈α
how␈α
worthwhile␈α
each␈α
task␈α
is.␈α
 A␈α
typical␈α
entry␈αon
␈↓ α,␈↓the agenda might look like this:
␈↓"␈↓ α,␈↓π␈↓ α\⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 	l⊃
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬      Fill in  the  EXAMPLES  facet of the  PRIMES  concept ␈↓π         ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬   ␈↓βReasons for ≡lling in this facet␈↓¬ ␈↓π         ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π↓␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬␈↓ ∧␈↓π⊂ααααααααααααααααααααααααααααααααααααα␈↓ λl␈↓ λl⊃       ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬␈↓ ∧␈↓π}␈↓¬ 1. No examples of primes are known so far.␈↓ λl␈↓π}␈↓¬ ␈↓π      ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬␈↓ ∧␈↓π}␈↓¬ 2. Focus of attention: AM just defined Primes.␈↓ λl␈↓π}␈↓¬ ␈↓π      ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\}␈↓ ∧%ααααααααααααααααααααααααααααααααααααα␈↓ λl␈↓ λl$       ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬   ␈↓βOverall value of these reasons␈↓¬ ␈↓π          ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π}␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                            ␈↓π↓␈↓¬ ␈↓π                                   ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬                           ␈↓↓250␈↓¬ ␈↓π                                  ␈↓ 	l}
␈↓"␈↓ α,␈↓π␈↓ α\%αααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 	l$

␈↓ α,␈↓While␈αthe␈αtop␈αtask␈α
is␈αbeing␈αexecuted,␈αsome␈αnew␈α
tasks␈αmight␈αget␈αproposed␈αand␈α
merged␈αinto
␈↓ α,␈↓the␈α∂agenda.␈α∂ Also,␈α∂some␈α∂new␈α∂concepts␈α⊂might␈α∂get␈α∂created,␈α∂and␈α∂this,␈α∂too,␈α∂would␈α⊂generate␈α∂a
␈↓ α,␈↓∨urry of new tasks.

␈↓ α,␈↓After␈α
AM␈α
stops␈α
≡lling␈α
in␈αentries␈α
for␈α
the␈α
facet␈α
speci≡ed␈αin␈α
the␈α
chosen␈α
task,␈α
it␈α
removes␈αthat
␈↓ α,␈↓task␈αfrom␈αthe␈αagenda,␈αand␈αmoves␈αon␈αto␈αwork␈αon␈αwhichever␈αtask␈αis␈αthe␈αhighest-rated␈αat␈αthat
␈↓ α,␈↓time.

␈↓ α,␈↓The␈α
reader␈α∞probably␈α
has␈α∞a␈α
dozen␈α∞good␈α
questions␈α
in␈α∞mind␈α
at␈α∞this␈α
point␈α∞(e.g.,␈α
How␈α∞do␈α
the
␈↓ α,␈↓reasons␈α∃get␈α∃rated?,␈α∃How␈α∃do␈α∃the␈α∃tasks␈α∃get␈α∃proposed?,␈α∃What␈α∃happens␈α∃after␈α∃a␈α∃task␈α∀is
␈↓ α,␈↓selected?,...).␈α∂ The␈α∂next␈α⊂section␈α∂should␈α∂answer␈α∂most␈α⊂of␈α∂these.␈α∂Some␈α∂more␈α⊂judgmental␈α∂ones
␈↓ α,␈↓(How␈αdare␈α
you␈αpropose␈αa␈α
numeric␈αcalculus␈α
of␈αplausible␈αreasoning?!␈α
 If␈αyou␈α
slightly␈αde-tune
␈↓ α,␈↓all␈αthose␈αnumbers,␈αdoes␈αthe␈αsystem's␈αperformance␈αfall␈αapart?...)␈αwill␈αbe␈αanswered␈αin␈αChapter
␈↓ α,␈↓7.




␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	11␈↓ε␈α	Past␈α	AI␈αλprograms␈α	(e.g.,␈α	[Samuel␈αλ67])␈α	have␈α	indicated␈α	that␈αλconstraining␈α	generation␈α	(1)␈αλis␈α	more␈α	important␈α	than␈αλsophisticated
␈↓ α,␈↓ε␈↓ βLordering of the resultant candidates (2). This was confirmed by the experiments performed on AM.
␈↓ α,␈↓␈↓εChapter 3␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε31␈↓-

␈↓ α,␈↓␈↓ ¬u␈↓∧␈↓&3.3. The Agenda␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|Creative energy is used mainly to ask the right question.

␈↓ α,␈↓¬␈↓ ε\-- Halmos




␈↓ α,␈↓␈↓ αL␈↓↓␈↓&3.3.1. Why an Agenda?␈↓)αβ␈↓

␈↓ α,␈↓This␈α
subsection␈α
provides␈α
motivation␈αfor␈α
the␈α
following␈α
one,␈αby␈α
arguing␈α
that␈α
a␈αjob-list␈α
scheme
␈↓ α,␈↓is␈α
a␈α
natural␈αmechanism␈α
to␈α
use␈α
to␈αmanage␈α
the␈α
task-selection␈αproblem␈α
AM␈α
faces.␈α
Recall␈αthat
␈↓ α,␈↓AM␈αmust␈αzero␈α
in␈αon␈αone␈αof␈α
the␈αbest␈αfew␈α
tasks␈αto␈αperform␈αnext,␈α
and␈αit␈αrepeatedly␈αmakes␈α
this
␈↓ α,␈↓choice.␈α At␈αeach␈αmoment,␈αthere␈αmight␈αbe␈αthousands␈αof␈αdirections␈αto␈αexplore␈α(plausible␈αtasks
␈↓ α,␈↓to consider).

␈↓ α,␈↓If␈α∞all␈α∞the␈α∞legal␈α∞tasks␈α∞were␈α∞written␈α∞out,␈α
and␈α∞reasons␈α∞were␈α∞thought␈α∞up␈α∞to␈α∞support␈α∞each␈α
one,
␈↓ α,␈↓then␈α
perhaps␈α
we␈α
could␈α
order␈α
them␈α
by␈α∞the␈α
strength␈α
of␈α
those␈α
reasons,␈α
and␈α
thereby␈α∞settle␈α
on
␈↓ α,␈↓the␈α"best"␈αtask␈αto␈αwork␈αon␈αnext.␈α In␈αorder␈αto␈αappear␈α"smart"␈αto␈αthe␈αhuman␈αuser,␈αAM␈αshould
␈↓ α,␈↓␈↓βnever␈↓ execute a task having no reasons attached.

␈↓ α,␈↓Some␈αmagical␈αfunction␈α
will␈αbe␈αassumed␈αto␈α
exist,␈αwhich␈αprovides␈α
a␈αnumeric␈αrating,␈αa␈α
priority
␈↓ α,␈↓value,␈α
for␈α
any␈α
given␈α
task.␈α
 The␈α
function␈αlooks␈α
at␈α
a␈α
given␈α
facet/concept␈α
pair,␈α
examines␈αall␈α
the
␈↓ α,␈↓associated␈α⊂reasons␈α⊂supporting␈α∂that␈α⊂task,␈α⊂and␈α∂computes␈α⊂an␈α⊂estimate␈α∂of␈α⊂how␈α⊂worthwhile␈α∂it
␈↓ α,␈↓would be for AM to spend some time now working on that facet of that concept.

␈↓ α,␈↓So␈αAM␈αwill␈αmaintain␈αa␈αlist␈αof␈αthose␈αlegal␈αtasks␈αwhich␈αhave␈αsome␈αgood␈αreasons␈αtacked␈αonto
␈↓ α,␈↓them,␈α∪which␈α∪justify␈α∪why␈α∪each␈α∪task␈α∪should␈α∩be␈α∪executed,␈α∪why␈α∪it␈α∪is␈α∪plausible.␈α∪ At␈α∩least
␈↓ α,␈↓implicitly, AM has a numeric rating for each task.

␈↓ α,␈↓Give␈αor␈αtake␈αa␈αfew␈αfeatures,␈αthis␈αnotion␈αof␈αa␈α"job-list"␈αis␈αthe␈αone␈αwhich␈αAM␈αuses.␈αIt␈αis␈αalso
␈↓ α,␈↓called␈α
an␈α␈↓βagenda␈↓.␈↓	12␈↓␈α
"A␈αtask␈α
on␈α
the␈αagenda"␈α
is␈αthe␈α
same␈αas␈α
"a␈α
job␈αon␈α
the␈αjob-list"␈α
is␈αthe␈α
same
␈↓ α,␈↓as␈α∞"a␈α∞facet/concept␈α∞pair␈α∞which␈α∞has␈α∞been␈α∞proposed"␈α∞is␈α∞the␈α∞same␈α∞as␈α∞"an␈α∞active␈α∞node␈α∂in␈α∞the
␈↓ α,␈↓search␈αspace".␈α Henceforth,␈α
I'll␈αuse␈αthe␈αfollowing␈α
all␈αinterchangeably:␈αtask,␈αfacet/concept␈α
pair,
␈↓ α,␈↓node, job.␈↓	13␈↓.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&3.3.2. Details of the Agenda scheme␈↓)αβ␈↓

␈↓ α,␈↓At␈αeach␈αmoment,␈αAM␈α
has␈αmany␈αplausible␈αtasks␈α
(hundreds␈αor␈αeven␈αthousands)␈α
which␈αhave
␈↓ α,␈↓been␈αsuggested␈αfor␈αsome␈αgood␈αreason␈αor␈αother,␈αbut␈αhaven't␈αbeen␈αcarried␈αout␈αyet.␈α Each␈αtask

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	12␈↓ε␈αBorrowed␈αfrom␈αKaplan's␈αterm␈αfor␈αthe␈αjob-list␈αpresent␈αin␈αKRL␈α(see␈α[Bobrow␈α&␈αWinograd␈α77]).␈αFor␈αan␈αearlier␈αgeneral
␈↓ α,␈↓ε␈↓ βLdiscussion of agendas, see [Knuth 68].
␈↓ α,␈↓ε␈↓	13␈↓ε␈α	Each␈αλof␈α	these␈αλterms␈α	will␈αλconjure␈α	up␈αλa␈α	slightly␈αλdifferent␈α	image:␈αλa␈α	"job"␈αλis␈α	something␈αλto␈α	do,␈αλa␈α	"node"␈αλis␈α	an␈αλitem␈α	in␈α	a␈αλsearch
␈↓ α,␈↓ε␈↓ βLspace, "facet/concept pair" reminds you of the format of a task.
␈↓ α,␈↓␈↓εChapter 3␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε32␈↓-

␈↓ α,␈↓is␈αat␈αthe␈αlevel␈α
of␈αworking␈αon␈αa␈αcertain␈α
facet␈αof␈αa␈αcertain␈αconcept:␈α
≡lling␈αit␈αin,␈αchecking␈αit,␈α
etc.
␈↓ α,␈↓Recall␈α
that␈αeach␈α
task␈αalso␈α
has␈αtacked␈α
onto␈α
it␈αa␈α
list␈αof␈α
symbolic␈αreasons␈α
explaining␈α
why␈αthe
␈↓ α,␈↓task is worth doing.

␈↓ α,␈↓In␈α
addition,␈α∞a␈α
number␈α∞(between␈α
0␈α∞and␈α
1000)␈α
is␈α∞attached␈α
␈↓βto␈α∞each␈α
reason␈↓,␈α∞representing␈α
some
␈↓ α,␈↓absolute␈α∪measure␈α∩of␈α∪the␈α∪value␈α∩of␈α∪that␈α∩reason␈α∪(at␈α∪the␈α∩moment).␈α∪One␈α∪global␈α∩formula␈↓	14␈↓
␈↓ α,␈↓combines␈α
all␈α
the␈αreasons'␈α
values␈α
into␈αa␈α
single␈α
priority␈αvalue␈α
for␈α
the␈αtask␈α
as␈α
a␈α
whole.␈α This
␈↓ α,␈↓overall␈αrating␈αis␈αtaken␈αto␈αindicate␈αhow␈αworthwhile␈αit␈αwould␈αbe␈αfor␈αAM␈αto␈αbother␈αexecuting
␈↓ α,␈↓that␈α∂task,␈α∂how␈α∂interesting␈α∂the␈α∂task␈α∂would␈α∂probably␈α∂turn␈α∂out␈α∂to␈α∂be.␈α∂ The␈α∂"intelligence"␈α∞of
␈↓ α,␈↓AM's␈α
selection␈α
of␈α
task␈α
is␈α
thus␈α
seen␈αto␈α
depend␈α
on␈α
this␈α
one␈α
formula.␈α
 Yet␈α
experiments␈αshow
␈↓ α,␈↓that␈αits␈αprecise␈αform␈αis␈αnot␈αimportant.␈αWe␈αconclude␈αthat␈αthe␈α"intelligence"␈αhas␈α
been␈αpushed
␈↓ α,␈↓down into the careful assigning of reasons (and ␈↓βtheir␈↓ values) for each proposed task.

␈↓ α,␈↓A typical entry on the agenda might look like this:
␈↓"␈↓ α,␈↓π␈↓ αL⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ≤⊃
␈↓"␈↓ α,␈↓π␈↓ αL} ␈↓¬ TASK: Fill-in examples of Sets ␈↓π                                          ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αL} ␈↓¬ PRIORITY: 300 ␈↓π                                                      ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αL} ␈↓¬ REASONS:  ␈↓π                                                         ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αL} ␈↓¬            100: No known examples for Sets so far. ␈↓π                           ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αL} ␈↓¬            100: Failed to fillin examples of Set-union, for lack of examples of Sets ␈↓π    ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αL} ␈↓¬            200: Focus of attention: AM recently worked on the concept of Set-union ␈↓π   ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αL%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ≤$

␈↓ α,␈↓Notice␈α
the␈αsimilarity␈α
of␈α
this␈αto␈α
the␈α
initial␈αfew␈α
lines␈α
which␈αAM␈α
types␈α
just␈αafter␈α
it␈α
selects␈αa␈α
job
␈↓ α,␈↓to work on.

␈↓ α,␈↓The␈αpriority␈αvalue␈αof␈αa␈αtask␈αserves␈αa␈αdual␈αpurpose:␈αFirst,␈αit␈αis␈αused␈αto␈αdetermine␈αwhich␈αtask
␈↓ α,␈↓on␈α
the␈α∞agenda␈α
is␈α
the␈α∞most␈α
promising␈α
one␈α∞to␈α
work␈α
on␈α∞next.␈α
 Second,␈α
once␈α∞a␈α
task␈α∞has␈α
been
␈↓ α,␈↓chosen,␈αthat␈αtask's␈αpriority␈αvalue␈αthen␈αis␈αused␈α
as␈αan␈αestimate␈αof␈αhow␈αmuch␈αtime␈αand␈αspace␈α
it
␈↓ α,␈↓deserves.␈αThe␈αsample␈αtask␈αabove␈αmight␈αrate␈α20␈αcpu␈αseconds,␈αand␈α200␈αlist␈αcells.␈αWhen␈αeither
␈↓ α,␈↓of␈αthese␈αresources␈αis␈αused␈αup,␈αAM␈αterminates␈αwork␈αon␈αthe␈αtask,␈αand␈αproceeds␈αto␈αpick␈αa␈αnew
␈↓ α,␈↓one.␈α∞ These␈α∞two␈α∂limits␈α∞will␈α∞be␈α∞referred␈α∂to␈α∞in␈α∞the␈α∞sequel␈α∂as␈α∞"␈↓βtime/space␈α∞quanta␈↓"␈α∂which␈α∞are
␈↓ α,␈↓allocated␈α
to␈α
the␈αchosen␈α
task.␈α
 Whenever␈αseveral␈α
techniques␈α
exist␈αfor␈α
satisfying␈α
some␈αtask,␈α
the
␈↓ α,␈↓remaining␈αtime/space␈αquanta␈αare␈αdivided␈αevenly␈αamong␈αthose␈αalternatives;␈αi.e.,␈αeach␈αmethod
␈↓ α,␈↓is␈α⊃tried␈α⊂for␈α⊃a␈α⊂small␈α⊃time.␈α⊂This␈α⊃policy␈α⊂of␈α⊃parceling␈α⊂out␈α⊃time␈α⊂and␈α⊃space␈α⊂quanta␈α⊃is␈α⊂called
␈↓ α,␈↓"activation␈α⊂energy"␈α⊂in␈α⊃[Hewitt␈α⊂76]␈α⊂and␈α⊂called␈α⊃"resource-limited␈α⊂processes"␈α⊂in␈α⊃[Norman␈α⊂&
␈↓ α,␈↓Bobrow␈α75].␈α In␈αthe␈αcase␈αof␈α≡lling␈αin␈αexamples␈αof␈αsets,␈αthe␈αspace␈αquantum␈α(200␈αcells)␈αwill␈αbe
␈↓ α,␈↓used up quickly (long before the 20 seconds expire).

␈↓ α,␈↓There are two big questions now:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ Exactly how is a task proposed and ranked?
␈↓ α,␈↓␈↓ ∧LHow is a plausible new task ≡rst formulated?
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	14␈↓ε␈αλHere␈αλis␈αλthat␈αλformula:␈αλWorth(J)␈αλ=␈αλ||SQRT(SUM␈αλR␈↓#vi␈↓#␈↓#
2␈↓#)||␈αλx␈α	[␈αλ0.2xWorth(A)␈αλ+␈αλ0.3xWorth(F)␈αλ+␈αλ0.5xWorth(C)],␈αλwhere␈αλJ␈αλ=␈αλjob␈α	to␈αλbe
␈↓ α,␈↓ε␈↓ βLjudged␈α	=␈αλ(Act␈α	A,␈α	Facet␈αλF,␈α	Concept␈αλC),␈α	and␈α	{R␈↓#vi␈↓#}␈αλare␈α	the␈αλratings␈α	of␈α	the␈αλreasons␈α	supporting␈αλJ.␈α	 For␈α	the␈αλsample
␈↓ α,␈↓ε␈↓ βLjob␈α∂pictured␈α∂in␈α∂the␈α∂box␈α∂below,␈α∞A=Fillin,␈α∂F=Examples,␈α∂C=Sets,␈α∂{R␈↓#vi␈↓#}={100,100,200}.␈α∂The␈α∂formula␈α∂will␈α∞be
␈↓ α,␈↓ε␈↓ βLrepeated -- and explained -- in Section 4.2, on page 39.
␈↓ α,␈↓␈↓εChapter 3␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε33␈↓-

␈↓ α,␈↓␈↓ ∧LHow do the supporting reasons for the task get assigned?
␈↓ α,␈↓␈↓ ∧LHow does each reason get assigned an absolute numeric rating?
␈↓ α,␈↓␈↓ ∧LDoes a task's priority value change? When and how?
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓ How does AM execute a task, once it's chosen?
␈↓ α,␈↓␈↓ ∧LExactly what can be done during a task's execution?

␈↓ α,␈↓The next chapter will deal with both of these questions.
␈↓ α,␈↓␈↓ ,-␈↓ε34␈↓-




␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬	␈↓∧Chapter 4.   Heuristic Rules␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓



␈↓ α,␈↓Assume␈α∂that␈α∂somehow␈α∂AM␈α∂has␈α∂selected␈α∂a␈α∂particular␈α∂task␈α∂from␈α∂the␈α∂agenda␈α∂¬␈α⊂say␈α∂"␈↓¬Fill-in
␈↓ α,␈↓¬Examples␈α∞of␈α∞Primes␈↓".␈α∂ What␈α∞precisely␈α∞does␈α∞AM␈α∂do,␈α∞in␈α∞order␈α∞to␈α∂execute␈α∞the␈α∞task?␈α∂How␈α∞␈↓βare␈↓
␈↓ α,␈↓examples of primes ≡lled in?

␈↓ α,␈↓The answer can be compactly stated as follows:
␈↓ α,␈↓β␈↓ αl"AM selects relevant heuristics, and executes them."

␈↓ α,␈↓This␈α∂really␈α∂just␈α∂splits␈α∂our␈α∂original␈α∂question␈α∂into␈α∂two␈α∂new␈α∂ones:␈α∂(i)␈α∂How␈α∂are␈α∂the␈α∞relevant
␈↓ α,␈↓heuristics␈αselected,␈α
and␈α(ii)␈α
What␈αdoes␈α
it␈αmean␈α
for␈αheuristics␈α
to␈αbe␈α
executed␈α(e.g.,␈α
how␈αdoes
␈↓ α,␈↓executing a heuristic rule help to ≡ll in examples of primes?).

␈↓ α,␈↓These␈α∞two␈α∞topics␈α∞(in␈α∞reverse␈α∞order)␈α∂are␈α∞the␈α∞two␈α∞major␈α∞subjects␈α∞of␈α∞this␈α∂chapter.␈α∞Although
␈↓ α,␈↓several examples of heuristics will be given, the complete list is relegated to Appendix 2. ␈↓	1␈↓

␈↓ α,␈↓The␈α
≡rst␈α
section␈α
explains␈αwhat␈α
heuristic␈α
rules␈α
look␈α
like␈α(their␈α
"syntax",␈α
as␈α
it␈α
were).␈α The␈α
next
␈↓ α,␈↓three␈α⊂sections␈α⊂illustrate␈α⊂how␈α⊂they␈α⊂can␈α⊂be␈α⊂executed␈α⊂to␈α⊂achieve␈α⊂their␈α⊂desired␈α⊃results␈α⊂(their
␈↓ α,␈↓"semantics").

␈↓ α,␈↓Section␈α
4.5␈α
explains␈αwhere␈α
the␈α
rules␈α
are␈αstored␈α
and␈α
how␈α
they␈αare␈α
accessed␈α
at␈αthe␈α
appropriate
␈↓ α,␈↓times.

␈↓ α,␈↓Finally,␈α
the␈αinitial␈α
body␈αof␈α
heuristics␈αis␈α
analyzed.␈α The␈α
informal␈αknowledge␈α
they␈α
contain␈αis
␈↓ α,␈↓categorized␈α⊗and␈α⊗described.␈α⊗ Unintentionally,␈α↔the␈α⊗distribution␈α⊗of␈α⊗heuristics␈α↔among␈α⊗the
␈↓ α,␈↓concepts is quite nonhomogeneous; this too is described in Section 4.6.


␈↓ α,␈↓␈↓ ¬π␈↓∧␈↓&4.1. Syntax of the Heuristics␈↓)αβ␈↓

␈↓ α,␈↓Let's␈αstart␈αby␈αseeing␈αwhat␈αa␈αheuristic␈αrule␈αlooks␈αlike.␈α In␈αgeneral␈α(see␈α[Davis␈α&␈αKing␈α75]␈αfor
␈↓ α,␈↓historical references to production rules), it will have the form

␈↓ α,␈↓¬␈↓ β,If ␈↓β<situational ∨uent>␈↓¬
␈↓ α,␈↓¬␈↓ β,Then ␈↓β<actions>␈↓¬

␈↓ α,␈↓As an illustration, here is a heuristic rule, relevant when checking examples of anything:



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε␈α	There␈α	they␈α	are␈α	condensed␈α	and␈α	phrased␈α	in␈α	English.␈α
 The␈α	reader␈α	wishing␈α	to␈α	see␈α	examples␈α	of␈α	the␈α	heuristics␈α	as␈α
they␈α	actually
␈↓ α,␈↓ε␈↓ βLwere coded in LISP should glance at Appendix 1.1.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε35␈↓-


␈↓ α,␈↓¬␈↓ αlIf the current task is to Check Examples of any concept X,
␈↓ α,␈↓¬␈↓ β,and (Forsome Y) Y is a generalization of X,
␈↓ α,␈↓¬␈↓ β,and Y has at least 10 examples,
␈↓ α,␈↓¬␈↓ β,and all examples of Y are also examples of X,
␈↓ α,␈↓¬␈↓ αlThen print the following conjecture: X is really no more specialized than Y,
␈↓ α,␈↓¬␈↓ β,and add it to the Examples facet of the concept named "Conjectures",
␈↓ α,␈↓¬␈↓ β,and␈αadd␈αthe␈αfollowing␈αtask␈αto␈αthe␈αagenda:␈α"Check␈αexamples␈αof␈αY",␈αfor␈αthe␈αreason:␈α"Just
␈↓ α,␈↓¬␈↓ ∧,as␈αY␈αwas␈αno␈αmore␈αgeneral␈αthan␈αX,␈αone-of␈αGeneralizations(Y)␈αmay␈αturn␈αout␈αto
␈↓ α,␈↓¬␈↓ ∧,be␈α
no␈α
more␈αgeneral␈α
than␈α
Y",␈α
with␈αa␈α
rating␈α
for␈αthat␈α
reason␈α
computed␈α
as␈αthe
␈↓ α,␈↓¬␈↓ ∧,average␈α6of:␈α6||Examples(Generalizations(Y))||,␈α6||Examples(Y)||,␈α5and
␈↓ α,␈↓¬␈↓ ∧,Priority(Current task).

␈↓ α,␈↓As␈αwith␈αproduction␈αrules,␈αand␈αformal␈αgrammatical␈αrules,␈αeach␈αof␈αAM's␈αheuristic␈αrules␈αhas␈αa
␈↓ α,␈↓left-hand-side␈α∩and␈α∩a␈α∩right-hand-side.␈α∩ On␈α∩the␈α∩left␈α∩is␈α∩a␈α∩test␈α∩to␈α∩see␈α∩whether␈α∩the␈α∩rule␈α∩is
␈↓ α,␈↓applicable,␈α
and␈α
on␈α
the␈α
right␈α
is␈α
a␈α
list␈α
of␈α
actions␈α
to␈α
take␈α
if␈α
the␈α
rule␈α
applies.␈α
 The␈α
left-hand-side
␈↓ α,␈↓will␈α
also␈α
be␈α
called␈α∞the␈α
IF-part,␈α
the␈α
predicate,␈α
the␈α∞preconditions,␈α
left␈α
side,␈α
or␈α∞the␈α
situational
␈↓ α,␈↓∨uent␈αof␈α
the␈αrule.␈α
 The␈αright-hand-side␈α
will␈αsometimes␈α
be␈αreferred␈α
to␈αas␈α
the␈αTHEN-part,␈α
the
␈↓ α,␈↓response, the right side, or the actions part of the rule.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.1.1. Syntax of the Left-hand Side␈↓)αβ␈↓

␈↓ α,␈↓The␈α
situational␈α
∨uent␈α
is␈α
a␈α
LISP␈α
predicate,␈α
a␈α
function␈α
which␈α
always␈α
returns␈α
True␈α∞or␈α
False
␈↓ α,␈↓(in␈α⊂LISP,␈α⊂it␈α⊂actually␈α⊃returns␈α⊂either␈α⊂the␈α⊂atom␈α⊂T␈α⊃or␈α⊂the␈α⊂atom␈α⊂NIL).␈α⊂ This␈α⊃predicate␈α⊂may
␈↓ α,␈↓investigate␈α
facets␈αof␈α
any␈αconcept␈α
(often␈αmerely␈α
to␈α
see␈αwhether␈α
they␈αare␈α
empty␈αor␈α
not),␈αuse␈α
the
␈↓ α,␈↓results␈α∞of␈α∞recent␈α∞tests␈α∞and␈α∞behaviors␈α∞(e.g.,␈α∂to␈α∞see␈α∞how␈α∞much␈α∞cpu␈α∞time␈α∞AM␈α∞spent␈α∂trying␈α∞to
␈↓ α,␈↓work on a certain task), etc.

␈↓ α,␈↓The␈α
left␈αside␈α
is␈αa␈α
conjunction␈αof␈α
the␈αform␈α
P1␈α∧␈α
P2␈α∧...␈α
 All␈αthe␈α
conjuncts,␈αexcept␈α
the␈αvery
␈↓ α,␈↓≡rst␈α→one,␈α→are␈α→arbitrary␈α→LISP␈α→predicates.␈α_ They␈α→are␈α→only␈α→constrained␈α→to␈α→obey␈α_two
␈↓ α,␈↓commandments:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ ␈↓βBe quick!␈↓ (return either True or False in under 0.1 cpu seconds)
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α
␈↓βHave␈αno␈α
side␈α
e≥ects!␈↓␈α(destroying␈α
or␈αcreating␈α
list␈α
structures␈αor␈α
Lisp␈αfunctions,␈α
resetting
␈↓ α,␈↓␈↓ β≤global variables)

␈↓ α,␈↓Here␈α
are␈α
some␈α
sample␈α
conjuncts␈α
that␈α
might␈α
appear␈α
inside␈α
a␈α
left-hand␈α
side␈α
(but␈α
␈↓βnot␈↓␈α
as␈α
the
␈↓ α,␈↓very ≡rst conjunct):

␈↓ α,␈↓␈↓ αl␈↓π# ␈↓α More than half of the current task's time quantum is already exhausted,...

␈↓ α,␈↓α␈↓ αl␈↓π# ␈↓α There are some known examples of Structures,...

␈↓ α,␈↓α␈↓ αl␈↓π#␈α∂␈↓α␈α∂Some␈α∂generalization␈α∂of␈α∂the␈α∂current␈α∞concept␈α∂(the␈α∂concept␈α∂mentioned␈α∂as␈α∂part␈α∂of␈α∞the
␈↓ α,␈↓α␈↓ β≤current task) has an empty Examples facet,...

␈↓ α,␈↓α␈↓ αl␈↓π#␈α␈↓α␈αThe␈αspace␈αquantum␈αof␈αthe␈αcurrent␈αtask␈α
is␈αgone,␈αbut␈αits␈αtime␈αallocation␈αis␈αless␈αthan␈α
10%
␈↓ α,␈↓α␈↓ β≤used up,...

␈↓ α,␈↓α␈↓ αl␈↓π#␈α
␈↓α␈α
A␈αtask␈α
recently␈α
selected␈αhad␈α
the␈α
form␈α
␈↓¬"Restructure␈αfacet␈α
F␈α
of␈αconcept␈α
X"␈↓α,␈α
where␈α
F␈αis
␈↓ α,␈↓α␈↓ β≤any facet, and X is the current concept,...
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε36␈↓-

␈↓ α,␈↓α␈↓ αl␈↓π# ␈↓α The user has used this system at least once before,...

␈↓ α,␈↓α␈↓ αl␈↓π# ␈↓α It's Tuesday,...

␈↓ α,␈↓The␈αvery␈α≡rst␈αconjunct␈αof␈αeach␈αleft-hand␈αside␈αis␈αspecial.␈αIts␈αsyntax␈αis␈αhighly␈αconstrained.␈α It
␈↓ α,␈↓speci≡es␈αthe␈α
domain␈αof␈αapplicability␈α
of␈αthe␈α
rule,␈αby␈αnaming␈α
a␈αparticular␈α
facet␈αof␈αa␈α
particular
␈↓ α,␈↓concept to which this rule is relevant.

␈↓ α,␈↓AM␈α∞uses␈α
this␈α∞≡rst␈α
conjunct␈α∞as␈α
a␈α∞fast␈α
"pre-precondition",␈α∞so␈α
that␈α∞the␈α
only␈α∞rules␈α∞whose␈α
left-
␈↓ α,␈↓hand␈α
sides␈αget␈α
evaluated␈αare␈α
already␈αknown␈α
to␈α
be␈αsomewhat␈α
relevant␈αto␈α
the␈αtask␈α
at␈αhand.␈α
In
␈↓ α,␈↓fact,␈α∩AM␈α∩physically␈α∩attaches␈α∩each␈α∩rule␈α∪to␈α∩the␈α∩facet␈α∩and␈α∩concept␈α∩mentioned␈α∩in␈α∪its␈α∩≡rst
␈↓ α,␈↓conjunct.␈↓	2␈↓

␈↓ α,␈↓This␈αcan␈αbe␈αviewed␈αas␈α
the␈α"syntax"␈αof␈αthe␈αvery␈α
≡rst␈αconjunct␈αon␈αeach␈αrule's␈α
left-hand␈αside.
␈↓ α,␈↓Here are two typical examples of allowable ≡rst conjuncts:

␈↓ α,␈↓␈↓ αl␈↓π#␈α∞␈↓α␈α
The␈α∞current␈α∞task␈α
(the␈α∞one␈α∞last␈α
selected␈α∞from␈α∞the␈α
agenda)␈α∞is␈α∞of␈α
the␈α∞form␈α∞␈↓¬"Check␈α
the
␈↓ α,␈↓¬␈↓ β≤Domain/range facet of concept X"␈↓α, where X is any operation

␈↓ α,␈↓α␈↓ αl␈↓π# ␈↓α The current task is of the form ␈↓¬"Fillin the examples facet of the Primes concept"

␈↓ α,␈↓These␈αare␈αthe␈α
only␈αguidelines␈αwhich␈αthe␈α
left-hand␈αside␈αof␈αa␈α
heuristic␈αrule␈αmust␈αsatisfy.␈α
 Any
␈↓ α,␈↓LISP␈α
predicate␈α
which␈α
satis≡es␈α
these␈α
constraints␈α
is␈α
a␈α
syntactically␈α
valid␈α
left-hand␈α
side␈α
for␈αa
␈↓ α,␈↓heuristic␈α∞rule.␈α
 It␈α∞turned␈α
out␈α∞later␈α
that␈α∞this␈α
excessive␈α∞freedom␈α
made␈α∞it␈α
di≠cult␈α∞for␈α∞AM␈α
to
␈↓ α,␈↓inspect␈α
and␈α
analyze␈α
and␈α
synthesize␈αits␈α
own␈α
heuristics;␈α
such␈α
a␈αneed␈α
was␈α
not␈α
foreseen␈α
at␈αthe
␈↓ α,␈↓time AM was designed.

␈↓ α,␈↓Because␈α
of␈α
this␈α
freedom,␈αthere␈α
is␈α
not␈α
much␈αmore␈α
to␈α
say␈α
about␈αthe␈α
left-hand␈α
sides␈α
of␈αrules.
␈↓ α,␈↓As␈α⊗the␈α⊗reader␈α⊗encounters␈α⊗heuristics␈α⊗in␈α∃the␈α⊗next␈α⊗few␈α⊗sections,␈α⊗he␈α⊗should␈α⊗notice␈α∃the
␈↓ α,␈↓(unfortunate) variety of conjuncts which may occur as part of their left-hand sides.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.1.2. Syntax of the Right-hand Side␈↓)αβ␈↓

␈↓ α,␈↓"Running"␈αthe␈α
left-hand-side␈αmeans␈αevaluating␈α
the␈αseries␈αof␈α
conjoined␈αlittle␈αpredicates␈α
there,
␈↓ α,␈↓to␈α
see␈α
if␈αthey␈α
all␈α
return␈α
True.␈αIf␈α
so,␈α
we␈α
say␈αthat␈α
the␈α
rule␈α
"triggers".␈αIn␈α
that␈α
case,␈α
the␈αright-
␈↓ α,␈↓hand-side␈αis␈α"run",␈αwhich␈αmeans␈αexecuting␈αall␈αthe␈αactions␈αspeci≡ed␈αthere.␈α A␈αsingle␈αheuristic
␈↓ α,␈↓rule␈α
may␈α
have␈αa␈α
list␈α
of␈α
several␈αactions␈α
as␈α
its␈α
right-hand-side.␈α The␈α
actions␈α
are␈α
executed␈αin
␈↓ α,␈↓order, and we then say the rule has ≡nished running.

␈↓ α,␈↓Only␈α
the␈αright-hand-side␈α
of␈αa␈α
heuristic␈αrule␈α
is␈αpermitted␈α
to␈αhave␈α
side␈αe≥ects.␈α
 The␈αright␈α
side
␈↓ α,␈↓of a rule is a series of little LISP functions, each of which is called an ␈↓βaction␈↓.

␈↓ α,␈↓Semantically,␈α
each␈αaction␈α
performs␈αsome␈α
processing␈αwhich␈α
is␈αappropriate␈α
in␈αsome␈α
way␈αto␈α
the
␈↓ α,␈↓kinds␈αof␈αsituations␈α
in␈αwhich␈αthe␈α
left-hand-side␈αwould␈αhave␈α
triggered.␈α The␈α≡nal␈α
value␈αthat
␈↓ α,␈↓the action function returns is irrelevant.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	2␈↓ε␈α
Sometimes,␈α
I␈α
will␈α
mention␈α
where␈α
a␈α
certain␈α
rule␈α
is␈α
attached;␈α
in␈α
that␈α
case,␈α
I␈α
can␈α
omit␈α
explicit␈α
mention␈α
of␈α
the␈αfirst␈α
conjunct.
␈↓ α,␈↓ε␈↓ βLConversely, if I include that conjunct, I needn't tell you where the rule is stored.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε37␈↓-

␈↓ α,␈↓Syntactically,␈αthere␈αis␈αonly␈αone␈αconstraint␈αwhich␈αeach␈αfunction␈αor␈α"action"␈αmust␈αsatisfy:␈αEach
␈↓ α,␈↓action has one of the following 3 side-e≥ects, and no other side-e≥ects:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ It suggests a new task for the agenda.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓ It causes a new concept to be created.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓ It adds (or deletes) a certain entry to a particular facet of a particular concept.

␈↓ α,␈↓To␈α∞repeat:␈α∞the␈α∞right␈α∞side␈α∞of␈α
a␈α∞rule␈α∞contains␈α∞a␈α∞list␈α∞of␈α
actions,␈α∞each␈α∞of␈α∞which␈α∞is␈α∞one␈α∞of␈α
the
␈↓ α,␈↓above␈αthree␈α
types.␈α A␈αsingle␈α
rule␈αmight␈αthus␈α
result␈αin␈αthe␈α
creation␈αof␈αseveral␈α
new␈αconcepts,
␈↓ α,␈↓the␈α∞addition␈α∂of␈α∞many␈α∂new␈α∞tasks␈α∞to␈α∂the␈α∞agenda,␈α∂and␈α∞the␈α∞≡lling␈α∂in␈α∞of␈α∂some␈α∞facets␈α∂of␈α∞some
␈↓ α,␈↓already-existing concepts.

␈↓ α,␈↓These three kinds of actions will now be discussed in the following three sections.


␈↓ α,␈↓␈↓ ∧U␈↓∧␈↓&4.2. Heuristics Suggest New Tasks␈↓)αβ␈↓

␈↓ α,␈↓This␈α
section␈α
discusses␈α
the␈α"proposing␈α
a␈α
new␈α
task"␈αkind␈α
of␈α
action.␈α
 Here␈αis␈α
the␈α
basic␈α
idea␈αin␈α
a
␈↓ α,␈↓nutshell:␈α∂The␈α∞left-hand-side␈α∂of␈α∂a␈α∞rule␈α∂triggers.␈α∂ Scattered␈α∞among␈α∂the␈α∂"things␈α∞to␈α∂do"␈α∂in␈α∞its
␈↓ α,␈↓right-hand-side␈α⊃are␈α⊃some␈α⊂suggestions␈α⊃for␈α⊃future␈α⊂tasks.␈α⊃These␈α⊃new␈α⊂tasks␈α⊃are␈α⊃then␈α⊂simply
␈↓ α,␈↓added to the agenda list.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.2.1. An Illustration: "Fill in Generalizations of Equality"␈↓)αβ␈↓

␈↓ α,␈↓If␈αa␈α
new␈αtask␈α
is␈αsuggested␈α
by␈αa␈α
heuristic␈αrule,␈α
then␈αthat␈α
rule␈αmust␈α
specify␈αhow␈α
to␈αassemble
␈↓ α,␈↓the␈α
new␈α
task,␈α∞how␈α
to␈α
get␈α∞reasons␈α
for␈α
it,␈α∞and␈α
how␈α
to␈α∞evaluate␈α
those␈α
reasons.␈α∞ For␈α
example,
␈↓ α,␈↓here␈α
is␈α
a␈αtypical␈α
heuristic␈α
rule␈α
which␈αproposes␈α
a␈α
new␈αtask␈α
to␈α
add␈α
to␈αthe␈α
agenda.␈α
It␈α
says␈αto
␈↓ α,␈↓generalize a predicate if it is very rarely␈↓	3␈↓ satis≡ed:

␈↓ α,␈↓¬␈↓ αlIf the current task was (Fill-in examples of X),
␈↓ α,␈↓¬␈↓ β,and X is a predicate,
␈↓ α,␈↓¬␈↓ β,and more than 100 items are known in the domain of X,
␈↓ α,␈↓¬␈↓ β,and at least 10 cpu seconds were spent trying to randomly instantiate X,
␈↓ α,␈↓¬␈↓ β,and the ratio of successes/failures is both >0 and less than .05
␈↓ α,␈↓¬␈↓ αlThen␈α∞add␈α∞the␈α∞following␈α∞task␈α∞to␈α∞the␈α∞agenda:␈α∞(Fill-in␈α∞generalizations␈α∞of␈α∞X),␈α∞for␈α∞the␈α
following
␈↓ α,␈↓¬␈↓ ∧,reason:
␈↓ α,␈↓¬␈↓ β,"X is rarely satisfied; a slightly less restrictive concept might be more interesting".
␈↓ α,␈↓¬␈↓ β,This␈α⊂reason's␈α⊂rating␈α⊂is␈α⊂computed␈α⊃as␈α⊂three␈α⊂times␈α⊂the␈α⊂ratio␈α⊃of␈α⊂nonexamples/examples
␈↓ α,␈↓¬␈↓ ∧,found.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	3␈↓ε␈α
The␈α
most␈α
suspicious␈α
part␈α
of␈α
the␈α	situational␈α
fluent␈α
(the␈α
IF-part)␈α
is␈α
the␈α
number␈α	".05".␈α
Where␈α
did␈α
it␈α
come␈α
from?␈α
Hint:␈α
if␈α	all
␈↓ α,␈↓ε␈↓ βLhumans␈αhad␈αf␈αfingers␈αper␈αhand,␈αh␈αhands,␈αt␈αtoes␈αper␈αfoot,␈αand␈αF␈αfeet,␈αthis␈αwould␈αprobably␈αbe␈α1/(fh+tF).
␈↓ α,␈↓ε␈↓ βLSeriously,␈α	one␈αλcan␈α	change␈αλthis␈α	value␈αλ(to␈α	.01␈αλor␈α	to␈α	.25)␈αλwith␈α	virtually␈αλno␈α	change␈αλin␈α	AM's␈αλbehavior.␈α	This␈α	is␈αλthe
␈↓ α,␈↓ε␈↓ βLconclusion␈αof␈αexperiment␈α3␈α(see␈αSection␈α6.2.3).␈αSuch␈αempirical␈αjustification␈αis␈αone␈αimportant␈α
reason␈αfor
␈↓ α,␈↓ε␈↓ βLactually writing and running large programs like AM.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε38␈↓-

␈↓ α,␈↓Even␈α
this␈α
is␈α
one␈α∞full␈α
step␈α
above␈α
the␈α
actual␈α∞LISP␈α
implementation,␈α
where␈α
␈↓¬"X␈α
is␈α∞a␈α
predicate"␈↓
␈↓ α,␈↓would␈α∪be␈α∪coded␈α∪as␈α∪␈↓¬"(MEMBER␈α∪X␈α∪(EXAMPLES␈α∪PREDICATE))"␈↓.␈α∪ The␈α∪function␈α∪␈↓¬EXAMPLES(X)␈↓
␈↓ α,␈↓rummages␈αabout␈αlooking␈αfor␈αalready-existing␈αexamples␈αof␈αX.␈α Also,␈αthe␈αLISP␈αcode␈αcontains
␈↓ α,␈↓information␈αfor␈αnormalizing␈αall␈αthe␈αnumbers␈αproduced,␈αso␈αthat␈αthey␈αwill␈αlie␈αin␈αthe␈αrange␈α0-
␈↓ α,␈↓1000.

␈↓ α,␈↓Let's␈α
examine␈α
an␈α
instance␈α
of␈αwhere␈α
this␈α
rule␈α
was␈α
used.␈αAt␈α
some␈α
point,␈α
AM␈α
chose␈α
the␈αtask
␈↓ α,␈↓␈↓¬"Fillin␈αexamples␈αof␈αList-equality"␈↓.␈α One␈αof␈αthe␈αways␈αit␈α≡lled␈αin␈αexamples␈αof␈αthis␈αpredicate␈αwas
␈↓ α,␈↓to␈α∂run␈α∞it␈α∂on␈α∂pairs␈α∞of␈α∂randomly-chosen␈α∞lists,␈α∂and␈α∂observe␈α∞whether␈α∂the␈α∞result␈α∂was␈α∂True␈α∞or
␈↓ α,␈↓False␈↓	4␈↓.␈α∂ Say␈α∂that␈α∂244␈α∂random␈α∂pairs␈α∂of␈α∞lists␈α∂were␈α∂tried,␈α∂and␈α∂only␈α∂twice␈α∂was␈α∂this␈α∞predicate
␈↓ α,␈↓satis≡ed.␈α Sometime␈αlater,␈α
the␈αIF␈αpart␈αof␈α
the␈αabove␈αheuristic␈α
is␈αexamined.␈α All␈αthe␈α
conditions
␈↓ α,␈↓are␈αmet,␈αso␈αit␈α"triggers".␈αFor␈αexample,␈αthe␈α"ratio␈αof␈αsuccesses␈αto␈αfailures"␈αis␈αjust␈α2/242,␈αwhich
␈↓ α,␈↓is␈αclearly␈αgreater␈αthan␈αzero␈αand␈αless␈αthan␈α0.05.␈α So␈αthe␈αright-hand␈αside␈α(THEN-part)␈αof␈αthe
␈↓ α,␈↓above␈α∪rule␈α∪is␈α∀executed.␈α∪The␈α∪right-hand␈α∀side␈α∪initiates␈α∪only␈α∀one␈α∪action:␈α∪the␈α∀task␈α∪"␈↓¬Fillin
␈↓ α,␈↓¬generalizations␈αof␈αList-equality␈↓"␈αis␈αadded␈αto␈αthe␈αagenda,␈αtagged␈αwith␈αthe␈αreason␈α"List-equality
␈↓ α,␈↓is␈α⊂rarely␈α⊂satis≡ed;␈α⊂a␈α⊂slightly␈α⊂less␈α⊂restrictive␈α⊂concept␈α⊂might␈α⊂be␈α⊂more␈α⊂interesting",␈α⊃and␈α⊂that
␈↓ α,␈↓reason is assigned a numeric rating of 3x(242/2) = 363.

␈↓ α,␈↓Notice␈α
that␈α
the␈α
heuristic␈α
rule␈α
above␈α
supplied␈α
a␈α
little␈α
function␈α
to␈α
compute␈α
the␈α
value␈α∞of␈α
the
␈↓ α,␈↓reason.␈α∃ That␈α∃formula␈α∃was:␈α∃"three␈α∃times␈α∃the␈α∃ratio␈α∃of␈α⊗examples/nonexamples␈α∃found".␈↓	5␈↓
␈↓ α,␈↓Functions␈α
of␈αthis␈α
type,␈αto␈α
compute␈αthe␈α
rating␈α
for␈αa␈α
reason,␈αsatisfy␈α
the␈αsame␈α
constraints␈αas␈α
the
␈↓ α,␈↓left-hand-side␈α∂did:␈α∂the␈α∂function␈α∂must␈α∂be␈α∂very␈α∂fast␈α∂and␈α∂it␈α∂must␈α∂have␈α∂no␈α∂side␈α⊂e≥ects.␈α∂The
␈↓ α,␈↓"intelligence"␈α
that␈α
AM␈α
exhibits␈αin␈α
selecting␈α
which␈α
task␈α
to␈αwork␈α
on␈α
ultimately␈α
depends␈αon␈α
the
␈↓ α,␈↓accuracy␈αof␈αthese␈αlocal␈αrule␈αevaluation␈αformulae.␈αEach␈αone␈αis␈αso␈αspecialized␈αthat␈αit␈αis␈α"easy"
␈↓ α,␈↓for␈αit␈αto␈α
give␈αa␈αvalid␈α
result;␈αthe␈αrange␈α
of␈αsituations␈αit␈α
must␈αjudge␈αis␈α
quite␈αnarrow.␈α Note␈α
that
␈↓ α,␈↓these␈α⊃little␈α⊂formulae␈α⊃were␈α⊂hand-written,␈α⊃individually,␈α⊃by␈α⊂the␈α⊃author.␈α⊂AM␈α⊃wasn't␈α⊃able␈α⊂to
␈↓ α,␈↓create new little reason-rating formulae.

␈↓ α,␈↓The␈α
reason-rating␈α∞function␈α
is␈α
evaluated␈α∞at␈α
the␈α
moment␈α∞the␈α
job␈α
is␈α∞suggested,␈α
and␈α∞only␈α
the
␈↓ α,␈↓numeric␈α
result␈α
is␈αremembered,␈α
not␈α
the␈αoriginal␈α
function.␈α
 In␈αother␈α
words,␈α
we␈αtack␈α
on␈α
a␈αlist␈α
of
␈↓ α,␈↓reasons␈αand␈αassociated␈αnumbers,␈αfor␈αeach␈αjob␈αon␈αthe␈αagenda.␈α The␈αagenda␈α␈↓βdoesn't␈↓␈αmaintain
␈↓ α,␈↓copies␈αof␈αthe␈αreason-rating␈αfunctions␈αwhich␈αgave␈αthose␈αnumbers.␈αThis␈αsimpli≡cation␈αis␈αused
␈↓ α,␈↓merely to save the system some space and time.

␈↓ α,␈↓Let's␈α⊂turn␈α⊂now␈α∂from␈α⊂the␈α⊂reason-rating␈α∂formulae␈α⊂to␈α⊂the␈α∂reasons␈α⊂themselves.␈α⊂ Each␈α∂reason
␈↓ α,␈↓supporting␈αa␈αnewly-suggested␈αjob␈αis␈αsimply␈αan␈αEnglish␈αsentence␈α(an␈αopaque␈αstring,␈αa␈αtoken).
␈↓ α,␈↓AM␈α
cannot␈α
do␈α
much␈α
intelligent␈α
processing␈α
on␈α
these␈α
reasons.␈α
AM␈α
is␈α
not␈α
allowed␈α
to␈αinspect
␈↓ α,␈↓parts␈αof␈αit,␈αparse␈αit,␈αtransform␈αit,␈αetc.␈αThe␈αmost␈αAM␈αcan␈αdo␈αis␈αcompare␈αtwo␈αsuch␈αtokens␈αfor
␈↓ α,␈↓equality.␈α Of␈αcourse,␈αit␈αis␈α
not␈αto␈αhard␈αto␈αimagine␈α
this␈αcapability␈αextended␈αto␈αpermit␈α
AM␈αto
␈↓ α,␈↓syntactically␈α
analyze␈α∞such␈α
strings,␈α
or␈α∞to␈α
trivially␈α
compute␈α∞some␈α
sort␈α
of␈α∞"di≥erence"␈α
between



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	4␈↓ε␈αλThe␈αλTrue␈αλones␈αλbecame␈αλexamples␈αλof␈αλList-equality,␈αλand␈αλthe␈αλpairs␈αλof␈αλlists␈αλwhich␈αλdidn't␈αλsatisfy␈αλthis␈αλpredicate␈αλbecame␈α	known␈αλas
␈↓ α,␈↓ε␈↓ βLnon-examples␈α(failures,␈α
foibles,...).␈α A␈αheuristic␈α
similar␈αto␈α
this␈α"random␈αinstantiation"␈α
one␈αis␈α
illustrated␈αin
␈↓ α,␈↓ε␈↓ βLSection 4.4, on page 46.
␈↓ α,␈↓ε␈↓	5␈↓ε In actuality, this would be checked to ensure that the result lies between 0 and 1000.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε39␈↓-

␈↓ α,␈↓two␈αgiven␈αreasons.␈↓	6␈↓␈αEach␈αreason␈αis␈αassumed␈αto␈αhave␈αsome␈αsemantic␈αimpact␈αon␈αthe␈αuser,␈αand
␈↓ α,␈↓is kept around partly for that purpose.

␈↓ α,␈↓Each␈α⊂reason␈α⊂will␈α⊂have␈α⊂a␈α∂numeric␈α⊂rating␈α⊂(a␈α⊂number␈α⊂between␈α∂0␈α⊂and␈α⊂1000)␈α⊂assigned␈α⊂to␈α∂it
␈↓ α,␈↓locally␈α
by␈α
the␈α
heuristic␈αrule␈α
which␈α
proposed␈α
the␈α
task␈αfor␈α
that␈α
reason.␈α
 One␈α
global␈αformula
␈↓ α,␈↓then combines all the reasons' ratings into one single priority value for the task.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.2.2. The Ratings Game␈↓)αβ␈↓

␈↓ α,␈↓In␈αgeneral,␈αa␈α
task␈αon␈αthe␈α
agenda␈αlist␈αwill␈αhave␈α
several␈αreasons␈αin␈α
support␈αof␈αit.␈α Each␈α
reason
␈↓ α,␈↓consists␈α
of␈αan␈α
English␈αphrase␈α
and␈αa␈α
numeric␈αrating.␈α
 How␈αcan␈α
a␈αtask␈α
have␈αmore␈α
than␈αone
␈↓ α,␈↓reason?␈α⊂ There␈α∂are␈α⊂two␈α⊂contributing␈α∂factors:␈α⊂(i)␈α∂A␈α⊂single␈α⊂heuristic␈α∂rule␈α⊂can␈α⊂have␈α∂several
␈↓ α,␈↓reasons␈αin␈α
support␈αof␈α
a␈αjob␈α
it␈αsuggests,␈α
and␈α(ii)␈α
When␈αa␈α
rule␈αsuggests␈α
a␈α"new"␈α
task,␈αthat␈α
very
␈↓ α,␈↓same␈αtask␈αmay␈αalready␈αexist␈αon␈αthe␈αagenda,␈αwith␈αquite␈αdistinct␈αreasons␈αtacked␈αon␈αthere.␈α In
␈↓ α,␈↓that case, the new reason(s) are added to the already-known ones.

␈↓ α,␈↓One␈αglobal␈α
formula␈αlooks␈α
at␈αall␈α
the␈αratings␈αfor␈α
the␈αreasons,␈α
and␈αcombines␈α
them␈αinto␈αa␈α
single
␈↓ α,␈↓priority value for the task as a whole.  Below is that formula, in all its gory detail:

␈↓ α,␈↓¬␈↓ αlWorth(J) = ||SQRT(SUM R␈↓#vi␈↓#␈↓#
2␈↓#)|| ␈↓εx␈↓¬ [ .2␈↓εx␈↓¬Worth(A) + .3␈↓εx␈↓¬Worth(F) + .5␈↓εx␈↓¬Worth(C)]

␈↓ α,␈↓¬␈↓ αlWhere J = job to be judged = (Act A, Facet F, Concept C)
␈↓ α,␈↓¬␈↓ αl     and {R␈↓#vi␈↓#} are the ratings of the reasons supporting J.


␈↓ α,␈↓For␈α⊃example,␈α⊃consider␈α⊃the␈α⊃job␈α⊂J␈α⊃=␈α⊃(Check␈α⊃examples␈α⊃of␈α⊂Primes).␈α⊃ The␈α⊃act␈α⊃A␈α⊃would␈α⊂be
␈↓ α,␈↓"Check",␈αwhich␈αhas␈αa␈αnumeric␈αworth␈αof␈α100.␈α The␈αfacet␈αF␈αwould␈αbe␈α"Examples",␈αwhich␈αhas
␈↓ α,␈↓a␈α∞numeric␈α
worth␈α∞of␈α∞700.␈α
 The␈α∞concept␈α
C␈α∞would␈α∞be␈α
"Primes",␈α∞which␈α
at␈α∞the␈α∞moment␈α
might
␈↓ α,␈↓have␈α∞Worth␈α∂of␈α∞800.␈α∞ Say␈α∂there␈α∞were␈α∞four␈α∂reasons,␈α∞having␈α∞values␈α∂200,␈α∞300,␈α∞200,␈α∂and␈α∞500.
␈↓ α,␈↓The␈α⊂double␈α⊂lines␈α⊃"||...||"␈α⊂indicate␈α⊂normalization,␈α⊃which␈α⊂means␈α⊂that␈α⊃the␈α⊂≡nal␈α⊂value␈α⊃of␈α⊂the
␈↓ α,␈↓square-root␈αmust␈α
be␈αbetween␈α
0␈αand␈α
1,␈αwhich␈αis␈α
done␈αby␈α
dividing␈αthe␈α
result␈αof␈α
the␈αSquare-
␈↓ α,␈↓root by 1000 and then truncating to 1.0 if the result exceeds unity.

␈↓ α,␈↓In␈α∂this␈α∂case,␈α∂we␈α∂≡rst␈α∂compute␈α∂Sqrt(200␈↓#
2␈↓#␈α∂+␈α∂300␈↓#
2␈↓#␈α∂+␈α∂200␈↓#
2␈↓#␈α∂+␈α∂500␈↓#
2␈↓#)␈α∂=␈α∂Sqrt(420,000),␈α∂which␈α∞is
␈↓ α,␈↓about␈α648.␈α
 After␈αnormalization,␈αthis␈α
becomes␈α0.648.␈α The␈α
expression␈αin␈αsquare␈α
brackets␈αin
␈↓ α,␈↓the␈α
formula␈↓	7␈↓␈α
is␈α
actually␈α
computed␈α
as␈α
the␈α
dot-product␈α
of␈α
two␈α
vectors␈↓	8␈↓;␈α
in␈α
this␈α
case␈α
it␈α
is␈αthe
␈↓ α,␈↓dot-product␈α⊂of␈α⊂(100␈α⊂700␈α∂800)␈α⊂and␈α⊂(.2␈α⊂.3␈α∂.5),␈α⊂which␈α⊂yields␈α⊂630.␈α∂This␈α⊂is␈α⊂multiplied␈α⊂by␈α∂the
␈↓ α,␈↓normalized Square-root value, 0.648, and we end up with a ≡nal priority rating of 408.

␈↓ α,␈↓The␈α⊃four␈α⊂reasons␈α⊃each␈α⊃have␈α⊂a␈α⊃fairly␈α⊃low␈α⊂priority,␈α⊃and␈α⊂the␈α⊃total␈α⊃priority␈α⊂of␈α⊃the␈α⊃task␈α⊂is
␈↓ α,␈↓therefore␈αnot␈αgreat.␈αIt␈αis,␈αhowever,␈αhigher␈αthan␈αany␈αsingle␈αreason␈αmultiplied␈αby␈α0.648.␈α This
␈↓ α,␈↓is␈αbecause␈αthere␈αare␈αmany␈α␈↓βdistinct␈↓␈αreasons␈αsupporting␈αit.␈α The␈αglobal␈αformula␈αuniting␈αthese

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	6␈↓ε␈αλIt␈α	is␈αλin␈αλfact␈α	trivial␈αλto␈αλIMAGINE␈α	it.␈αλOf␈αλcourse␈α	DOING␈αλit␈α	is␈αλquite␈αλa␈α	bit␈αλless␈αλtrivial.␈α	In␈αλfact,␈αλit␈α	probably␈αλis␈αλthe␈α	toughest␈αλof␈α	all␈αλthe
␈↓ α,␈↓ε␈↓ βL"open research problems" I'll propose.
␈↓ α,␈↓ε␈↓	7␈↓ε Namely, [ 0.2xWorth(A) + 0.3xWorth(F) + 0.5xWorth(C) ].
␈↓ α,␈↓ε␈↓	8␈↓ε␈α	Namely,␈α	<Worth(A),␈α	Worth(F),␈α	Worth(C)>␈α	and␈α	<␈α	.2,␈α	.3,␈α	.5␈α	>.␈α	The␈α	dot-product␈α	of␈α	<a1␈α	a2␈α	a3...>␈α	and␈α	<b1␈α	b2␈α	b3...>␈α	is␈α	defined
␈↓ α,␈↓ε␈↓ βLas (a1 x b1) + (a2 x b2) + (a3 x b3) +...
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε40␈↓-

␈↓ α,␈↓reasons'␈α⊂values␈α⊂does␈α⊂not␈α⊂simply␈α∂take␈α⊂the␈α⊂largest␈α⊂of␈α⊂them␈α∂(ignoring␈α⊂the␈α⊂rest),␈α⊂nor␈α⊂does␈α∂it
␈↓ α,␈↓simply add them up.

␈↓ α,␈↓The␈α
above␈α
formula␈αwas␈α
intended␈α
originally␈α
as␈αa␈α
≡rst␈α
pass,␈α
an␈α␈↓βad␈α
hoc␈↓␈α
guess,␈α
which␈αI␈α
expected
␈↓ α,␈↓I'd␈αhave␈αto␈αmodify␈αlater.␈αSince␈αit␈αhas␈αworked␈αsuccessfully,␈αI␈αhave␈αnot␈αmessed␈αwith␈αit.␈α There
␈↓ α,␈↓is␈αno␈α
reason␈αbehind␈α
it,␈αno␈α
justi≡cation␈αfor␈αtaking␈α
dot-products␈αof␈α
vectors,␈αetc.␈α
 I␈αconcluded,
␈↓ α,␈↓and␈α∃recent␈α∃experiments␈α∃tend␈α∃to␈α∃con≡rm,␈α∃that␈α∃the␈α∃particular␈α∃form␈α∃of␈α∃the␈α∃formula␈α∀is
␈↓ α,␈↓unimportant; only some general characteristics need be present:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α
The␈α
priority␈αvalue␈α
of␈α
a␈αtask␈α
is␈α
a␈αmonotone␈α
increasing␈α
function␈αof␈α
each␈α
of␈αits␈α
reasons'
␈↓ α,␈↓␈↓ β≤ratings.␈α∞ If␈α∞a␈α∞new␈α∞supporting␈α∞reason␈α∂is␈α∞found,␈α∞the␈α∞task's␈α∞value␈α∞is␈α∂increased.␈α∞ The
␈↓ α,␈↓␈↓ β≤better that new reason, the bigger the increase.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α
If␈α
an␈α∞already-known␈α
supporting␈α
reason␈α∞is␈α
re-proposed,␈α
the␈α
value␈α∞of␈α
the␈α
task␈α∞is␈α
␈↓βnot␈↓
␈↓ α,␈↓␈↓ β≤increased␈α⊃(at␈α∩least,␈α⊃it's␈α∩not␈α⊃increased␈α⊃very␈α∩much).␈α⊃ Like␈α∩humans,␈α⊃AM␈α∩is␈α⊃fooled
␈↓ α,␈↓␈↓ β≤whenever the same reason reappears in disguised form.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α∪The␈α∪priority␈α∪of␈α∩a␈α∪task␈α∪involving␈α∪concept␈α∩C␈α∪should␈α∪be␈α∪a␈α∪monotone␈α∩increasing
␈↓ α,␈↓␈↓ β≤function␈α∞of␈α∞the␈α∞overall␈α∂worth␈α∞of␈α∞C.␈α∞Two␈α∂similar␈α∞tasks␈α∞dealing␈α∞with␈α∂two␈α∞di≥erent
␈↓ α,␈↓␈↓ β≤concepts,␈αeach␈αsupported␈αby␈αthe␈αsame␈αlist␈αof␈αreasons␈αand␈αreason␈αratings,␈α
should␈αbe
␈↓ α,␈↓␈↓ β≤ordered by the worth of those two concepts.

␈↓ α,␈↓I␈α∞believe␈α∞that␈α∞all␈α∞of␈α
these␈α∞criteria␈α∞are␈α∞absolutely␈α∞essential␈α
to␈α∞good␈α∞behavior␈α∞of␈α∞the␈α
system.
␈↓ α,␈↓Several␈α⊃of␈α⊃the␈α⊃experiments␈α⊃discussed␈α⊃later␈α⊂bear␈α⊃on␈α⊃this␈α⊃question␈α⊃(See␈α⊃Section␈α⊃6.2,␈α⊂page
␈↓ α,␈↓94).␈α∂ Note␈α∞that␈α∂the␈α∂messy␈α∞formula␈α∂given␈α∂on␈α∞the␈α∂last␈α∞page␈α∂does␈α∂incorporate␈α∞all␈α∂3␈α∂of␈α∞these
␈↓ α,␈↓constraints.␈α In␈αaddition,␈αthere␈α
are␈αa␈αfew␈αfeatures␈αof␈α
that␈αformula␈αwhich,␈αwhile␈αprobably␈α
not
␈↓ α,␈↓necessary or even desirable, the reader should be informed of explicitly:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α
The␈α
task's␈α
value␈α
does␈α
not␈α
depend␈α
on␈α
the␈α
order␈α
in␈α
which␈α
the␈α
reasons␈αwere␈α
discovered.
␈↓ α,␈↓␈↓ β≤This␈α⊃is␈α⊂not␈α⊃true␈α⊂psychologically␈α⊃of␈α⊂people,␈α⊃but␈α⊂it␈α⊃is␈α⊂a␈α⊃feature␈α⊂of␈α⊃the␈α⊂particular
␈↓ α,␈↓␈↓ β≤priority-estimating formula initially selected.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α∂Two␈α∞reasons␈α∂are␈α∂either␈α∞considered␈α∂identical␈α∞or␈α∂unrelated.␈α∂ No␈α∞attempt␈α∂is␈α∂made␈α∞to
␈↓ α,␈↓␈↓ β≤reduce␈α~the␈α→priority␈α~value␈α→because␈α~several␈α→of␈α~the␈α→reasons␈α~are␈α→overlapping
␈↓ α,␈↓␈↓ β≤semantically or even just syntacticaly. This, too, is no doubt a mistake.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α
There␈α
is␈α
no␈α
need␈α
to␈α
keep␈α
around␈α
all␈α
the␈α
individual␈α
reasons'␈α
rating␈α∞numbers.␈α
 The
␈↓ α,␈↓␈↓ β≤addition␈α
of␈αa␈α
new␈αreason␈α
will␈α
demand␈αonly␈α
the␈αknowledge␈α
of␈αthe␈α
␈↓βnumber␈↓␈α
of␈αother
␈↓ α,␈↓␈↓ β≤reasons, and the old priority value of the task.
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓␈αA␈αtask␈αwith␈αno␈αreasons␈αgets␈αan␈αabsolute␈αzero␈αrating.␈α As␈αnew␈αreasons␈αare␈αadded,␈αthe
␈↓ α,␈↓␈↓ β≤priority␈α∂slowly␈α∂increases␈α∂toward␈α∂an␈α∞absolute␈α∂maximum␈α∂which␈α∂is␈α∂dependent␈α∞upon
␈↓ α,␈↓␈↓ β≤the overall worth of the concept and facet involved.

␈↓ α,␈↓There␈αis␈αone␈αtopic␈αof␈αpassing␈αinterest␈αwhich␈αshould␈αbe␈αcovered␈αhere.␈α Each␈αpossible␈αAct␈αA
␈↓ α,␈↓(e.g.,␈αFillin,␈αCheck,␈αApply)␈αand␈αeach␈αpossible␈αfacet␈αF␈α(e.g.,␈αExamples,␈αDe≡nition,␈αName(s))␈αis
␈↓ α,␈↓assigned␈αa␈α≡xed␈αnumeric␈αvalue␈α(by␈αhand,␈αby␈αthe␈αauthor).␈α These␈αvalues␈αare␈αused␈αinside␈αthe
␈↓ α,␈↓formula␈αon␈αthe␈αlast␈αpage,␈αwhere␈αit␈αsays␈α`Worth(A)'␈αand␈α`Worth(F)'.␈αThey␈αare␈αfairly␈αresistant
␈↓ α,␈↓to␈α⊂change,␈α⊂but␈α⊂certain␈α⊂orderings␈α⊂should␈α⊂be␈α⊂maintained␈α⊂for␈α⊂best␈α⊂results.␈α⊃E.g.,␈α⊂"Examples"
␈↓ α,␈↓should␈α∞be␈α∞rated␈α∂higher␈α∞than␈α∞"Specializations",␈α∞or␈α∂else␈α∞AM␈α∞may␈α∞whirl␈α∂away␈α∞on␈α∞a␈α∂cycle␈α∞of
␈↓ α,␈↓specialization␈αlong␈αafter␈αthe␈α
concept␈αhas␈αbeen␈αconstrained␈α
into␈αvacuousness.␈αAs␈αfor␈αthe␈α
Acts,
␈↓ α,␈↓their precise values turned out to be even less important than the Facets'.

␈↓ α,␈↓Now␈αthat␈αwe've␈αseen␈αhow␈αto␈αcompute␈αthis␈αpriority␈αvalue␈αfor␈αany␈αgiven␈αtask,␈αlet's␈αnot␈αforget
␈↓ α,␈↓what it's used for. The overall rating has two functions:
␈↓ α,␈↓␈↓ αl(i)␈αThese␈αratings␈αdetermine␈αwhich␈αtask␈αto␈αexecute␈αnext.␈αThis␈αis␈αnot␈αan␈αironclad␈αpolicy:
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε41␈↓-

␈↓ α,␈↓␈↓ β≤In␈α∩reality,␈α∪AM␈α∩prints␈α∪out␈α∩the␈α∪top␈α∩few␈α∪tasks,␈α∩and␈α∪the␈α∩user␈α∪has␈α∩the␈α∪option␈α∩of
␈↓ α,␈↓␈↓ β≤interrupting␈αand␈α
directing␈αAM␈α
to␈αwork␈α
on␈αone␈αof␈α
those␈αother␈α
tasks␈αinstead␈α
of␈αthe
␈↓ α,␈↓␈↓ β≤very top one.
␈↓ α,␈↓␈↓ αl(ii)␈α∞Once␈α∞a␈α∂task␈α∞is␈α∞chosen,␈α∞its␈α∂rating␈α∞determines␈α∞how␈α∞much␈α∂time␈α∞and␈α∞space␈α∂AM␈α∞will
␈↓ α,␈↓␈↓ β≤expend␈αon␈αit␈αbefore␈αquitting␈αand␈α
moving␈αon␈αto␈αa␈αnew␈αtask.␈α The␈α
precise␈αformulae
␈↓ α,␈↓␈↓ β≤are␈α
unimportant.␈α∞ Roughly,␈α
the␈α∞0-1000␈α
rating␈α
is␈α∞divided␈α
by␈α∞ten␈α
to␈α∞determine␈α
how
␈↓ α,␈↓␈↓ β≤much␈αtime␈αto␈α
allow,␈αin␈αcpu␈α
seconds.␈αThe␈αrating␈α
is␈αdivided␈αby␈α
two␈αto␈αdetermine␈α
how
␈↓ α,␈↓␈↓ β≤much space to allow, in list cells.


␈↓ α,␈↓␈↓ ∧E␈↓∧␈↓&4.3. Heuristics Create New Concepts␈↓)αβ␈↓

␈↓ α,␈↓Recall that a heuristic rule's actions are of three types:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ Suggest new tasks and add them to the agenda.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓ Create a new concept.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓ Fill in some entries for a facet of a concept.

␈↓ α,␈↓This subsection discusses the second activity.

␈↓ α,␈↓Here␈αis␈αthe␈α
basic␈αidea␈αin␈α
a␈αnutshell:␈αScattered␈α
among␈αthe␈α"things␈α
to␈αdo"␈αin␈α
the␈αright-hand-
␈↓ α,␈↓side␈α
of␈α
a␈α
rule␈αare␈α
some␈α
requests␈α
to␈αcreate␈α
speci≡c␈α
new␈α
concepts.␈αFor␈α
each␈α
such␈α
request,␈αthe
␈↓ α,␈↓heuristic␈α⊃rule␈α⊃must␈α⊃specify␈α⊃how␈α⊃to␈α⊃construct␈α⊃it.␈α⊃At␈α⊃least,␈α⊃the␈α⊃rule␈α⊃must␈α⊃specify␈α∩ways␈α⊃of
␈↓ α,␈↓assembling␈α
enough␈α
facets␈α
of␈α
the␈α
new␈α
concept␈α
to␈α
disambiguate␈α
it␈α
from␈α
all␈α
the␈α∞other␈α
known
␈↓ α,␈↓concepts.␈αTypically,␈αthe␈αrule␈αwill␈αexplain␈αhow␈αto␈α≡ll␈αin␈αthe␈αDe≡nition␈αof␈α¬␈αor␈αan␈αAlgorithm
␈↓ α,␈↓for␈α¬␈αthe␈αnew␈αconcept.␈α After␈αexecuting␈αthese␈αinstructions,␈αthe␈αnew␈αconcept␈αwill␈α"exist",␈αand
␈↓ α,␈↓a␈α
few␈α
of␈αits␈α
facets␈α
will␈αbe␈α
≡lled␈α
in,␈αand␈α
a␈α
few␈αnew␈α
jobs␈α
will␈αprobably␈α
exist␈α
on␈α
the␈αagenda,
␈↓ α,␈↓indicating␈αthat␈αAM␈α
might␈αwant␈αto␈α
≡ll␈αin␈αcertain␈αother␈α
facets␈αof␈αthis␈α
new␈αconcept␈αin␈αthe␈α
near
␈↓ α,␈↓future.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.3.1. An Illustration: Discovering Primes␈↓)αβ␈↓

␈↓ α,␈↓Here is a heuristic rule that results in a new concept being created:

␈↓ α,␈↓¬␈↓ αlIf the current task was (Fill-in examples of F),
␈↓ α,␈↓¬␈↓ β,and F is an operation from domain space A into range space B,
␈↓ α,␈↓¬␈↓ β,and more than 100 items are known examples of A (in the domain of F),
␈↓ α,␈↓¬␈↓ β,and more than 10 range items (in B) were found by applying F to these domain items,
␈↓ α,␈↓¬␈↓ β,and at least 1 of these range items is a distinguished member (esp: extremum)␈↓	9␈↓¬ of B,
␈↓ α,␈↓¬␈↓ αlThen (for each such distinguished member `b'εB) create the following new concept:




␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	9␈↓ε␈αλThis␈α	is␈αλhandled␈α	as␈αλfollows:␈α	AM␈αλtakes␈α	the␈αλgiven␈α	list␈αλof␈αλrange␈α	items.␈αλIt␈α	eliminates␈αλany␈α	which␈αλare␈α	not␈αλinteresting␈α	(according␈αλto
␈↓ α,␈↓ε␈↓ βLInterests(B))␈α
or␈α
extreme␈α(an␈α
entry␈α
on␈α
B.Exs-Bdy,␈αthe␈α
boundary␈α
examples␈α
of␈αB).␈α
Finally,␈α
all␈αthose␈α
extreme
␈↓ α,␈↓ε␈↓ βLrange␈α
items␈α
are␈α
moved␈α
to␈αthe␈α
front␈α
of␈α
this␈α
list.␈α
 AM␈αbegins␈α
walking␈α
down␈α
this␈α
list,␈α
creating␈αnew␈α
concepts
␈↓ α,␈↓ε␈↓ βLaccording␈α
to␈α
the␈α
rule.␈α
Sooner␈α
or␈α
later,␈α
a␈α
timer␈α
(or␈α
a␈α
storage-space-watcher)␈α
will␈α
terminate␈α
this␈α
costly
␈↓ α,␈↓ε␈↓ βLactivity.␈α	 Only␈α
the␈α	frontmost␈α	few␈α
range␈α	items␈α
on␈α	the␈α	list␈α
will␈α	have␈α	generated␈α
new␈α	concepts.␈α
So␈α	"especially"
␈↓ α,␈↓ε␈↓ βLreally just means priority consideration.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε42␈↓-

␈↓"␈↓ α,␈↓π␈↓ β<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ β<} ␈↓¬ Name: F-Inverse-of-b ␈↓π                                             ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ β<} ␈↓¬ Definition: λ (x) ( F(x) is b ) ␈↓π                                        ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ β<} ␈↓¬ Generalization: A ␈↓π                                                ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ β<} ␈↓¬ Worth: Average(Worth(A), Worth(F), Worth(B), Worth(b), ||Examples(B)||) ␈↓π      ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ β<} ␈↓¬ Interest: Any conjecture involving both this concept and either F or Inverse(F) ␈↓π  ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ β<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$

␈↓ α,␈↓¬␈↓ αlIn␈αcase␈αthe␈α
user␈αasks,␈αthe␈α
reason␈αfor␈αdoing␈α
this␈αis:␈α"Worthwhile␈α
investigating␈αthose␈αA's␈α
which
␈↓ α,␈↓¬␈↓ ∧,have an unusual F-value, namely, those whose F-value is b"
␈↓ α,␈↓¬␈↓ αlThe total amount of time to spend right now on all of these new concepts is computed as:
␈↓ α,␈↓¬␈↓ β,Half the remaining cpu time in the current task's time quantum.
␈↓ α,␈↓¬␈↓ αlThe total amount of space to spend right now on each of these new concepts is computed as:
␈↓ α,␈↓¬␈↓ β,90% of the remaining space quantum for the current task.




␈↓ α,␈↓Heuristics␈αfor␈αthe␈αnew␈αconcept␈αare␈αquite␈αhard␈αto␈α≡ll␈αin.␈αThis␈αwas␈αone␈αof␈αAM's␈αmost␈αserious
␈↓ α,␈↓limitations,␈α
in␈α∞fact␈α
(see␈α
Chapter␈α∞7).␈α
 Above,␈α
we␈α∞see␈α
a␈α
trivial␈α∞kind␈α
of␈α
"heuristic␈α∞schema"␈α
or
␈↓ α,␈↓template,␈α∞which␈α∂gets␈α∞instantiated␈α∂to␈α∞provide␈α∞one␈α∂new,␈α∞specialized␈α∂heuristic␈α∞about␈α∂the␈α∞new
␈↓ α,␈↓concept.␈α That␈αnew␈αheuristic␈αtells␈αhow␈αto␈αjudge␈αthe␈αinterestingness␈αof␈αany␈αconjecture␈αwhich
␈↓ α,␈↓crops␈α
up␈α
involving␈α
this␈α
new␈α
concept.␈α
As␈α
new␈α
conjectures␈α
get␈α
proposed,␈α
they␈α∞are␈α
evaluated
␈↓ α,␈↓by calling on a set of heuristics including this one.

␈↓ α,␈↓Any␈α∞facet␈α∂X␈α∞which␈α∂didn't␈α∞get␈α∞≡lled␈α∂in␈α∞"right␈α∂now"␈α∞will␈α∞probably␈α∂cause␈α∞a␈α∂new␈α∞task␈α∂to␈α∞be
␈↓ α,␈↓added␈αto␈αthe␈αagenda,␈αof␈αthe␈αform:␈α"␈↓¬Fillin␈αfacet␈α␈↓&X␈↓)αβ␈αof␈αconcept␈α␈↓&F-inverse-of-b␈↓)αβ␈↓".␈α Eventually,␈αAM
␈↓ α,␈↓would choose that task, and spend a large quantum of time working on that single facet.

␈↓ α,␈↓Now␈α⊃let's␈α⊂look␈α⊃at␈α⊃an␈α⊂instance␈α⊃of␈α⊃when␈α⊂this␈α⊃heuristic␈α⊂was␈α⊃used.␈α⊃At␈α⊂one␈α⊃point,␈α⊃AM␈α⊂was
␈↓ α,␈↓working on the task "␈↓¬Fill-in examples of Divisors-of␈↓".

␈↓ α,␈↓This␈αheuristic's␈α
IF-part␈αwas␈α
triggered␈αbecause:␈α
Divisors-of␈αis␈α
an␈αoperation␈α
(from␈αNumbers␈α
to
␈↓ α,␈↓Sets␈α
of␈α
numbers),␈αand␈α
far␈α
more␈αthan␈α
100␈α
di≥erent␈αnumbers␈α
are␈α
known,␈αand␈α
more␈α
than␈α10
␈↓ α,␈↓di≥erent␈α⊂sets␈α⊂of␈α∂factors␈α⊂were␈α⊂found␈α∂altogether,␈α⊂and␈α⊂some␈α∂of␈α⊂them␈α⊂were␈α⊂distinguished␈α∂by
␈↓ α,␈↓being extreme kinds of sets: empty-sets, singletons, doubletons and tripletons.

␈↓ α,␈↓After␈αits␈αleft␈αside␈αtriggered,␈αthe␈αright␈αside␈αof␈αthe␈αheuristic␈αrule␈αwas␈αexecuted.␈α Namely,␈αfour
␈↓ α,␈↓new concepts were created immediately. Here is one of them:
␈↓"␈↓ α,␈↓π␈↓ α|⊂ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ,⊃
␈↓"␈↓ α,␈↓π␈↓ α|} ␈↓¬ Name: Divisors-of-Inverse-of-Doubleton ␈↓π                                 ␈↓ ,}
␈↓"␈↓ α,␈↓π␈↓ α|} ␈↓¬ Definition: λ (x) ( Divisors-of(x) is a Doubleton ) ␈↓π                           ␈↓ ,}
␈↓"␈↓ α,␈↓π␈↓ α|} ␈↓¬ Generalization: Numbers ␈↓π                                             ␈↓ ,}
␈↓"␈↓ α,␈↓π␈↓ α|} ␈↓¬ Worth: 100 ␈↓π                                                       ␈↓ ,}
␈↓"␈↓ α,␈↓π␈↓ α|} ␈↓¬ Interest: Any conjecture involving both this concept and either Divisors-of or Times ␈↓π␈↓ ,}
␈↓"␈↓ α,␈↓π␈↓ α|%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ,$

␈↓ α,␈↓This␈α⊃is␈α⊂a␈α⊃concept␈α⊂representing␈α⊃a␈α⊂certain␈α⊃class␈α⊂of␈α⊃numbers,␈α⊂in␈α⊃fact␈α⊂the␈α⊃numbers␈α⊃we␈α⊂call
␈↓ α,␈↓␈↓βprimes␈↓.␈α The␈αheuristic␈αresets␈αa␈αcertain␈αvariable,␈αso␈αthat␈αin␈αcase␈αthe␈αuser␈αinterrupts␈αand␈αasks
␈↓ α,␈↓␈↓βWhy?␈↓, AM informs him:
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε43␈↓-

␈↓ α,␈↓␈↓ α]"This concept was created because it's worthwhile investigating those numbers which
␈↓ α,␈↓␈↓ αKhave an extreme divisors-of value; in this case, numbers which have only two divisors".


␈↓ α,␈↓AM␈αwas␈αwilling␈αto␈αspend␈αhalf␈αthe␈αremaining␈αquantum␈αof␈αtime␈αallotted␈αto␈α␈↓¬"Fillin␈αexamples␈αof
␈↓ α,␈↓¬Divisors-of"␈↓ on these four new concepts␈↓	10␈↓.

␈↓ α,␈↓The␈α
heuristic␈αrule␈α
is␈α
applicable␈αto␈α
any␈α
operation,␈αnot␈α
just␈α
numeric␈αones.␈α
 For␈αexample,␈α
when
␈↓ α,␈↓AM␈α
was␈α
≡lling␈α
in␈α
examples␈αof␈α
Set-Intersection,␈α
it␈α
was␈α
noticed␈αthat␈α
some␈α
pairs␈α
of␈α
sets␈αwere
␈↓ α,␈↓mapped␈α
into␈α∞the␈α
extreme␈α
kind␈α∞of␈α
set␈α
Empty-set.␈α∞The␈α
above␈α
rule␈α∞then␈α
had␈α
AM␈α∞de≡ne␈α
the
␈↓ α,␈↓concept of ␈↓βDisjointness␈↓: pairs of sets having empty intersection.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.3.2. The Theory of Creating New Concepts␈↓)αβ␈↓

␈↓ α,␈↓All␈α∪the␈α∩heuristic␈α∪rule␈α∪must␈α∩do␈α∪is␈α∪to␈α∩≡ll␈α∪in␈α∪enough␈α∩facets␈α∪so␈α∪that␈α∩the␈α∪new␈α∪concept␈α∩is
␈↓ α,␈↓disambiguated␈αfrom␈αall␈αthe␈αothers,␈αso␈αthat␈αit␈αis␈α"de≡ned"␈αclearly.␈α Should␈αAM␈αpause␈αand␈α≡ll
␈↓ α,␈↓in␈αlots␈αof␈αfacets␈αat␈αthat␈αtime?␈α After␈α
all,␈αseveral␈αpieces␈αof␈αinformation␈αare␈αtrivial␈αto␈αobtain␈α
at
␈↓ α,␈↓this␈αmoment,␈αbut␈αmay␈αbe␈αhard␈αto␈αreconstruct␈αlater␈α(e.g.,␈αthe␈αreason␈αwhy␈αC␈αwas␈αcreated).␈α On
␈↓ α,␈↓the␈αother␈αhand,␈α≡lling␈αin␈αanything␈α
without␈αa␈αgood␈αreason␈αis␈αa␈α
bad␈αidea␈α(it␈αuses␈αup␈αtime␈α
and
␈↓ α,␈↓space, and it won't dazzle the user as a brilliant choice of activity).

␈↓ α,␈↓So␈α
the␈αuniversal␈α
motto␈αof␈α
AM␈αis␈α
to␈α≡ll␈α
in␈αfacets␈α
of␈αa␈α
new␈αconcept␈α
if␈α¬␈α
and␈αonly␈α
if␈α
¬␈αthat
␈↓ α,␈↓≡lling-in activity will be much easier at that moment than later on.

␈↓ α,␈↓In␈α
almost␈αall␈α
cases,␈α
the␈αfollowing␈α
facets␈↓	11␈↓␈α
will␈αbe␈α
speci≡ed␈α
explicitly␈αin␈α
the␈α
heuristic␈αrule,␈α
and
␈↓ α,␈↓thus␈α
will␈α
get␈α
≡lled␈α
in␈α
right␈α
away:␈α
De≡nitions,␈α
Algorithms,␈α
Domain/range,␈α
Worth,␈α
plus␈α
a␈αtie␈α
to
␈↓ α,␈↓some␈α∞related␈α
concept␈α∞(e.g.,␈α
if␈α∞the␈α
new␈α∞concept␈α
is␈α∞a␈α
generalization␈α∞of␈α
Equality,␈α∞then␈α∞we␈α
can
␈↓ α,␈↓trivially ≡ll in an entry on its Specializations facet: "Equality".)

␈↓ α,␈↓On␈αthe␈αother␈αhand,␈αthe␈αfollowing␈αfacets␈αwill␈α␈↓βnot␈↓␈αbe␈αtrivial␈αto␈α≡ll␈αin:␈αConjectures,␈αExamples,
␈↓ α,␈↓Generalizations,␈α∨Specializations,␈α∨and␈α Interestingness.␈α∨ For␈α∨example,␈α∨≡lling␈α in␈α∨the
␈↓ α,␈↓Specializations␈α
facet␈α
of␈αa␈α
new␈α
concept␈αmay␈α
involve␈α
creating␈αsome␈α
new␈α
concepts;␈α≡nding␈α
some
␈↓ α,␈↓entries␈α∞for␈α∞its␈α∂Conjectures␈α∞facet␈α∞may␈α∂involve␈α∞a␈α∞great␈α∂deal␈α∞of␈α∞experimenting;␈α∂≡nding␈α∞some
␈↓ α,␈↓Examples␈α∞of␈α∞it␈α∞may␈α∞involve␈α∞twisting␈α∞its␈α∞de≡nition␈α∞around␈α∞or␈α∞searching.␈α∞ None␈α∞of␈α∞these␈α
is
␈↓ α,␈↓easier␈α
to␈α
do␈α
at␈α
time␈αof␈α
creation␈α
than␈α
any␈α
other␈α
time,␈αso␈α
it's␈α
deferred␈α
until␈α
some␈α
reason␈αfor
␈↓ α,␈↓doing it exists.

␈↓ α,␈↓For␈αeach␈αsuch␈α"time-consuming"␈αfacet␈αF,␈αof␈α
the␈αnew␈αconcept␈αC,␈αone␈αnew␈αtask␈αgets␈α
added␈αto

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	10␈↓ε␈α
Some␈α
trivial␈α
details:␈αOne-eighth␈α
of␈α
the␈α
remaining␈αtime␈α
is␈α
spent␈α
on␈αeach␈α
of␈α
these␈α
4␈αconcepts:␈α
Numbers-with-0-divisors,
␈↓ α,␈↓ε␈↓ βLNumbers-with-1-divisor,␈αNumbers-with-2-divisors,␈αNumbers-with-3-divisors.␈αThe␈αoriginal␈αtime/space␈αlimits
␈↓ α,␈↓ε␈↓ βLwere␈α
in␈α
reality␈α
about␈α	25␈α
cpu␈α
seconds␈α
and␈α
800␈α	list␈α
cells,␈α
and␈α
at␈α
the␈α	moment␈α
this␈α
heuristic␈α
was␈α
called,␈α	only
␈↓ α,␈↓ε␈↓ βLabout␈α	10␈α	seconds␈α	and␈α	600␈α	cells␈α	remained,␈α	so␈α
e.g.␈α	the␈α	concept␈α	Primes␈α	was␈α	allotted␈α	only␈α	1.2␈α	cpu␈α
seconds␈α	to
␈↓ α,␈↓ε␈↓ βL"get␈α	off␈αλthe␈α	ground".␈α	This␈αλwas␈α	no␈α	problem,␈αλas␈α	it␈α	used␈αλfar␈α	less␈α	than␈αλthat.␈α	The␈α	heuristic␈αλrule␈α	states␈α	that␈αλeach
␈↓ α,␈↓ε␈↓ βLof␈α	the␈αλfour␈α	new␈α	concepts␈αλmay␈α	use␈αλup␈α	to␈α	90%␈αλof␈α	the␈αλremaining␈α	space␈α	allocation␈αλ(540␈α	out␈αλof␈α	600␈α	cells),␈αλand,
␈↓ α,␈↓ε␈↓ βLe.g., Primes needed only a fraction of that initially.
␈↓ α,␈↓ε␈↓	11␈↓ε␈α
The␈α
reader␈αmay␈α
wish␈α
to␈α
glance␈αahead␈α
to␈α
Section␈α
5.2,␈αpage␈α
63␈α
to␈α
note␈αthe␈α
full␈α
range␈α
of␈αfacets␈α
that␈α
any␈αconcept␈α
may
␈↓ α,␈↓ε␈↓ βLpossess: what their names are, and the kind of information that is stored in each.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε44␈↓-

␈↓ α,␈↓the␈α∞agenda,␈α
of␈α∞the␈α
form␈α∞␈↓¬"Fill␈α
in␈α∞entries␈α
for␈α∞facet␈α
F␈α∞of␈α
concept␈α∞C"␈↓,␈α
with␈α∞reasons␈α
of␈α∞the␈α
form
␈↓ α,␈↓"Because␈αC␈αwas␈αjust␈αcreated,"␈αand␈αalso␈α"No␈αentries␈αexist␈αso␈αfar␈αon␈αC.F"␈↓	12␈↓.␈αMost␈αof␈αthe␈αtasks
␈↓ α,␈↓generated␈α∂this␈α⊂way␈α∂will␈α⊂have␈α∂low␈α⊂priority␈α∂ratings,␈α⊂and␈α∂may␈α⊂stay␈α∂near␈α⊂the␈α∂bottom␈α⊂of␈α∂the
␈↓ α,␈↓agenda until/unless they are re-suggested for a new reason.

␈↓ α,␈↓Using␈α⊂the␈α⊃Primes␈α⊂example,␈α⊃from␈α⊂the␈α⊃last␈α⊂subsection,␈α⊃we␈α⊂see␈α⊃that␈α⊂a␈α⊃new␈α⊂task␈α⊃like␈α⊂"␈↓¬Fillin
␈↓ α,␈↓¬specializations␈αof␈αPrimes␈↓"␈αwas␈αsuggested␈αwith␈αa␈αlow␈αrating,␈αand␈α"␈↓¬Fillin␈αexamples␈αof␈αPrimes␈↓"␈αwas
␈↓ α,␈↓suggested␈αwith␈α
a␈αmediocre␈↓	13␈↓␈αrating.␈α
 The␈αratings␈αof␈α
these␈αtasks␈αincrease␈α
later␈αon,␈α
when␈αthe
␈↓ α,␈↓same tasks are re-proposed for new reasons.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.3.3. Another Illustration: Squaring a number␈↓)αβ␈↓

␈↓ α,␈↓Let's take another simple (but not atypical) illustration of how new concepts get created.

␈↓ α,␈↓Assume␈α∪that␈α∩AM␈α∪has␈α∩recently␈α∪discovered␈α∪the␈α∩concept␈α∪of␈α∩multiplication,␈α∪which␈α∪it␈α∩calls
␈↓ α,␈↓"TIMES," and AM decides that it is very interesting. A heuristic rule exists which says:␈↓	14␈↓

␈↓ α,␈↓¬␈↓ β,If a newly-interesting operation F(x,y) takes a pair of N's as arguments,
␈↓ α,␈↓¬␈↓ β,Then␈α
create␈α∞a␈α
new␈α
concept,␈α∞a␈α
specialization␈α
of␈α∞F,␈α
called␈α
F-Itself,␈α∞taking␈α
just␈α
one␈α∞N␈α
as
␈↓ α,␈↓¬␈↓ ∧,argument, defined as F(x,x), with initial worth Worth(F).

␈↓ α,␈↓In␈αthe␈αcase␈α
of␈αF␈α=␈α
TIMES,␈αwe␈αsee␈α
that␈αF␈αtakes␈αa␈α
pair␈αof␈αnumbers␈α
as␈αits␈αarguments,␈α
so␈αthe
␈↓ α,␈↓heuristic␈α
rule␈α
would␈α
have␈αAM␈α
create␈α
a␈α
new␈αconcept␈α
called␈α
TIMES-Itself,␈α
de≡ned␈αas␈α
TIMES-
␈↓ α,␈↓Itself(x)␈α
␈↓¬≡␈↓␈α
TIMES(x,x).␈α
 That␈αis,␈α
create␈α
the␈α
new␈α
concept␈αwhich␈α
is␈α
the␈α
operation␈α
of␈α␈↓βsquaring␈α
a
␈↓ α,␈↓βnumber.␈↓

␈↓ α,␈↓What␈α∞would␈α∂AM␈α∞do␈α∂in␈α∞this␈α∞situation?␈α∂ The␈α∞global␈α∂list␈α∞of␈α∞concepts␈α∂would␈α∞be␈α∂enlarged␈α∞to
␈↓ α,␈↓include␈αthe␈αnew␈αatom␈α"TIMES-Itself",␈αand␈αthe␈αfacets␈αof␈αthis␈αnew␈αconcept␈αwould␈αbegin␈αto␈αbe
␈↓ α,␈↓≡lled in.  The following facets would get ≡lled in almost instantly:








␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	12␈↓ε C.F is an abbreviation for facet F of concept C
␈↓ α,␈↓ε␈↓	13␈↓ε␈α	Not␈αλas␈α	low␈αλa␈α	rating␈αλas␈α	the␈αλtask␈α	just␈αλmentioned.␈α	Why?␈αλ Each␈α	possible␈αλfacet␈α	has␈αλa␈α	worth␈αλrating␈α	which␈αλis␈α	fixed␈αλonce␈α	and␈αλfor
␈↓ α,␈↓ε␈↓ βLall.␈α	As␈α	an␈α	illustration,␈α	we␈αλmention␈α	that␈α	the␈α	facet␈α	Examples␈α	is␈αλrated␈α	much␈α	higher␈α	than␈α	Specializations.␈α	 Why␈αλis
␈↓ α,␈↓ε␈↓ βLthis?␈α
 Because␈α
looking␈αfor␈α
examples␈α
of␈αa␈α
concept␈α
is␈α
often␈αa␈α
good␈α
expenditure␈αof␈α
time,␈α
producing␈αthe␈α
raw
␈↓ α,␈↓ε␈↓ βLdata␈αλon␈αλwhich␈αλempirical␈αλinduction␈αλthrives.␈αλ On␈αλthe␈αλother␈αλhand,␈αλeach␈αλspecialization␈αλof␈αλthe␈αλnew␈αλconcept␈α	C␈αλwould
␈↓ α,␈↓ε␈↓ βLitself␈α
be␈α
a␈α
brand␈α	new␈α
concept.␈α
So␈α
filling␈α
in␈α	entries␈α
for␈α
the␈α
Specializations␈α	facet␈α
would␈α
be␈α
a␈α
very␈α	explosive
␈↓ α,␈↓ε␈↓ βLprocess.
␈↓ α,␈↓ε␈↓	14␈↓ε␈α
By␈α
glancing␈α
back␈α
at␈α
the␈α
Primes␈α
example,␈α
two␈α
subsections␈α
ago,␈α
page␈α
41,␈α
you␈α
can␈α
imagine␈α
what␈α
this␈α
rule␈αactually␈α
looked
␈↓ α,␈↓ε␈↓ βLlike.␈α∞There␈α∞is␈α∞nothing␈α
to␈α∞be␈α∞gained␈α∞by␈α∞stretching␈α
it␈α∞out␈α∞in␈α∞all␈α∞its␈α
glory,␈α∞hence␈α∞I've␈α∞taken␈α∞the␈α
liberty
␈↓ α,␈↓ε␈↓ βLcondensing it, inserting pronouns, etc.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε45␈↓-

␈↓"␈↓ α,␈↓π␈↓ βl⊂ααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 
⊃
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ␈↓π                                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ NAME: TIMES-Itself ␈↓π                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ␈↓π                                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ DEFINITIONS: ␈↓π                                    ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬            ORIGIN: λ(x,y) [TIMES.DEFN(x,x,y)] ␈↓π             ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ␈↓π                                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ALGORITHMS: λ(x) [TIMES.ALG(x,x)] ␈↓π                   ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ␈↓π                                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ DOMAIN/RANGE: Number → Number ␈↓π                   ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ␈↓π                                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ GENERALIZATIONS: TIMES ␈↓π                          ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ␈↓π                                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ WORTH: 600 ␈↓π                                     ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl} ␈↓¬ ␈↓π                                               ␈↓ 
}
␈↓"␈↓ α,␈↓π␈↓ βl%ααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 
$

␈↓ α,␈↓The␈α∞name,␈α∂de≡nition,␈α∞domain/range,␈α∂generalizations,␈α∞and␈α∞worth␈α∂are␈α∞speci≡ed␈α∂explicitly␈α∞by
␈↓ α,␈↓the heuristic rule.

␈↓ α,␈↓The␈α∞lambda␈α∞expression␈α
stored␈α∞under␈α∞the␈α
de≡nition␈α∞facet␈α∞is␈α
an␈α∞executable␈α∞LISP␈α
predicate,
␈↓ α,␈↓which␈αaccepts␈αtwo␈αarguments␈αand␈αthen␈αtests␈αthem␈αto␈αsee␈αwhether␈αthe␈αsecond␈αone␈αis␈αequal␈αto
␈↓ α,␈↓TIMES-Itself␈α⊃of␈α⊂the␈α⊃≡rst␈α⊂argument.␈α⊃ It␈α⊂performs␈α⊃this␈α⊂test␈α⊃by␈α⊂calling␈α⊃upon␈α⊃the␈α⊂predicate
␈↓ α,␈↓stored␈αunder␈αthe␈αde≡nition␈αfacet␈αof␈αthe␈αTIMES␈αconcept.␈α Thus␈αTIMES-Itself.Defn(4,16)␈αwill
␈↓ α,␈↓call␈αon␈αTIMES.Defn(4,4,16),␈αand␈αreturn␈αwhatever␈αvalue␈α␈↓βthat␈↓␈αpredicate␈αreturns␈α(in␈α
this␈αcase,
␈↓ α,␈↓it returns True, since 4x4 does equal 16).

␈↓ α,␈↓A␈α∀trivial␈α∃transformation␈α∀of␈α∀this␈α∃de≡nition␈α∀provides␈α∀an␈α∃algorithm␈α∀for␈α∃computing␈α∀this
␈↓ α,␈↓operation.␈αThe␈αalgorithm␈αsays␈αto␈αcall␈αon␈αthe␈αAlgorithms␈αfacet␈αof␈αthe␈αconcept␈αTIMES.␈αThus
␈↓ α,␈↓TIMES-Itself.Alg(4)␈α∂is␈α⊂computed␈α∂by␈α⊂calling␈α∂on␈α∂TIMES.Alg(4,4)␈α⊂and␈α∂returning␈α⊂␈↓βthat␈↓␈α∂value
␈↓ α,␈↓(namely, 16).

␈↓ α,␈↓The␈αworth␈αof␈αTIMES␈αwas␈α600␈α
at␈αthe␈αmoment␈αTIMES-Itself␈αwas␈αcreated,␈αand␈α
this␈αbecomes
␈↓ α,␈↓the worth of TIMES-Itself.

␈↓ α,␈↓TIMES-Itself␈αis␈αby␈αde≡nition␈αa␈αspecialization␈αof␈αTIMES,␈αso␈αthe␈αSPECIALIZATIONS␈αfacet
␈↓ α,␈↓of␈α&TIMES␈α&is␈α&incremented␈α&to␈α%point␈α&to␈α&this␈α&new␈α&concept.␈α& Likewise,␈α%the
␈↓ α,␈↓GENERALIZATIONS facet of TIMES-Itself points to TIMES.

␈↓ α,␈↓Note␈α
how␈α
easy␈α
it␈α
was␈α
to␈α
≡ll␈α
in␈α
these␈αfacets␈α
now,␈α
but␈α
how␈α
di≠cult␈α
it␈α
might␈α
be␈α
later␈α
on,␈α"out␈α
of
␈↓ α,␈↓context".␈αBy␈α
way␈αof␈αcontrast,␈α
the␈αtask␈αof,␈α
e.g.,␈α≡lling␈αin␈α
␈↓βSpecializations␈↓␈αof␈α
TIMES-Itself␈αwill
␈↓ α,␈↓be␈α
no␈αharder␈α
later␈αon␈α
than␈αit␈α
is␈αright␈α
now,␈α
so␈αwe␈α
may␈αas␈α
well␈αdefer␈α
it␈αuntil␈α
there's␈α
a␈αgood
␈↓ α,␈↓reason␈α
for␈α∞it.␈α
This␈α
task␈α∞will␈α
probably␈α
be␈α∞added␈α
to␈α
the␈α∞agenda␈α
with␈α
so␈α∞low␈α
a␈α∞priority␈α
that
␈↓ α,␈↓AM will never get around to it, unless some new reasons for it emerge.

␈↓ α,␈↓The␈α
task␈α
␈↓¬"Fill-in␈α
examples␈α∞of␈α
TIMES-Itself"␈↓␈α
is␈α
probably␈α∞worthwhile␈α
doing␈α
soon,␈α
but␈α∞again␈α
it
␈↓ α,␈↓won't␈α∞be␈α∞any␈α∞harder␈α∞to␈α∞do␈α∞at␈α∞a␈α∞later␈α∞time␈α∂than␈α∞it␈α∞is␈α∞right␈α∞now.␈α∞ So␈α∞it␈α∞is␈α∞not␈α∞done␈α∂at␈α∞the
␈↓ α,␈↓moment; rather, it is added to the agenda (with a fairly high priority).
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε46␈↓-

␈↓ α,␈↓Incidentally,␈αthe␈α
reader␈αmay␈αbe␈α
interested␈αto␈αknow␈α
that␈αthe␈αnext␈α
few␈αtasks␈αAM␈α
selected␈α(in
␈↓ α,␈↓reality)␈α∞were␈α∞to␈α∂create␈α∞the␈α∞inverse␈α∂of␈α∞this␈α∞operation␈α∞(i.e.,␈α∂integer␈α∞square-root),␈α∞and␈α∂then␈α∞to
␈↓ α,␈↓create␈α∂a␈α∂new␈α∞kind␈α∂of␈α∂number,␈α∂the␈α∞ones␈α∂which␈α∂can␈α∂be␈α∞produced␈α∂by␈α∂squaring␈α∂(i.e.,␈α∞perfect
␈↓ α,␈↓squares).␈α Perfect␈αsquares␈αwere␈αdeemed␈αworth␈αhaving␈αaround␈αbecause␈αInteger-square-root␈αis
␈↓ α,␈↓␈↓βde≡ned␈↓ precisely on that set of integers.


␈↓ α,␈↓␈↓ βL␈↓∧␈↓&4.4. Heuristics Fill in Entries for a Specific Facet␈↓)αβ␈↓

␈↓ α,␈↓The␈α
last␈α
two␈α
subsections␈α
dealt␈α
with␈α
how␈α
a␈α
heuristic␈α
rule␈α
is␈α
able␈α
to␈α
propose␈α
new␈α∞tasks␈α
and
␈↓ α,␈↓create␈α
new␈αconcepts.␈α
 This␈α
section␈αwill␈α
illustrate␈αhow␈α
a␈α
rule␈αcan␈α
≡nd␈α
some␈αentries␈α
for␈αa␈α
given
␈↓ α,␈↓facet of a speci≡c concept.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.4.1. An Illustration: "Fill in Examples of Set-union"␈↓)αβ␈↓

␈↓ α,␈↓Typically,␈α
the␈α∞facet/concept␈α
involved␈α
will␈α∞be␈α
the␈α∞one␈α
mentioned␈α
in␈α∞the␈α
current␈α∞task␈α
which
␈↓ α,␈↓was␈α
chosen␈αfrom␈α
the␈α
agenda.␈α If␈α
the␈α
task␈α"␈↓¬Fillin␈α
Examples␈α
of␈αSet-union␈↓"␈α
were␈α
plucked␈αfrom␈α
the
␈↓ α,␈↓agenda,␈α∂then␈α∂the␈α∂"relevant"␈α∂heuristics␈α∂would␈α∂be␈α∂those␈α∂useful␈α∂for␈α∂≡lling␈α∂in␈α∂entries␈α∂for␈α∞the
␈↓ α,␈↓Examples facet of the Set-union concept.

␈↓ α,␈↓Here's one such relevant heuristic, attached to the concept Activity:

␈↓ α,␈↓¬␈↓ β,If the current task is to fill in examples of the activity␈↓	15␈↓¬ F,
␈↓ α,␈↓¬␈↓ β,One way to get them is to run F on randomly chosen examples of the domain of F.

␈↓ α,␈↓Of␈α⊃course,␈α⊃in␈α⊃the␈α⊂LISP␈α⊃implementation,␈α⊃this␈α⊃situation-action␈α⊂rule␈α⊃is␈α⊃not␈α⊃coded␈α⊃quite␈α⊂so
␈↓ α,␈↓neatly.  It would be more faithfully translated as follows:

␈↓ α,␈↓¬␈↓ αlIf CURRENT-TASK matches (FILLIN EXAMPLES F←anything)),
␈↓ α,␈↓¬␈↓ β,and F isa Activity,
␈↓ α,␈↓¬␈↓ αlThen carry out the following procedure:
␈↓ α,␈↓¬␈↓ β,1.  Find the domain of F, and call it D;
␈↓ α,␈↓¬␈↓ β,2.  Find examples of D, and call them E;
␈↓ α,␈↓¬␈↓ β,3.  Find an algorithm to compute F, and call it A;
␈↓ α,␈↓¬␈↓ β,4.  Repeatedly:
␈↓ α,␈↓¬␈↓ ∧,4a. Choose any member of E, and call it E1.
␈↓ α,␈↓¬␈↓ ∧,4b. Run A on E1, and call the result X.
␈↓ α,␈↓¬␈↓ ∧,4c. Check whether <E1,X> satisfies the definition of F.
␈↓ α,␈↓¬␈↓ ∧,4d. If so, then add <E1 → X> to the Examples facet of F.
␈↓ α,␈↓¬␈↓ ∧,4e. If not, then add <E1 → X> to the Non-examples facet of F.

␈↓ α,␈↓When␈α⊂the␈α⊂current␈α⊃task␈α⊂is␈α⊂␈↓¬"Fillin␈α⊃examples␈α⊂of␈α⊂Set-union"␈↓,␈α⊂the␈α⊃left-hand-side␈α⊂of␈α⊂the␈α⊃rule␈α⊂is
␈↓ α,␈↓satis≡ed, so the right-hand-side is run.

␈↓ α,␈↓Step␈α
(1)␈α
says␈α∞to␈α
locate␈α
the␈α∞domain␈α
of␈α
Set-union.␈α
The␈α∞facet␈α
labelled␈α
Domain/Range,␈α∞on␈α
the
␈↓ α,␈↓Set-union␈α
concept,␈αcontains␈α
the␈αentry␈α
(SET␈αSET␈α
→␈α
SET),␈αwhich␈α
indicates␈αthat␈α
the␈αdomain␈α
is
␈↓ α,␈↓a␈αpair␈αof␈αsets.␈α That␈αis,␈αSet-union␈αis␈αan␈αoperation␈αwhich␈αaccepts␈α(as␈αits␈αarguments)␈αa␈αpair␈αof
␈↓ α,␈↓sets, and returns (as its value) some new set.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	15␈↓ε "Activity" is a general concept which includes operations, predicates, relations, functions, etc.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε47␈↓-

␈↓ α,␈↓Since␈α⊃the␈α⊂domain␈α⊃elements␈α⊃are␈α⊂sets,␈α⊃step␈α⊂(2)␈α⊃says␈α⊃to␈α⊂locate␈α⊃examples␈α⊂of␈α⊃sets.␈α⊃ The␈α⊂facet
␈↓ α,␈↓labelled␈α⊂Examples,␈α⊂on␈α∂the␈α⊂Sets␈α⊂concept,␈α∂points␈α⊂to␈α⊂a␈α∂list␈α⊂of␈α⊂about␈α∂30␈α⊂di≥erent␈α⊂sets.␈α∂ This
␈↓ α,␈↓includes {Z}, {A,B,C,D,E}, {}, {A,{{B}}},...

␈↓ α,␈↓Step␈α↔(3)␈α⊗involves␈α↔nothing␈α↔more␈α⊗than␈α↔accessing␈α↔some␈α⊗randomly-chosen␈α↔entry␈α↔on␈α⊗the
␈↓ α,␈↓Algorithms␈α∀facet␈α∀of␈α∃Set-union.␈α∀One␈α∀such␈α∃entry␈α∀is␈α∀a␈α∃recursive␈α∀LISP␈α∀function␈α∃of␈α∀two
␈↓ α,␈↓arguments,␈α∂which␈α∞halts␈α∂when␈α∂the␈α∞≡rst␈α∂argument␈α∂is␈α∞the␈α∂empty␈α∂set,␈α∞and␈α∂otherwise␈α∂pulls␈α∞an
␈↓ α,␈↓element␈αout␈αof␈αthat␈αset␈αand␈αSET-INSERT's␈αit␈αinto␈αthe␈αsecond␈αargument,␈αand␈αthen␈αrecurs␈α
on
␈↓ α,␈↓the new values of the two sets.  For convenience, we'll refer to this algorithm as UNION.

␈↓ α,␈↓We␈α
then␈α
enter␈α
the␈α
loop␈α∞of␈α
Step␈α
(4).␈α
 Step␈α
(4a)␈α
has␈α∞us␈α
choose␈α
one␈α
pair␈α
of␈α
our␈α∞examples␈α
of
␈↓ α,␈↓sets,␈αsay␈αthe␈α≡rst␈αtwo␈α{Z}␈αand␈α{A,B,C,D,E}.␈α Step␈α(4b)␈αhas␈αus␈αrun␈αUNION␈αon␈αthese␈αtwo␈αsets.
␈↓ α,␈↓The␈α∞result␈α∞is␈α
{A,B,C,D,E,Z}.␈α∞ Step␈α∞(4c)␈α
has␈α∞us␈α∞grab␈α
an␈α∞entry␈α∞from␈α
the␈α∞De≡nitions␈α∞facet␈α
of
␈↓ α,␈↓Set-union, and run it. A typical de≡nition is this formal one:

␈↓ α,␈↓¬␈↓ α\(λ (S1 S2 S3)
␈↓ α,␈↓¬␈↓ α\        (AND
␈↓ α,␈↓¬␈↓ α\                (For all x in S1, x is in S3)
␈↓ α,␈↓¬␈↓ α\                (For all x in S2, x is in S3)
␈↓ α,␈↓¬␈↓ α\                (For all x in S3, x is in S1 or x is in S2)
␈↓ α,␈↓¬␈↓ α\        )
␈↓ α,␈↓¬␈↓ α\)  )



␈↓ α,␈↓It␈α
is␈α
run␈α
on␈α
the␈α
three␈α
arguments␈α
S1={Z},␈α
S2={A,B,C,D,E},␈α
S3={A,B,C,D,E,Z}.␈α
 Since␈α
it␈α
returns
␈↓ α,␈↓"True",␈α
we␈α
proceed␈α
to␈α
Step␈α
(4d).␈α
 The␈α
construct␈α
<{Z},␈α
{A,B,C,D,E}␈α
→␈α
{A,B,C,D,E,Z}>␈α
is␈α
added
␈↓ α,␈↓to the Examples facet of Set-union.

␈↓ α,␈↓At␈α
this␈α
stage,␈α
control␈α
returns␈α
to␈α
the␈α
beginning␈α
of␈α
the␈α
Step␈α
(4)␈α
loop.␈α
 A␈α
new␈α
pair␈α
of␈α∞sets␈α
is
␈↓ α,␈↓chosen, and so on.

␈↓ α,␈↓But␈α∂when␈α∂would␈α∂this␈α∂loop␈α∂stop?␈α∂Recall␈α∂that␈α∞each␈α∂task␈α∂has␈α∂a␈α∂time␈α∂and␈α∂a␈α∂space␈α∞allotment
␈↓ α,␈↓(based␈αon␈αits␈αpriority␈αvalue).␈αIf␈αthere␈αare␈αmany␈αdi≥erent␈αrules␈αall␈αclaiming␈αto␈αbe␈αrelevant␈αto
␈↓ α,␈↓the␈α⊂current␈α∂task,␈α⊂then␈α∂each␈α⊂one␈α∂is␈α⊂allocated␈α∂a␈α⊂small␈α∂fraction␈α⊂of␈α∂those␈α⊂time/space␈α∂quanta.
␈↓ α,␈↓When␈αeither␈αof␈αthese␈αresources␈αis␈αexhausted,␈αAM␈αwould␈αbreak␈αaway␈αat␈αa␈α"clean"␈αpoint␈α(just
␈↓ α,␈↓after␈α≡nishing␈αa␈αcycle␈α
of␈αthe␈αStep␈α(4)␈α
loop)␈αand␈αwould␈αmove␈αon␈α
to␈αa␈αnew␈αheuristic␈α
rule␈αfor
␈↓ α,␈↓≡lling in examples of Set-union.

␈↓ α,␈↓This␈α
concludes␈αthe␈α
demonstration␈αthat␈α
a␈αheuristic␈α
rule␈α
really␈αcan␈α
be␈αexecuted␈α
to␈αproduce␈α
the
␈↓ α,␈↓kinds of entities requested by the current task.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.4.2. Heuristics Propose New Conjectures␈↓)αβ␈↓

␈↓ α,␈↓We␈α
saw␈α∞in␈α
the␈α
sample␈α∞excerpt␈α
(Chapter␈α∞2)␈α
that␈α
AM␈α∞occasionally␈α
notices␈α∞some␈α
unexpected
␈↓ α,␈↓relationship,␈α
and␈αformulates␈α
it␈α
into␈αa␈α
precise␈α
conjecture.␈α Below␈α
is␈α
an␈αexample␈α
of␈α
how␈αthis␈α
is
␈↓ α,␈↓done.␈α As␈αyou␈αmight␈αguess␈αfrom␈αthe␈αplacement␈αof␈αthis␈αsubsection,␈αthe␈αmechanism␈αis␈αour␈αold
␈↓ α,␈↓friend the heuristic rule which ≡lls in entries for certain facets.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε48␈↓-

␈↓ α,␈↓In fact, a conjecture evolves through four stages:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈αA␈αheuristic␈αrule␈αlooks␈αfor␈αa␈α
particular␈αkind␈αof␈αrelationship.␈α This␈αwill␈αtypically␈αbe␈α
of
␈↓ α,␈↓␈↓ β≤the␈α∞form␈α∞"X␈α∞is␈α∞a␈α∞Generalization␈α∞of␈α∞Y",␈α∞or␈α∞"X␈α∞is␈α∞an␈α∞example␈α∞of␈α∞Y",␈α∞or␈α∞"X␈α∞is␈α
the
␈↓ α,␈↓␈↓ β≤same␈αas␈αY",␈αor␈α"F1.Defn(X,Y)"␈αwhere␈αF1␈αis␈αan␈αactive␈αconcept␈αAM␈αknows␈αabout,␈αor
␈↓ α,␈↓␈↓ β≤"F1.Defn(Y,X)"␈↓	16␈↓.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈αOnce␈αfound,␈αthe␈αrelationship␈αis␈αchecked,␈αusing␈αsupporting␈αcontacts.␈α A␈αgreat␈αdeal␈αof
␈↓ α,␈↓␈↓ β≤empirical␈α_evidence␈α_must␈α_favor␈α_it,␈α↔and␈α_any␈α_contradictory␈α_evidence␈α_must␈α↔be
␈↓ α,␈↓␈↓ β≤"explained away" somehow.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈αNow␈αit␈αis␈α
believed,␈αand␈αAM␈αprints␈αit␈α
out␈αto␈αthe␈αuser.␈αIt␈α
is␈αadded␈αas␈αa␈αnew␈α
entry␈αto
␈↓ α,␈↓␈↓ β≤the␈α
Conjecs␈α∞facet␈α
of␈α∞both␈α
concepts␈α∞X␈α
and␈α∞Y.␈α
 It␈α∞is␈α
also␈α∞added␈α
as␈α∞an␈α
entry␈α∞to␈α
the
␈↓ α,␈↓␈↓ β≤Examples facet of the Conjecture concept.
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓␈αEventually,␈αAM␈αwill␈αget␈αaround␈αto␈αthe␈αtask␈α"␈↓¬Check␈αExamples␈αof␈αConjecture␈↓",␈αor␈αto␈αthe
␈↓ α,␈↓␈↓ β≤task␈α∞"␈↓¬Check␈α
Conjecs␈α∞of␈α∞X␈↓".␈α
 If␈α∞AM␈α
had␈α∞any␈α∞concepts␈α
for␈α∞proving␈α∞conjectures,␈α
they
␈↓ α,␈↓␈↓ β≤would␈α⊃then␈α⊃be␈α⊃invoked.␈α⊃In␈α⊃the␈α⊃current␈α⊃LISP␈α⊃implementation,␈α⊃these␈α∩are␈α⊃absent.
␈↓ α,␈↓␈↓ β≤Nevertheless,␈αseveral␈α
"checks"␈αare␈αperformed:␈α
(␈↓βi␈↓)␈αsee␈αif␈α
any␈αnew␈α
empirical␈αevidence
␈↓ α,␈↓␈↓ β≤(pro␈α
or␈α
con)␈α∞has␈α
appeared␈α
recently;␈α∞(␈↓βii␈↓)␈α
see␈α
if␈α∞this␈α
conjecture␈α
can␈α∞be␈α
strengthened;
␈↓ α,␈↓␈↓ β≤(␈↓βiii␈↓)␈α∞check␈α∞it␈α∞for␈α∞extreme␈α∞cases,␈α∞and␈α∞modify␈α∞it␈α∞if␈α∞necessary;␈α∞(␈↓βiv␈↓)␈α∞Modify␈α∞the␈α
worth
␈↓ α,␈↓␈↓ β≤ratings of the concepts involved in the conjecture.

␈↓ α,␈↓The␈α
left-hand-side␈α
of␈α
such␈α
a␈α
heuristic␈α
rule␈α
will␈α
be␈α
longer␈α
and␈α
more␈α
complex␈α
than␈αmost␈α
other
␈↓ α,␈↓kinds,␈α∞but␈α∂the␈α∞basic␈α∂activities␈α∞of␈α∂the␈α∞right-hand-side␈α∞will␈α∂still␈α∞be␈α∂≡lling␈α∞in␈α∂an␈α∞entry␈α∂for␈α∞a
␈↓ α,␈↓particular facet.

␈↓ α,␈↓The␈αentries␈α≡lled␈αin␈αwill␈αinclude:␈α(␈↓βi␈↓)␈αa␈αnew␈αexample␈αof␈αConjectures,␈α(␈↓βii␈↓)␈αa␈αnew␈αentry␈αfor␈αthe
␈↓ α,␈↓Conjec␈αfacet␈αof␈αeach␈αconcept␈αinvolved␈α
in␈αthe␈αconjecture,␈α(␈↓βiii␈↓)␈αif␈αwe're␈αclaiming␈α
that␈αconcept
␈↓ α,␈↓X␈αis␈α
a␈αgeneralization␈αof␈α
concept␈αY,␈αthen␈α
"X"␈αwould␈α
be␈αadded␈αto␈α
the␈αGeneralizations␈αfacet␈α
of
␈↓ α,␈↓Y,␈α∞and␈α
"Y"␈α∞added␈α
to␈α∞the␈α
Specializations␈α∞facet␈α
of␈α∞X,␈α
(␈↓βiv␈↓)␈α∞if␈α
X␈α∞is␈α
an␈α∞Example␈α
of␈α∞Y,␈α∞"X"␈α
is
␈↓ α,␈↓added to the Examples facet of Y, and "Y" is added to the ISA facet of X.

␈↓ α,␈↓The␈α∪right-hand-side␈α∀may␈α∪also␈α∪involve␈α∀adding␈α∪new␈α∪tasks␈α∀to␈α∪the␈α∪agenda,␈α∀creating␈α∪new
␈↓ α,␈↓concepts,␈αand␈αmodifying␈αentries␈αof␈αparticular␈αfacets␈αof␈αparticular␈αconcepts.␈α As␈αis␈αtrue␈αof␈αall
␈↓ α,␈↓heuristic␈α⊃rules,␈α∩both␈α⊃sides␈α⊃of␈α∩this␈α⊃type␈α∩of␈α⊃conjecture-perceiving␈α⊃rule␈α∩may␈α⊃run␈α∩any␈α⊃little
␈↓ α,␈↓functions␈α∩they␈α∪want␈α∩to:␈α∪any␈α∩functions␈α∪which␈α∩are␈α∩quick␈α∪and␈α∩have␈α∪no␈α∩side␈α∪e≥ects␈α∩(e.g.,
␈↓ α,␈↓FORALL tests, PRINT functions, accesses to a speci≡ed facet of some concept).


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.4.3. An Illustration: "All primes except 2 are odd"␈↓)αβ␈↓

␈↓ α,␈↓As an illustration, here is a heuristic rule, relevant when checking examples of any concept:









␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	16␈↓ε These last two say that F1(X)=Y, and that F1(Y)=X, respectively.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε49␈↓-


␈↓ α,␈↓¬␈↓ αlIf the current task is to Check Examples of X,
␈↓ α,␈↓¬␈↓ β,and (Forsome Y) Y is a generalization of X,
␈↓ α,␈↓¬␈↓ β,and Y has at least 10 examples,
␈↓ α,␈↓¬␈↓ β,and all examples of Y (ignoring boundary cases) are also examples of X,
␈↓ α,␈↓¬␈↓ αlThen print the following conjecture: X is really no more specialized than Y,
␈↓ α,␈↓¬␈↓ β,and add it to the Examples facet of Conjectures,
␈↓ α,␈↓¬␈↓ β,and␈α
if␈α
the␈α
user␈α
asks,␈α
inform␈α
him␈α
that␈α
the␈α
evidence␈α
for␈α
this␈α
was␈α
that␈α
all␈α
||Examples(Y)||␈α
Y's
␈↓ α,␈↓¬␈↓ ∧,(ignoring boundary examples of Y's) turned out to be X's as well,
␈↓ α,␈↓¬␈↓ β,and Check the truth of this conjecture on boundary examples of Y,
␈↓ α,␈↓¬␈↓ β,and add "X" to the Generalizations facet of Y,
␈↓ α,␈↓¬␈↓ β,and add "Y" to the Specializations facet of X,
␈↓ α,␈↓¬␈↓ β,and␈α(if␈αthere␈αis␈αan␈αentry␈αin␈αthe␈α
Generalizations␈αfacet␈αof␈αY)␈αadd␈αthe␈αfollowing␈αtask␈αto␈α
the
␈↓ α,␈↓¬␈↓ ∧,agenda␈α⊂"Check␈α⊂examples␈α⊂of␈α⊂Y",␈α⊂for␈α⊂the␈α⊂reason:␈α⊂"Just␈α⊂as␈α⊂Y␈α⊂was␈α⊂no␈α⊂more
␈↓ α,␈↓¬␈↓ ∧,general␈α∪than␈α∩X,␈α∪one-of␈α∪Generalizations(Y)␈α∩may␈α∪turn␈α∪out␈α∩to␈α∪be␈α∪no␈α∩more
␈↓ α,␈↓¬␈↓ ∧,general than Y", with a rating for that reason computed as:
␈↓ α,␈↓¬␈↓ ¬,0.4x||Examples(Generalizations(Y))|| +
␈↓ α,␈↓¬␈↓ ¬,0.3x||Examples(Y)|| +
␈↓ α,␈↓¬␈↓ ¬,0.3xPriority(Current task).

␈↓ α,␈↓Let's␈α∂take␈α⊂a␈α∂particular␈α∂instance␈α⊂where␈α∂this␈α∂rule␈α⊂would␈α∂be␈α∂useful.␈α⊂Say␈α∂the␈α∂current␈α⊂task␈α∂is
␈↓ α,␈↓␈↓¬"Check␈αexamples␈αof␈αOdd-primes"␈↓.␈α The␈αleft-hand-side␈αof␈αthe␈αrule␈αis␈αrun,␈αand␈αis␈αsatis≡ed␈αwhen
␈↓ α,␈↓the generalization Y is the concept "Primes".  Let's see why this is satis≡ed.

␈↓ α,␈↓One␈α
of␈α
entries␈α
of␈α
the␈α
Generalization␈α∞facet␈α
of␈α
Odd-primes␈α
is␈α
"Primes".␈α
 AM␈α
grabs␈α∞hold␈α
of
␈↓ α,␈↓the␈α30␈αexamples␈αof␈αprimes␈α(located␈αon␈αthe␈αExamples␈αfacet␈αof␈αPrimes),␈αand␈αremoves␈αthe␈αones
␈↓ α,␈↓which␈αare␈αtagged␈αas␈αboundary␈αexamples␈α("2"␈αand␈α"3").␈α A␈αde≡nition␈αof␈αOdd-prime␈αnumbers
␈↓ α,␈↓is␈α
obtained␈α
(De≡nitions␈αfacet␈α
of␈α
Odd-primes),␈α
and␈αit␈α
is␈α
run␈αon␈α
each␈α
remaining␈α
example␈αof
␈↓ α,␈↓primes␈α⊂(5,␈α⊂7,␈α⊃11,␈α⊂13,␈α⊂17,␈α⊃...).␈α⊂ Sure␈α⊂enough,␈α⊃they␈α⊂all␈α⊂satisfy␈α⊃the␈α⊂de≡nition.␈α⊂ So␈α⊃all␈α⊂primes
␈↓ α,␈↓(ignoring boundary cases) appear to be odd.  The left-hand-side of the rule is satis≡ed.

␈↓ α,␈↓At␈αthis␈αpoint,␈αthe␈α
user␈αsees␈αa␈αmessage␈α
of␈αthe␈αform␈α"Odd-primes␈α
is␈αreally␈αno␈αmore␈α
specialized
␈↓ α,␈↓than␈α
Primes".␈α If␈α
he␈αinterrupts␈α
and␈αasks␈α
about␈αit,␈α
he␈αis␈α
told␈αthat␈α
the␈αevidence␈α
for␈α
this␈αwas
␈↓ α,␈↓that all 30 primes (ignoring boundary examples of primes) turned out to be Odd-primes.

␈↓ α,␈↓Of␈αthe␈α
boundary␈αexamples␈α(the␈α
numbers␈α2␈αand␈α
3),␈αonly␈αthe␈α
integer␈α"2"␈αfails␈α
to␈αbe␈α
an␈αodd-
␈↓ α,␈↓prime,␈αso␈αthe␈αthe␈αuser␈αis␈αnoti≡ed␈αof␈αthe␈α≡nalized␈αconjecture:␈α"All␈αprimes␈α(other␈αthan␈α`2')␈αare
␈↓ α,␈↓also␈α
odd-primes".␈α
 This␈α
is␈α
added␈α
as␈α
an␈αentry␈α
on␈α
the␈α
Examples␈α
facet␈α
of␈α
the␈α
concept␈αnamed
␈↓ α,␈↓`Conjectures.'

␈↓ α,␈↓Before␈αbeginning␈αall␈αthis,␈αthe␈αGeneralizations␈αfacet␈αof␈αOdd-primes␈αpointed␈αto␈αPrimes.␈αNow,
␈↓ α,␈↓this␈αrule␈αhas␈α
us␈αadd␈α"Primes"␈αas␈α
an␈αentry␈αon␈αthe␈α
Specializations␈αfacet␈αof␈αOdd-primes.␈α
 Thus
␈↓ α,␈↓Primes␈αis␈αboth␈αa␈αgeneralization␈αand␈αa␈αspecialization␈αof␈αOdd-primes␈α(to␈αwithin␈αa␈αsingle␈αstray
␈↓ α,␈↓exception),␈αand␈αAM␈αwill␈α
be␈αable␈αto␈αtreat␈α
these␈αtwo␈αconcepts␈αas␈α
if␈αthey␈αwere␈αmerged␈α
together.
␈↓ α,␈↓They␈α∞are␈α∞still␈α
kept␈α∞separate,␈α∞however,␈α
in␈α∞case␈α∞AM␈α
ever␈α∞needs␈α∞to␈α
know␈α∞precisely␈α∞what␈α
the
␈↓ α,␈↓di≥erence between them is, or in case later evidence shows the conjecture to be false␈↓	17␈↓.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	17␈↓ε␈α	When␈α	space␈α	is␈α	exhausted,␈αλone␈α	emergency␈α	measure␈α	AM␈α	takes␈αλis␈α	to␈α	destructively␈α	coalesce␈α	a␈αλpair␈α	of␈α	concepts␈α	X,Y␈α	where␈αλX
␈↓ α,␈↓ε␈↓ βLis both a generalization of and a specialization of Y, even if there are a couple "boundary" exceptions.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε50␈↓-

␈↓ α,␈↓The␈α
≡nal␈αaction␈α
of␈αthe␈α
right-hand-side␈αof␈α
this␈αrule␈α
is␈αto␈α
propose␈αa␈α
new␈αtask␈α
(if␈α
there␈αexist
␈↓ α,␈↓some␈αgeneralizations␈αof␈αthe␈αconcept␈αY,␈αwhich␈αin␈αour␈αcase␈αis␈α"Primes").␈α So␈αAM␈αaccesses␈αthe
␈↓ α,␈↓Generalizations␈αfacet␈αof␈αPrimes,␈αwhich␈αis␈α"(Numbers)".␈α
A␈αnew␈αtask␈αis␈αtherefore␈αadded␈αto␈α
the
␈↓ α,␈↓agenda:␈α␈↓¬"Check␈αexamples␈αof␈αPrimes"␈↓,␈αwith␈αan␈αassociated␈αreason:␈α"Just␈αas␈αPrimes␈αwas␈αno␈αmore
␈↓ α,␈↓general␈α
than␈α
Odd-primes,␈α∞so␈α
Numbers␈α
may␈α∞turn␈α
out␈α
to␈α
be␈α∞no␈α
more␈α
general␈α∞than␈α
Primes".
␈↓ α,␈↓The reason is rated according to the formula given in the rule; say it gets the value 500.

␈↓ α,␈↓To␈αmake␈α
this␈αexample␈α
a␈αlittle␈α
more␈αinteresting,␈α
let's␈αsuppose␈α
that␈αthe␈α
task␈α␈↓¬"Check␈αexamples␈α
of
␈↓ α,␈↓¬Primes"␈↓␈αalready␈αexisted␈αon␈αthe␈αagenda,␈αbut␈αfor␈αthe␈αreason␈α"Many␈αexamples␈αof␈αPrimes␈αhave
␈↓ α,␈↓been␈αfound,␈αbut␈αnever␈αchecked",␈αwith␈αa␈αrating␈α
for␈αthe␈αreason␈αof␈α100,␈αand␈αfor␈αthe␈αtask␈α
as␈αa
␈↓ α,␈↓whole␈α
of␈α
200.␈α
 The␈αglobal␈α
task-rating␈α
formula␈α
then␈α
assigns␈αthe␈α
task␈α
a␈α
new␈α
overall␈αpriority␈α
of
␈↓ α,␈↓600, because of the new, fairly good reason supporting it.

␈↓ α,␈↓When␈αthat␈αtask␈αis␈αeventually␈αchosen,␈αthe␈αheuristic␈αrule␈αpictured␈αabove␈α(at␈αthe␈αbeginning␈αof
␈↓ α,␈↓this␈α
subsection)␈αwill␈α
trigger␈α
and␈αwill␈α
be␈α
run␈αagain,␈α
with␈α
X=Primes␈αand␈α
Y=Numbers.␈αThat␈α
is,
␈↓ α,␈↓AM␈αwill␈αbe␈αconsidering␈α
whether␈α(almost)␈αall␈αnumbers␈α
are␈αprimes.␈α The␈αleft-hand-side␈αof␈α
the
␈↓ α,␈↓heuristic␈α∂rule␈α∂will␈α∂quickly␈α∂fail,␈α∂since,␈α∂e.g.,␈α∞"6"␈α∂is␈α∂an␈α∂example␈α∂of␈α∂Numbers␈α∂which␈α∂does␈α∞not
␈↓ α,␈↓satisfy the de≡nition of Primes.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.4.4. Another illustration: Discovering Unique Factorization␈↓)αβ␈↓

␈↓ α,␈↓Below␈αis␈αa␈αheuristic␈αrule␈αwhich␈αis␈αa␈αkey␈αagent␈αin␈αthe␈αprocess␈αof␈α"noticing"␈αthe␈αfundamental
␈↓ α,␈↓theorem of arithmetic␈↓	18␈↓.

␈↓ α,␈↓¬␈↓ αlIf F(a) is unexpectedly a B,
␈↓ α,␈↓¬␈↓ αlThen maybe (∀x) F(x) is a B.

␈↓ α,␈↓Below,␈α⊂the␈α⊂same␈α⊂rule␈α⊂is␈α⊂given␈α⊂in␈α⊂more␈α∂detail.␈α⊂ The␈α⊂≡rst␈α⊂conjunct␈α⊂on␈α⊂the␈α⊂IF-part␈α⊂of␈α∂the
␈↓ α,␈↓heuristic␈αrule␈αindicates␈αthat␈αit's␈αrelevant␈αto␈αchecking␈αexamples␈αof␈αany␈αgiven␈αoperation␈αF.␈α A
␈↓ α,␈↓typical␈α
example␈α
is␈α
selected␈α
at␈αrandom,␈α
say␈α
F(x)=y.␈α
Then␈α
y␈αis␈α
examined,␈α
to␈α
see␈α
if␈α
it␈αsatis≡es
␈↓ α,␈↓any␈αmore␈αstringent␈αproperties␈αthan␈αthose␈αspeci≡ed␈αin␈αthe␈αDomain/range␈αfacet␈αof␈αF.␈α That␈αis,
␈↓ α,␈↓the␈α
Domain/range␈α
facet␈α
of␈αF␈α
contains␈α
an␈α
entry␈α
of␈αthe␈α
form␈α
A→B;␈α
so␈α
if␈αx␈α
is␈α
an␈α
A,␈α
then␈αall␈α
we
␈↓ α,␈↓are␈α
guaranteed␈α
about␈α
y␈α
is␈α
that␈α
it␈α
is␈α
an␈αexample␈α
of␈α
a␈α
B.␈α
 But␈α
now,␈α
this␈α
heuristic␈α
is␈α
asking␈αif␈α
y
␈↓ α,␈↓isn't␈α∂really␈α∂an␈α∂example␈α∂of␈α⊂a␈α∂much␈α∂more␈α∂specialized␈α∂concept␈α∂than␈α⊂B.␈α∂ If␈α∂it␈α∂is␈α∂(say␈α⊂it's␈α∂an
␈↓ α,␈↓example␈αof␈αa␈α
BB),␈αthen␈αthe␈αrest␈α
of␈αthe␈αexamples␈αof␈α
F␈αare␈αexamined␈αto␈α
see␈αif␈αthey␈αtoo␈α
satisfy
␈↓ α,␈↓this␈α∞same␈α∞property.␈α∞If␈α
all␈α∞examples␈α∞appear␈α∞to␈α
map␈α∞from␈α∞domain␈α∞set␈α
A␈α∞into␈α∞range␈α∞set␈α
BB
␈↓ α,␈↓(where␈αBB␈αis␈αmuch␈αmore␈αrestricted␈αthan␈αthe␈αset␈αB␈αspeci≡ed␈αoriginally␈αin␈αthe␈αDomain/range
␈↓ α,␈↓facet␈α
of␈αF),␈α
then␈α
a␈αnew␈α
conjecture␈α
is␈αmade:␈α
the␈α
domain/range␈αof␈α
F␈α
is␈αreally␈α
A→BB,␈αnot␈α
A→B.
␈↓ α,␈↓Here is that rule, in crisper notation:







␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	18␈↓ε␈αThe␈αunique␈αfactorization␈αconjecture:␈αany␈αpositive␈αinteger␈αn␈αcan␈αbe␈αrepresented␈αas␈αthe␈αproduct␈αof␈αprime␈αnumbers␈αin
␈↓ α,␈↓ε␈↓ βLprecisely␈α∞one␈α∞way␈α∂(to␈α∞within␈α∞reorderings␈α∂of␈α∞those␈α∞prime␈α∂factors).␈α∞ Thus␈α∞28␈α∂=␈α∞2x2x7,␈α∞and␈α∂we␈α∞don't
␈↓ α,␈↓ε␈↓ βLdistinguish between the factorization (2 2 7) and (2 7 2).
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε51␈↓-


␈↓ α,␈↓¬␈↓ αlIf the current task is to Check Examples of the operation F,
␈↓ α,␈↓¬␈↓ β,and F is an operation from domain A into range B,
␈↓ α,␈↓¬␈↓ β,and F has at least 10 examples,
␈↓ α,␈↓¬␈↓ β,and a typical one of these examples is "<x→y>" (so xεA and yεB),
␈↓ α,␈↓¬␈↓ β,and (Forsome Specialization BB of B), y is a BB.
␈↓ α,␈↓¬␈↓ β,and ␈↓&all␈↓)αβ examples of F (ignoring boundary cases) turn out to be BB's,
␈↓ α,␈↓¬␈↓ αlThen print the following conjecture: "F(a) is always a BB, not simply a B",
␈↓ α,␈↓¬␈↓ β,and add it to the Examples facet of Conjectures concept,
␈↓ α,␈↓¬␈↓ β,and␈α∩add␈α∩"<A␈α∩→␈α∩BB>"␈α∩as␈α∩a␈α∩new␈α∩entry␈α∩to␈α∩the␈α∩Domain/range␈α∩facet␈α∩of␈α∩F,␈α⊃replacing
␈↓ α,␈↓¬␈↓ ∧,"<A→B>",
␈↓ α,␈↓¬␈↓ β,and␈α
if␈α
the␈α
user␈α∞asks,␈α
inform␈α
him␈α
that␈α
the␈α∞evidence␈α
for␈α
this␈α
was␈α
that␈α∞all␈α
||Examples(F)||
␈↓ α,␈↓¬␈↓ ∧,examples of F (ignoring boundary examples) turned out to be BB's,
␈↓ α,␈↓¬␈↓ β,and check the truth of this conjecture by running F on boundary examples of A.

␈↓ α,␈↓Let's␈α⊂see␈α⊂how␈α⊂this␈α⊂rule␈α⊂was␈α⊂used␈α⊂in␈α⊂one␈α⊂instance.␈α⊂In␈α⊂Task␈α⊂79␈α⊂in␈α⊂the␈α⊂sample␈α⊂excerpt␈α⊂in
␈↓ α,␈↓Chapter␈α∂2,␈α∞AM␈α∂de≡ned␈α∞the␈α∂concept␈α∞Prime-times,␈α∂which␈α∞was␈α∂a␈α∞function␈α∂transforming␈α∞any
␈↓ α,␈↓number␈α∃n␈α∃into␈α∃the␈α∀set␈α∃of␈α∃all␈α∃factorizations␈α∃of␈α∀n␈α∃into␈α∃primes.␈α∃ For␈α∃example,␈α∀Prime-
␈↓ α,␈↓times(12)={(2␈α
2␈α
3)},␈α
Prime-times(13)={(13)}.␈α∞ The␈α
domain␈α
of␈α
F=Prime-times␈α
was␈α∞the␈α
concept
␈↓ α,␈↓Numbers.␈α The␈αrange␈αwas␈αSets.␈αMore␈αprecisely,␈αthe␈αrange␈αwas␈αSets-of-Bags-of-Numbers,␈αbut
␈↓ α,␈↓AM didn't know that concept at that time.

␈↓ α,␈↓The␈α
above␈α∞heuristic␈α
rule␈α∞was␈α
applicable.␈α∞F␈α
was␈α
Prime-times,␈α∞A␈α
was␈α∞Numbers,␈α
and␈α∞B␈α
was
␈↓ α,␈↓Sets.␈α∞ There␈α∞were␈α∞far␈α∞more␈α∞than␈α∞10␈α∞known␈α∞examples␈α∞of␈α∞Prime-times␈α∞in␈α∞action.␈α∞ A␈α
typical
␈↓ α,␈↓example␈α⊃was␈α⊃this␈α⊃one:␈α⊃<21␈α⊂→␈α⊃{(3,7)}>.␈α⊃ The␈α⊃rule␈α⊃now␈α⊂asked␈α⊃that␈α⊃{(3,7)}␈α⊃be␈α⊃fed␈α⊃to␈α⊂each
␈↓ α,␈↓specialization␈αof␈αSets,␈αto␈αsee␈αif␈αit␈αsatis≡ed␈αany␈αof␈αtheir␈αde≡nitions.␈α The␈αSpecializations␈αfacet
␈↓ α,␈↓of␈α
Sets␈α
was␈α
acccessed,␈α
and␈α
each␈α
concept␈α
pointed␈αto␈α
was␈α
run␈α
(its␈α
de≡nition␈α
was␈α
run)␈α
on␈αthe
␈↓ α,␈↓argument␈α∂"{(3,7)}".␈α∂ It␈α∂turned␈α∂out␈α∂that␈α∂Singleton␈α∂and␈α∂Set-of-doubletons␈α∂were␈α∂the␈α∂only␈α∞two
␈↓ α,␈↓specializations␈αof␈αSets␈αsatis≡ed␈αby␈αthis␈αexample.␈α At␈αthis␈αmoment,␈αAM␈αhad␈α
narrowed␈αdown
␈↓ α,␈↓the potential conjectures to these two:

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ Prime-times(x) is always a singleton set.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓ Prime-times(x) is always a set of doubletons.

␈↓ α,␈↓Each␈αexample␈αof␈αPrime-times␈αwas␈αexamined,␈αuntil␈αone␈αwas␈αfound␈αto␈αrefute␈αeach␈αconjecture
␈↓ α,␈↓(for␈α∂example,␈α⊂<8→{(2,2,2)}>␈α∂destroys␈α∂conjecture␈α⊂2).␈α∂ But␈α∂no␈α⊂example␈α∂was␈α∂able␈α⊂to␈α∂disprove
␈↓ α,␈↓conjecture␈α∞1.␈α∂So␈α∞the␈α∂heuristic␈α∞rule␈α∞plunged␈α∂forward,␈α∞and␈α∂printed␈α∞out␈α∞to␈α∂the␈α∞user␈α∂"A␈α∞new
␈↓ α,␈↓conjecture:␈α→Prime-times(␈↓βn␈↓)␈α→is␈α→always␈α→a␈α→singleton-set,␈α→not␈α→simply␈α→a␈α→set".␈α~ The␈α→entry
␈↓ α,␈↓<Numbers→Singleton-sets>␈α
was␈α∞added␈α
to␈α∞the␈α
Domain/range␈α∞facet␈α
of␈α∞Prime-times,␈α
replacing
␈↓ α,␈↓the old entry <Numbers→Sets>.

␈↓ α,␈↓Let's␈α
digress␈α
for␈α∞a␈α
moment␈α
to␈α
discuss␈α∞the␈α
robustness␈α
of␈α
the␈α∞system.␈α
 What␈α
if␈α∞this␈α
heuristic
␈↓ α,␈↓were␈αto␈αbe␈αexcised?␈α Could␈αAM␈αstill␈αpropose␈αunique␈αfactorization?␈α The␈αanswer␈αis␈αyes,␈α
there
␈↓ α,␈↓are␈αother␈α
ways␈αto␈α
notice␈αit.␈α
If␈αAM␈αhas␈α
the␈αconcept␈α
of␈αa␈α
Function␈↓	19␈↓,␈αthen␈α
a␈αheuristic␈αrule␈α
like
␈↓ α,␈↓the␈α∂one␈α∞in␈α∂the␈α∞previous␈α∂subsection␈α∞(page␈α∂48)␈α∞will␈α∂cause␈α∞AM␈α∂to␈α∞ask␈α∂if␈α∞Prime-times␈α∂is␈α∞not
␈↓ α,␈↓merely a Relation, but also a Function.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	19␈↓ε␈α	A␈α	single-valued␈α	relation.␈α
 That␈α	is,␈α	for␈α	any␈α	domain␈α
element␈α	x,␈α	F(x)␈α	contains␈α	precisely␈α
one␈α	member.␈α	 It␈α	is␈α	never␈α
empty␈α	(i.e.,
␈↓ α,␈↓ε␈↓ βLundefined), nor is it ever larger than a singleton (i.e., multiple-valued).
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε52␈↓-

␈↓ α,␈↓The␈α
past␈αfew␈α
sections␈αshould␈α
have␈αconvinced␈α
the␈αreader␈α
that␈αisolated␈α
heuristic␈α
rules␈αreally
␈↓ α,␈↓can␈αdo␈αall␈αkinds␈αof␈αthings:␈αpropose␈αnew␈αtasks,␈αcreate␈αnew␈αconcepts,␈α≡ll␈αin␈αentries␈αfor␈αspeci≡c
␈↓ α,␈↓facets␈α∞(goal-driven),␈α
and␈α∞look␈α
for␈α∞conjectures␈α
(data-driven␈α∞empirical␈α
induction).␈α∞ The␈α
rules
␈↓ α,␈↓appear␈α∃fairly␈α∃general␈↓	20␈↓␈α∃¬␈α∃though␈α∃that␈α∀must␈α∃be␈α∃later␈α∃veri≡ed␈α∃empirically.␈α∃ They␈α∀are
␈↓ α,␈↓redundant␈α⊂in␈α∂a␈α⊂pleasing␈α∂way:␈α⊂some␈α⊂of␈α∂the␈α⊂most␈α∂"important",␈α⊂well-known,␈α⊂and␈α∂interesting
␈↓ α,␈↓conjectures␈α∃can␈α∃(apparently)␈α∀be␈α∃derived␈α∃in␈α∀many␈α∃ways.␈α∃ Again,␈α∀we␈α∃must␈α∃justify␈α∀this
␈↓ α,␈↓experimentally.


␈↓ α,␈↓␈↓ ∧T␈↓∧␈↓&4.5. Gathering Relevant Heuristics␈↓)αβ␈↓

␈↓ α,␈↓Each␈αconcept␈α
has␈αfacets␈αwhich␈α
(may)␈αcontain␈αsome␈α
heuristics.␈α Some␈αof␈α
these␈αare␈α
for␈α≡lling
␈↓ α,␈↓in,␈αsome␈αfor␈αchecking,␈αsome␈αfor␈αdeciding␈αinterestingness,␈αsome␈αfor␈αnoticing␈αnew␈αconjectures,
␈↓ α,␈↓etc.

␈↓ α,␈↓AM␈α
contains␈α
hundreds␈α
of␈α
these␈α
heuristics.␈α
 In␈α
order␈α
to␈α
save␈α
time␈α
(and␈α
to␈α
make␈α
AM␈αmore
␈↓ α,␈↓rational),␈α∞each␈α∞heuristic␈α∞should␈α∞only␈α∞be␈α∞tried␈α∞in␈α∞situations␈α∞where␈α∞it␈α∞might␈α∞apply,␈α∂where␈α∞it
␈↓ α,␈↓makes sense.

␈↓ α,␈↓How␈α∞is␈α∞AM␈α∞able␈α
to␈α∞zero␈α∞in␈α∞on␈α
the␈α∞relevant␈α∞heuristic␈α∞rules,␈α
once␈α∞a␈α∞task␈α∞has␈α∞been␈α
selected
␈↓ α,␈↓from the agenda?


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.5.1. Domain of Applicability␈↓)αβ␈↓

␈↓ α,␈↓The␈α∩secret␈α⊃is␈α∩that␈α∩each␈α⊃heuristic␈α∩rule␈α⊃is␈α∩stored␈α∩somewhere␈α⊃␈↓βa␈α∩propos␈↓␈α⊃to␈α∩its␈α∩"domain␈α⊃of
␈↓ α,␈↓applicability".␈α
This␈α
"proper␈αplace"␈α
is␈α
determined␈αby␈α
the␈α
≡rst␈αconjunct␈α
in␈α
the␈α
left-hand␈αside
␈↓ α,␈↓of the rule.

␈↓ α,␈↓What does this mean?  Consider this heuristic:

␈↓ α,␈↓¬␈↓ αlIf the current task is to fill in examples of the operation F, ␈↓π <====␈↓¬
␈↓ α,␈↓¬␈↓ β,and some examples of the domain of F are known,
␈↓ α,␈↓¬␈↓ αlThen␈αone␈αway␈αto␈αget␈αexamples␈αof␈αF␈αis␈αto␈αrun␈αF␈αon␈αrandomly␈αchosen␈αexamples␈αof␈αthe␈αdomain
␈↓ α,␈↓¬␈↓ ∧,of F.

␈↓ α,␈↓This␈αis␈αa␈αreasonable␈αthing␈αto␈αtry␈α¬␈αbut␈αonly␈αin␈αcertain␈αsituations.␈α Should␈αit␈αbe␈αtried␈αwhen
␈↓ α,␈↓the␈αcurrent␈αtask␈αis␈αto␈αcheck␈αthe␈αWorth␈αfacet␈αof␈αthe␈αSets␈αconcept?␈αNo,␈αit␈αwould␈αbe␈αirrational.
␈↓ α,␈↓Of␈α
course,␈α
even␈α
if␈α
it␈α
were␈α
tried␈αthen,␈α
the␈α
left-hand-side␈α
would␈α
fail␈α
very␈α
quickly.␈α
 Yet␈α␈↓βsome␈↓
␈↓ α,␈↓cpu␈αtime␈αwould␈α
have␈αbeen␈αused,␈α
and␈αif␈αthe␈α
user␈αwere␈αwatching,␈α
his␈αopinion␈αof␈α
AM␈αwould
␈↓ α,␈↓decrease.␈↓	21␈↓

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	20␈↓ε␈α	i.e.,␈α	applicable␈α	in␈α	many␈α	situations.␈α	It␈α	would␈α	be␈α	worse␈α	than␈α	useless␈α	if␈α	a␈α	rule␈α	existed␈α	which␈α	could␈α	lead␈α	to␈α	a␈α
single␈α	discovery
␈↓ α,␈↓ε␈↓ βLlike␈α
"Fibonacci␈α
series"␈α
but␈α
never␈α
lead␈α
to␈α
any␈α
other␈α
discoveries.␈α
 The␈α
reasons␈α
for␈α
demanding␈α
generality␈α
are
␈↓ α,␈↓ε␈↓ βLnot␈α	only␈α	"fairness",␈α
but␈α	the␈α	insights␈α
which␈α	occur␈α	when␈α
it␈α	is␈α	observed␈α
that␈α	several␈α	disparate␈α
concepts␈α	were
␈↓ α,␈↓ε␈↓ βLall motivated by the same general principle (e.g., "looking for the inverse image of extrema").
␈↓ α,␈↓ε␈↓	21␈↓ε␈α	This␈α	notion␈α	of␈α	worrying␈α	about␈α	a␈α	human␈α	user␈α	who␈α	is␈αλobserving␈α	AM␈α	run␈α	in␈α	real␈α	time␈α	may␈α	appear␈α	to␈α	be␈α	quite␈α	language-␈αλand
␈↓ α,␈↓ε␈↓ βLmachine-dependent.␈α∂An␈α∂increase␈α∂in␈α⊂speed␈α∂of␈α∂a␈α∂couple␈α∂orders␈α⊂of␈α∂magnitude␈α∂would␈α∂radically␈α⊂alter␈α∂the
␈↓ α,␈↓ε␈↓ βLqualitative␈α⊂appearance␈α⊂of␈α⊂AM.␈α⊂In␈α⊂Chapter␈α⊂7,␈α⊂however,␈α⊂the␈α⊂reader␈α⊂will␈α⊂grasp␈α⊂how␈α⊂difficult␈α⊂it␈α⊃is␈α⊂to
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε53␈↓-

␈↓ α,␈↓That␈αparticular␈αheuristic␈αhas␈αa␈αprecise␈αdomain␈αof␈αapplicability:␈αAM␈αshould␈αuse␈αit␈αwhenever
␈↓ α,␈↓the current task is to ≡ll in examples of an operation, and only in those kinds of situations.

␈↓ α,␈↓The␈α⊂key␈α⊂observation␈α⊂is␈α⊂that␈α⊂a␈α⊂heuristic␈α∂typically␈α⊂applies␈α⊂to␈α⊂␈↓βall␈α⊂examples␈α⊂of␈α⊂a␈α∂particular
␈↓ α,␈↓βconcept␈αC␈↓.␈α In␈αthe␈αcase␈αwe␈αwere␈αconsidering,␈αC=Operation.␈α Intuitively,␈αwe'd␈αlike␈αto␈αtack␈α
that
␈↓ α,␈↓heuristic␈α
onto␈αthe␈α
Examples␈αfacet␈α
of␈α
the␈αconcept␈α
Operation,␈αso␈α
it␈α
would␈αonly␈α
"come␈αto␈α
mind"
␈↓ α,␈↓in appropriate situations.  This is in fact precisely where the heuristic rule ␈↓βis␈↓ stored.

␈↓ α,␈↓Initially,␈αthe␈αauthor␈αidenti≡ed␈αthe␈αproper␈αconcept␈αC␈αand␈αfacet␈αF␈αfor␈αeach␈αheuristic␈αH␈αwhich
␈↓ α,␈↓AM␈αpossessed,␈α
and␈αtacked␈α
H␈αonto␈α
C.F␈↓	22␈↓.␈α This␈α
was␈αall␈α
preparation,␈αcompleted␈α
long␈αbefore
␈↓ α,␈↓AM␈α⊃started␈α⊃up.␈α⊃ Each␈α⊃heuristic␈α⊃was␈α⊃tacked␈α⊃onto␈α⊃the␈α⊃facet␈α⊃which␈α⊃uniquely␈α⊃indicates␈α⊂its
␈↓ α,␈↓domain␈αof␈αapplicability.␈α The␈α≡rst␈αconjunct␈αof␈αthe␈αIF-part␈αof␈αeach␈αheuristic␈αindicates␈αwhere
␈↓ α,␈↓it␈αis␈αstored␈αand␈α
where␈αit␈αis␈αapplicable.␈α
Notice␈αthe␈αlittle␈αarrow␈α
(␈↓π<==␈↓)␈αpointing␈αto␈αthat␈α
conjunct
␈↓ α,␈↓above.␈↓	23␈↓.

␈↓ α,␈↓While␈α∞AM␈α
is␈α∞running,␈α
it␈α∞will␈α∞choose␈α
a␈α∞task␈α
dealing␈α∞with,␈α∞say,␈α
facet␈α∞F␈α
of␈α∞concept␈α∞C.␈α
 AM
␈↓ α,␈↓must␈α⊂quickly␈α∂locate␈α⊂the␈α∂heuristic␈α⊂rules␈α∂which␈α⊂are␈α∂relevant␈α⊂to␈α∂satisfying␈α⊂that␈α⊂chosen␈α∂task.
␈↓ α,␈↓AM␈α∂simply␈α⊂locates␈α∂all␈α⊂concepts␈α∂which␈α∂claim␈α⊂C␈α∂as␈α⊂an␈α∂example.␈α∂ If␈α⊂the␈α∂current␈α⊂task␈α∂were
␈↓ α,␈↓"␈↓¬Check␈α
the␈α
Domain/range␈α
of␈α
Union␈↓εo␈↓¬Union␈↓"␈↓	24␈↓,␈αthen␈α
C␈α
would␈α
be␈α
Union␈↓εo␈↓Union.␈α
Which␈αconcepts
␈↓ α,␈↓claim␈αC␈αas␈αan␈αexample?␈α They␈αinclude␈αCompose-with-Self,␈αComposition,␈αOperation,␈αActive,
␈↓ α,␈↓Any-concept,␈α∂and␈α∞Anything.␈α∂ AM␈α∂then␈α∞collects␈α∂the␈α∞heuristics␈α∂tacked␈α∂onto␈α∞facet␈α∂F␈α∂(in␈α∞this
␈↓ α,␈↓case,␈α
F␈α
is␈αDomain/range)␈α
of␈α
each␈αof␈α
those␈α
concepts.␈αAll␈α
such␈α
heuristics␈αwill␈α
be␈α
relevant.␈αIn
␈↓ α,␈↓the␈α
current␈α
case,␈α
some␈α
relevant␈α
heuristics␈α
might␈α
be␈α
garnered␈α
from␈α
the␈α
Domain/range␈αfacet␈α
of
␈↓ α,␈↓the␈α
concept␈αOperation.␈α
 Any␈αheuristic␈α
which␈αcan␈α
deal␈αwith␈α
the␈αDomain/range␈α
facet␈α
of␈α␈↓βany␈↓
␈↓ α,␈↓operation␈α↔can␈α↔certainly␈α↔deal␈α_with␈α↔Union␈↓εo␈↓Union's␈α↔Domain/range.␈α↔ A␈α↔typical␈α_rule␈α↔on
␈↓ α,␈↓Operation.Domain/range.Check␈↓	25␈↓ would be this one:

␈↓ α,␈↓¬␈↓ β,If a Dom/ran entry of F is of the form <D D D...D → R>, and R is a generalization of D,
␈↓ α,␈↓¬␈↓ β,Then test whether the range might not be simply D.

␈↓ α,␈↓Suppose␈α∨one␈α≡entry␈α∨on␈α≡Union␈↓εo␈↓Union.Dom/ran␈α∨was␈α∨`<Nonempty-sets␈α≡Nonempty-sets
␈↓ α,␈↓Nonempty-sets␈α∂→␈α∞Sets>'.␈α∂Then␈α∞this␈α∂last␈α∞heuristic␈α∂rule␈α∞would␈α∂be␈α∞relevant,␈α∂and␈α∂would␈α∞have
␈↓ α,␈↓AM␈αask␈αthe␈αplausible␈αquestion:␈αIs␈αthe␈αunion␈αof␈αthree␈αnonempty␈αsets␈αalways␈αnonempty?␈α
 The
␈↓ α,␈↓answer␈α≤is␈α≤a≠rmative,␈α≠empirically,␈α≤so␈α≤AM␈α≠modi≡es␈α≤that␈α≤Domain/range␈α≤entry␈α≠for
␈↓ α,␈↓Union␈↓εo␈↓Union.␈α∩ AM␈α∩would␈α∩ask␈α∪the␈α∩same␈α∩question␈α∩for␈α∩Intersect␈↓εo␈↓Intersect.␈α∪ Although␈α∩the
␈↓ α,␈↓answer␈αthen␈αwould␈α
be␈α`␈↓βNo␈↓',␈αit's␈α
still␈αa␈αrational␈α
inquiry.␈α If␈αAM␈α
called␈αon␈αthis␈α
heuristic␈αrule
␈↓ α,␈↓when␈αthe␈αcurrent␈αtask␈αwas␈α
"␈↓¬Fillin␈αspecializations␈αof␈αBags␈↓",␈αit␈α
would␈αclearly␈αbe␈αan␈αirrational␈α
act.
␈↓ α,␈↓The␈αdomain␈αof␈αapplicability␈αof␈αthe␈αrule␈αis␈αclear,␈αand␈αis␈αprecisely␈α≡tted␈αto␈αthe␈αslot␈αwhere␈αthe
␈↓ α,␈↓rule is stored (tacked onto Operation.Domain/range).
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓ βLobjectively␈α
rate␈α
a␈α
system␈α
like␈α
AM.␈α
For␈α
that␈α
reason,␈α
all␈α
measures␈α
of␈α
judgment␈α
must␈α
be␈α
respected.␈α Also,␈α
to
␈↓ α,␈↓ε␈↓ βLthe actual human being using the system this really is one of the most important measures.
␈↓ α,␈↓ε␈↓	22␈↓ε Recall that C.F is an abbreviation for facet F of concept C
␈↓ α,␈↓ε␈↓	23␈↓ε␈αIn␈αthe␈αLISP␈αimplementation,␈αthese␈αfirst␈αconjuncts␈αare␈αomitted,␈αsince␈αthe␈α␈↓βplacement␈↓ε␈αof␈αa␈αheuristic␈αserves␈αthe␈αsame
␈↓ α,␈↓ε␈↓ βLpurpose as if it had some "pre-preconditions" (like these first conjuncts) to determine relevance quickly.
␈↓ α,␈↓ε␈↓	24␈↓ε␈α	This␈α	operation␈α	is␈α	defined␈αλas:␈α	␈↓Union␈↓εo␈↓Union␈↓ε(x,y,z)␈α	≡␈α	(x␈α	∪␈α	y)␈αλ∪␈α	z.␈α	 It␈α	accepts␈α	3␈αλsets␈α	as␈α	arguments,␈α	and␈α	returns␈α	a␈αλnew
␈↓ α,␈↓ε␈↓ βLset as its value.
␈↓ α,␈↓ε␈↓	25␈↓ε the `Check' subfacet of the `Domain/range' facet of the `Operation' concept.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε54␈↓-

␈↓ α,␈↓To␈αrecap␈αthe␈αbasic␈αidea:␈αwhen␈αdealing␈αwith␈αa␈αtask␈α"Do␈αact␈αA␈αon␈αfacet␈αF␈αof␈αconcept␈αC",␈αAM
␈↓ α,␈↓must␈α
locate␈α
all␈α∞the␈α
concepts␈α
X␈α
claiming␈α∞C␈α
as␈α
an␈α
example.␈α∞AM␈α
then␈α
gathers␈α∞the␈α
heuristics
␈↓ α,␈↓tacked␈α
onto␈α
X.F.A,␈α
for␈α∞each␈α
such␈α
general␈α
concept␈α∞X.␈α
 All␈α
of␈α
them␈α∞¬␈α
and␈α
only␈α
they␈α∞¬␈α
are
␈↓ α,␈↓relevant to satisfying that task.

␈↓ α,␈↓So␈α∞the␈α∞whole␈α∞problem␈α∞of␈α∞locating␈α∞relevant␈α∞heuristics␈α∞has␈α∞been␈α∞reduced␈α∞to␈α∞the␈α∞problem␈α
of
␈↓ α,␈↓e≠ciently␈α∂≡nding␈α⊂all␈α∂concepts␈α⊂of␈α∂which␈α⊂C␈α∂is␈α⊂an␈α∂example␈α⊂(for␈α∂a␈α⊂given␈α∂concept␈α⊂C).␈α∂ This
␈↓ α,␈↓process␈αis␈α
called␈α␈↓β"rippling␈α
away␈αfrom␈α
C␈αin␈αthe␈α
ISA␈αdirection"␈↓,␈α
and␈αforms␈α
the␈αsubject␈α
of␈αthe
␈↓ α,␈↓next subsection.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.5.2. Rippling␈↓)αβ␈↓

␈↓ α,␈↓Given a concept C, how can AM ≡nd all the concepts which claim C as an example?

␈↓ α,␈↓The␈α
most␈α
obvious␈αscheme␈α
is␈α
to␈α
store␈αthis␈α
information␈α
explicitly.␈α
 So␈αthe␈α
Examples␈α
facet␈αof␈α
C
␈↓ α,␈↓would␈αpoint␈α
to␈αall␈αknown␈α
examples␈αof␈αC,␈α
and␈αthe␈αIsa␈α
facet␈αof␈αC␈α
would␈αpoint␈αto␈α
all␈αknown
␈↓ α,␈↓concepts␈α∂claiming␈α∂C␈α∂as␈α∂one␈α∂of␈α∂their␈α⊂examples.␈α∂ Why␈α∂not␈α∂just␈α∂do␈α∂this?␈α∂ Because␈α⊂one␈α∂can
␈↓ α,␈↓substitute␈α⊃a␈α⊃modest␈α⊃amount␈α∩of␈α⊃processing␈α⊃time␈α⊃(via␈α∩chasing␈α⊃links␈α⊃around)␈α⊃for␈α∩the␈α⊃vast
␈↓ α,␈↓amount of storage space that would be needed to have "everything point to everything".

␈↓ α,␈↓Each␈αfacet␈αcontains␈α
only␈αenough␈αpointers␈αso␈α
that␈αthe␈αentire␈α
graph␈αof␈αExs/Isa␈αand␈α
Spec/Genl
␈↓ α,␈↓links␈α∪could␈α∩be␈α∪␈↓βreconstructed␈↓␈α∩if␈α∪needed.␈α∩ Since␈α∪"Genl"␈α∩is␈α∪a␈α∩transitive␈α∪relation,␈α∪AM␈α∩can
␈↓ α,␈↓compute␈α∩that␈α∪Numbers␈α∩is␈α∪a␈α∩generalization␈α∪of␈α∩Mersenne-primes,␈α∪if␈α∩the␈α∪facet␈α∩Mersenne-
␈↓ α,␈↓primes.Genl␈α∂contains␈α∞the␈α∂entry␈α∞"Odd-primes",␈α∂and␈α∞Odd-primes.Genl␈α∂contains␈α∞a␈α∂pointer␈α∞to
␈↓ α,␈↓"Primes",␈αand␈αPrimes.Genl␈αpoints␈αto␈α"Numbers".␈α This␈αkind␈αof␈α"␈↓βrippling␈↓"␈αactivity␈αis␈αused␈αto
␈↓ α,␈↓e≠ciently␈α∞locate␈α∂all␈α∞concepts␈α∂related␈α∞to␈α∂a␈α∞given␈α∂one␈α∞X.␈α∂ In␈α∞particular,␈α∂AM␈α∞knows␈α∂how␈α∞to
␈↓ α,␈↓"ripple␈αupward␈αin␈αthe␈αIsa␈α
direction",␈αand␈αquickly␈↓	26␈↓␈αlocate␈αall␈α
concepts␈αwhich␈αclaim␈αX␈αas␈α
one
␈↓ α,␈↓of their examples.

␈↓ α,␈↓It␈αturns␈αout␈αthat␈αAM␈αcannot␈αsimply␈αcall␈αfor␈αX.Isa,␈αthen␈αthe␈αIsa␈αfacets␈αof␈αthose␈αconcepts,␈αetc.,
␈↓ α,␈↓because␈αIsa␈α
is␈αnot␈α
transitive␈↓	27␈↓.␈α For␈α
the␈αinterested␈αreader,␈α
the␈αalgorithm␈α
AM␈αuses␈α
to␈αcollect
␈↓ α,␈↓Isa's of X is given below.␈↓	28␈↓

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α
All␈α∞generalizations␈α
of␈α
the␈α∞given␈α
concept␈α∞X␈α
are␈α
located.␈α∞ AM␈α
accesses␈α∞X.Genl,␈α
then
␈↓ α,␈↓␈↓ β≤the Genl facets of ␈↓βthose␈↓ concepts, etc.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓ The "Isa" facet of each of those concepts is accessed.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α∞AM␈α∞locates␈α
all␈α∞generalizations␈α∞of␈α∞these␈α
newly-found␈α∞higher-level␈α∞concepts.␈α∞ This␈α
is
␈↓ α,␈↓␈↓ β≤the list of all known concepts which claim X as one of their examples.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	26␈↓ε␈α
With␈α
about␈α	200␈α
known␈α
concepts,␈α	with␈α
each␈α
Isa␈α	facet␈α
and␈α
each␈α	Genl␈α
facet␈α
pointing␈α	to␈α
about␈α
3␈α	other␈α
concepts,␈α
about␈α	25
␈↓ α,␈↓ε␈↓ βLlinks␈αλwill␈α	be␈αλtraced␈α	along␈αλin␈α	order␈αλto␈α	locate␈αλabout␈αλa␈α	dozen␈αλfinal␈α	concepts,␈αλeach␈α	of␈αλwhich␈α	claims␈αλthe␈α	given␈αλone
␈↓ α,␈↓ε␈↓ βLas␈α
an␈α∞example.␈α
 This␈α∞whole␈α
rippling␈α∞process,␈α
tracing␈α
25␈α∞linkages,␈α
uses␈α∞less␈α
than␈α∞.01␈α
cpu␈α∞seconds,␈α
in
␈↓ α,␈↓ε␈↓ βLcompiled Interlisp, on a KI-10 type PDP-10.
␈↓ α,␈↓ε␈↓	27␈↓ε␈α	If␈α	x␈α	isa␈α	y,␈α	and␈α
y␈α	isa␈α	z,␈α	then␈α	x␈α	is␈α
(generally)␈α	␈↓&NOT␈↓)αβ␈α	a␈α	z.␈α	This␈α	is␈α
due␈α	to␈α	the␈α	intransitivity␈α	of␈α	"member-of".␈α
 Generalization␈α	is
␈↓ α,␈↓ε␈↓ βLtransitive, on the other hand, because "subset-of" is transitive.
␈↓ α,␈↓ε␈↓	28␈↓ε␈α∞For␈α∞the␈α∞␈↓&very␈↓)αβ␈α∞interested␈α∞reader,␈α∞it␈α∞is␈α∞explained␈α∞in␈α∞great␈α∞detail␈α∞in␈α∞file␈α∞RIPPLE[dis,dbl]␈α∞at␈α∞SAIL.␈α∞This␈α∞file␈α∞has␈α
been
␈↓ α,␈↓ε␈↓ βLpermanently archived at SAIL.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε55␈↓-

␈↓ α,␈↓In␈α regular␈α form,␈α∨one␈α might␈α express␈α this␈α∨rippling␈α recipe␈α more␈α compactly␈α∨as:
␈↓ α,␈↓␈↓¬Genl␈↓#
*␈↓#(Isa(Genl␈↓#
*␈↓#(X)))␈↓.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.5.3. Ordering the Relevant Heuristics␈↓)αβ␈↓

␈↓ α,␈↓Now␈α⊂that␈α⊃all␈α⊂these␈α⊃relevant␈α⊂heuristics␈α⊃have␈α⊂been␈α⊃assembled,␈α⊂in␈α⊃what␈α⊂order␈α⊃should␈α⊂AM
␈↓ α,␈↓execute␈αthem?␈↓	29␈↓␈αIt␈αis␈αimportant␈αto␈αnote␈αthat␈αthe␈αheuristics␈αtacked␈αonto␈αvery␈αgeneral␈αconcepts
␈↓ α,␈↓will␈αbe␈αapplicable␈α
frequently,␈αyet␈αwill␈αnot␈α
be␈αvery␈αpowerful.␈α For␈α
example,␈αhere␈αis␈α
a␈αtypical
␈↓ α,␈↓heuristic␈α∂rule␈α∞which␈α∂is␈α∂tacked␈α∞onto␈α∂the␈α∂Examples␈α∞facet␈α∂of␈α∂the␈α∞very␈α∂general␈α∂concept␈α∞Any-
␈↓ α,␈↓concept:

␈↓ α,␈↓¬␈↓ β,If the current task is to fill in examples of any concept X,
␈↓ α,␈↓¬␈↓ β,Then one way to get them is to symbolically instantiate␈↓	30␈↓¬ a definition of X.

␈↓ α,␈↓It␈α∩takes␈α∩a␈α∩tremendous␈α∩amount␈α∩of␈α∩inference␈α∩to␈α∩squeeze␈α∩a␈α∩couple␈α∩awkward␈α∩examples␈α∩of
␈↓ α,␈↓Intersect␈↓εo␈↓Intersect out that concept's de≡nition.  Much time could be wasted doing so␈↓	31␈↓.

␈↓ α,␈↓Just␈α∞as␈α∞general␈α
heuristics␈α∞are␈α∞weak␈α
but␈α∞often␈α∞relevant,␈α
speci≡c␈α∞heuristics␈α∞are␈α∞powerful␈α
but
␈↓ α,␈↓rarely␈α
relevant.␈α
 Consider␈α
this␈α
heuristic␈α
rule,␈αwhich␈α
is␈α
attached␈α
to␈α
the␈α
very␈α
speci≡c␈αconcept
␈↓ α,␈↓Compose-with-Self:

␈↓ α,␈↓¬␈↓ β,If the current task is to fill in examples of the composition F␈↓εo␈↓¬F,
␈↓ α,␈↓¬␈↓ β,Then include any fixed-points of F.


␈↓ α,␈↓For␈α↔example,␈α↔since␈α↔Intersect(␈↓βphi␈↓,X)␈α↔equals␈α↔␈↓βphi␈↓,␈α↔so␈α↔must␈α↔Intersect␈↓εo␈↓Intersect(␈↓βphi␈↓,X,Y).␈↓	32␈↓.
␈↓ α,␈↓Assuming␈α⊂that␈α⊂such␈α⊂examples␈α⊃exist␈α⊂already␈α⊂on␈α⊂Intersect,␈α⊂this␈α⊃heuristic␈α⊂will␈α⊂≡ll␈α⊂in␈α⊃a␈α⊂few
␈↓ α,␈↓examples␈α∀of␈α∀Intersect␈↓εo␈↓Intersect␈α∀with␈α∀essentially␈α∀no␈α∀processing␈α∀required.␈α∀ Of␈α∀course␈α∪the
␈↓ α,␈↓domain of applicability of this heuristic is minuscule.

␈↓ α,␈↓As␈αwe␈αexpected,␈αthe␈αnarrower␈αits␈αdomain␈αof␈αapplicability,␈αthe␈αmore␈αpowerful␈αand␈αe≠cient␈αa
␈↓ α,␈↓heuristic␈αis,␈αand␈αthe␈αless␈αfrequently␈αit's␈αuseful.␈α Thus␈αin␈αany␈αgiven␈αsituation,␈αwhere␈αAM␈αhas
␈↓ α,␈↓gathered␈αmany␈αheuristic␈αrules,␈αit␈αwill␈αprobably␈αbe␈αbest␈αto␈αexecute␈αthe␈αmost␈αspeci≡c␈αones␈α≡rst,
␈↓ α,␈↓and execute the most general ones last.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	29␈↓ε␈αλThe␈α	discussion␈αλbelow␈α	assumes␈αλthat␈α	the␈αλheuristics␈α	don't␈αλinteract␈α	with␈αλeach␈α	other;␈αλi.e.,␈α	that␈αλeach␈α	one␈αλmay␈α	act␈αλindependently
␈↓ α,␈↓ε␈↓ βLof␈αλall␈α	others;␈αλi.e.,␈α	they␈αλare␈α	not␈αλ`strongly␈α	coupled'.␈αλThe␈α	validity␈αλof␈α	this␈αλsimplification␈α	is␈αλtested␈α	empirically␈αλ(see
␈↓ α,␈↓ε␈↓ βLChapter 6) and discussed theoretically (see Chapter 7) later.
␈↓ α,␈↓ε␈↓	30␈↓ε␈α"Symbolic␈αinstantiation"␈αis␈αa␈αeuphemism␈αfor␈αa␈αbag␈α
of␈αtricks␈αwhich␈αtransform␈αa␈αdeclarative␈αdefinition␈αof␈αa␈αconcept␈α
into
␈↓ α,␈↓ε␈↓ βLparticular␈α
entities␈α
satisfying␈α
that␈α
definition.␈α
The␈α
only␈α
constraint␈α
on␈α
the␈α
tricks␈α
is␈α
that␈α
they␈α
not␈α
actually␈α
␈↓&run␈↓)αβ
␈↓ α,␈↓ε␈↓ βLthe␈α	definition.␈α	 One␈α	such␈α	trick␈α	might␈α	be:␈α	if␈α	the␈α	definition␈α	is␈α	recursive,␈α	merely␈α	find␈α	some␈α	entity␈α	that␈αλsatisfies
␈↓ α,␈↓ε␈↓ βLthe␈αλbase␈αλstep.␈αλAM's␈αλsymbolic␈α	instantiation␈αλtricks␈αλare␈αλtoo␈αλhand-crafted␈αλto␈α	be␈αλof␈αλgreat␈αλinterest,␈αλhence␈α	this␈αλwill
␈↓ α,␈↓ε␈↓ βLnot␈α∞be␈α∞covered␈α∞any␈α∞more␈α
deeply␈α∞here.␈α∞ The␈α∞interested␈α∞reader␈α
is␈α∞directed␈α∞to␈α∞the␈α∞pioneering␈α∞work␈α
by
␈↓ α,␈↓ε␈↓ βL[Lombardi␈α
&␈α
Raphael␈α64],␈α
or␈α
the␈α
more␈αrecent␈α
literature␈α
on␈α
these␈αtechniques␈α
applied␈α
to␈αautomatic␈α
program
␈↓ α,␈↓ε␈↓ βLverification (e.g., [Moore 75], [Balzer 78]).
␈↓ α,␈↓ε␈↓	31␈↓ε Incidentally, this illustrates why no single heuristic should be allowed to monopolize the processing of any one task.
␈↓ α,␈↓ε␈↓	32␈↓ε␈α	␈↓βphi␈↓␈α	␈↓εis␈α	another␈α	name␈α	for␈α	the␈α	empty␈α	set,␈α	also␈α	written␈α	{}.␈α	 This␈α	last␈α	sentence␈α	thus␈α	says␈α	that␈α	since␈α	{}␈α	∩␈α	X␈α	=␈α	{},␈α	then␈α
({}␈α	∩
␈↓ α,␈↓ε␈↓ βLX) ∩ Y must also equal {}.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε56␈↓-

␈↓ α,␈↓Below␈α∩are␈α∩summarized␈α∩the␈α∩three␈α∩main␈α∪points␈α∩that␈α∩make␈α∩up␈α∩AM's␈α∩scheme␈α∪for␈α∩≡nding
␈↓ α,␈↓relevant heuristics in a "natural" way and then using them:

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α∂Each␈α∂heuristic␈α∂is␈α∂tacked␈α∂onto␈α∂the␈α∂most␈α∂general␈α∂concept␈α∂for␈α∂which␈α∂it␈α∂applies:␈α⊂it␈α∂is
␈↓ α,␈↓␈↓ β≤given␈α∀as␈α∀large␈α∀a␈α∀domain␈α∀of␈α∪applicability␈α∀as␈α∀possible.␈α∀This␈α∀will␈α∀maximize␈α∪its
␈↓ α,␈↓␈↓ β≤generality,␈α∩but␈α⊃leave␈α∩its␈α⊃power␈α∩untouched.␈α∩ This␈α⊃brings␈α∩it␈α⊃closer␈α∩to␈α∩the␈α⊃"ideal"
␈↓ α,␈↓␈↓ β≤tradeo≥ point between these two quantities.

␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈αWhen␈αthe␈αcurrent␈αtask␈αdeals␈αwith␈αconcept␈αC,␈αAM␈αripples␈αaway␈αfrom␈αC␈αand␈αquickly
␈↓ α,␈↓␈↓ β≤locates␈α⊃all␈α⊂the␈α⊃concepts␈α⊂of␈α⊃which␈α⊃C␈α⊂is␈α⊃an␈α⊂example.␈α⊃ Each␈α⊂of␈α⊃them␈α⊃will␈α⊂contain
␈↓ α,␈↓␈↓ β≤heuristics relevant to dealing with C.

␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α
AM␈α
then␈α
applies␈αthose␈α
heuristics␈α
in␈α
order␈αof␈α
increasing␈α
generality.␈α
 You␈αmay␈α
wonder
␈↓ α,␈↓␈↓ β≤how␈α
AM␈α
orders␈αthe␈α
heuristics␈α
by␈αgenerality.␈α
It␈α
turns␈αout␈α
that␈α
the␈α
rippling␈αprocess
␈↓ α,␈↓␈↓ β≤automatically␈α∩gathers␈α⊃heuristics␈α∩in␈α∩order␈α⊃of␈α∩increasing␈α⊃generality.␈α∩ In␈α∩the␈α⊃LISP
␈↓ α,␈↓␈↓ β≤system,␈αeach␈αrule␈αis␈αtherefore␈αexecuted␈αas␈αsoon␈αas␈αit's␈αfound.␈α So␈αAM␈αnevers␈αwastes
␈↓ α,␈↓␈↓ β≤time gathering heuristics it won't have time to execute.


␈↓ α,␈↓␈↓ ¬ε␈↓∧␈↓&4.6. AM's Starting Heuristics␈↓)αβ␈↓

␈↓ α,␈↓This␈α∂section␈α∂will␈α∂brie∨y␈α∂characterize␈α∂the␈α∂collection␈α∂of␈α∂242␈α∂heuristic␈α∂rules␈α∂which␈α⊂AM␈α∂was
␈↓ α,␈↓originally␈α∪given.␈α∩ A␈α∪complete␈α∩listing␈α∪of␈α∪those␈α∩rules␈α∪is␈α∩found␈α∪in␈α∩Appendix␈α∪2;␈α∪the␈α∩rule
␈↓ α,␈↓numbers below refer to the numbering given in that appendix.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.6.1. Heuristics Grouped by the Knowledge They Embody␈↓)αβ␈↓

␈↓ α,␈↓Many␈α
heuristics␈α
embody␈αthe␈α
belief␈α
that␈αmathematics␈α
is␈α
an␈αempirical␈α
inquiry.␈α
 That␈α
is,␈αone
␈↓ α,␈↓approach␈αto␈αdiscovery␈αis␈αsimply␈αto␈αperform␈αexperiments,␈αobserve␈αthe␈αresults,␈αthereby␈αgather
␈↓ α,␈↓statistically␈αsigni≡cant␈αamounts␈αof␈αdata,␈αinduce␈αfrom␈αthat␈αdata␈αsome␈αnew␈αconjectures␈αor␈αnew
␈↓ α,␈↓concepts␈α
worth␈αisolating,␈α
and␈αthen␈α
repeat␈αthis␈α
whole␈α
process␈αagain.␈α
 Some␈αof␈α
the␈αrules␈α
which
␈↓ α,␈↓capture␈α∪this␈α∩spirit␈α∪are␈α∩numbers␈α∪21,␈α∩43-57,␈α∪91,␈α∩136-139,␈α∪146-148,␈α∩153-154,␈α∪212-216,␈α∩225,
␈↓ α,␈↓and␈α⊃241.␈α⊃ As␈α⊃one␈α∩might␈α⊃expect,␈α⊃most␈α⊃of␈α⊃these␈α∩are␈α⊃"Suggest"␈α⊃type␈α⊃rules.␈α∩ They␈α⊃indicate
␈↓ α,␈↓plausible␈αmoves␈αfor␈αAM␈αto␈αmake,␈αpromising␈αnew␈αtasks␈αto␈αtry,␈αnew␈αconcepts␈αworth␈αstudying.
␈↓ α,␈↓Almost␈αall␈αthe␈αrest␈αare␈α"Fillin"␈αtype␈αrules,␈αproviding␈αempirical␈αmethods␈αto␈α≡nd␈αentries␈αfor␈αa
␈↓ α,␈↓speci≡ed facet.

␈↓ α,␈↓Another␈α∂large␈α∂set␈α∂of␈α∂heuristics␈α∂is␈α∂used␈α∂to␈α∂embody␈α∂¬␈α∂or␈α∂to␈α∂modify␈α∂¬␈α∂what␈α∂can␈α∂be␈α∞called
␈↓ α,␈↓"focus␈α∞of␈α∞attention".␈α∂When␈α∞should␈α∞AM␈α∞keep␈α∂on␈α∞the␈α∞same␈α∞track,␈α∂and␈α∞when␈α∞not?␈α∂The␈α∞≡rst
␈↓ α,␈↓rules␈α⊂expressing␈α⊂varying␈α⊂nuances␈α⊂of␈α⊂this␈α⊂idea␈α⊂are␈α⊂numbers␈α⊂1-5.␈α⊂The␈α⊂last␈α⊂such␈α⊂rules␈α⊂are
␈↓ α,␈↓numbers␈α
209-216.␈α
 Some␈α
of␈α
these␈α
rules␈α
are␈α
akin␈α
to␈α
goal-setting␈α
mechanisms␈α
(e.g.,␈α
rule␈α
141).
␈↓ α,␈↓In␈αaddition,␈αmany␈αof␈αthe␈α"Interest"␈αtype␈αrules␈αhave␈αsome␈αrelation␈αto␈αkeeping␈αAM␈αinterested
␈↓ α,␈↓in␈α
recently-chosen␈α
concepts␈α
(or:␈α
in␈α
concepts␈αrelated␈α
to␈α
them,␈α
e.g.␈α
by␈α
Analogy,␈α
by␈αGenl/Spec,
␈↓ α,␈↓by Isa/Exs,...).

␈↓ α,␈↓The␈α
remaining␈α
"Interest"␈α
rules␈α
are␈α
generally␈α
some␈α
re-echoing␈α
of␈α
the␈α
following␈α
notion:␈α
X␈αis
␈↓ α,␈↓interesting␈α
if␈α
F(X)␈α∞has␈α
an␈α
unexpected␈α∞(interesting)␈α
value.␈α
 For␈α∞example,␈α
in␈α
rule␈α∞26,␈α
"F(X)"
␈↓ α,␈↓is␈α∪just␈α∩"Generalizations␈α∪of␈α∩X".␈α∪ In␈α∪slightly␈α∩more␈α∪detail,␈α∩the␈α∪principle␈α∪characteristics␈α∩of
␈↓ α,␈↓interestingness are:
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε57␈↓-

␈↓ α,␈↓␈↓ αl␈↓π#␈↓ symmetry (e.g., in an expanding analogy)
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ coincidence (e.g., in a concept being re-discovered often)
␈↓ α,␈↓␈↓ αl␈↓π#␈↓␈α∩appropriateness␈α∩(e.g.,␈α∩in␈α∩choosing␈α∩an␈α∪operation␈α∩H␈α∩so␈α∩that␈α∩G␈↓εo␈↓H␈α∩will␈α∪have␈α∩nicer
␈↓ α,␈↓␈↓ β≤Domain/Range characteristics than G itself did)
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ recency (see the previous paragraph on focus of attention)
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ individuality (e.g., the ≡rst entity observed which satis≡es some property)
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ usefulness (e.g., there are many conjectures involving it)
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ association (i.e., the given concept is related to an interesting one)



␈↓ α,␈↓One␈α∪group␈α∪of␈α∪heuristic␈α∪rules␈α∩embeds␈α∪syntactic␈α∪tricks␈α∪for␈α∪generalizing␈α∪de≡nitions␈α∩(Lisp
␈↓ α,␈↓predicates),␈α⊂specializing␈α⊂them,␈α⊃instantiating␈α⊂them,␈α⊂symbolically␈α⊂evaluating␈α⊃them,␈α⊂inverting
␈↓ α,␈↓them,␈α⊂rudimentarily␈α⊂analyzing␈α⊂them,␈α⊂etc.␈α⊂For␈α⊃example,␈α⊂see␈α⊂rules␈α⊂31␈α⊂and␈α⊂89.␈α⊃ Some␈α⊂rules
␈↓ α,␈↓serve␈α∞other␈α∞syntactic␈α∞functions,␈α∞like␈α∞ensuring␈α∞that␈α∞various␈α∞limits␈α∞aren't␈α∞exceeded␈α∂(e.g.,␈α∞rule
␈↓ α,␈↓15),␈α∂that␈α∂the␈α∂format␈α∂for␈α⊂each␈α∂facet␈α∂is␈α∂adhered␈α∂to␈α∂(e.g.,␈α⊂rule␈α∂16),␈α∂that␈α∂the␈α∂entries␈α⊂on␈α∂each
␈↓ α,␈↓facet␈α∂are␈α∂used␈α∂as␈α∂they␈α∂are␈α∂meant␈α∂to␈α∂be␈α∂(e.g.,␈α∂rules␈α∂9␈α∂and␈α∂59),␈α∂etc.␈α∂ Many␈α∂of␈α⊂the␈α∂"Check"
␈↓ α,␈↓type heuristics fall into this category.

␈↓ α,␈↓Finally,␈α
AM␈α∞possesses␈α
a␈α∞mass␈α
of␈α
miscellaneous␈α∞rules␈α
which␈α∞evade␈α
categorization.␈α∞ See,␈α
e.g.,
␈↓ α,␈↓rules␈α185␈α
and␈α236.␈αThese␈α
range␈αfrom␈αgenuine␈α
math␈αheuristics␈α(rules␈α
which␈αlead␈αto␈α
discovery
␈↓ α,␈↓frequently) to simple data management hacks.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&4.6.2. Heuristics Grouped by How Speci≡c They Are␈↓)αβ␈↓

␈↓ α,␈↓Another␈α⊂dimension␈α∂of␈α⊂distribution␈α∂of␈α⊂heuristics,␈α∂aside␈α⊂from␈α∂the␈α⊂above␈α∂␈↓βfunctional␈↓␈α⊂one,␈α∂is
␈↓ α,␈↓simply␈α⊃that␈α⊃of␈α∩how␈α⊃high␈α⊃up␈α⊃in␈α∩the␈α⊃Genl/Spec␈α⊃tree␈α⊃they␈α∩are␈α⊃located.␈α⊃ The␈α∩table␈α⊃below
␈↓ α,␈↓summarizes how the rules were distributed in that tree:

␈↓ α,␈↓π⊂ααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓α␈↓π}␈↓ε␈↓&LEVEL␈↓)αβ␈↓ αl␈↓ ∧¬␈↓&e.g.␈↓)αβ ␈↓ ∧,␈↓ ∧S␈↓&# Con's␈↓)αβ␈↓ ¬≤␈↓ ¬2␈↓&# w/Heur␈↓)αβ␈↓ ε␈↓ ε0␈↓&# Heurs␈↓)αβ␈↓ ε|␈↓ π0␈↓&Avg␈↓)αβ␈↓π ␈↓ε ␈↓ πl ␈↓&Avg w/Heur␈↓)αβ␈↓π ␈↓ε␈↓&# Fillin␈↓)αβ␈↓ 	,␈↓ 	H␈↓&# Sugg␈↓)αβ␈↓ 
␈↓ 
∨␈↓&# Check␈↓)αβ␈↓ 
l␈↓ ↔␈↓&# Int␈↓)αβ ␈↓π}␈↓α 
␈↓"␈↓ α,␈↓α␈↓π}␈↓α ␈↓ α\0␈↓ αl␈↓ β=Anything␈↓ ∧,␈↓ ¬1␈↓ ¬≤␈↓ ¬|1␈↓ ε␈↓ ε\10␈↓ ε|␈↓ π610.0␈↓ πl␈↓ λ⊗10.0␈↓ λL␈↓ 	≤0␈↓ 	,␈↓ 	|5␈↓ 
␈↓ 
\0␈↓ 
l␈↓ 25 ␈↓π}␈↓α 
␈↓"␈↓ α,␈↓α␈↓π}␈↓α ␈↓ α\1␈↓ αl␈↓ β	Any-Concept␈↓ ∧,␈↓ ¬1␈↓ ¬≤␈↓ ¬|1␈↓ ε␈↓ εL110␈↓ ε|␈↓ π&110.0␈↓ πl␈↓ λε110.0␈↓ λL␈↓ 	39␈↓ 	,␈↓ 	l30␈↓ 
␈↓ 
L20␈↓ 
l␈↓ "21 ␈↓π}␈↓α 
␈↓"␈↓ α,␈↓α␈↓π}␈↓α ␈↓ α\2␈↓ αl␈↓ β↑Active␈↓ ∧,␈↓ ¬2␈↓ ¬≤␈↓ ¬|2␈↓ ε␈↓ ε\24␈↓ ε|␈↓ π612.0␈↓ πl␈↓ λ⊗12.0␈↓ λL␈↓ 	≤7␈↓ 	,␈↓ 	l10␈↓ 
␈↓ 
\4␈↓ 
l␈↓ 23 ␈↓π}␈↓α 
␈↓"␈↓ α,␈↓α␈↓π}␈↓α ␈↓ α\3␈↓ αl␈↓ β3Operation␈↓ ∧,␈↓ ¬6␈↓ ¬≤␈↓ ¬|3␈↓ ε␈↓ ε\31␈↓ ε|␈↓ πF5.2␈↓ πl␈↓ λ⊗10.3␈↓ λL␈↓ 	11␈↓ 	,␈↓ 	|3␈↓ 
␈↓ 
\3␈↓ 
l␈↓ "14 ␈↓π}␈↓α 
␈↓"␈↓ α,␈↓α␈↓π}␈↓α ␈↓ αN≥4␈↓ αl␈↓ βgUnion␈↓ ∧,␈↓ ∧l100␈↓ ¬≤␈↓ ¬l11␈↓ ε␈↓ ε\63␈↓ ε|␈↓ πF0.6␈↓ πl␈↓ λ&5.7␈↓ λL␈↓ 	26␈↓ 	,␈↓ 	l15␈↓ 
␈↓ 
\8␈↓ 
l␈↓ "16 ␈↓π}␈↓α 
␈↓"␈↓ α,␈↓π%ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$ 



␈↓ α,␈↓Here is a key to the column headings:
␈↓ α,␈↓␈↓ αlLEVEL: How far down the Genl/Spec tree of concepts we are looking.
␈↓ α,␈↓␈↓ αle.g.: A sample concept at that level.
␈↓ α,␈↓␈↓ αl# Con's: The total number of concepts at that level.
␈↓ α,␈↓␈↓ αl# w/Heur: How many of them have ␈↓βsome␈↓ heuristics.
␈↓ α,␈↓␈↓ αl# Heurs: The total number of heuristics attached to concepts at that level.
␈↓ α,␈↓␈↓ αlAvg:␈α
(#␈αHeurs)␈α
/␈α(#␈α
Concepts);␈αi.e.,␈α
the␈αmean␈α
number␈αof␈α
heuristics␈αper␈α
concept,␈α
at␈αthat
␈↓ α,␈↓␈↓ β,level.
␈↓ α,␈↓␈↓ αlAvg w/Heur: (# Heurs) / (# w. Heurs)
␈↓ α,␈↓␈↓ αl# Fillin: Total number of "Fillin" type heuristics at that level.
␈↓ α,␈↓␈↓εChapter 4␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε58␈↓-

␈↓ α,␈↓␈↓ αl# Sugg: Total number of "Suggest" type heuristics at that level.
␈↓ α,␈↓␈↓ αl# Check: Total number of "Check" type heuristics at that level.
␈↓ α,␈↓␈↓ αl# Int: Total number of "Interestingness" type heuristics at that level.



␈↓ α,␈↓The␈α
heuristic␈αrules␈α
are␈αseen␈α
␈↓βnot␈↓␈αto␈α
be␈αdistributed␈α
uniformly,␈αhomogeneously␈α
among␈α
all␈αthe
␈↓ α,␈↓initial␈αconcepts.␈αThe␈αextent␈α
of␈αthis␈αskewing␈αwas␈αnot␈α
realized␈αby␈αthe␈αauthor␈αuntil␈α
the␈αabove
␈↓ α,␈↓table␈α∂was␈α∂constructed.␈α∞ A␈α∂surprising␈α∂proportion␈α∞of␈α∂rules␈α∂are␈α∞attached␈α∂to␈α∂the␈α∂very␈α∞general
␈↓ α,␈↓concepts.␈α⊃ The␈α⊃top␈α⊃10%␈α∩of␈α⊃the␈α⊃concepts␈α⊃contain␈α⊃73%␈α∩of␈α⊃all␈α⊃the␈α⊃heuristics.␈α∩ One␈α⊃notable
␈↓ α,␈↓exception␈αis␈αthe␈α"Interest"␈αtype␈αheuristics:␈αthey␈αseem␈αmore␈αevenly␈αdistributed␈αthroughout␈αthe
␈↓ α,␈↓tree␈α∪of␈α∪initial␈α∀concepts.␈α∪ This␈α∪tends␈α∀to␈α∪suggest␈α∪that␈α∀future␈α∪work␈α∪on␈α∀providing␈α∪"meta-
␈↓ α,␈↓heuristics"␈α
should␈α
concentrate␈α
on␈α
how␈α
to␈α
automatically␈α
synthesize␈α
those␈α
Interest␈α
heuristics␈α
for
␈↓ α,␈↓newly-created concepts.
␈↓ α,␈↓␈↓ ,-␈↓ε59␈↓-




␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬⊃␈↓∧Chapter 5.   AM's Concepts␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓




␈↓ α,␈↓␈↓ ∧␈␈↓∧␈↓&5.1. Motivation and Overview␈↓)αβ␈↓

␈↓ α,␈↓Each␈α
concept␈α
consists␈α
merely␈α
of␈α
a␈α
bundle␈α
of␈α
facets.␈α
 The␈α
facets␈α
represent␈α
the␈α
di≥erent␈α
aspects
␈↓ α,␈↓of each concept, the kinds of questions one might want to ask about the concept:

␈↓ α,␈↓␈↓ αlHow valuable is this concept?
␈↓ α,␈↓␈↓ αlWhat is its de≡nition?
␈↓ α,␈↓␈↓ αlIf it's an operation, what is legally in its domain?
␈↓ α,␈↓␈↓ αlWhat are some generalizations of this concept?
␈↓ α,␈↓␈↓ αlHow can you separate the interesting instances of this concept from the dull ones?
␈↓ α,␈↓␈↓ αletc.

␈↓ α,␈↓Since␈α
each␈α
concept␈α
represents␈α
a␈α
mathematical␈α∞entity,␈α
the␈α
kinds␈α
of␈α
questions␈α
one␈α∞might␈α
ask
␈↓ α,␈↓are␈α
fairly␈αconstant␈α
from␈α
concept␈αto␈α
concept.␈α This␈α
set␈α
of␈αquestions␈α
might␈α
change␈αsomewhat
␈↓ α,␈↓for a new domain of concept.

␈↓ α,␈↓One␈α"natural"␈αrepresentation␈α
for␈αa␈αconcept␈α
in␈αLISP␈αis␈α
as␈αa␈αset␈α
of␈αattribute/value␈αpairs.␈α
That
␈↓ α,␈↓is,␈αeach␈αconcept␈αis␈αmaintained␈αas␈αan␈αatom␈αwith␈αa␈αproperty␈αlist.␈αThe␈αnames␈αof␈αthe␈αproperties
␈↓ α,␈↓(Worth,␈α
De≡nitions,␈αDomain/Range,␈α
Generalizations,␈α
Interestingness,␈αetc.)␈α
correspond␈α
to␈αthe
␈↓ α,␈↓questions␈α
above,␈α
and␈α
the␈α
value␈α
stored␈α
under␈α
property␈α
F␈α
of␈α
atom␈α
C␈α
is␈α
simply␈α
the␈α∞value␈α
of
␈↓ α,␈↓the␈αF-facet␈αof␈αthe␈αC-concept.␈α This␈αvalue␈αcan␈αalso␈αbe␈αviewed␈αas␈αthe␈αanswer␈αwhich␈αexpert␈αC
␈↓ α,␈↓would give, if asked question F. Or, it can be viewed as the contents of slot F of frame C.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.1.1. A Glimpse of a Typical Concept␈↓)αβ␈↓

␈↓ α,␈↓As␈αan␈αexample,␈αhere␈αis␈α
a␈αstylized␈αrendition␈αof␈αthe␈αSETS␈α
concept.␈αThis␈αis␈αa␈αconcept␈αwhich␈α
is
␈↓ α,␈↓meant␈αto␈αcorrespond␈αto␈αthe␈αnotion␈αof␈αa␈αset␈αof␈αelements.␈α For␈αexample,␈αaccording␈αto␈αthe␈αbox
␈↓ α,␈↓below,␈α⊂"Singleton"␈α⊂is␈α⊂one␈α⊂entry␈α⊂on␈α⊂the␈α⊂Specializations␈α⊂facet␈α⊂of␈α⊂Sets␈α⊂¬␈α⊂i.e.,␈α⊃singletons␈α⊂are
␈↓ α,␈↓speci≡c kinds of sets..
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε60␈↓-

␈↓"␈↓ α,␈↓π␈↓ α\⊂ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ≤⊃
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Name(s): Set, Class, Collection ␈↓π                                       ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Definitions: ␈↓π                                                      ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Recursive: λ (S) [S={} or Set.Definition (Remove(Any-member(S),S))] ␈↓π     ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Recursive quick: λ (S) [S={} or Set.Definition (CDR(S))] ␈↓π                ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Quick: λ (S) [Match S with {...} ] ␈↓π                                 ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Specializations: Empty-set, Nonempty-set, Set-of-structures, Singleton ␈↓π         ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Generalizations: Unordered-Structure, No-multiple-elements-Structure ␈↓π         ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Examples: ␈↓π                                                       ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Typical: {{}},  {A},  {A,B},  {3} ␈↓π                                  ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Barely: {},    {A, B, {C, { { { A, C, (3,3,3,9), <4,1,A,{B},A>}}}}} ␈↓π         ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Not-quite: {A,A}, (), {B,A} ␈↓π                                      ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Foible: <4,1,A,1> ␈↓π                                             ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Conjec's: All unordered-structures are sets. ␈↓π                             ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Intu's: ␈↓π                                                          ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Geometric: Venn diagram.  {See [Venn 89], or [Skemp 71].} ␈↓π            ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Analogs: bag, list, oset ␈↓π                                             ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Worth: 600 ␈↓π                                                      ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    View: ␈↓π                                                          ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Predicate: λ (P) {xεDomain(P) | P(x)} ␈↓π                              ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬            Structure: λ (S) Enclose-in-braces(Sort(Remove-multiple-elements(S))) ␈↓π   ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    Suggest: If P is an interesting predicate over X, consider {xεX | P(x)}. ␈↓π          ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    In-domain-of: Union, Intersection, Set-difference, Set-equality, Subset, Member ␈↓π  ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\} ␈↓¬    In-range-of: Union, Intersection, Set-difference, Satisfying ␈↓π                  ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ α\%ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ≤$

␈↓ α,␈↓To␈αdecipher␈αthe␈αDe≡nitions␈αfacet,␈αthere␈αare␈αa␈αfew␈αthings␈αyou␈αmust␈αknow.␈α An␈αexpression␈αof
␈↓ α,␈↓the␈α
form␈α
"(λ␈α(x)␈α
E)"␈α
is␈αcalled␈α
a␈α
Lambda␈αexpression␈α
after␈α
Church␈↓	1␈↓,␈αand␈α
may␈α
be␈αconsidered
␈↓ α,␈↓an␈α∞executable␈α∞procedure.␈α∞ it␈α∞accepts␈α∞one␈α∞argument,␈α
binds␈α∞the␈α∞variable␈α∞"x"␈α∞to␈α∞the␈α∞value␈α
of
␈↓ α,␈↓that␈α∞argument,␈α∞and␈α∂then␈α∞evaluates␈α∞"E"␈α∞(which␈α∂is␈α∞probably␈α∞some␈α∞expression␈α∂involving␈α∞the
␈↓ α,␈↓variable␈α
x).␈α For␈α
example,␈α"(λ␈α
(x)␈α(x+5))"␈α
is␈αa␈α
function␈αwhich␈α
adds␈α5␈α
to␈αany␈α
number;␈αif␈α
given
␈↓ α,␈↓the argument 3, this lambda expression will return the value 8.

␈↓ α,␈↓The␈α∀second␈α∀thing␈α∀you␈α∀must␈α∀know␈α∀is␈α∪that␈α∀facet␈α∀F␈α∀of␈α∀concept␈α∀C␈α∀will␈α∀occasionally␈α∪be
␈↓ α,␈↓abbreviated␈α∞as␈α∞C.F.␈α∞In␈α∞those␈α∞cases␈α∞where␈α∂F␈α∞is␈α∞"executable",␈α∞the␈α∞notation␈α∞C.F␈α∞will␈α∂refer␈α∞to
␈↓ α,␈↓applying␈αthe␈αcorresponding␈αfunction.␈α
 So␈αthe␈α≡rst␈αentry␈α
in␈αthe␈αDe≡nitions␈αfacet␈α
is␈αrecursive
␈↓ α,␈↓because␈α∂it␈α∂contains␈α⊂an␈α∂embedded␈α∂call␈α⊂on␈α∂the␈α∂function␈α⊂Set.De≡nition.␈α∂ Notice␈α∂that␈α⊂we␈α∂are
␈↓ α,␈↓implying that the ␈↓βname␈↓ of that lambda expression itself is "Set.De≡nition".

␈↓ α,␈↓There␈α
are␈α
some␈α
unusual␈α
implications␈α
of␈αthis:␈α
since␈α
there␈α
are␈α
three␈α
separate␈α
but␈αequivalent
␈↓ α,␈↓de≡nitions,␈αAM␈αmay␈αchoose␈αwhichever␈αone␈αit␈αwants␈αwhen␈αit␈αrecurs.␈αAM␈αcan␈αchoose␈αone␈αvia
␈↓ α,␈↓a␈αrandom␈αselection␈αscheme,␈αor␈αalways␈αtry␈αto␈αrecur␈αinto␈αthe␈αsame␈αde≡nition␈αas␈αit␈αwas␈αjust␈αin,
␈↓ α,␈↓or perhaps suit its choice to the form of the argument at the moment.

␈↓ α,␈↓For␈α∞example,␈α∞one␈α∞de≡nition␈α∞might␈α∞be␈α∞great␈α∞for␈α
arguments␈α∞of␈α∞size␈α∞10␈α∞or␈α∞less,␈α∞but␈α∞slow␈α
for

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε Before and during Church, it's called a function. See [Church 41].
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε61␈↓-

␈↓ α,␈↓bigger␈α⊂ones,␈α⊂and␈α∂another␈α⊂de≡nition␈α⊂might␈α∂be␈α⊂mediocre␈α⊂for␈α∂all␈α⊂size␈α⊂arguments;␈α⊂then␈α∂AM
␈↓ α,␈↓should␈α
use␈αthe␈α
mediocre␈α
de≡nition␈αover␈α
and␈αover␈α
again,␈α
until␈αthe␈α
argument␈α
becomes␈αsmall
␈↓ α,␈↓enough,␈αand␈αfrom␈αthen␈αon␈αrecur␈αonly␈αinto␈αthe␈αfast␈αde≡nition.␈α Although␈αAM␈αembodies␈αthis
␈↓ α,␈↓"smart"␈αscheme,␈α
the␈αlittle␈α
comments␈αnecessary␈α
to␈αsee␈αhow␈α
it␈αdoes␈α
so␈αhave␈α
be␈αexcised␈αfrom␈α
the
␈↓ α,␈↓version shown above in the box.  This will be explained later in this chapter, on page 80.

␈↓ α,␈↓All␈αconcepts␈αpossess␈αexecutable␈αde≡nitions,␈αthough␈αnot␈αnecessarily␈αe≥ective␈αones.␈α
They␈αeach
␈↓ α,␈↓have␈α∞a␈α∞LISP␈α
predicate,␈α∞but␈α∞that␈α∞predicate␈α
is␈α∞not␈α∞guaranteed␈α
to␈α∞terminate.␈α∞Notice␈α∞that␈α
the
␈↓ α,␈↓de≡nitions for Sets are all de≡nitions of ≡nite sets.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.1.2. The main constraint: Fixed set of facets␈↓)αβ␈↓

␈↓ α,␈↓One␈αimportant␈αconstraint␈αon␈αthe␈α
representation␈αis␈αthat␈αthe␈αset␈α
of␈αfacets␈αbe␈α≡xed␈αfor␈α
all␈αthe
␈↓ α,␈↓concepts.␈α
An␈α∞additional␈α
constraint␈α
is␈α∞that␈α
this␈α∞set␈α
of␈α
facets␈α∞not␈α
grow,␈α
that␈α∞it␈α
be␈α∞≡xed␈α
once
␈↓ α,␈↓and␈αfor␈αall.␈α So␈αthere␈αis␈αone␈α≡xed,␈αuniversal␈αlist␈αof␈αtwo␈αdozen␈αtypes␈αof␈αfacets.␈α Any␈αfacet␈αof
␈↓ α,␈↓any␈α∩concept␈α∪␈↓βmust␈↓␈α∩have␈α∩one␈α∪of␈α∩those␈α∪standard␈α∩names.␈α∩ All␈α∪concepts␈α∩which␈α∪have␈α∩some
␈↓ α,␈↓examples␈α∩must␈α∩store␈α∪them␈α∩as␈α∩entries␈α∪on␈α∩a␈α∩facet␈α∩called␈α∪Examples;␈α∩they␈α∩can't␈α∪call␈α∩them
␈↓ α,␈↓Instances,␈αor␈αCases,␈αor␈αG00037's.␈α This␈α
constraint␈αis␈αknown␈αas␈αthe␈α"␈↓¬Beings␈↓␈α
constraint"␈↓	2␈↓,␈αand
␈↓ α,␈↓has three important consequences:

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α
OUTLINE:␈α
First,␈α
it␈α
provides␈α
a␈α
nice,␈α
distributed,␈α
universal␈α
framework␈α
on␈α
which␈α
to
␈↓ α,␈↓␈↓ β≤display␈αall␈αthat␈αis␈αknown␈αabout␈αa␈αgiven␈αconcept.␈α AM␈αcan␈αinstantly␈αtell␈αwhat␈αfacets
␈↓ α,␈↓␈↓ β≤are␈α
not␈αyet␈α
≡lled␈αin␈α
for␈α
any␈αgiven␈α
concept,␈αand␈α
this␈α
will␈αin␈α
turn␈αsuggest␈α
new␈αtasks␈α
to
␈↓ α,␈↓␈↓ β≤perform.␈α
 In␈α
other␈α
words,␈α
this␈α
constraint␈αhelps␈α
de≡ne␈α
the␈α
"space"␈α
which␈α
AM␈αmust
␈↓ α,␈↓␈↓ β≤explore,␈α
and␈α
makes␈α
it␈α
obvious␈α
what␈α
parts␈α
of␈α
each␈α
concept␈α
have␈α
and␈α
have␈α
not␈α
yet
␈↓ α,␈↓␈↓ β≤been investigated.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α∞STRUCTURE:␈α∞The␈α
constraint␈α∞speci≡es␈α∞that␈α∞there␈α
be␈α∞a␈α∞␈↓βset␈↓␈α
of␈α∞facets,␈α∞not␈α∞just␈α
one.
␈↓ α,␈↓␈↓ β≤This␈αset␈αwas␈αmade␈αlarge␈αenough␈αthat␈αall␈αthe␈αe≠ciency␈αadvantages␈αof␈αa␈α"structured"
␈↓ α,␈↓␈↓ β≤representation␈α∃are␈α⊗preserved␈α∃(unlike␈α∃totally␈α⊗uniform␈α∃representations,␈α⊗e.g.␈α∃pure
␈↓ α,␈↓␈↓ β≤production systems with simple memories as data structures, or predicate calculus).
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈αUNIFORMITY:␈α
When␈αAM␈αwishes␈α
a␈αpiece␈αof␈α
information,␈αit␈αmust␈α
ask␈αa␈α
concept␈αa
␈↓ α,␈↓␈↓ β≤"question"␈α∂¬␈α∂i.e.,␈α∞mention␈α∂a␈α∂particular␈α∞facet␈α∂by␈α∂name.␈α∞ The␈α∂bene≡t␈α∂of␈α∂the␈α∞␈↓¬Beings␈↓
␈↓ α,␈↓␈↓ β≤constraint␈α∞is␈α∞that␈α∞there␈α∞is␈α∞a␈α∞≡xed,␈α∞small␈α∞repertoire␈α∞of␈α∞questions␈α∞it␈α∞may␈α∞ask,␈α
hence
␈↓ α,␈↓␈↓ β≤there␈αwill␈α
be␈αno␈α
long␈αsearching,␈α
no␈αmisunderstandings.␈α
 This␈αis␈α
the␈αsame␈α
advantage
␈↓ α,␈↓␈↓ β≤that always accrues when everyone uses a common language.

␈↓ α,␈↓We␈αshall␈αillustrate␈αthe␈αlast␈αtwo␈αadvantages␈αby␈αusing␈αthe␈αSets␈αconcept␈αpictured␈αin␈αthe␈αbox␈αa
␈↓ α,␈↓couple␈αpages␈α
ago.␈α How␈α
does␈αAM␈α
handle␈αa␈αtask␈α
of␈αthis␈α
form:␈α"␈↓¬Check␈α
examples␈αof␈αSets␈↓"?␈α
 AM
␈↓ α,␈↓accesses␈αthe␈αexamples␈αfacet␈αof␈αthe␈αSets␈αconcept,␈αand␈αobtains␈αa␈αbunch␈αof␈αitems␈αwhich␈αare␈αall
␈↓ α,␈↓probably␈α∞sets.␈α∞ If␈α∞any␈α∞isn't␈α
a␈α∞set,␈α∞AM␈α∞would␈α∞like␈α∞to␈α
make␈α∞it␈α∞one,␈α∞if␈α∞that␈α∞involves␈α
nothing
␈↓ α,␈↓di≠cult.␈α⊃AM␈α⊃locates␈α⊃all␈α⊃the␈α⊃generalizations␈α∩of␈α⊃Sets␈↓	3␈↓,␈α⊃and␈α⊃comes␈α⊃up␈α⊃with␈α⊃the␈α∩list␈α⊃<Sets,
␈↓ α,␈↓Unordered-Structures,␈α_No-multiple-elements-Structures,␈α_Structures,␈α→Objects,␈α_Any-concept,
␈↓ α,␈↓Anything>.␈α
Next,␈αthe␈α
"Check"␈αfacet␈α
of␈αeach␈α
of␈αthese␈α
is␈αexamined,␈α
and␈αall␈α
its␈α
heuristics␈αare

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	2␈↓ε See [Lenat 75b]. Historically, each concept module was called a "BEING".
␈↓ α,␈↓ε␈↓	3␈↓ε by "rippling" upward from Sets, in the Genl direction
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε62␈↓-

␈↓ α,␈↓collected.␈α∪ For␈α∪example,␈α∪the␈α∩Check␈α∪facet␈α∪of␈α∪the␈α∪No-multiple-elements-Structures␈α∩concept
␈↓ α,␈↓contains␈α
the␈αfollowing␈α
entry:␈α"Eliminate␈α
multiple␈α
occurrences␈αof␈α
each␈αelement"␈α
(of␈αcourse␈α
this
␈↓ α,␈↓is␈αpresent␈αnot␈αas␈αan␈αEnglish␈αsentence␈αbut␈αrather␈αas␈αa␈αlittle␈αLISP␈αfunction).␈α So␈αeven␈αthough
␈↓ α,␈↓Sets␈α∞has␈α∞no␈α∞entries␈α
for␈α∞its␈α∞Check␈α∞facet,␈α
several␈α∞little␈α∞functions␈α∞will␈α
be␈α∞gathered␈α∞up␈α∞by␈α
the
␈↓ α,␈↓rippling␈α∞process.␈α∞Each␈α∞potential␈α∞set␈α∞would␈α∂be␈α∞subjected␈α∞to␈α∞all␈α∞those␈α∞checks,␈α∞and␈α∂might␈α∞be
␈↓ α,␈↓modi≡ed or discarded as a result.

␈↓ α,␈↓There␈αis␈αenough␈α"structure"␈αaround␈αto␈αkeep␈αthe␈αheuristic␈αrules␈αrelevant␈αto␈αthis␈αtask␈αisolated
␈↓ α,␈↓from␈α∂very␈α∂irrelevant␈α∂rules,␈α∂and␈α∂there␈α∞is␈α∂enough␈α∂"uniformity"␈α∂to␈α∂make␈α∂≡nding␈α∂those␈α∞rules
␈↓ α,␈↓very easy.

␈↓ α,␈↓The␈αsame␈α
rippling␈αwould␈α
be␈αdone␈αto␈α
≡nd␈αpredicates␈α
which␈αtell␈αwhether␈α
a␈αset␈α
is␈αinteresting
␈↓ α,␈↓or␈α∞dull.␈α
 For␈α∞example,␈α
one␈α∞entry␈α
on␈α∞the␈α
Interestingness␈α∞facet␈α
of␈α∞the␈α
Structure␈α∞concept␈α
says
␈↓ α,␈↓that␈αa␈α
structure␈αis␈α
interesting␈αif␈αall␈α
pairs␈αof␈α
members␈αsatisfy␈αthe␈α
same␈αrare␈α
predicate␈αP(x,y)
␈↓ α,␈↓[for␈α∪any␈α∩such␈α∪P].␈α∩ So␈α∪a␈α∩set,␈α∪all␈α∩pairs␈α∪of␈α∩whose␈α∪members␈α∩satisfy␈α∪"Equality,"␈α∪would␈α∩be
␈↓ α,␈↓considered␈α∩interesting.␈α∩In␈α⊃fact,␈α∩every␈α∩Singleton␈α⊃is␈α∩an␈α∩interesting␈α⊃␈↓βStructure␈↓␈α∩for␈α∩just␈α⊃that
␈↓ α,␈↓reason.␈α A␈αsingleton␈αmight␈αbe␈αan␈αinteresting␈α
␈↓βAnything␈↓␈αbecause␈αit␈αtakes␈αonly␈αa␈αfew␈α
characters
␈↓ α,␈↓to type it out (thereby satisfying a criterion on Anything.Interest).

␈↓ α,␈↓To␈α
locate␈αall␈α
the␈αspecializations␈α
of␈α
Sets,␈αthe␈α
rippling␈αwould␈α
go␈α
in␈αthe␈α
opposite␈αdirection.␈α
 For
␈↓ α,␈↓example,␈αone␈αof␈αthe␈αentries␈αon␈αthe␈αSpecializations␈αfacet␈αof␈αSets␈αis␈αSet-of-structures;␈αone␈αif␈α␈↓βits␈↓
␈↓ α,␈↓Specialization␈α∂entries␈α∂is␈α∂Set-of-sets.␈α∂So␈α∂this␈α∂latter␈α∂concept␈α∂will␈α∂be␈α∂caught␈α∂in␈α∂the␈α∂net␈α∞when
␈↓ α,␈↓rippling away from Sets in the Specializations direction.

␈↓ α,␈↓If␈αAM␈α
wants␈αlots␈α
of␈αexamples␈α
of␈αsets,␈αit␈α
has␈αonly␈α
to␈αripple␈α
in␈αthe␈α
Specializations␈αdirection,
␈↓ α,␈↓gathering␈α
Examples␈α∞of␈α
each␈α∞concept␈α
it␈α
encounters.␈α∞ Examples␈α
of␈α∞Sets-of-sets␈α
(like␈α∞this␈α
one:
␈↓ α,␈↓{{A},{{C,D}}})␈αwill␈αbe␈αcaught␈αin␈αthis␈αway,␈αas␈αwill␈αexamples␈αof␈αSets-of-numbers␈α(like␈αthis␈αone:
␈↓ α,␈↓{1,4,5}), because two specializations of Sets are Sets-of-Sets and Sets-of-Numbers␈↓	4␈↓.

␈↓ α,␈↓In␈α∂addition␈α∂to␈α∂the␈α∂three␈α∂main␈α∂reasons␈α∂for␈α∂keeping␈α∂the␈α∂set␈α∂of␈α∂facets␈α∂the␈α∂same␈α∂for␈α⊂all␈α∂the
␈↓ α,␈↓concepts␈α
(see␈α
previous␈α
page),␈αwe␈α
claimed␈α
there␈α
were␈α
also␈αreasons␈α
for␈α
keeping␈α
that␈α
set␈α≡xed
␈↓ α,␈↓once␈αand␈αfor␈αall.␈α Why␈αnot␈αdynamically␈αenlarge␈αit?␈α To␈αadd␈αa␈αnew␈αfacet,␈αits␈αvalue␈αhas␈αto␈αbe
␈↓ α,␈↓≡lled␈αin␈αfor␈αlots␈αof␈αconcepts.␈α How␈αcould␈α
AM␈αdevelop␈αthe␈αhuge␈αbody␈αof␈αheuristics␈αneeded␈α
to
␈↓ α,␈↓guide␈α∞such␈α∞≡lling-in␈α∞and␈α∞checking␈α∞activities?␈α∞Also,␈α∞the␈α∞number␈α∞of␈α∞facets␈α∞is␈α∞small␈α∂to␈α∞begin
␈↓ α,␈↓with␈α∪because␈α∩people␈α∪don't␈α∪seem␈α∩to␈α∪use␈α∩more␈α∪than␈α∪a␈α∩few␈α∪tens␈α∩of␈α∪such␈α∪"properties"␈α∩in
␈↓ α,␈↓classifying␈α
knowledge␈α
about␈α
a␈α∞concept␈↓	5␈↓.␈α
 If␈α
the␈α
viability␈α
of␈α∞AM␈α
seemed␈α
to␈α
depend␈α∞on␈α
this
␈↓ α,␈↓ability,␈αI␈αwould␈αhave␈αworked␈αon␈αit.␈αAM␈αgot␈αalong␈α≡ne␈αwithout␈αbeing␈αable␈αto␈αenlarge␈αits␈αset
␈↓ α,␈↓of␈αfacets,␈αso␈αno␈αtime␈αwas␈αever␈αspent␈αon␈αthat␈αproblem.␈α I␈αleave␈αit␈αas␈αa␈αchallenging,␈αambitious
␈↓ α,␈↓"open research problem".


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.1.3. BEINGs Representation of Knowledge␈↓)αβ␈↓

␈↓ α,␈↓Before␈α
discussing␈αeach␈α
facet␈α
in␈αdetail,␈α
let's␈α
interject␈αa␈α
brief␈α
historic␈αdigression,␈α
to␈αexplain␈α
the
␈↓ α,␈↓origins of this modular representation scheme.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	4␈↓ε We are assuming that AM has run for some time, and already discovered Numbers, and already defined Sets-of-Numbers.
␈↓ α,␈↓ε␈↓	5␈↓ε␈α	This␈α	data␈α	is␈α
gathered␈α	from␈α	introspection␈α	by␈α	myself␈α
and␈α	a␈α	few␈α	others,␈α	and␈α
should␈α	probably␈α	be␈α	tested␈α	by␈α
performing␈α	some
␈↓ α,␈↓ε␈↓ βLpsychological experiments.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε63␈↓-

␈↓ α,␈↓The␈αideas␈αarose␈αin␈αan␈αautomatic␈αprogramming␈αcontext,␈αwhile␈αworking␈αout␈αa␈αsolution␈αto␈αthe
␈↓ α,␈↓problem␈α∀of␈α∀constructing␈α∪a␈α∀computer␈α∀system␈α∀capable␈α∪of␈α∀synthesizing␈α∀a␈α∀simple␈α∪concept-
␈↓ α,␈↓discrimination␈α⊃program␈α⊃(similar␈α⊃to␈α⊃[Winston␈α⊃70]).␈α⊃ The␈α⊃scenario␈α⊃envisioned␈α⊃was␈α⊃one␈α⊂of
␈↓ α,␈↓mutual␈α∞cooperation␈α∞among␈α
a␈α∞group␈α∞of␈α
a␈α∞hundred␈α∞or␈α
so␈α∞experts,␈α∞each␈α
a␈α∞specialist␈α∞in␈α
some
␈↓ α,␈↓minute␈α
detail␈α
of␈α
coding,␈α
concept␈α
formation,␈α
debugging,␈α
communicating,␈α
etc.␈α
 Each␈α
expert␈α
was
␈↓ α,␈↓modelled␈α∂by␈α∂one␈α∂module,␈α∂one␈α∂BEING.␈α∂Each␈α∂BEING␈α∂had␈α∂the␈α∂same␈α∂kinds␈α∂of␈α∂slots␈α∞(parts,
␈↓ α,␈↓facets),␈α
and␈αeach␈α
slot␈αwas␈α
interpreted␈αas␈α
a␈α␈↓βquestion␈↓␈α
which␈αthat␈α
BEING␈αcould␈α
answer.␈α The
␈↓ α,␈↓community␈α
of␈α
experts␈αcarried␈α
on␈α
a␈αround-table␈α
discussion␈α
of␈αa␈α
programming␈α
task␈αwhich␈α
was
␈↓ α,␈↓speci≡ed␈α⊗by␈α⊗a␈α⊗human␈α⊗user.␈α⊗ Eventually,␈α⊗by␈α⊗cooperating␈α⊗and␈α⊗answering␈α⊗each␈α⊗other's
␈↓ α,␈↓questions, they hammered out the program he desired.  See [Lenat 75b] for details.

␈↓ α,␈↓The␈α∃≡nal␈α∃system,␈α∃called␈α∃PUP6,␈α∀did␈α∃actually␈α∃synthesize␈α∃several␈α∃large␈α∃LISP␈α∀programs,
␈↓ α,␈↓including␈α∂many␈α∂variants␈α∂of␈α∞the␈α∂concept-learning␈α∂program.␈α∂ This␈α∞is␈α∂described␈α∂in␈α∂detail␈α∞in
␈↓ α,␈↓[Lenat␈α∩75a].␈α∪ Unfortunately,␈α∩PUP6␈α∪had␈α∩virtually␈α∪no␈α∩natural␈α∪language␈α∩ability␈α∪and␈α∩was
␈↓ α,␈↓therefore unusable by an untrained human. Its modal output was "␈↓βEh?␈↓".

␈↓ α,␈↓The␈αsearch␈α
for␈αa␈α
new␈αproblem␈α
domain␈αwhere␈α
this␈αcommunication␈α
di≠culty␈αwouldn't␈α
be␈αso
␈↓ α,␈↓severe led to consideration of elementary mathematics.

␈↓ α,␈↓The␈α∞other␈α∞main␈α∞defect␈α∂of␈α∞PUP6␈α∞was␈α∞its␈α∞narrowness,␈α∂the␈α∞small␈α∞range␈α∞of␈α∂`target'␈α∞programs
␈↓ α,␈↓which␈α∞could␈α∞be␈α∂synthesized.␈α∞PUP6␈α∞had␈α∂been␈α∞designed␈α∞with␈α∞just␈α∂one␈α∞target␈α∞in␈α∂mind,␈α∞and
␈↓ α,␈↓almost␈αall␈αit␈αcould␈αdo␈αwas␈αto␈αhit␈α
that␈αtarget.␈α The␈αsecond␈αconstraint␈αon␈αthe␈αnew␈αtask␈α
domain
␈↓ α,␈↓was␈αthen␈αone␈αof␈αhaving␈αa␈αnon-speci≡c␈αtarget,␈αa␈αvery␈αbroad␈αor␈αdi≥use␈αgoal.␈α This␈αpointed␈αto
␈↓ α,␈↓an automated researcher, rather than a problem-solver.

␈↓ α,␈↓These␈α
two␈α∞constraints␈α
then␈α∞were␈α
(i)␈α
elementary␈α∞math,␈α
because␈α∞of␈α
communication␈α∞ease,␈α
and
␈↓ α,␈↓(ii)␈α⊂self-guided␈α∂exploration,␈α⊂because␈α∂of␈α⊂the␈α∂danger␈α⊂of␈α∂too␈α⊂speci≡c␈α∂a␈α⊂goal.␈α⊂ Together,␈α∂they
␈↓ α,␈↓directed the author to an investigation which ultimately resulted in the AM project.


␈↓ α,␈↓␈↓ ε*␈↓∧␈↓&5.2. Facets␈↓)αβ␈↓

␈↓ α,␈↓How␈α⊂␈↓βis␈↓␈α∂each␈α⊂concept␈α⊂represented?␈α∂ Without␈α⊂claiming␈α∂that␈α⊂this␈α⊂is␈α∂the␈α⊂"best"␈α⊂or␈α∂preferred
␈↓ α,␈↓scheme, this section will treat in detail AM's representation of this knowledge.

␈↓ α,␈↓We␈α
have␈αseen␈α
that␈α
the␈αrepresentation␈α
of␈αa␈α
concept␈α
can␈αloosely␈α
be␈α
described␈αas␈α
a␈αcollection␈α
of
␈↓ α,␈↓facet/value␈α∂pairs,␈α∂where␈α∂the␈α∂facets␈α∞are␈α∂drawn␈α∂from␈α∂a␈α∂≡xed␈α∞set␈α∂of␈α∂about␈α∂25␈α∂total␈α∞possible
␈↓ α,␈↓facets.␈α Some␈α
facets␈αcome␈α
in␈αseveral␈α∨avors␈α
(e.g.,␈αthere␈α
are␈αreally␈αfour␈α
separate␈αfacets␈α
¬␈αnot
␈↓ α,␈↓just one ¬ which point to Examples: boundary, typical, just-barely-failing, foibles).
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε64␈↓-


␈↓ α,␈↓This␈αsection␈α
will␈αcover␈α
each␈αfacet␈αin␈α
turn.␈α Let's␈α
begin␈αby␈αlisting␈α
each␈αof␈α
them.␈αFor␈αa␈α
change
␈↓ α,␈↓of pace, we'll show a typical question that each one might answer about concept C:␈↓	6␈↓

␈↓ α,␈↓␈↓ αlName: What shall we call C when communicating with the user?
␈↓ α,␈↓␈↓ αlGeneralizations: Which other concepts have less restrictive de≡nitions than C?
␈↓ α,␈↓␈↓ αlSpecializations: Which concepts satisfy C's de≡nition plus some additional constraints?
␈↓ α,␈↓␈↓ αlExamples: What are some things that satisfy C's de≡nition?
␈↓ α,␈↓␈↓ αlIsa's: Which concepts' de≡nitions does C itself satisfy?␈↓	7␈↓
␈↓ α,␈↓␈↓ αlIn-domain-of: Which operations can be performed on C's?
␈↓ α,␈↓␈↓ αlIn-range-of: Which operations result in values which are C's?
␈↓ α,␈↓␈↓ αlViews: How can we view some other kind of entity as if it were a C?
␈↓ α,␈↓␈↓ αlIntu's: What is an abstract, analogic representation for C?
␈↓ α,␈↓␈↓ αlAnalogies: Are there similar (though formally unrelated) concepts?
␈↓ α,␈↓␈↓ αlConjec's: What are some potential theorems involving C?
␈↓ α,␈↓␈↓ αlDe≡nitions: How can we tell if x is an example of C?
␈↓ α,␈↓␈↓ αlAlgorithms: How can we execute the operation C on a given argument?
␈↓ α,␈↓␈↓ αlDomain/Range:␈α⊃What␈α⊃kinds␈α⊃of␈α⊃arguments␈α⊃can␈α⊃operation␈α⊃C␈α⊃be␈α⊃executed␈α∩on?␈α⊃What
␈↓ α,␈↓␈↓ β≤kinds of values will it return?
␈↓ α,␈↓␈↓ αlWorth: How valuable is C? (overall, aesthetic, utility, etc.)
␈↓ α,␈↓␈↓ αlInterestingness: What special features make a C especially interesting?

␈↓ α,␈↓In␈α⊂addition,␈α⊂each␈α⊂facet␈α⊂F␈α⊂of␈α⊂concept␈α⊃C␈α⊂can␈α⊂possess␈α⊂a␈α⊂few␈α⊂little␈α⊂subfacets␈α⊃which␈α⊂contain
␈↓ α,␈↓heuristics for dealing with that facet of C's:

␈↓ α,␈↓␈↓ αlC.F.Fillin:␈α
How␈α
can␈αentries␈α
on␈α
C.F␈α
be␈α≡lled␈α
in?␈α
 These␈α
heuristics␈αget␈α
called␈α
on␈αwhen␈α
the
␈↓ α,␈↓␈↓ β≤current task is ␈↓¬"Fillin facet F of concept X"␈↓, where X is a C.
␈↓ α,␈↓␈↓ αlC.F.Check: How can potential entries on C.F be checked and patched up?
␈↓ α,␈↓␈↓ αlC.F.Suggest:␈α⊂If␈α∂AM␈α⊂gets␈α∂bogged␈α⊂down,␈α⊂what␈α∂are␈α⊂some␈α∂new␈α⊂tasks␈α∂(related␈α⊂to␈α⊂C.F)␈α∂it
␈↓ α,␈↓␈↓ β≤might consider?


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.1. Generalizations/Specializations␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|Generalization␈α
makes␈α
possible␈α
conscious,␈αcontrolled,␈α
and␈α
accurate␈α
accomodation␈αof
␈↓ α,␈↓β␈↓ α|one's␈α∞existing␈α∂schemas,␈α∞not␈α∞only␈α∂in␈α∞response␈α∞to␈α∂the␈α∞demands␈α∞for␈α∂assimilation␈α∞of
␈↓ α,␈↓β␈↓ α|new␈α∞situations␈α∞as␈α∞they␈α∞are␈α∞encountered,␈α∞but␈α∞␈↓&ahead␈↓)αβ␈α∞of␈α∞these␈α∞demands,␈α∞seeking␈α
or
␈↓ α,␈↓β␈↓ α|creating new examples to ≡t the enlarged concept.

␈↓ α,␈↓¬␈↓ ε\-- Skemp




␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	6␈↓ε␈α
In␈α
this␈α
discussion,␈α
"C"␈α
represents␈α
the␈α	name␈α
of␈α
the␈α
concept␈α
whose␈α
facet␈α
is␈α	being␈α
discussed,␈α
and␈α
may␈α
be␈α
read␈α
"the␈α	given
␈↓ α,␈↓ε␈↓ βLconcept".
␈↓ α,␈↓ε␈↓	7␈↓ε Notice that C will therefore be an example of each member of Isa's(C).
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε65␈↓-

␈↓ α,␈↓We␈α
say␈α
concept␈α
A␈α"␈↓βis␈α
a␈α
generalization␈α
of␈↓"␈αconcept␈α
B␈α
i≥␈α
every␈α
example␈αof␈α
B␈α
is␈α
an␈αexample␈α
of
␈↓ α,␈↓A.␈αEquivalently,␈αthis␈αis␈αtrue␈αi≥␈αthe␈αde≡nition␈αof␈αB␈αcan␈αbe␈αphrased␈αas␈α"λ␈α(x)␈α[A.Defn(x)␈αand
␈↓ α,␈↓P(x)]";␈α∪that␈α∩is,␈α∪for␈α∪x␈α∩to␈α∪satisfy␈α∩B's␈α∪de≡nition,␈α∪it␈α∩must␈α∪satisfy␈α∩A's␈α∪de≡nition␈α∪plus␈α∩some
␈↓ α,␈↓additional␈α⊃predicate␈α⊃P.␈α⊃ The␈α⊃Generalizations␈α⊃facet␈α⊃of␈α⊃concept␈α⊃C␈α⊃will␈α⊃be␈α⊃abbreviated␈α⊂as
␈↓ α,␈↓C.Genl.

␈↓ α,␈↓C.Genl␈α∞does␈α∞not␈α∞contain␈α∞␈↓βall␈↓␈α∞generalizations␈α∞of␈α∞C;␈α∞rather,␈α∞just␈α∞the␈α∞"immediate"␈α∞ones.␈α∞More
␈↓ α,␈↓formally,␈α
if␈α
A␈α
is␈α
a␈α
generalization␈α
of␈α
B,␈α
and␈αB␈α
of␈α
C,␈α
then␈α
C.Genl␈α
will␈α
␈↓βnot␈↓␈α
contain␈α
a␈αpointer␈α
to
␈↓ α,␈↓A.  Instead, C will point to B

␈↓ α,␈↓Here␈α∃are␈α∃the␈α∃recursive␈α∃equations␈α∃which␈α∃permit␈α∃a␈α∃search␈α∃process␈α∃to␈α∃quickly␈α∃≡nd␈α∀all
␈↓ α,␈↓generalizations or specializations of a given concept X:

␈↓ α,␈↓¬␈↓ β,Generalizations(X) = Genl␈↓#
*␈↓#(X) = {X} ∪ Generalizations(X.Genl)
␈↓ α,␈↓¬␈↓ β,Specializations(X) = Spec␈↓#
*␈↓#(X) = {X} ∪ Specializations(X.Spec)

␈↓ α,␈↓For␈αthe␈α
reader's␈αconvenience,␈αhere␈α
are␈αthe␈α
similar␈αequations,␈αpresented␈α
elsewhere␈αin␈αthe␈α
text,
␈↓ α,␈↓for ≡nding all examples of ¬ and Isa's of ¬ X:

␈↓ α,␈↓¬␈↓ β,Examples(X) = Spec␈↓#
*␈↓#(Exs(Spec␈↓#
*␈↓#(X)))
␈↓ α,␈↓¬␈↓ β,Isa's(X) = Genl␈↓#
*␈↓#(Isa(Genl␈↓#
*␈↓#(X)))

␈↓ α,␈↓The␈α
format␈α
of␈α
the␈α
Generalizations␈α
facet␈α
is␈α∞quite␈α
simple:␈α
it␈α
is␈α
a␈α
list␈α
of␈α
concept␈α∞names.␈α
The
␈↓ α,␈↓Generalizations facet for Odd-primes might be:

␈↓ α,␈↓¬␈↓ β,(Odd-numbers Primes)
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε66␈↓-


␈↓ α,␈↓Here␈α
is␈α
a␈α
small␈α
diagram␈α
representing␈α
generalization␈α
relationships.␈α
The␈α
lines␈α
drawn␈α
represent
␈↓ α,␈↓the pointers found in the Genl facets of these concepts:

␈↓"␈↓ α,␈↓π␈↓ ¬<Object
␈↓"␈↓ α,␈↓π␈↓ ¬<   \
␈↓"␈↓ α,␈↓π␈↓ ¬<    \
␈↓"␈↓ α,␈↓π␈↓ ¬<     \
␈↓"␈↓ α,␈↓π␈↓ ¬<      \
␈↓"␈↓ α,␈↓π␈↓ ¬<       \
␈↓"␈↓ α,␈↓π␈↓ ¬<        \
␈↓"␈↓ α,␈↓π␈↓ ¬<        Number
␈↓"␈↓ α,␈↓π␈↓ ¬<         / \
␈↓"␈↓ α,␈↓π␈↓ ¬<        /   \
␈↓"␈↓ α,␈↓π␈↓ ¬<       /     \
␈↓"␈↓ α,␈↓π␈↓ ¬<      /       \
␈↓"␈↓ α,␈↓π␈↓ ¬<     /         \
␈↓"␈↓ α,␈↓π␈↓ ¬<    /           \
␈↓"␈↓ α,␈↓π␈↓ ¬<Odd-numbers    Primes
␈↓"␈↓ α,␈↓π␈↓ ¬<    \           / \
␈↓"␈↓ α,␈↓π␈↓ ¬<     \         /   \
␈↓"␈↓ α,␈↓π␈↓ ¬<      \       /     \
␈↓"␈↓ α,␈↓π␈↓ ¬<       \     /       \
␈↓"␈↓ α,␈↓π␈↓ ¬<        \   /         \
␈↓"␈↓ α,␈↓π␈↓ ¬<         \ /           \
␈↓"␈↓ α,␈↓π␈↓ ¬<      Odd-primes   Even-primes
␈↓"␈↓ α,␈↓π␈↓ ¬<           \
␈↓"␈↓ α,␈↓π␈↓ ¬<            \
␈↓"␈↓ α,␈↓π␈↓ ¬<             \
␈↓"␈↓ α,␈↓π␈↓ ¬<              \
␈↓"␈↓ α,␈↓π␈↓ ¬<               \
␈↓"␈↓ α,␈↓π␈↓ ¬<                \
␈↓"␈↓ α,␈↓π␈↓ ¬<         Mersenne-primes


␈↓ α,␈↓Each␈αof␈αthose␈αlines␈αrepresents␈αan␈αarrow␈αwhich␈αslants␈αupwards,␈αindicating␈αa␈αGenl␈αlink.␈α For
␈↓ α,␈↓example,␈α⊂we␈α⊂see␈α⊂that␈α⊂the␈α⊂Generalizations␈α⊂facet␈α⊂of␈α⊂Odd-primes␈α⊂contains␈α⊂pointers␈α⊃to␈α⊂both
␈↓ α,␈↓Odd-numbers␈α∞and␈α∞to␈α∞Primes.␈α∞ There␈α∞is␈α
no␈α∞pointer␈α∞from␈α∞Odd-primes␈α∞upward␈α∞to␈α
Number,
␈↓ α,␈↓because␈α⊃there␈α⊃is␈α⊃an␈α⊃"intermediate"␈α⊃concept␈α⊃(namely,␈α⊃Primes).␈α⊃ There␈α⊃is␈α⊃no␈α⊃pointer␈α⊂from
␈↓ α,␈↓Mersenne-primes to Object, since a chain of intermediate concepts links them.

␈↓ α,␈↓The␈α∪reason␈α∩for␈α∪these␈α∩strange␈α∪constraints␈α∩is␈α∪so␈α∩that␈α∪the␈α∩total␈α∪number␈α∩of␈α∪links␈α∪can␈α∩be
␈↓ α,␈↓minimized.␈α∞There␈α∞is␈α∞no␈α∞harm␈α∞if␈α∞a␈α∞few␈α∞redundant␈α∞ones␈α∞sneak␈α∞in.␈α∞ In␈α∞fact,␈α∞frequently-used
␈↓ α,␈↓paths are granted the status of single links, as we shall soon see.

␈↓ α,␈↓We've␈α⊂been␈α∂talking␈α⊂about␈α∂both␈α⊂Specializations␈α∂and␈α⊂Generalizations␈α∂as␈α⊂if␈α∂they␈α⊂were␈α∂very
␈↓ α,␈↓similar to each other. It's time to make that more explicit:
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε67␈↓-

␈↓ α,␈↓Specializations␈αare␈αthe␈αconverse␈αof␈αGeneralizations.␈αThe␈αformat␈αis␈αthe␈αsame,␈αand␈α(hopefully)
␈↓ α,␈↓A is an entry on B's Specializations facet i≥ B is an entry on A's Generalizations facet.

␈↓ α,␈↓The uses of these two facets are many:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α∞AM␈α∞can␈α∞sometimes␈α∞establish␈α∞independently␈α∞that␈α∞A␈α∞is␈α∞both␈α∞a␈α∞generalization␈α∂and␈α∞a
␈↓ α,␈↓␈↓ β≤specialization␈α
of␈α
B;␈αin␈α
that␈α
case,␈αAM␈α
would␈α
like␈αto␈α
recognize␈α
that␈αfact␈α
easily,␈α
so␈αit
␈↓ α,␈↓␈↓ β≤can␈α⊂conjecture␈α∂that␈α⊂A␈α⊂and␈α∂B␈α⊂specify␈α⊂equivalent␈α∂concepts.␈α⊂Such␈α⊂coincidences␈α∂are
␈↓ α,␈↓␈↓ β≤easily␈α⊃detected␈α⊃as␈α⊃␈↓β␈↓&cycles␈↓)αβ␈↓␈α⊃in␈α⊃the␈α∩Genl␈α⊃(or␈α⊃Spec)␈α⊃graph.␈α⊃In␈α⊃these␈α⊃cases,␈α∩AM␈α⊃may
␈↓ α,␈↓␈↓ β≤physically merge A and B (and all the other concepts in the cycle) into one concept.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈αSometimes,␈αAM␈αwants␈αto␈αassemble␈αa␈αlist␈αof␈αall␈αspecializations␈α(or␈α
generalizations)␈αof
␈↓ α,␈↓␈↓ β≤X,␈αso␈αthat␈αit␈α
can␈αtest␈αwhether␈αsome␈α
statement␈αwhich␈αis␈αjust␈α
barely␈αtrue␈α(or␈αfalse)␈α
for
␈↓ α,␈↓␈↓ β≤X will hold for any of those specializations of X.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α∞Sometimes,␈α∞the␈α∞list␈α∞of␈α∞generalizations␈α∞is␈α∂used␈α∞to␈α∞assemble␈α∞a␈α∞list␈α∞of␈α∞isa's;␈α∞the␈α∂list␈α∞of
␈↓ α,␈↓␈↓ β≤specializations helps assemble a list of examples.
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓␈αA␈αcommon␈αand␈αcrucial␈αuse␈αof␈αthe␈αlist␈αof␈αgeneralizations␈αis␈αto␈αlocate␈αall␈αthe␈αheuristic
␈↓ α,␈↓␈↓ β≤rules␈αwhich␈αare␈αrelevant␈αto␈αa␈αgiven␈αconcept.␈αTypically,␈αthe␈αrelevant␈αrules␈α
are␈αthose
␈↓ α,␈↓␈↓ β≤tacked␈α∞onto␈α∞Isa's␈α
of␈α∞that␈α∞concept,␈α
and␈α∞the␈α∞list␈α∞of␈α
Isa's␈α∞is␈α∞built␈α
up␈α∞from␈α∞the␈α∞list␈α
of
␈↓ α,␈↓␈↓ β≤generalizations of that concept.  This was also mentioned on page 54.
␈↓ α,␈↓␈↓ αl␈↓¬5.␈↓␈α⊗To␈α⊗incorporate␈α⊗new␈α∃knowledge.␈α⊗If␈α⊗AM␈α⊗learns,␈α∃conjectures,␈α⊗etc.␈α⊗that␈α⊗A␈α⊗is␈α∃a
␈↓ α,␈↓␈↓ β≤specialization␈αof␈αB,␈αthen␈αall␈αthe␈αmachinery␈α(all␈αthe␈αtheorems,␈αalgorithms,␈αetc.)␈αfor␈αB
␈↓ α,␈↓␈↓ β≤become available for working with A.

␈↓ α,␈↓Here␈αis␈αa␈αlittle␈αtrick␈α
that␈αdeserves␈αa␈αcouple␈αparagraphs␈αof␈α
its␈αown.␈αAM␈αstores␈αthe␈αanswers␈α
to
␈↓ α,␈↓common␈α∪questions␈α∪(like␈α∪"What␈α∪are␈α∩␈↓βall␈↓␈α∪the␈α∪specializations␈α∪of␈α∪Operation")␈α∪explicitly,␈α∩by
␈↓ α,␈↓intentionally␈αpermitting␈αredundant␈αlinks␈αto␈αbe␈αmaintained.␈α If␈αtwo␈αrequests␈αarrive␈αclosely␈αin
␈↓ α,␈↓time,␈αto␈αtest␈αwhether␈αA␈α
is␈αa␈αgeneralization␈αof␈αB,␈αthen␈α
the␈αresult␈αis␈αstored␈αby␈αadding␈α
"A"␈αas
␈↓ α,␈↓an␈α∪entry␈α∪on␈α∪the␈α∪Generalizations␈α∪facet␈α∪of␈α∪B,␈α∪and␈α∪adding␈α∪"B"␈α∪as␈α∪a␈α∪new␈α∪entry␈α∀on␈α∪the
␈↓ α,␈↓Specializations␈α∂facet␈α∞of␈α∂A.␈α∞The␈α∂slight␈α∞extra␈α∂space␈α∞is␈α∂more␈α∞than␈α∂recompensed␈α∞in␈α∂cpu␈α∞time
␈↓ α,␈↓saved.

␈↓ α,␈↓If␈αthe␈αresult␈αwere␈αFalse␈α(A␈αturned␈αout␈αnot␈αto␈αbe␈αa␈αgeneralization␈αof␈αB)␈αthen␈αthe␈αlinks␈αwould
␈↓ α,␈↓specify␈α∂that␈α∂≡nding␈α∂explicitly,␈α∂so␈α∂that␈α∂the␈α∞next␈α∂request␈α∂would␈α∂not␈α∂generate␈α∂a␈α∂long␈α∞search
␈↓ α,␈↓again.␈α∞ Such␈α
failures␈α∞are␈α
recorded␈α∞on␈α∞two␈α
additional␈α∞facets:␈α
Genl-not␈α∞and␈α∞Spec-not.␈α
 Since
␈↓ α,␈↓most␈αconcept␈αpairs␈αA/B␈αare␈αrelated␈αby␈αSpec-not␈αand␈αby␈αGenl-not,␈αthe␈αonly␈αentries␈αwhich␈αget
␈↓ α,␈↓recorded␈αhere␈α
are␈αthe␈α
ones␈αwhich␈α
were␈αfrequently␈αcalled␈α
for␈αby␈α
AM.␈αIf␈α
space␈αever␈αgets␈α
tight,
␈↓ α,␈↓all such facets can be wiped clean with no permanent damage done.

␈↓ α,␈↓These␈α
two␈α
"shadow"␈α
facets␈α
(Genl-not␈α
and␈α
Spec-not)␈α
are␈α
not␈α
useful␈α
or␈α
interesting␈α
in␈α
their␈α
own
␈↓ α,␈↓right.␈α If␈αAM␈αever␈αwished␈αto␈αknow␈αall␈αthe␈αconcepts␈αwhich␈αare␈α␈↓βnot␈↓␈αgeneralizations␈αof␈αC,␈αthe
␈↓ α,␈↓fastest␈αway␈αwould␈αbe␈αto␈αtake␈αthe␈αset-di≥erence␈αof␈αall␈αconcepts␈αand␈αGeneralizations(C).␈α Since
␈↓ α,␈↓they␈αare␈α
quite␈αincomplete,␈αGenl-not␈α
and␈αSpec-not␈αare␈α
used␈αmore␈αlike␈α
a␈αcache␈α
memory:␈αthey
␈↓ α,␈↓save␈α∩time␈α∩whenever␈α∪they␈α∩are␈α∩applicable,␈α∪and␈α∩don't␈α∩really␈α∪cost␈α∩much␈α∩when␈α∪they␈α∩aren't
␈↓ α,␈↓applicable.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.2. Examples/Isa's␈↓)αβ␈↓
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε68␈↓-



␈↓ α,␈↓β␈↓ α|Usually,␈αto␈α
show␈αthat␈α
a␈αde≡nition␈α
implies␈αno␈α
contradiction,␈αwe␈α
proceed␈α␈↓&by␈α
example␈↓)αβ,
␈↓ α,␈↓β␈↓ α|we␈αtry␈αto␈αmake␈αan␈α␈↓&example␈↓)αβ␈αof␈αa␈αthing␈αsatisfying␈αthe␈αde≡nition.␈αWe␈αwish␈αto␈α
de≡ne
␈↓ α,␈↓β␈↓ α|a␈α∂notion␈α∞A,␈α∂and␈α∂we␈α∞say␈α∂that,␈α∞by␈α∂de≡nition,␈α∂an␈α∞A␈α∂is␈α∞anything␈α∂for␈α∂which␈α∞certain
␈↓ α,␈↓β␈↓ α|postulates␈α∞are␈α∞true.␈α∞If␈α∞we␈α∞can␈α
demonstrate␈α∞directly␈α∞that␈α∞all␈α∞these␈α∞postulates␈α
are
␈↓ α,␈↓β␈↓ α|true␈α
of␈α
a␈α
certain␈α
object␈α∞B,␈α
the␈α
de≡nition␈α
will␈α
be␈α∞justi≡ed;␈α
the␈α
object␈α
B␈α
will␈α∞be␈α
an
␈↓ α,␈↓β␈↓ α|␈↓&example␈↓)αβ of an A.

␈↓ α,␈↓¬␈↓ ε\-- Poincare'



␈↓ α,␈↓Following␈α∩Poincare',␈α∩we␈α∩say␈α∩"␈↓βconcept␈α⊃A␈α∩is␈α∩an␈α∩example␈α∩of␈α⊃concept␈α∩B␈↓"␈α∩i≥␈α∩A␈α∩satis≡es␈α⊃B's
␈↓ α,␈↓de≡nition.␈↓	8␈↓␈αEquivalently,␈αwe␈αsay␈αthat␈α"␈↓βA␈αisa␈αB␈↓".␈α
 It␈αwould␈αbe␈αlegal␈α(in␈αthat␈αsituation)␈αfor␈α
"A"
␈↓ α,␈↓to␈α
be␈α
an␈αentry␈α
on␈α
B.Exs␈α(the␈α
Examples␈α
facet␈αof␈α
concept␈α
B)␈αand␈α
for␈α
"B"␈αto␈α
be␈α
an␈α
entry␈αon
␈↓ α,␈↓A.Isa␈α(the␈αIsa's␈αfacet␈αof␈αconcept␈αA).␈α Some␈αearlier␈αmention␈αof␈αthe␈αExamples␈αand␈αIsa's␈αfacets
␈↓ α,␈↓can be seen in Chapter 4, page 54.

␈↓ α,␈↓The␈α
Examples␈αfacet␈α
of␈α
C␈αdoes␈α
not␈αcontain␈α
␈↓βall␈↓␈α
examples␈αof␈α
C;␈αrather,␈α
just␈α
the␈α"immediate"
␈↓ α,␈↓ones.␈α⊂The␈α⊂examples␈α⊂facet␈α⊂of␈α⊂Numbers␈α⊂will␈α⊂not␈α⊂contain␈α⊂"11"␈α⊂since␈α⊂it␈α⊂is␈α⊂contained␈α⊂in␈α∂the
␈↓ α,␈↓examples␈α⊃facet␈α⊃of␈α⊃Odd-primes.␈α∩ A␈α⊃"rippling"␈α⊃procedure␈α⊃is␈α⊃used␈α∩to␈α⊃acquire␈α⊃a␈α⊃list␈α∩of␈α⊃all
␈↓ α,␈↓examples of a given concept. The basic equation is:

␈↓ α,␈↓¬␈↓ β,Examples(x) = Specializations(Exs(Specializations(x)))

␈↓ α,␈↓where␈α
Exs(x)␈α
is␈αthe␈α
contents␈α
of␈α
the␈αexamples␈α
facet␈α
of␈α
x.␈α Examples(x)␈α
represents␈α
the␈α≡nal␈α
list
␈↓ α,␈↓of␈αall␈αknown␈αitems␈αwhich␈αsatisfy␈αthe␈αde≡nition␈αof␈αX.␈αExamples(x)␈αthus␈αmust␈αinclude␈αExs(x).
␈↓ α,␈↓Specializations(x)␈αmight␈αbe␈αmore␈αregularly␈αwritten␈αSpec␈↓#
*␈↓#(x).␈αThat␈αis,␈αall␈αmembers␈αof␈αx.Spec,
␈↓ α,␈↓all␈α
members␈α
of␈α␈↓βtheir␈↓␈α
Spec␈α
facet,␈αetc.␈α
 Note␈α
the␈αsimilarity␈α
of␈α
this␈αto␈α
the␈α
formula␈α
for␈αIsa's(x),
␈↓ α,␈↓given on page 54.  We could also write the above equation as follows:

␈↓ α,␈↓¬␈↓ β,Examples(x) = Spec␈↓#
*␈↓#(Exs(Spec␈↓#
*␈↓#(x)))

␈↓ α,␈↓As␈α⊃an␈α⊃illustration,␈α⊃we␈α⊃shall␈α⊂show␈α⊃how␈α⊃AM␈α⊃would␈α⊃recognize␈α⊂that␈α⊃"3"␈α⊃is␈α⊃an␈α⊃example␈α⊂of
␈↓ α,␈↓Object:









␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	8␈↓ε␈α	What␈α	does␈α	this␈α	mean?␈α	B.Defn␈α	is␈α	a␈αλLisp␈α	predicate,␈α	a␈α	Lambda␈α	expression.␈α	If␈α	it␈α	is␈αλfed␈α	A␈α	as␈α	its␈α	argument,␈α	and␈α	it␈α	returns␈αλTrue,
␈↓ α,␈↓ε␈↓ βLwe␈α	say␈α	that␈α	A␈α	is␈α	a␈α	B,␈α	or␈α
that␈α	A␈α	satisfies␈α	B's␈α	definition.␈α	If␈α	B.Defn␈α	returns␈α
NIL,␈α	we␈α	say␈α	that␈α	A␈α	is␈α	not␈α	a␈α
B,␈α	or
␈↓ α,␈↓ε␈↓ βLthat␈α
A␈αfails␈α
B's␈αdefinition.␈α
If␈α
B.Defn␈αruns␈α
out␈αof␈α
time␈αbefore␈α
returning␈α
a␈αT/NIL␈α
value,␈αthere␈α
is␈αno␈α
definite
␈↓ α,␈↓ε␈↓ βLstatement of this form we can make.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε69␈↓-


␈↓"␈↓ α,␈↓π␈↓ ¬<Object
␈↓"␈↓ α,␈↓π␈↓ ¬<   \
␈↓"␈↓ α,␈↓π␈↓ ¬<    \
␈↓"␈↓ α,␈↓π␈↓ ¬<     \
␈↓"␈↓ α,␈↓π␈↓ ¬<      \
␈↓"␈↓ α,␈↓π␈↓ ¬<       \
␈↓"␈↓ α,␈↓π␈↓ ¬<        \
␈↓"␈↓ α,␈↓π␈↓ ¬<        Number
␈↓"␈↓ α,␈↓π␈↓ ¬<         / \
␈↓"␈↓ α,␈↓π␈↓ ¬<        /   \
␈↓"␈↓ α,␈↓π␈↓ ¬<       /     \
␈↓"␈↓ α,␈↓π␈↓ ¬<      /       \
␈↓"␈↓ α,␈↓π␈↓ ¬<     /         \
␈↓"␈↓ α,␈↓π␈↓ ¬<    /           \
␈↓"␈↓ α,␈↓π␈↓ ¬<Odd-numbers    Primes
␈↓"␈↓ α,␈↓π␈↓ ¬<    \           /
␈↓"␈↓ α,␈↓π␈↓ ¬<     \         /
␈↓"␈↓ α,␈↓π␈↓ ¬<      \       /
␈↓"␈↓ α,␈↓π␈↓ ¬<       \     /
␈↓"␈↓ α,␈↓π␈↓ ¬<        \   /
␈↓"␈↓ α,␈↓π␈↓ ¬<         \ /
␈↓"␈↓ α,␈↓π␈↓ ¬<      Odd-primes
␈↓"␈↓ α,␈↓π␈↓ ¬<           \
␈↓"␈↓ α,␈↓π␈↓ ¬<            \
␈↓"␈↓ α,␈↓π␈↓ ¬<             \
␈↓"␈↓ α,␈↓π␈↓ ¬<              \
␈↓"␈↓ α,␈↓π␈↓ ¬<               \
␈↓"␈↓ α,␈↓π␈↓ ¬<                \
␈↓"␈↓ α,␈↓π␈↓ ¬<         Mersenne-primes
␈↓"␈↓ α,␈↓π␈↓ ¬<                ⊗
␈↓"␈↓ α,␈↓π␈↓ ¬<                ⊗
␈↓"␈↓ α,␈↓π␈↓ ¬<                ⊗
␈↓"␈↓ α,␈↓π␈↓ ¬<                3


␈↓ α,␈↓As␈α
the␈α
graph␈α
above␈α∞shows,␈α
AM␈α
would␈α
ripple␈α
in␈α∞the␈α
Spec␈α
direction␈α
4␈α
times,␈α∞moving␈α
from
␈↓ α,␈↓Object␈α
all␈αthe␈α
way␈αto␈α
Mersenne-primes;␈αthen␈α
descend␈αonce␈α
in␈αthe␈α
Exs␈αdirection,␈α
to␈αreach␈α
"3";
␈↓ α,␈↓then␈α⊂ripple␈α⊂0␈α⊂more␈α⊂times␈α∂in␈α⊂the␈α⊂Spec␈α⊂direction.␈α⊂ Thus␈α⊂"3"␈α∂is␈α⊂seen␈α⊂to␈α⊂be␈α⊂an␈α⊂example␈α∂of
␈↓ α,␈↓Object,␈α∞according␈α∞to␈α∞the␈α∞above␈α∞formula.␈α∞ Similarly,␈α
we␈α∞see␈α∞that␈α∞"3"␈α∞is␈α∞also␈α∞an␈α∞example␈α
of
␈↓ α,␈↓Number,␈α⊗of␈α∃Primes,␈α⊗of␈α∃Odd-number,␈α⊗of␈α∃Odd-primes,␈α⊗and␈α∃of␈α⊗course␈α∃an␈α⊗example␈α∃of
␈↓ α,␈↓Mersenne-primes.

␈↓ α,␈↓As␈αwith␈αGeneralizations/Specializations,␈αthe␈α
reasons␈αbehind␈αthe␈αincomplete␈αpointer␈α
structure
␈↓ α,␈↓is␈α⊂simply␈α⊃to␈α⊂save␈α⊃space,␈α⊂and␈α⊃to␈α⊂minimize␈α⊂the␈α⊃di≠culty␈α⊂of␈α⊃updating␈α⊂the␈α⊃graph␈α⊂structure
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε70␈↓-

␈↓ α,␈↓whenever␈αnew␈αlinks␈αare␈αfound.␈α Suppose␈αa␈αnew␈αMersenne␈αprime␈↓	9␈↓␈αis␈αcomputed.␈α
Wouldn't␈αit
␈↓ α,␈↓be␈α∞nice␈α∞simply␈α∞to␈α∞add␈α∞a␈α∞single␈α∞entry␈α∞to␈α∞the␈α∞Exs␈α∞facet␈α∞of␈α∞Mersenne-primes,␈α∞rather␈α∞than␈α∞to
␈↓ α,␈↓have␈α⊃to␈α⊃update␈α⊃the␈α⊃Exs␈α∩pointers␈α⊃from␈α⊃a␈α⊃dozen␈α⊃concepts?␈α∩ There␈α⊃is␈α⊃no␈α⊃harm␈α⊃if␈α∩a␈α⊃few
␈↓ α,␈↓redundant links sneak in.

␈↓ α,␈↓"Isa's"␈α∞is␈α∞the␈α∞converse␈α
of␈α∞"Examples".␈α∞The␈α∞format␈α
is␈α∞the␈α∞same,␈α∞and␈α
(if␈α∞A␈α∞and␈α∞B␈α∞are␈α
both
␈↓ α,␈↓concepts)␈αA␈αis␈αan␈αentry␈αon␈αB.Isa␈αi≥␈αB␈αis␈αan␈αentry␈αon␈αA.Exs.␈αIn␈αother␈αwords,␈αA␈αis␈αa␈αmember
␈↓ α,␈↓of Examples(B) i≥ B is a member of Isa's(A).

␈↓ α,␈↓The ␈↓βuses␈↓ of the Exs/Isa's facets are similar to those for Genl/Spec (see previous subsection).

␈↓ α,␈↓Their␈α
formats␈αare␈α
quite␈α
a␈αbit␈α
more␈α
complicated␈αthan␈α
the␈α
Genl/Spec␈αfacets'␈α
formats,␈αwhen␈α
we
␈↓ α,␈↓≡nally␈α
get␈α
to␈α
the␈α
implementation␈α
level,␈αhowever.␈α
 There␈α
are␈α
really␈α
a␈α
cluster␈α
of␈αdi≥erent␈α
facets
␈↓ α,␈↓all related to Examples:

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈αTYPICAL:␈α
This␈αis␈α
a␈αlist␈α
of␈αaverage␈α
examples.␈αCare␈α
must␈αbe␈α
taken␈αto␈α
include␈αa␈α
wide
␈↓ α,␈↓␈↓ β≤spectrum␈α∂of␈α∂allowable␈α∂kinds␈α∂of␈α∂examples.␈α∂For␈α∂"Sets",␈α∂these␈α∂would␈α∂include␈α∂sets␈α∞of
␈↓ α,␈↓␈↓ β≤varying size, nesting, complexity, type of elements, etc.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α⊃BOUNDARY:␈α⊂Items␈α⊃which␈α⊂just␈α⊃barely␈α⊂pass␈α⊃the␈α⊂de≡nition␈α⊃of␈α⊂this␈α⊃concept.␈α⊂This
␈↓ α,␈↓␈↓ β≤might␈α
include␈α
items␈α∞which␈α
satisfy␈α
the␈α∞base␈α
step␈α
of␈α∞a␈α
recursive␈α
de≡nition,␈α∞or␈α
items
␈↓ α,␈↓␈↓ β≤which␈αwere␈α
intuitively␈αbelieved␈α
to␈αbe␈α␈↓βnon␈↓-examples␈α
of␈αthe␈α
concept.␈αFor␈α
"Sets",␈αthis
␈↓ α,␈↓␈↓ β≤might include the empty set.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈αBOUNDARY-NOT:␈αItems␈αwhich␈αjust␈αbarely␈αfail␈αthe␈αde≡nition.␈αThis␈αmight␈αinclude
␈↓ α,␈↓␈↓ β≤an␈αitem␈αwhich␈αhad␈αto␈αbe␈αslightly␈αmodi≡ed␈αduring␈αchecking,␈αlike␈α{A,B,A}␈αbecoming
␈↓ α,␈↓␈↓ β≤{A,B}.
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓␈α∩FOIBLES:␈α⊃Total␈α∩failures.␈α⊃Items␈α∩which␈α∩are␈α⊃completely␈α∩against␈α⊃the␈α∩grain␈α∩of␈α⊃this
␈↓ α,␈↓␈↓ β≤concept. For "Sets", this might include the operation "Compose".
␈↓ α,␈↓␈↓ αl␈↓¬5.␈↓␈α∪NOT:␈α∪This␈α∪is␈α∪the␈α∪"cache"␈α∪trick␈α∪used␈α∪to␈α∪store␈α∪the␈α∪answers␈α∪to␈α∩frequently-asked
␈↓ α,␈↓␈↓ β≤questions.␈αIf␈αAM␈α
frequently␈αwants␈αto␈α
know␈αwhether␈αX␈α
is␈αan␈αexample␈α
of␈αY,␈αand␈α
the
␈↓ α,␈↓␈↓ β≤answer␈α
is␈α
␈↓βNo␈↓,␈α
then␈α
much␈αtime␈α
can␈α
be␈α
saved␈α
by␈α
adding␈αX␈α
as␈α
an␈α
entry␈α
to␈αthe␈α
Exs-not
␈↓ α,␈↓␈↓ β≤facet of Y.

␈↓ α,␈↓An␈α∩individual␈α∪item␈α∩on␈α∩these␈α∪facets␈α∩may␈α∪just␈α∩be␈α∩a␈α∪concept␈α∩name,␈α∩or␈α∪it␈α∩may␈α∪be␈α∩more
␈↓ α,␈↓complicated.␈α
 In␈α
the␈α
case␈α
of␈α
an␈α
operation,␈α
it␈α
is␈α
an␈α
item␈α
of␈α
the␈α
form␈α
<a␈↓#v1␈↓#a␈↓#v2␈↓#...→v>;␈α
i.e.,␈α
actual
␈↓ α,␈↓arguments␈α
and␈α
the␈α
value␈α
returned.␈α
In␈α
the␈α
case␈α
of␈α
objects,␈α
it␈α
is␈α
an␈α
object␈α
of␈α
that␈α
form.␈αAn
␈↓ α,␈↓Exs facet of the concept Sets might contain {a} as one entry.

␈↓ α,␈↓Here␈α∂is␈α⊂a␈α∂more␈α⊂detailed␈α∂illustration.␈α⊂Consider␈α∂the␈α⊂Examples␈α∂facet␈α⊂of␈α∂Set-union.␈α⊂It␈α∂might
␈↓ α,␈↓appear thus:






␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	9␈↓ε␈α	"Mersenne␈α	prime",␈α	without␈α	a␈α	hyphen,␈α	refers␈α	to␈αλa␈α	number␈α	satisfying␈α	certain␈α	properties.␈α	"Mersenne-primes",␈α	with␈α	a␈αλhyphen,
␈↓ α,␈↓ε␈↓ βLrefers␈α	to␈α	one␈α	specific␈α	AM␈α	concept,␈α	a␈α	data␈α	structure␈α	with␈α	facets.␈α	Each␈α	Mersenne␈α	prime␈α	is␈α	an␈α	example␈α	of␈αλthe
␈↓ α,␈↓ε␈↓ βLconcept Mersenne-primes.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε71␈↓-


␈↓ α,␈↓¬␈↓ β,TYPICAL: {A}∪{A,B}→{A,B};
␈↓ α,␈↓¬␈↓ ∧≤{A,B}∪{A,B}→{A,B};
␈↓ α,␈↓¬␈↓ ∧≤{A,<3,4,3>,{A,B}}∪{3,A}→{A,<3,4,3>,{A,B},3}.
␈↓ α,␈↓¬␈↓ β,BOUNDARY: {}∪X→X ␈↓	10␈↓¬
␈↓ α,␈↓¬␈↓ β,BOUNDARY-NOT: {A,B}∪{A,C}→{A,B,A,C};
␈↓ α,␈↓¬␈↓ ¬<{A,B,C,D}∪{E,F,G,H,I,J}→{A,B,C,E,F,G,H,I,J}
␈↓ α,␈↓¬␈↓ β,FOIBLES: <2,A,2>
␈↓ α,␈↓¬␈↓ β,NOT: no entries

␈↓ α,␈↓The␈αformat␈α
for␈αIsa's␈αare␈α
much␈αsimpler:␈α
there␈αare␈αonly␈α
two␈αkinds␈αof␈α
links,␈αand␈α
they're␈αeach
␈↓ α,␈↓merely a list of concept names.  Here is the Isa facet of Set-union:

␈↓ α,␈↓¬␈↓ β,ISA: (Operation␈↓	11␈↓¬ Domain=Range-op)
␈↓ α,␈↓¬␈↓ β,ISA-NOT: (Structure Composition Predicate)

␈↓ α,␈↓At␈αsome␈αtime,␈αsome␈αrule␈αasked␈αwhether␈αSet-union␈α␈↓&isa␈↓)αβ␈αComposition.␈αAs␈αa␈αresult,␈αthe␈α
negative
␈↓ α,␈↓response␈α∩was␈α∩recorded␈α⊃by␈α∩adding␈α∩"Composition"␈α⊃to␈α∩the␈α∩Isa-not␈α⊃facet␈α∩of␈α∩Set-union,␈α⊃and
␈↓ α,␈↓adding␈α_"Set-union"␈α_to␈α_the␈α_Exs-not␈α_subfacet␈α↔of␈α_the␈α_Examples␈α_facet␈α_of␈α_the␈α↔concept
␈↓ α,␈↓Composition␈α∞(indicating␈α∞that␈α∞Set-union␈α∞was␈α
de≡nitely␈α∞not␈α∞an␈α∞example␈α∞of␈α∞Composition,␈α
yet
␈↓ α,␈↓there was no reason to consider it a foible).


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.3. In-Domain-of/In-Range-of␈↓)αβ␈↓

␈↓ α,␈↓We shall say that A is in the domain of B (written "A In-dom-of B") i≥
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ A and B are concepts
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓ B isa Operation
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α
A␈αis␈α
equal␈αto␈α
(or␈α
at␈αleast␈α
a␈αspecialization␈α
of)␈α
one␈αof␈α
the␈αdomain␈α
components␈α
of␈αthe
␈↓ α,␈↓␈↓ β≤operation␈α∂B.␈α∂That␈α∂is,␈α∂B␈α∂can␈α∂be␈α⊂executed␈α∂using␈α∂any␈α∂example␈α∂of␈α∂A␈α∂as␈α∂one␈α⊂of␈α∂its
␈↓ α,␈↓␈↓ β≤arguments.

␈↓ α,␈↓The␈α⊃semantic␈α⊃content␈α⊃of␈α⊃"In-dom-of"␈α⊃is:␈α⊃what␈α⊃can␈α⊃be␈α⊃done␈α⊃to␈α⊃any␈α⊃example␈α⊃of␈α⊃a␈α⊂given
␈↓ α,␈↓concept␈α∞C?␈α∂ Given␈α∞an␈α∂example␈α∞of␈α∞concept␈α∂C,␈α∞what␈α∂operations␈α∞can␈α∞be␈α∂run␈α∞on␈α∂that␈α∞thing?
␈↓ α,␈↓E.g.,␈α"Primes␈αIn-dom-of␈αSquaring"␈αtells␈αus␈αthat␈αwe␈αcan␈αapply␈αthe␈αoperation␈αSquaring␈αto␈αany
␈↓ α,␈↓particular prime number we wish.

␈↓ α,␈↓Let␈αus␈αnow␈αturn␈αfrom␈αIn-dom-of␈αto␈αthe␈αrelated␈αfacet␈αIn-ran-of.␈α We␈αsay␈αthat␈αconcept␈αA␈αis␈αin
␈↓ α,␈↓the range of B i≥ B is an Activity and A is a specialization of the range of B.

␈↓ α,␈↓Semantically,␈α⊂the␈α⊂In-ran-of␈α⊂relation␈α⊂between␈α⊂A␈α⊂and␈α⊂B␈α⊂means␈α⊂that␈α⊂one␈α⊂might␈α⊂be␈α⊂able␈α∂to
␈↓ α,␈↓produce␈αexamples␈αof␈α
A␈αby␈αrunning␈α
operation␈αB.␈α Aha!␈α This␈α
is␈αa␈αpotential␈α
mechanism␈αfor
␈↓ α,␈↓≡nding␈αexamples␈α
of␈αa␈α
concept␈αA.␈α All␈α
you␈αneed␈α
do␈αis␈αget␈α
hold␈αof␈α
In-ran-of(A),␈αand␈αrun␈α
each

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	10␈↓ε␈α	Actually,␈αλAM␈α	is␈α	not␈αλquite␈α	smart␈α	enough␈αλto␈α	use␈α	the␈αλvariable␈α	X␈α	as␈αλshown␈α	in␈α	the␈αλboundary␈α	examples.␈α	It␈αλwould␈α	simply␈α	store␈αλa
␈↓ α,␈↓ε␈↓ βLfew␈α∞instances␈α
of␈α∞this␈α∞general␈α
rule,␈α∞plus␈α∞have␈α
an␈α∞entry␈α
of␈α∞the␈α∞form␈α
<Equivalent:␈α∞Identity(X)␈α∞and␈α
Set-
␈↓ α,␈↓ε␈↓ βLunion(X,{})>␈αon␈αthe␈αExs␈αfacet␈αof␈αConjectures.␈αNotice␈αthat␈αbecause␈αof␈αthe␈αasymmetric␈αway␈αSet-union␈αwas
␈↓ α,␈↓ε␈↓ βLdefined,␈αonly␈α
one␈αlopsided␈α
boundary␈αexample␈α
was␈αfound.␈α
If␈αanother␈α
definition␈αwere␈α
supplied,␈αthe␈α
converse
␈↓ α,␈↓ε␈↓ βLkind of boundary examples would be found.
␈↓ α,␈↓ε␈↓	11␈↓ε This entry is redundant.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε72␈↓-

␈↓ α,␈↓of␈α∞those␈α∞operations.␈α∞Even␈α
more␈α∞expeditious␈α∞is␈α∞to␈α∞examine␈α
the␈α∞Examples␈α∞facets␈α∞of␈α∞each␈α
of
␈↓ α,␈↓those␈α
operations,␈αfor␈α
already-run␈αexamples␈α
whose␈αvalues␈α
should␈αbe␈α
tested␈αusing␈α
A.Defn,␈αto
␈↓ α,␈↓see␈α∞if␈α∞they␈α∞are␈α∞examples␈α∞of␈α∞A's.␈α∞ AM␈α∞relies␈α
on␈α∞this␈α∞in␈α∞times␈α∞of␈α∞high␈α∞motivation;␈α∞it␈α∞is␈α
too
␈↓ α,␈↓"blind" a method to use heavily all the time.

␈↓ α,␈↓This␈α⊗facet␈α⊗is␈α⊗also␈α⊗useful␈α⊗for␈α⊗generating␈α⊗situations␈α⊗to␈α⊗investigate.␈α⊗ Suppose␈α⊗that␈α∃the
␈↓ α,␈↓Domain/range␈α
facet␈α
of␈α
Doubling␈α
contains␈α
only␈α
one␈α
entry:␈α
<␈α
Numbers␈α
→␈α
Numbers␈α∞>.␈α
Then
␈↓ α,␈↓syntactically,␈α∂Odd-numbers␈α⊂is␈α∂in␈α∂the␈α⊂range␈α∂of␈α∂Doubling.␈α⊂Eventually␈α∂a␈α∂heuristic␈α⊂rule␈α∂may
␈↓ α,␈↓have␈α
AM␈α∞spend␈α
some␈α
time␈α∞looking␈α
for␈α
an␈α∞example␈α
of␈α
Doubling,␈α∞where␈α
the␈α
result␈α∞was␈α
an
␈↓ α,␈↓odd␈αnumber.␈α
 If␈αnone␈α
is␈αquickly␈α
found,␈αAM␈α
conjectures␈αthat␈α
it␈α␈↓βnever␈↓␈α
will␈αbe␈α
found.␈α Since
␈↓ α,␈↓one␈α∞de≡nition␈α
of␈α∞Odd-number(x)␈α∞is␈α
"Number(x)␈α∞and␈α
Not(Even-number(x))",␈α∞the␈α∞only␈α
non-
␈↓ α,␈↓odd␈α∀numbers␈α∃are␈α∀even␈α∃numbers.␈α∀ So␈α∀AM␈α∃will␈α∀increment␈α∃the␈α∀Domain/range␈α∃facet␈α∀of
␈↓ α,␈↓Doubling␈αwith␈αthe␈αentry␈α<Numbers→Even-numbers>,␈αand␈αremove␈αthe␈αold␈αentry.␈αThus␈αOdd-
␈↓ α,␈↓numbers␈α∩will␈α∩no␈α∩longer␈α∩be␈α∩In-dom-of␈α∩Doubling.␈α∩ AM␈α∩can␈α∩of␈α∩course␈α∩chance␈α∩upon␈α⊃this
␈↓ α,␈↓conjecture␈α
in␈α
a␈αmore␈α
positive␈α
way,␈α
by␈αnoticing␈α
that␈α
all␈αknown␈α
examples␈α
of␈α
Doubling␈αhave
␈↓ α,␈↓results which are examples of Even-numbers.␈↓	12␈↓.

␈↓ α,␈↓A␈α∞more␈α∞productive␈α∞result␈α∞is␈α∂suggested␈α∞by␈α∞examining␈α∞the␈α∞cases␈α∂where␈α∞Odd-perfect-squares
␈↓ α,␈↓are␈αthe␈αresult␈α
of␈αcubing.␈α The␈α
smallest␈αsuch␈αodd␈α
numbers␈αare␈α1,␈α
729,␈αand␈α15625.␈α In␈α
general,
␈↓ α,␈↓these␈αnumbers␈α
are␈αall␈α
those␈αof␈αthe␈α
form␈α(2n+1)␈↓#
6␈↓#.␈α
 How␈αcould␈αAM␈α
notice␈αsuch␈α
an␈αawkward
␈↓ α,␈↓relationship?

␈↓ α,␈↓The␈α
general␈α
question␈α
to␈α
ask,␈α
when␈α
A␈α
In-ran-of␈αB,␈α
is␈α
"What␈α
is␈α
the␈α
set␈α
of␈α
domain␈αitems␈α
whose
␈↓ α,␈↓values␈α(under␈αthe␈αoperation␈αB)␈αare␈αA's?"␈αIn␈αcase␈αthe␈αanswer␈αis␈α"All"␈αor␈α"None",␈αsome␈αspecial
␈↓ α,␈↓modi≡cations␈α∞can␈α∂be␈α∞made␈α∂to␈α∞the␈α∞Domain/range␈α∂facets␈α∞and␈α∂In-dom-of,␈α∞In-ran-of␈α∂facets␈α∞of
␈↓ α,␈↓various␈αconcepts,␈αand␈αa␈αnew␈αconjecture␈αcan␈α
be␈αprinted.␈α In␈αother␈αcases,␈αa␈αnew␈αconcept␈α
might
␈↓ α,␈↓get␈αcreated,␈αrepresenting␈αprecisely␈αthe␈αset␈αof␈αall␈αarguments␈αto␈αB␈αwhich␈αyield␈αvalues␈αin␈αA.␈α If
␈↓ α,␈↓you␈α
will,␈α
this␈α
is␈αthe␈α
inverse␈α
image␈α
of␈α
A,␈αunder␈α
operation␈α
B.␈α
 In␈αthe␈α
case␈α
of␈α
B␈α
a␈αpredicate,
␈↓ α,␈↓this might be the set of all arguments which satisfy the predicate.

␈↓ α,␈↓In␈α∞the␈α∞case␈α∂of␈α∞B=Cubing␈α∞and␈α∞A=Odd-perfect-squares,␈α∂the␈α∞heuristic␈α∞mentioned␈α∂above␈α∞will
␈↓ α,␈↓have␈α∪AM␈α∀create␈α∪a␈α∪new␈α∀concept:␈α∪the␈α∪inverse␈α∀image␈α∪of␈α∪Odd-perfect-squares␈α∀under␈α∪the
␈↓ α,␈↓operation␈α∂of␈α∂Cubing.␈α⊂ That␈α∂is,␈α∂≡nd␈α∂numbers␈α⊂whose␈α∂cubes␈α∂are␈α∂Odd-perfect-squares.␈α⊂ It␈α∂is
␈↓ α,␈↓quickly␈α
noticed␈α
that␈α
such␈α
numbers␈α
are␈α
precisely␈α
the␈α
set␈α
of␈α
Odd-perfect-squares␈α
themselves!
␈↓ α,␈↓So␈α∞The␈α∞Domain/range␈α∞facet␈α∞of␈α∂Cubing␈α∞might␈α∞get␈α∞this␈α∞new␈α∞entry:␈α∂<Odd-perfect-squares␈α∞→
␈↓ α,␈↓Odd-perfect-squares>.␈α∞ But␈α
not␈α∞all␈α
squares␈α∞can␈α
be␈α∞reached␈α
by␈α∞cubing,␈α
only␈α∞a␈α
few␈α∞of␈α
them
␈↓ α,␈↓can.␈α AM␈αwill␈αnotice␈αthis,␈αand␈αthe␈αnew␈α
range␈αwould␈αthen␈αbe␈αisolated␈αand␈αmight␈αbe␈α
renamed
␈↓ α,␈↓by␈αthe␈αuser␈α"Perfect-sixth-powers".␈α Note␈αthat␈αall␈αthis␈αwas␈αbrought␈αon␈αby␈αexamining␈αthe␈αIn-
␈↓ α,␈↓ran-of␈α⊃facet␈α⊃of␈α∩Odd-perfect-squares.␈α⊃ "Cubing"␈α⊃was␈α⊃just␈α∩one␈α⊃of␈α⊃the␈α⊃seven␈α∩entries␈α⊃there.
␈↓ α,␈↓There are six more stories to tell in this tiny nook of AM's activities.

␈↓ α,␈↓How␈α⊃exactly␈α⊃does␈α⊃AM␈α⊃go␈α⊃about␈α⊂gathering␈α⊃the␈α⊃In-ran-of␈α⊃and␈α⊃In-dom-of␈α⊃lists?␈α⊃ Given␈α⊂a
␈↓ α,␈↓concept␈αC,␈αAM␈αcan␈αscan␈αdown␈αthe␈αglobal␈αtree␈αof␈αoperations␈α(the␈αExs␈αand␈αSpec␈αlinks␈αbelow
␈↓ α,␈↓the␈α⊂concept␈α⊃`Active').␈α⊂ For␈α⊃if␈α⊂C␈α⊂is␈α⊃not␈α⊂In-dom-of␈α⊃F,␈α⊂it␈α⊂certainly␈α⊃won't␈α⊂be␈α⊃In-dom-of␈α⊂any

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	12␈↓ε This positive approach is in fact the way AM noticed this particular relationship.
␈↓ α,␈↓ε␈↓	6␈↓ε Wrong. That was an exponent, not a footnote!
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε73␈↓-

␈↓ α,␈↓specialization␈α∂of␈α∂F.␈α∂Similarly,␈α∂if␈α∂it␈α∂can't␈α∂be␈α∂produced␈α∂by␈α∂F,␈α∂it␈α∂won't␈α∂be␈α∂produced␈α⊂by␈α∂any
␈↓ α,␈↓specialization␈αof␈αF.␈αIf␈αyou␈αcan't␈αget␈αx␈α
using␈αDoubling␈αyou'll␈αnever␈αget␈αit␈αby␈αQuadrupling.␈α
So
␈↓ α,␈↓AM␈αsimply␈αripples␈αaround,␈αas␈αusual.␈αThe␈αprecise␈αcode␈αfor␈αthis␈αalgorithm␈αis␈αof␈αlittle␈αinterest.
␈↓ α,␈↓There␈αare␈αnot␈αthat␈αmany␈αoperations,␈αand␈αit␈αis␈αcheap␈αto␈αtell␈αwhether␈αX␈αis␈αa␈αspecialization␈αof
␈↓ α,␈↓a␈α
given␈α
concept,␈α
so␈α
even␈α
an␈α
exhaustive␈α
search␈α
wouldn't␈α
be␈α
prohibitive.␈α
Finally,␈α∞recall␈α
that
␈↓ α,␈↓such␈αa␈αsearch␈αis␈αnot␈αdone␈αall␈αthe␈αtime.␈α It␈αwill␈αbe␈αdone␈αinitially,␈αperhaps,␈αbut␈αafter␈αthat␈αthe
␈↓ α,␈↓In-dom-of and In-ran-of networks will only need slight updating now and then.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.4. Views␈↓)αβ␈↓

␈↓ α,␈↓Often,␈α∞two␈α∞concepts␈α∞A␈α∞and␈α∞B␈α∞will␈α∞be␈α∞inequivalent,␈α∞yet␈α∞there␈α∞will␈α∞be␈α∞a␈α∞"natural"␈α∞bijection
␈↓ α,␈↓between␈αone␈αand␈α(a␈αsubset␈αof)␈αthe␈αother.␈α For␈αexample,␈αconsider␈αa␈α≡nite␈αset␈αS␈αof␈αatoms,␈αand
␈↓ α,␈↓consider␈αthe␈αset␈αof␈αall␈αits␈αsubsets,␈α2␈↓#
S␈↓#,␈αalso␈αcalled␈αthe␈α␈↓βpower␈αset␈↓␈αof␈αS.␈α Now␈αS␈αis␈αa␈αmember␈αof,
␈↓ α,␈↓but␈αnot␈αa␈α␈↓βsubset␈↓␈αof,␈α2␈↓#
S␈↓#␈α
(e.g.,␈αif␈αS={x,y,...},␈αthen␈αx␈αis␈αnot␈α
a␈αmember␈αof␈α2␈↓#
S␈↓#).␈αOn␈αthe␈αother␈α
hand,
␈↓ α,␈↓we␈αcan␈αidentify␈αor␈αview␈αS␈αas␈αa␈αsubset␈αby␈αthe␈αmapping␈αv→{v}.␈αThen␈αS␈αis␈αassociated␈αwith␈αthe
␈↓ α,␈↓following␈αsubset␈α
of␈α2␈↓#
S␈↓#:␈α
{␈α{x},␈α
{y},...␈α}.␈α
Why␈αwould␈α
we␈αwant␈α
to␈αdo␈α
this?␈αWell,␈α
it␈αshows␈α
that␈αS␈α
is
␈↓ α,␈↓identi≡ed␈α∃with␈α∀a␈α∃␈↓βproper␈↓␈α∃subset␈α∀of␈α∃2␈↓#
S␈↓#,␈α∃and␈α∀indicates␈α∃that␈α∃S␈α∀has␈α∃a␈α∃lower␈α∀cardinality
␈↓ α,␈↓(remember: all sets are ≡nite).

␈↓ α,␈↓As␈α
another␈αexample,␈α
most␈αof␈α
us␈α
would␈αagree␈α
that␈αthe␈α
set␈α
{x,␈α{y},␈α
z}␈αcan␈α
be␈α
associated␈αwith
␈↓ α,␈↓the␈α
following␈α
bag:␈α
(x,␈α
{y},␈α
z).␈α
Each␈α
of␈α
them␈α
can␈α
be␈α
viewed␈α
as␈α
the␈α
other.␈α
Sometimes␈α
such␈α
a
␈↓ α,␈↓viewing␈α∞is␈α∞not␈α
perfectly␈α∞natural,␈α∞or␈α
isn't␈α∞really␈α∞a␈α∞bijection:␈α
how␈α∞could␈α∞the␈α
bag␈α∞(2,␈α∞2,␈α∞3)␈α
be
␈↓ α,␈↓viewed as a set? Is {2,3} better or worse than {2,{2},3)?

␈↓ α,␈↓The␈αView␈αfacet␈αof␈αa␈αconcept␈αC␈αdescribes␈αhow␈αto␈αview␈αinstances␈αof␈αanother␈αconcept␈αD␈αas␈αif
␈↓ α,␈↓they␈αwere␈αC's.␈αFor␈αexample,␈αthis␈αentry␈αon␈αthe␈αView␈αfacet␈αof␈αSets␈αexplains␈αhow␈αto␈α
view␈αany
␈↓ α,␈↓given structure as if it were a Set:

␈↓ α,␈↓¬␈↓ β,Structure: λ (x) Enclose-in-braces(Sort(Remove-multiple-elements(x)))

␈↓ α,␈↓If␈α∩given␈α∩the␈α∩list␈α∩<z,a,c,a>,␈α∪this␈α∩little␈α∩program␈α∩would␈α∩remove␈α∩multiple␈α∪elements␈α∩(leaving
␈↓ α,␈↓<z,a,c>),␈α∞sort␈α∞the␈α∞structure␈α∞(making␈α∞it␈α∞<a,c,z>),␈α∞and␈α∞replace␈α∞the␈α∞"<...>"␈α∞by␈α∞"{...}",␈α∂leaving␈α∞the
␈↓ α,␈↓≡nal␈α⊂value␈α⊂as␈α⊂{a,c,z}.␈α⊂Note␈α⊂that␈α⊂this␈α∂transformation␈α⊂is␈α⊂not␈α⊂1-1;␈α⊂the␈α⊂list␈α⊂<a,c,z>␈α⊂would␈α∂get
␈↓ α,␈↓transformed␈α⊗into␈α⊗this␈α⊗same␈α↔set.␈α⊗ On␈α⊗the␈α⊗other␈α⊗hand,␈α↔it␈α⊗may␈α⊗be␈α⊗more␈α↔useful␈α⊗than
␈↓ α,␈↓transforming␈α⊂the␈α⊃original␈α⊂list␈α⊃into␈α⊂{z,{a,{c,{a}}}}␈α⊃which␈α⊂retains␈α⊃the␈α⊂ordering␈α⊃and␈α⊂multiple
␈↓ α,␈↓element␈αinformation.␈α Both␈αof␈αthose␈αtransformations␈αmay␈αbe␈αpresent␈αas␈αentries␈αon␈αthe␈αView
␈↓ α,␈↓facet of Sets.

␈↓ α,␈↓As you've no doubt guessed, there is a general formula explaining this:

␈↓ α,␈↓¬␈↓ ¬.Views(X) ≡ View(Specializations(X))

␈↓ α,␈↓Thus,␈α
to␈α
≡nd␈α
all␈α∞the␈α
ways␈α
of␈α
viewing␈α∞something␈α
as␈α
a␈α
C,␈α
AM␈α∞ripples␈α
away␈α
from␈α
C␈α∞in␈α
the
␈↓ α,␈↓Spec␈α
direction,␈α
gathering␈α
all␈α
the␈α
View␈α
facets␈α
along␈α
the␈α
way.␈α
 All␈α
of␈α
their␈α
entries␈α∞are␈α
valid
␈↓ α,␈↓entries for C.View as well.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε74␈↓-


␈↓ α,␈↓In␈αaddition␈αto␈αthese␈αbuilt-in␈αways␈αof␈α
using␈αthe␈αViews␈αfacets,␈αsome␈αspecial␈αuses␈αare␈α
made␈αin
␈↓ α,␈↓individual␈α∂heuristic␈α∂rules.␈α∂ Here␈α∂is␈α∂a␈α∂heuristic␈α∂rule␈α∂which␈α∂employs␈α∂the␈α∂Viewing␈α∂facets␈α∞of
␈↓ α,␈↓relevant concepts in order to ≡nd some examples of a given concept C:

␈↓ α,␈↓¬␈↓ αlIF the current task is to Fill-in Examples of C,
␈↓ α,␈↓¬␈↓ β,and C has some entries on its View facet,
␈↓ α,␈↓¬␈↓ β,and one of those entries <X,P> indicates a concept X which has some known Examples,
␈↓ α,␈↓¬␈↓ αlTHEN run the associated program P on each member of Examples(X),
␈↓ α,␈↓¬␈↓ β,and␈α∂add␈α∂the␈α∂following␈α∂task␈α∂to␈α∂the␈α∂agenda:␈α∂"Check␈α∂Examples␈α∂of␈α∂C",␈α∂for␈α∂the␈α∂following
␈↓ α,␈↓¬␈↓ ∧,reason:␈α"Some␈αvery␈αrisky␈αtechniques␈αwere␈αused␈αto␈αfind␈αexamples␈αof␈αC",␈αand
␈↓ α,␈↓¬␈↓ ∧,that␈α∞reason's␈α
rating␈α∞is␈α∞computed␈α
as:␈α∞Average(Worth(X),␈α
||the␈α∞examples␈α∞of␈α
C
␈↓ α,␈↓¬␈↓ ∧,found in this manner||).

␈↓ α,␈↓Say␈α∂the␈α∂task␈α∂selected␈α∂from␈α∞the␈α∂agenda␈α∂was␈α∂"Fill-in␈α∂Examples␈α∞of␈α∂Sets".␈α∂ We␈α∂saw␈α∂that␈α∞one
␈↓ α,␈↓entry␈α
on␈α∞Sets.View␈α
was␈α
␈↓¬Structure:␈α∞λ(x)␈α
Enclose-in-braces(x)␈↓.␈α∞ Thus␈α
it␈α
is␈α∞of␈α
the␈α∞form␈α
<X,P>,
␈↓ α,␈↓with␈α∞X=Structure.␈α
The␈α∞above␈α∞heuristic␈α
rule␈α∞will␈α
trigger␈α∞if␈α∞any␈α
examples␈α∞of␈α∞Structures␈α
are
␈↓ α,␈↓known.␈α
The␈α
rule␈αwill␈α
then␈α
use␈α
the␈αView␈α
facet␈α
of␈αSets␈α
to␈α
≡nd␈α
some␈αexamples␈α
of␈α
Sets.␈α So␈α
AM
␈↓ α,␈↓will␈α⊃go␈α⊃o≥,␈α⊃gathering␈α∩all␈α⊃the␈α⊃examples␈α⊃of␈α⊃structures.␈α∩ Since␈α⊃Lists␈α⊃is␈α⊃a␈α∩Specialization␈α⊃of
␈↓ α,␈↓Structure,␈α∂the␈α∞computation␈α∂of␈α∞Examples(Structures)␈α∂will␈α∞eventually␈α∂ripple␈α∂downwards␈α∞and
␈↓ α,␈↓ask␈αfor␈αExamples␈αof␈αLists.␈αIf␈αthe␈αExamples␈αfacet␈αof␈αLists␈αcontains␈αthe␈αentry␈α<z,a,c,a,a>,␈αthen
␈↓ α,␈↓this␈α
will␈α
be␈α
retrieved␈α
as␈α
one␈α
of␈α
the␈α
members␈α
of␈α
Examples(Structure).␈α
The␈α
heuristic␈α
rule␈α
takes
␈↓ α,␈↓each␈α∂such␈α∂member␈α⊂in␈α∂turn,␈α∂and␈α∂feeds␈α⊂it␈α∂to␈α∂Set.View's␈α∂little␈α⊂program␈α∂P.␈α∂In␈α∂this␈α⊂case,␈α∂the
␈↓ α,␈↓program replaces the list brackets with set braces, thus converting <z,a,c,a,a> to {z,a,c,a,a}.

␈↓ α,␈↓In␈αthis␈α
manner,␈αall␈α
the␈αexisting␈α
structures␈αwill␈α
be␈αconverted␈α
into␈αsets,␈α
to␈αprovide␈αexamples␈α
of
␈↓ α,␈↓sets.␈α
 After␈α
all␈α
such␈α
conversions␈α
take␈α
place,␈αa␈α
great␈α
number␈α
of␈α
potential␈α
examples␈α
of␈αSets␈α
will
␈↓ α,␈↓exist.␈α
The␈α
≡nal␈α
action␈α
of␈α
the␈α
right␈α
side␈α
of␈αthe␈α
above␈α
heuristic␈α
rule␈α
is␈α
to␈α
add␈α
the␈α
new␈αtask
␈↓ α,␈↓"␈↓¬Check␈α⊂examples␈α∂of␈α⊂Sets␈↓"␈α⊂to␈α∂the␈α⊂agenda.␈α⊂When␈α∂this␈α⊂gets␈α⊂selected,␈α∂all␈α⊂the␈α⊂"slightly␈α∂wrong"
␈↓ α,␈↓examples will be ≡xed up. For example, {z,a,c,a,a} will be converted to {a,c,z}.

␈↓ α,␈↓If␈α∂any␈α∂reliance␈α⊂is␈α∂made␈α∂on␈α⊂those␈α∂unchecked␈α∂examples,␈α∂there␈α⊂is␈α∂the␈α∂danger␈α⊂of␈α∂incorrectly
␈↓ α,␈↓rejecting␈α∞a␈α∞valid␈α∞conjecture.␈α∞This␈α∞is␈α∞not␈α
too␈α∞serious,␈α∞since␈α∞the␈α∞very␈α∞≡rst␈α∞such␈α∞reliance␈α
will
␈↓ α,␈↓boost␈α
the␈α
priority␈α
of␈α
the␈α
task␈α
"␈↓¬Check␈α
examples␈α
of␈α
Sets␈↓",␈α
and␈α
it␈α
would␈α
then␈α
probably␈α
be␈α
the
␈↓ α,␈↓very next task chosen.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.5. Intuitions␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|The␈α∞mathematician␈α∂does␈α∞not␈α∂work␈α∞like␈α∂a␈α∞machine;␈α∂we␈α∞cannot␈α∂overemphasize␈α∞the
␈↓ α,␈↓β␈↓ α|fundamental␈αrole␈αplayed␈αin␈αhis␈α
research␈αby␈αa␈αspecial␈αintuition␈α(frequently␈α
wrong),
␈↓ α,␈↓β␈↓ α|which␈α⊂is␈α⊂not␈α⊂common-sense,␈α⊂but␈α⊃rather␈α⊂a␈α⊂divination␈α⊂of␈α⊂the␈α⊂regular␈α⊃behavior␈α⊂he
␈↓ α,␈↓β␈↓ α|expects of mathematical beings.

␈↓ α,␈↓¬␈↓ ε\-- Bourbaki



␈↓ α,␈↓This␈αfacet␈αturned␈αout␈αto␈α
be␈αa␈α"dud",␈αand␈αwas␈αlater␈α
excised␈αfrom␈αall␈αthe␈αconcepts.␈αIt␈α
will␈αbe
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε75␈↓-

␈↓ α,␈↓described␈α
below␈α
anyway,␈α
for␈α
the␈α
bene≡t␈α
of␈α
future␈α
researchers.␈α
 Feel␈α
free␈α
to␈α
skip␈α
directly␈α
to
␈↓ α,␈↓the next subsection.

␈↓ α,␈↓The␈α
initial␈α
idea␈α
was␈α
to␈α
have␈α
a␈α
set␈αof␈α
a␈α
few␈α
(3-10)␈α
large,␈α
global,␈α
opaque␈α
LISP␈αfunctions.␈α
Each
␈↓ α,␈↓of␈α∞these␈α
functions␈α∞would␈α∞be␈α
termed␈α∞an␈α
"Intuition"␈α∞and␈α∞would␈α
have␈α∞some␈α∞suggestive␈α
name
␈↓ α,␈↓like␈α∩"jigsaw-puzzle",␈α⊃"see-saw",␈α∩"archery",␈α⊃etc.␈α∩ Each␈α⊃function␈α∩would␈α⊃somehow␈α∩model␈α⊃the
␈↓ α,␈↓particular␈α∞activity␈α∞implied␈α
by␈α∞its␈α∞name.␈α
There␈α∞would␈α∞be␈α
a␈α∞multitude␈α∞of␈α∞parameters␈α
which
␈↓ α,␈↓could␈α⊃be␈α⊃speci≡ed␈α⊃by␈α∩the␈α⊃"caller"␈α⊃as␈α⊃if␈α⊃they␈α∩were␈α⊃the␈α⊃arguments␈α⊃of␈α⊃the␈α∩function.␈α⊃ The
␈↓ α,␈↓function␈α∞would␈α∞then␈α∞work␈α∞to␈α∞≡ll␈α∞in␈α∞values␈α∞for␈α∞any␈α∞unspeci≡ed␈α∞parameters.␈α∞ That's␈α∂all␈α∞the
␈↓ α,␈↓function␈α∀does.␈α∃ The␈α∀caller␈α∃would␈α∀also␈α∃have␈α∀to␈α∃specify␈α∀which␈α∃parameters␈α∀were␈α∃to␈α∀be
␈↓ α,␈↓considered as the "results" of the function.

␈↓ α,␈↓For␈α
the␈α
see-saw,␈α
the␈αcaller␈α
might␈α
provide␈α
the␈α
weight␈αof␈α
the␈α
left-hand-side␈α
sitter,␈α
and␈αthe␈α
≡nal
␈↓ α,␈↓position␈αof␈αthe␈αsee-saw,␈αand␈αask␈αfor␈αthe␈αweight␈αof␈αthe␈αright-hand␈αsitter.␈αThe␈αfunction␈αwould
␈↓ α,␈↓then␈α∂compute␈α∞that␈α∂weight␈α∂(as␈α∞any␈α∂random␈α∂number␈α∞greater/less-than␈α∂the␈α∂left-hand␈α∞weight,
␈↓ α,␈↓depending␈α∞on␈α
the␈α∞desired␈α∞tilt␈α
of␈α∞the␈α
board).␈α∞ Or,␈α∞the␈α
caller␈α∞might␈α
specify␈α∞the␈α∞two␈α
weights
␈↓ α,␈↓and ask for the ≡nal position.

␈↓ α,␈↓The␈α
See-saw␈α
function␈α
is␈α
an␈α
expert␈α
on␈α
this␈α
subject;␈α
it␈α
has␈α
e≠cient␈α
code␈α
for␈α∞computing␈α
any
␈↓ α,␈↓values␈α∞which␈α∂can␈α∞be␈α∂computed,␈α∞and␈α∞for␈α∂randomly␈α∞instantiating␈α∂any␈α∞variables␈α∂which␈α∞may
␈↓ α,␈↓take␈αon␈αany␈αvalue␈α(e.g.,␈αthe␈α≡rst␈αnames␈αof␈αthe␈αpeople␈αdoing␈αthe␈αsitting).␈αWhen␈αan␈α
individual
␈↓ α,␈↓call␈α
is␈αmade␈α
on␈α
this␈αfunction,␈α
the␈α
caller␈αis␈α
not␈α
told␈αhow␈α
the␈α
≡nal␈αvalues␈α
of␈α
the␈αvariables␈α
were
␈↓ α,␈↓computed, only what those values end up as.

␈↓ α,␈↓So␈α
the␈α
Intuitions␈α∞were␈α
to␈α
be␈α∞experimental␈α
laboratories␈α
for␈α
AM,␈α∞wherein␈α
it␈α
could␈α∞get␈α
some
␈↓ α,␈↓(simulated)␈α
real-world␈αempirical␈α
data.␈α
 If␈αthe␈α
seesaw␈αwere␈α
the␈α
Intuition␈αfor␈α
">",␈α
and␈αweight
␈↓ α,␈↓corresponded␈α
to␈α
Numbers,␈α
then␈α
several␈α
relationships␈α
might␈α
be␈α
visualized␈α
intuitively␈α(like␈α
the
␈↓ α,␈↓anti-symmetry␈αof␈α">").␈α
This␈αis␈αa␈αnice␈α
idea,␈αbut␈αin␈αpractice␈α
the␈αonly␈αrelationships␈α
derived␈αin
␈↓ α,␈↓this␈αway␈αwere␈αthe␈αones␈αthat␈αwere␈αthought␈αup␈αwhile␈αtrying␈αto␈αencode␈αthe␈αIntuition␈αfunctions.
␈↓ α,␈↓This␈α⊂shameful␈α⊂behavior␈α⊂led␈α∂to␈α⊂the␈α⊂excision␈α⊂of␈α∂the␈α⊂Intuitions␈α⊂facets␈α⊂completely␈α⊂from␈α∂the
␈↓ α,␈↓system.

␈↓ α,␈↓As␈αanother␈αexample,␈αsuppose␈αAM␈αis␈αconsidering␈αcomposing␈αtwo␈αrelations␈αR␈αand␈αS.␈α If␈αthey
␈↓ α,␈↓have␈αno␈αcommon␈αIntuition␈αreference,␈αthen␈αperhaps␈αthey're␈αnot␈αmeaningfully␈αcomposable.␈α If
␈↓ α,␈↓they␈α∞do␈α∂both␈α∞tie␈α∂into␈α∞the␈α∂same␈α∞Intuition␈α∂function,␈α∞then␈α∂perhaps␈α∞that␈α∂function␈α∞can␈α∂tell␈α∞us
␈↓ α,␈↓something␈α∂about␈α∂the␈α∞composition.␈α∂This␈α∂is␈α∂a␈α∞nice␈α∂idea,␈α∂but␈α∞in␈α∂practice␈α∂very␈α∂few␈α∞prunings
␈↓ α,␈↓were accomplished this way, and no unanticipated combinations were fused.

␈↓ α,␈↓Each␈α⊃Intuition␈α∩entry␈α⊃is␈α⊃like␈α∩a␈α⊃"way␈α⊃in"␈α∩to␈α⊃one␈α⊃of␈α∩the␈α⊃few␈α⊃global␈α∩scenarios.␈α⊃ It␈α∩can␈α⊃be
␈↓ α,␈↓characterized as follows:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈αOne␈αof␈αthe␈αsalient␈α
features␈αof␈αthese␈αentries␈α¬␈αand␈α
of␈αthe␈αscenarios␈α¬␈αis␈αthat␈α
AM␈αis
␈↓ α,␈↓␈↓ β≤absolutely␈αforbidden␈αto␈αlook␈αinside␈αthem,␈αto␈αtry␈αto␈αanalyze␈αthem.␈α They␈αare␈α␈↓β␈↓&opaque␈↓)αβ␈↓.
␈↓ α,␈↓␈↓ β≤Most␈αIntuition␈αfunctions␈α
use␈αnumbers␈αand␈αarithmetic,␈α
and␈αit␈αwould␈αbe␈α
pointless␈αto
␈↓ α,␈↓␈↓ β≤say that AM discovered such concepts if it had access to those algorithms all along.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α⊃The␈α⊂second␈α⊃characteristic␈α⊂of␈α⊃an␈α⊂Intuition␈α⊃is␈α⊂that␈α⊃it␈α⊂be␈α⊃␈↓β␈↓&fallible␈↓)αβ␈↓.␈α⊂ As␈α⊃with␈α⊂human
␈↓ α,␈↓␈↓ β≤intuition,␈α∀there␈α∀is␈α∀no␈α∀guarantee␈α∀that␈α∀what␈α∀is␈α∀suggested␈α∀will␈α∀be␈α∃veri≡ed␈α∀even
␈↓ α,␈↓␈↓ β≤empirically,␈α∪let␈α∪alone␈α∩formally.␈α∪ Not␈α∪only␈α∪does␈α∩this␈α∪make␈α∪the␈α∪programming␈α∩of
␈↓ α,␈↓␈↓ β≤Intuition␈α
functions␈α
easier,␈α
it␈α∞was␈α
meant␈α
to␈α
provide␈α∞a␈α
degree␈α
of␈α
"fairness"␈α∞to␈α
them.
␈↓ α,␈↓␈↓ β≤AM␈αwasn't␈αcheating␈αquite␈αas␈αmuch␈αif␈αthe␈αSee-saw␈αfunction␈αwas␈αonly␈αantisymmetric
␈↓ α,␈↓␈↓ β≤90% of the time.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε76␈↓-

␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈αNevertheless,␈αthe␈αintuitions␈αare␈α
very␈α␈↓β␈↓&suggestive␈↓)αβ␈↓.␈α Many␈αconjectures␈αcan␈α
be␈αproposed
␈↓ α,␈↓␈↓ β≤only␈α
via␈α
them.␈α
Some␈α
analogies␈α
(see␈α
the␈α
next␈α
subsection)␈α
can␈α
also␈α
be␈α
suggested␈α
via
␈↓ α,␈↓␈↓ β≤common intuitions.

␈↓ α,␈↓After␈αthey␈αwere␈αcoded␈αand␈αrunning,␈αI␈αdecided␈αthat␈αthe␈αintuition␈αfunctions␈αwere␈αunfair;␈αthey
␈↓ α,␈↓contained␈α∩some␈α∩major␈α∩discoveries␈α∩"built-in"␈α∩to␈α∩them.␈α∩ They␈α∩had␈α∩the␈α∩power␈α∩to␈α∩propose
␈↓ α,␈↓otherwise-obscure␈α
new␈αconcepts␈α
and␈α
potential␈αrelationships.␈α
 They␈α
contributed␈αnothing␈α
other
␈↓ α,␈↓than␈α⊃what␈α⊂was␈α⊃originally␈α⊃programmed␈α⊂into␈α⊃them;␈α⊂␈↓βthey␈α⊃were␈α⊃not␈α⊂synergetic␈↓.␈α⊃ Due␈α⊃to␈α⊂this
␈↓ α,␈↓dubious␈αcharacter␈αof␈αthe␈αcontributions␈αby␈αAM's␈αfew␈αIntuition␈αfunctions,␈αthey␈αwere␈αremoved
␈↓ α,␈↓from␈α
the␈α
system.␈α
 All␈α
the␈α
examples␈α
and␈α
all␈α
the␈α
discoveries␈α
listed␈α
in␈α
this␈α
document␈αwere␈α
made
␈↓ α,␈↓without their assistance.

␈↓ α,␈↓We␈α∞shall␈α∞now␈α∞drop␈α∞this␈α∞de-implemented␈α∞idea.␈α∞ I␈α∞think␈α∞there␈α∞is␈α∞some␈α∞real␈α∂opportunity␈α∞for
␈↓ α,␈↓research␈α∞here.␈α∞ For␈α∂the␈α∞bene≡t␈α∞of␈α∞any␈α∂future␈α∞researchers␈α∞in␈α∞this␈α∂area,␈α∞let␈α∞me␈α∞point␈α∂to␈α∞the
␈↓ α,␈↓excellent discussion of analogic representations in [Sloman 71].


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.6. Analogies␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|The␈αwhole␈αidea␈αof␈αanalogy␈αis␈αthat␈α`E≥ects',␈αviewed␈αas␈αa␈αfunction␈αof␈αsituation,␈αis␈αa
␈↓ α,␈↓β␈↓ α|␈↓&continuous␈↓)αβ function.

␈↓ α,␈↓¬␈↓ ε\-- Poincare'



␈↓ α,␈↓As␈α⊂with␈α⊂Views␈α⊂and␈α⊂Intuitions,␈α⊂this␈α⊂facet␈α⊃is␈α⊂useful␈α⊂for␈α⊂shifting␈α⊂between␈α⊂one␈α⊂part␈α⊃of␈α⊂the
␈↓ α,␈↓universe␈α∩and␈α∩another.␈α∩ Views␈α∩dealt␈α⊃with␈α∩transformations␈α∩between␈α∩two␈α∩speci≡c␈α⊃concepts;
␈↓ α,␈↓Intuitions␈α∂dealt␈α⊂with␈α∂transformations␈α∂between␈α⊂a␈α∂bunch␈α∂of␈α⊂concepts␈α∂and␈α∂a␈α⊂large␈α∂standard
␈↓ α,␈↓scenario␈α⊂which␈α∂was␈α⊂carefully␈α∂hand-crafted␈α⊂in␈α∂advance.␈α⊂ In␈α∂contrast,␈α⊂this␈α∂facet␈α⊂deals␈α∂with
␈↓ α,␈↓transforming between a list of concepts and another list of concepts.

␈↓ α,␈↓Analogies␈α
operate␈α
on␈αa␈α
much␈α
grander␈αscale␈α
than␈α
Views.␈α Rather␈α
than␈α
simply␈αtransforming␈α
a
␈↓ α,␈↓few␈αisolated␈αitems,␈α
they␈αinitiate␈αthe␈α
creation␈αof␈αmany␈α
new␈αconcepts.␈α Unlike␈α
Intuitions,␈αthey
␈↓ α,␈↓are not limited in scope beforehand, nor are they opaque. They are dynamically proposed.

␈↓ α,␈↓The␈αconcept␈α
of␈α"prime␈αnumbers"␈α
is␈α␈↓βanalogous␈↓␈αto␈α
the␈αnotion␈αof␈α
"simple␈αgroups".␈α
 While␈αnot
␈↓ α,␈↓isomorphic,␈α⊂you␈α⊂might␈α⊂guess␈α⊂at␈α⊂a␈α∂few␈α⊂relationships␈α⊂involving␈α⊂simple␈α⊂groups␈α⊂just␈α⊂by␈α∂my
␈↓ α,␈↓telling you this fact: simple groups are to groups what primes are to numbers.␈↓	13␈↓

␈↓ α,␈↓Let's take 3 elementary examples, involving very fundamental concepts.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	13␈↓ε␈α	If␈αλa␈α	group␈αλis␈α	not␈α	simple,␈αλit␈α	can␈αλbe␈α	factored.␈α	 Unfortunately,␈αλthe␈α	factorization␈αλof␈α	a␈α	group␈αλinto␈α	simple␈αλgroups␈α	is␈α	not␈αλunique.
␈↓ α,␈↓ε␈↓ βLAnother␈αanalogizing␈α
contact:␈αFor␈α
each␈αprime␈α
p,␈αwe␈αcan␈α
associate␈αthe␈α
cyclic␈αgroup␈α
of␈αorder␈α
p,␈αwhich␈αis␈α
of
␈↓ α,␈↓ε␈↓ βLcourse␈αsimple.␈α AM␈αnever␈αcame␈αup␈αwith␈αthe␈αconcept␈αof␈αsimple␈αgroups;␈αthis␈αis␈αjust␈αan␈αillustration␈αfor␈α
the
␈↓ α,␈↓ε␈↓ βLreader.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε77␈↓-

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ AM was told how to ␈↓&View␈↓)αβ a set as if it were a bag.

␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓ AM was told it could ␈↓&Intuit␈↓)αβ the relation "␈↓¬≥␈↓" as the predetermined "See-saw" function.

␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓ AM, by itself, once ␈↓&Analogize␈↓)αβd that these two lists correspond:
␈↓ α,␈↓␈↓ αl␈↓ βL<Bags␈↓ ¬≤Same-length␈↓ εlOperations-on-and-into Bags>
␈↓ α,␈↓␈↓ αl␈↓ βL<Bags-of-T's␈↓ ¬≤Equality␈↓ εlThose operations restricted to Bags-of-T's>

␈↓ α,␈↓The␈α∩concept␈α⊃of␈α∩a␈α⊃bag,␈α∩all␈α⊃of␈α∩whose␈α⊃elements␈α∩are␈α⊃"T"'s,␈α∩is␈α⊃the␈α∩unary␈α∩representation␈α⊃of
␈↓ α,␈↓␈↓βnumbers␈↓␈αdiscovered␈αby␈αAM.␈α When␈αthe␈αabove␈αanalogy␈α(#3)␈αis␈α≡rst␈αproposed,␈αthere␈αare␈αmany
␈↓ α,␈↓known␈αBag-operations␈↓	14␈↓,␈αbut␈α
there␈αare␈αas␈α
yet␈αno␈αnumeric␈α
operations␈↓	15␈↓.␈α This␈αtriggers␈αone␈α
of
␈↓ α,␈↓AM's␈α∀heuristic␈α∃rules,␈α∀which␈α∃spurs␈α∀AM␈α∃on␈α∀to␈α∃≡nding␈α∀the␈α∃analogues␈α∀of␈α∃speci≡c␈α∀Bag-
␈↓ α,␈↓operations.␈αThat␈αis,␈αwhat␈αspecial␈αproperties␈αdo␈αthe␈αbag-operations␈αhave␈αwhen␈αtheir␈αdomains
␈↓ α,␈↓and/or␈α
ranges␈α
are␈α
restricted␈α
from␈α
Bags␈α
to␈α
Bags-of-T's␈α
(i.e,␈α
Numbers).␈α
 In␈α
this␈α
way,␈α
in␈α
fact,
␈↓ α,␈↓AM␈α
discovers␈αAddition␈α
(by␈αrestricting␈α
Bag-union␈αto␈α
the␈αDomain/range␈α
<Bags-of-T's␈αBags-
␈↓ α,␈↓of-T's → Bags-of-T's>), plus many other nice arithmetic functions.

␈↓ α,␈↓Well,␈αif␈αit␈αleads␈αto␈αthe␈αdiscovery␈αof␈αAddition,␈αthat␈αanalogy␈αis␈αcertainly␈αworth␈αhaving.␈α How
␈↓ α,␈↓would␈α∀an␈α∀analogy␈α∀like␈α∃that␈α∀be␈α∀proposed?␈α∀ As␈α∀the␈α∃reader␈α∀might␈α∀expect␈α∀by␈α∃now,␈α∀the
␈↓ α,␈↓mechanism␈α
is␈α
simply␈α
some␈α
heuristic␈α
rule␈α
adding␈α
it␈α
as␈α
an␈α
entry␈α
to␈α
the␈α
Analogies␈α
facet␈α
of␈α
a
␈↓ α,␈↓certain concept.  For example:

␈↓ α,␈↓¬␈↓ αlIF␈αthe␈αcurrent␈αtask␈αhas␈αjust␈αcreated␈αa␈αcanonical␈αspecialization␈αC2␈αof␈αconcept␈αC1,␈αwith␈α
respect
␈↓ α,␈↓¬␈↓ ∧,to␈αoperations␈αF1␈αand␈αF2,␈α[i.e.,␈αtwo␈αmembers␈αof␈αC2␈αsatisfy␈αF1␈αiff␈αthey␈αsatisfy
␈↓ α,␈↓¬␈↓ ∧,F2],
␈↓ α,␈↓¬␈↓ β,THEN add the following entry to the Analogies facet of C2:
␈↓ α,␈↓¬␈↓ β,␈↓ ¬≤<C1␈↓ ¬|F1␈↓ ε\Operations-on-and-into(C1)>
␈↓ α,␈↓¬␈↓ β,␈↓ ¬≤<C2␈↓ ¬|F2␈↓ ε\Those operations restricted to C2's>

␈↓ α,␈↓After␈α⊗generalizing␈α⊗"Equality"␈α⊗into␈α⊗the␈α⊗operation␈α⊗"Same-length",␈α⊗AM␈α⊗seeks␈α⊗to␈α⊗≡nd␈α∃a
␈↓ α,␈↓canonical␈↓	16␈↓␈α
representation␈α
for␈α
Bags.␈α
That␈α
is,␈α
AM␈α
seeks␈α
a␈α
canonizing␈α
function␈α
f,␈α∞such␈α
that
␈↓ α,␈↓(for any two bags x,y)

␈↓ α,␈↓¬␈↓ β,Same-length(x,y) iff Equal( f(x), f(y) ).

␈↓ α,␈↓Then␈αthe␈αrange␈αof␈αf␈αwould␈αdelineate␈αthe␈αset␈αof␈α"canonical"␈αBags.␈α AM␈α≡nds␈αsuch␈αan␈α
f␈αand
␈↓ α,␈↓such␈α
a␈α
set␈α∞of␈α
canonical␈α
bags:␈α
the␈α∞operation␈α
f␈α
involves␈α
replacing␈α∞each␈α
element␈α
of␈α
a␈α∞bag␈α
by
␈↓ α,␈↓"T",␈α
and␈α∞the␈α
canonical␈α
bags␈α∞are␈α
those␈α∞whose␈α
elements␈α
are␈α∞all␈α
T's.␈α
 In␈α∞this␈α
case,␈α∞the␈α
above
␈↓ α,␈↓rule␈αtriggers,␈αwith␈αC1=Bags,␈αC2=Bags-of-T's,␈αF1=Same-length,␈αF2=Equality,␈αand␈αthe␈α
analogy
␈↓ α,␈↓which is produced is the one shown as example #3 above.

␈↓ α,␈↓The␈αAnalogy␈α
facets␈αare␈αnot␈α
implemented␈αin␈αfull␈α
generality␈αin␈αthe␈α
existing␈αLISP␈α
version␈αof

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	14␈↓ε i.e., all entries on In-dom-of(Bag) and In-ran-of(Bag); a few of these are: Bag-insert, Bag-union, Bag-intersection
␈↓ α,␈↓ε␈↓	15␈↓ε Examples of Operation whose domain/range contains "Number".
␈↓ α,␈↓ε␈↓	16␈↓ε␈α
A␈α
natural,␈α
standard␈α
form.␈α
 All␈α
bags␈α
differing␈αin␈α
only␈α
"unimportant"␈α
ways␈α
should␈α
be␈α
transformed␈α
into␈α
the␈αsame␈α
canonical
␈↓ α,␈↓ε␈↓ βLform.  Two bags B1 and B2 which have the same length should get transformed into the same canonical bag.
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε78␈↓-

␈↓ α,␈↓AM,␈αand␈αfor␈αthat␈αreason␈αI␈αshall␈αrefrain␈αfrom␈αdelving␈αdeeper␈αinto␈αtheir␈αformat.␈α Since␈αgood
␈↓ α,␈↓research␈α∞has␈α
already␈α∞been␈α
done␈α∞on␈α
reasoning␈α∞by␈α
analogy␈↓	17␈↓,␈α∞I␈α
did␈α∞not␈α
view␈α∞it␈α
as␈α∞a␈α
central
␈↓ α,␈↓feature of my work. Very little space will be devoted to it in this document.

␈↓ α,␈↓An␈αimportant␈αtype␈αof␈αanalogy␈αwhich␈αwas␈αuntapped␈αby␈αAM␈αwas␈αthat␈αbetween␈αheuristics.␈α If
␈↓ α,␈↓two␈αsituations␈αwere␈αsimilar,␈αthen␈αconceivably␈αthe␈αheuristics␈αuseful␈αin␈αone␈αsituation␈αmight␈αbe
␈↓ α,␈↓useful␈α∂(or␈α∂have␈α∂useful␈α∂analogues)␈α∂in␈α∂the␈α∂new␈α∂situation.␈α∂ Perhaps␈α∂this␈α∂is␈α∂a␈α∂viable␈α∂way␈α∂of
␈↓ α,␈↓enlarging␈α∪the␈α∪known␈α∪heuristics.␈α∪ Such␈α∀"meta-level"␈α∪activities␈α∪were␈α∪kept␈α∪to␈α∀a␈α∪minimum
␈↓ α,␈↓throughout AM, and this proved to be a serious limitation.

␈↓ α,␈↓Let␈αme␈αstress␈αthat␈αthe␈αfailure␈αof␈αthe␈αIntuitions␈αfacets␈αto␈αbe␈αnontrivial␈αwas␈αdue␈αto␈αthe␈αlack␈αof
␈↓ α,␈↓spontaneity␈α
which␈α∞they␈α
possessed.␈α
 Analogies␈α∞facets␈α
were␈α
useful␈α∞and␈α
"fair"␈α
since␈α∞their␈α
uses
␈↓ α,␈↓were not predetermined by the author.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.7. Conjec's␈↓)αβ␈↓

␈↓ α,␈↓The␈α∞"Conjec"␈α∞facet␈α∞of␈α∞a␈α∞concept␈α∞C␈α∞is␈α∞a␈α∞list␈α∞of␈α∞relationships␈α∞which␈α∞involve␈α∞C.␈α∞ There␈α
are
␈↓ α,␈↓several uses for C.Conjec:

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈αStore␈αawkwardly-phrased␈α
conjectures:␈αthis␈αwasn't␈α
really␈αuseful,␈αsince␈αmost␈α
conjectures
␈↓ α,␈↓␈↓ β≤fell␈αout␈αnaturally␈αas␈αsimple␈αrelationships,␈αexpressable,␈α
e.g.,␈αas␈αa␈αsingle␈αGenl␈αarc,␈αor␈α
a
␈↓ α,␈↓␈↓ β≤single Isa arc.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α⊂Store␈α∂∨imsy␈α⊂conjectures:␈α∂apparent␈α⊂relationships␈α∂worth␈α⊂remembering␈α∂but␈α⊂not␈α∂quite
␈↓ α,␈↓␈↓ β≤believed yet.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓ Hold heuristics which notice and check conjectures.
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓␈α⊂Obviate␈α⊂the␈α∂need␈α⊂for␈α⊂many␈α∂nearly-indistinguishable␈α⊂concepts:␈α⊂Collapse␈α⊂the␈α∂entire
␈↓ α,␈↓␈↓ β≤essence␈α⊃of␈α⊃a␈α⊂concept␈α⊃like␈α⊃"Odd-primes"␈α⊃into␈α⊂one␈α⊃or␈α⊃two␈α⊃relationships␈α⊂involving
␈↓ α,␈↓␈↓ β≤"Primes"; then discard "Odd-primes".
␈↓ α,␈↓␈↓ αl␈↓¬5.␈↓␈α∃Untangling␈α∃paradoxes:␈α∃a␈α∃historic␈α∃record,␈α∃to␈α∃facilitate␈α∃backtracking␈α∃in␈α∃case␈α∃of
␈↓ α,␈↓␈↓ β≤catastrophe, which (luckily!) wasn't ever needed.
␈↓ α,␈↓␈↓ αl␈↓¬6.␈↓␈αImprove␈αexisting␈αalgorithms␈α(e.g.,␈αonce␈αyou␈αknow␈αprimes␈αare␈αodd,␈αhunt␈αonly␈α
through
␈↓ α,␈↓␈↓ β≤odd numbers), improve testing procedures, representations, etc.
␈↓ α,␈↓␈↓ αl␈↓¬7.␈↓␈α⊃Display␈α⊃AM's␈α⊃most␈α⊃impressive␈α⊃observed␈α⊂relationships␈α⊃in␈α⊃a␈α⊃form␈α⊃which␈α⊃is␈α⊂easily
␈↓ α,␈↓␈↓ β≤inspectable by the user.

␈↓ α,␈↓The␈αsyntax␈αof␈αthis␈αfacet␈αis␈αsimply␈αa␈αlist␈αof␈αconjectures,␈αwhere␈αeach␈αconjecture␈αhas␈αthe␈αform
␈↓ α,␈↓of␈αa␈αrelationship:␈α
(R␈αa␈αb␈α
c...d).␈αR␈αis␈α
the␈αname␈αof␈α
a␈αknown␈αoperation␈α
(in␈αwhich␈αcase,␈αabc...␈α
are
␈↓ α,␈↓its␈α⊃arguments␈α⊂and␈α⊃we␈α⊂claim␈α⊃that␈α⊃d␈α⊂is␈α⊃its␈α⊂value).␈α⊃ For␈α⊂instance,␈α⊃"␈↓¬(Same-size␈α⊃Insert(S,S)␈α⊂S
␈↓ α,␈↓¬False)␈↓"␈αis␈αa␈αconjecture␈αthat␈αinserting␈αa␈αset␈αinto␈αitself␈αwill␈αalways␈αgive␈αyou␈αa␈αset␈αof␈αa␈αdi≥erent
␈↓ α,␈↓length.  This conjecture happens to be true for ≡nite sets.





␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	17␈↓ε␈α	An␈α	excellent␈α	discussion␈α	of␈α	reasoning␈α	by␈α	analogy␈α	is␈α	found␈α	in␈α	[Polya␈α	54].␈α	Some␈α	early␈α	work␈α	on␈α	emulating␈α	this␈α	was␈αλreported
␈↓ α,␈↓ε␈↓ βLin [Evans 68]; a more recent thesis on this topic is [Kling 71].
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε79␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.8. De≡nitions␈↓)αβ␈↓

␈↓ α,␈↓A␈αtypical␈αway␈αto␈αdisambiguate␈αa␈αconcept␈αfrom␈αall␈αothers␈αis␈αto␈αprovide␈αa␈α"de≡nition"␈αfor␈αit.␈↓	18␈↓
␈↓ α,␈↓Almost␈α⊂every␈α⊂concept␈α⊂had␈α⊂some␈α⊂entries␈α∂initially␈α⊂supplied␈α⊂on␈α⊂its␈α⊂"De≡nitions"␈α⊂facet.␈α∂ The
␈↓ α,␈↓format␈α
of␈α
this␈α
facet␈αis␈α
a␈α
list␈α
of␈αentries,␈α
each␈α
one␈α
describing␈αa␈α
separate␈α
de≡nition.␈α
 A␈αsingle
␈↓ α,␈↓entry will have the following parts:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α∃Descriptors:␈α∀Recursive/Linear/Iterative,␈α∃Quick/Slow,␈α∃Opaque/Transparent,␈α∀Once-
␈↓ α,␈↓␈↓ β≤only/Early/Late, Destructive/Nondestructive.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α∞Relators:␈α∞Reducing␈α∞to␈α
the␈α∞de≡nition␈α∞of␈α∞concept␈α
X,␈α∞Same␈α∞as␈α∞Y␈α∞except...,␈α
Specialized
␈↓ α,␈↓␈↓ β≤version of Z, Using the de≡nition of W, etc.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α∞Predicate:␈α∂A␈α∞small,␈α∂executable␈α∞piece␈α∂of␈α∞LISP␈α∞code,␈α∂to␈α∞tell␈α∂if␈α∞any␈α∂given␈α∞item␈α∂is␈α∞an
␈↓ α,␈↓␈↓ β≤example of this concept.

␈↓ α,␈↓Here is a typical entry from the De≡nitions facet of the Set-union concept:
␈↓"␈↓ α,␈↓π␈↓ αl⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ≤⊃
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬ ␈↓π                                                                ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬ Descriptors: Slow, Recursive, Transparent ␈↓π                                ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬ ␈↓π                                                                ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬ Relators: Uses the algorithm for Set-insert, Uses the definition of Empty-set, ␈↓π     ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬            Uses the definition of Set-equal, Uses the algorithm for Some-member, ␈↓π   ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬            Uses the algorithm for Set-delete, Uses the definition of Set-union ␈↓π      ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬ ␈↓π                                                                ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬ Code: λ (A B C) ␈↓π                                                   ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬            IF   Empty-set.Defn(A)  THEN  Set-equal.Defn(B,C)   ELSE ␈↓π            ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬                    X ← Some-member.Alg(A) ␈↓π                                ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬                    A ← Set-delete.Alg(X,A) ␈↓π                                 ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬                    B ← Set-insert.Alg(X,B) ␈↓π                                  ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl} ␈↓¬                    Set-union.Defn(A,B,C) ␈↓π                                   ␈↓ ≤}
␈↓"␈↓ α,␈↓π␈↓ αl%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ ≤$

␈↓ α,␈↓Let me stress that this is just one entry from one facet of one concept.

␈↓ α,␈↓This␈α∂particular␈α⊂de≡nition␈α∂is␈α⊂not␈α∂very␈α⊂e≠cient,␈α∂but␈α⊂it␈α∂is␈α⊂described␈α∂as␈α⊂Transparent.␈α∂ That
␈↓ α,␈↓means␈α⊃it␈α⊃is␈α⊃very␈α⊃well␈α⊃suited␈α⊃to␈α⊃analysis␈α⊃and␈α⊃modi≡cation␈α⊃by␈α⊃AM␈α⊃itself.␈α∩ Suppose␈α⊃some
␈↓ α,␈↓heuristic␈α
rule␈α
wants␈αto␈α
generalize␈α
this␈α
de≡nition.␈αIt␈α
can␈α
peer␈α
inside␈αit,␈α
and,␈α
e.g.,␈α
replace␈αthe
␈↓ α,␈↓base step call on Set-equal, by a call on a generalization of Set-equal (say "Same-length"␈↓	19␈↓).

␈↓ α,␈↓How␈α∩could␈α∩␈↓βdi≥erent␈↓␈α∩de≡nitions␈α∩help␈α∩here?␈α∩Suppose␈α∩there␈α∩were␈α∩a␈α∩de≡nition␈α∩which␈α∩≡rst
␈↓ α,␈↓checked␈α⊃to␈α∩see␈α⊃if␈α⊃the␈α∩three␈α⊃arguments␈α∩were␈α⊃Set-equal␈α⊃to␈α∩each␈α⊃other,␈α⊃and␈α∩if␈α⊃so␈α∩then␈α⊃it
␈↓ α,␈↓instantly␈αreturned␈αT␈αas␈αthe␈αvalue␈αof␈αthe␈αde≡nition␈αpredicate;␈αotherwise,␈αit␈αrecurred␈αinto␈αSet-
␈↓ α,␈↓union.Defn␈αagain.␈α This␈αmight␈αbe␈αa␈αgood␈αalgorithm␈αto␈αtry␈αat␈αthe␈αvery␈αbeginning,␈αbut␈αif␈αthe

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	18␈↓ε␈α
As␈α	EPAM␈α
studies␈α	showed␈α
[Feigenbaum␈α	63],␈α
one␈α	can␈α
never␈α	be␈α
sure␈α	that␈α
this␈α	definition␈α
will␈α	specify␈α
the␈α
concept␈α	uniquely
␈↓ α,␈↓ε␈↓ βLfor␈αall␈αtime.␈α In␈αthe␈αdistant␈αfuture,␈αsome␈αnew␈α
concept␈αmay␈αdiffer␈αin␈αways␈αthought␈αto␈αbe␈αignorable␈αat␈α
the
␈↓ α,␈↓ε␈↓ βLpresent time.
␈↓ α,␈↓ε␈↓	19␈↓ε For disjoint sets, the new definition would specify the operation which we call "addition".
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε80␈↓-

␈↓ α,␈↓Equality␈α⊂test␈α⊂fails,␈α⊂we␈α⊂don't␈α⊂want␈α⊂to␈α⊂keep␈α⊂recurring␈α⊂into␈α⊂this␈α⊂de≡nition.␈α⊂ This␈α∂algorithm
␈↓ α,␈↓should thus have a descriptor labelling it ONCE-ONLY EARLY.

␈↓ α,␈↓There are three purposes to having descriptors and relators hanging around:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈αFor␈αthe␈αbene≡t␈αof␈αthe␈αuser.␈αAM␈αappears␈αmore␈αintelligent␈αbecause␈αit␈αcan␈α␈↓βdescribe␈↓␈αthe
␈↓ α,␈↓␈↓ β≤kind of de≡nition it is using ¬ and why.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈αFor␈α
the␈αsake␈αof␈α
e≠ciency.␈αWhen␈αall␈α
AM␈αwants␈αto␈α
do␈αis␈αto␈α
evaluate␈αSet-union(A,B),
␈↓ α,␈↓␈↓ β≤it's␈α∂best␈α⊂just␈α∂to␈α∂grab␈α⊂a␈α∂␈↓βfast␈↓␈α⊂de≡nition.␈α∂ When␈α∂trying␈α⊂to␈α∂generalize␈α⊂Set-union,␈α∂it's
␈↓ α,␈↓␈↓ β≤more␈α∞appropriate␈α∞to␈α∞modify␈α∞a␈α∞very␈α∞clean,␈α∞transparent␈α∞de≡nition␈α∞¬␈α∞even␈α∞if␈α∞it␈α∞is␈α
a
␈↓ α,␈↓␈↓ β≤slow one.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α∞For␈α∂the␈α∞bene≡t␈α∂of␈α∞the␈α∂heuristic␈α∞rules.␈α∂ Often,␈α∞a␈α∂left-␈α∞or␈α∂a␈α∞right-hand-side␈α∂will␈α∞ask
␈↓ α,␈↓␈↓ β≤about␈α⊂a␈α⊂certain␈α∂kind␈α⊂of␈α⊂de≡nition.␈α∂ For␈α⊂example,␈α⊂␈↓¬"If␈α∂a␈α⊂transparent␈α⊂definition␈α⊂of␈α∂X
␈↓ α,␈↓¬␈↓ β≤exists, then try to specialize X"␈↓.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.9. Algorithms␈↓)αβ␈↓

␈↓ α,␈↓Earlier,␈α∞we␈α∞said␈α
that␈α∞each␈α∞concept␈α
can␈α∞have␈α∞any␈α
facets␈α∞from␈α∞the␈α
universal␈α∞≡xed␈α∞set␈α∞of␈α
25
␈↓ α,␈↓facets.␈α
 This␈α
is␈α
not␈α
strictly␈α
true.␈α
Sometimes,␈αa␈α
whole␈α
class␈α
of␈α
concepts␈α
will␈α
possess␈α
a␈αcertain
␈↓ α,␈↓type␈α∞of␈α∞facet␈α∞which␈α∞no␈α
others␈α∞may␈α∞meaningfully␈α∞have.␈α∞That␈α
is,␈α∞there␈α∞will␈α∞be␈α∞a␈α∞␈↓βdomain␈α
of
␈↓ α,␈↓βapplicability␈↓␈α∞for␈α∞the␈α∞facet,␈α∞just␈α∞as␈α∞we␈α∞de≡ned␈α∞such␈α∞domains␈α∞of␈α∞applicability␈α∂for␈α∞heuristics.
␈↓ α,␈↓For example, consider the "Domain/Range" facet. It is meaningful only to "operations".

␈↓ α,␈↓Let's␈αview␈αthis␈αin␈αa␈αmore␈αgeneral␈αlight.␈α For␈αeach␈αfacet␈αf,␈αthe␈αonly␈αconcepts␈αwhich␈αcan␈αhave
␈↓ α,␈↓entries␈α∩for␈α∩facet␈α∩f␈α∩are␈α∩examples␈α∩of␈α∩some␈α∩particular␈α∩concept␈α∩D(f)␈α∩¬␈α∩the␈α∩"J"␈α∪stands␈α∩for
␈↓ α,␈↓"jargon".␈α D(f)␈αis␈αthe␈α
domain␈αof␈αapplicability␈αof␈αfacet␈α
f.␈α If␈αC␈αis␈α
any␈αconcept␈αwhich␈αis␈αnot␈α
an
␈↓ α,␈↓example␈α∞of␈α∞D(f),␈α∞then␈α∞it␈α∞can␈α∞never␈α∞meaningfully␈α∞possess␈α∞any␈α∞entries␈α∞for␈α∞that␈α∞facet␈α∞f.␈α∞ For
␈↓ α,␈↓almost␈αall␈α
facets␈αf,␈α
D(f)␈αis␈α
"Any-concept".␈αThus␈α
any␈αconcept␈α
can␈αpossess␈α
almost␈αany␈αfacet.␈α
For
␈↓ α,␈↓example,␈α&D(Defn)="Any-concept",␈α%so␈α&any␈α%concept␈α&may␈α%have␈α&de≡nitions.␈α% But
␈↓ α,␈↓D(Domain/range)="Operation". So only operations can have domain/range facets.

␈↓ α,␈↓Similarly,␈α→D(Algorithms)="Actives".␈α→This␈α~facet␈α→is␈α→the␈α~subject␈α→of␈α→this␈α~section.␈α→ The
␈↓ α,␈↓Algorithms␈α⊗facet␈α⊗is␈α⊗present␈α⊗for␈α⊗all␈α∃¬␈α⊗but␈α⊗only␈α⊗for␈α⊗¬␈α⊗Actives␈α⊗(predicates,␈α∃relations,
␈↓ α,␈↓operations).

␈↓ α,␈↓The␈α
representation␈α
is,␈α
as␈α
usual,␈α
a␈α∞list␈α
of␈α
entries,␈α
each␈α
one␈α
describing␈α
a␈α∞separate␈α
algorithm,
␈↓ α,␈↓and␈αeach␈αone␈αhaving␈αthree␈αparts:␈αDescriptors,␈αRelators,␈αProgram.␈α Note␈αthe␈αsimilarity␈αto␈αthe
␈↓ α,␈↓format␈α∂for␈α∂the␈α⊂De≡nitions␈α∂facets␈α∂of␈α⊂concepts.␈α∂ Instead␈α∂of␈α⊂a␈α∂LISP␈α∂predicate,␈α⊂however,␈α∂the
␈↓ α,␈↓Algorithms␈αfacets␈αpossess␈αa␈αLISP␈αfunction␈α(an␈αexecutable␈αpiece␈αof␈αcode␈αwhose␈αvalue␈αwill␈αin
␈↓ α,␈↓general be other than True/False).

␈↓ α,␈↓All␈α
the␈α∞details␈α
about␈α∞understanding␈α
the␈α∞descriptors␈α
and␈α∞relators␈α
are␈α∞embedded␈α
in␈α∞the␈α
≡ne
␈↓ α,␈↓structure␈α∀of␈α∀the␈α∀heuristic␈α∀rules.␈α∀A␈α∀left-hand-side␈α∀may␈α∀test␈α∀whether␈α∀a␈α∀certain␈α∀kind␈α∪of
␈↓ α,␈↓algorithm␈αexists␈αfor␈αa␈αgiven␈αconcept.␈αA␈αright-hand-side␈αwhich␈α≡lls␈αin␈αa␈αnew␈αalgorithm␈αmust
␈↓ α,␈↓also␈α
worry␈α
about␈α
≡lling␈α
in␈α
the␈α∞appropriate␈α
descriptors␈α
and␈α
relators.␈α
As␈α
with␈α∞newly␈α
created
␈↓ α,␈↓concepts,␈α∂such␈α∂information␈α∂is␈α∂trivial␈α∂to␈α∂≡ll␈α∂in␈α∂at␈α∂the␈α∂time␈α∂of␈α∂creation,␈α∂but␈α∂becomes␈α∂much
␈↓ α,␈↓harder after the fact.

␈↓ α,␈↓Here␈αis␈αa␈αtypical␈αheuristic␈αrule␈αwhich␈αresults␈αin␈αa␈αnew␈αentry␈αbeing␈αadded␈αto␈αthe␈αAlgorithms
␈↓ α,␈↓facet of the newly-created concept named Compose-Set-Intersect&Set-Intersect:
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε81␈↓-

␈↓ α,␈↓¬␈↓ αlIF the task is to Fillin Algorithms for F,
␈↓ α,␈↓¬␈↓ β,and F is an example of Composition
␈↓ α,␈↓¬␈↓ β,and F has a definition of the form F≡GoH,
␈↓ α,␈↓¬␈↓ β,and F has no transparent, nonrecursive algorithm,
␈↓ α,␈↓¬␈↓ αlTHEN add a new entry to the Algorithms facet of F,
␈↓ α,␈↓¬␈↓ β,with Descriptors: Transparent, Non-recursive
␈↓ α,␈↓¬␈↓ β,with Relators: Reducing to G.Alg and H.Alg, Using the Definition of <G.Domain>
␈↓ α,␈↓¬␈↓ β,with Program: λ (||<G.Domain>||,||<H.Domain>||-1,X)
␈↓ α,␈↓¬␈↓ ε(SETQ X (H.Alg ||<G.Domain>||))
␈↓ α,␈↓¬␈↓ ε(AND
␈↓ α,␈↓¬␈↓ ε<(<G.Domain>.Defn X)
␈↓ α,␈↓¬␈↓ ε<(G.Alg X ||<H.Domain>||-1))

␈↓ α,␈↓The␈αintent␈αof␈αthe␈αlittle␈αprogram␈αwhich␈αgets␈αcreated␈αis␈αto␈αapply␈αthe␈α≡rst␈αoperator,␈αcheck␈αthat
␈↓ α,␈↓the␈α∩result␈α∩is␈α∩in␈α∩the␈α∩domain␈α∩of␈α∩the␈α∩second,␈α∩and␈α∩then␈α∩apply␈α∩the␈α∩second␈α∩operator.␈α∩ The
␈↓ α,␈↓expression␈α||<G.Domain>||␈αmeans␈α≡nd␈α
a␈αdomain/range␈αentry␈αfor␈α
G,␈αcount␈αhow␈αmany␈α
domain
␈↓ α,␈↓components␈α⊃there␈α⊃are,␈α⊃and␈α⊃form␈α⊃a␈α⊃list␈α⊃that␈α⊃long␈α⊃from␈α⊃randomly-chosen␈α∩variable␈α⊃names
␈↓ α,␈↓(u,v,w,x,y,z).

␈↓ α,␈↓For␈α∂the␈α∂case␈α∂mentioned␈α∂above,␈α∂F␈α∂=␈α∂Compose-Set-Intersect&Set-Intersect,␈α∂G␈α∂=␈α∞Set-Intersect,
␈↓ α,␈↓and␈α
H␈α
=␈α
Set-Intersect.␈α
The␈α
domain␈α∞of␈α
G␈α
is␈α
a␈α
pair␈α
of␈α∞Sets,␈α
so␈α
||<G.Domain>||␈α
is␈α
a␈α
list␈α∞of␈α
2
␈↓ α,␈↓variables,␈α∞say␈α∞(u␈α
v).␈α∞Similarly,␈α∞||<H.Domain>||-1␈α∞is␈α
a␈α∞list␈α∞of␈α∞1␈α
variable,␈α∞say␈α∞(w).␈α∞Putting␈α
all
␈↓ α,␈↓this␈α∂together,␈α∂we␈α∂see␈α∂that␈α∂the␈α∂new␈α∂de≡nition␈α∂entry␈α∂created␈α∂for␈α∂Compose-Set-Intersect&Set-
␈↓ α,␈↓Intersect would look like this:
␈↓"␈↓ α,␈↓π␈↓ β,⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 
\⊃
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬ ␈↓π                                                        ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬ Descriptors: Non-Recursive, Transparent ␈↓π                         ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬ ␈↓π                                                        ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬ Relators: Reducing to Set-Intersect.Alg, Using the definition of Sets ␈↓π    ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬ ␈↓π                                                        ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬ Code: λ (u,v,w,X) ␈↓π                                          ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬            (SETQ X (Set-Intersect.Alg u v)) ␈↓π                        ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬            (AND ␈↓π                                            ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬               (Sets.Defn X) ␈↓π                                     ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,} ␈↓¬               (Set-Intersect.Alg X w) ␈↓π                             ␈↓ 
\}
␈↓"␈↓ α,␈↓π␈↓ β,%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 
\$

␈↓ α,␈↓At␈αtimes,␈αAM␈αwill␈αbe␈αcapable␈αof␈αproducing␈αonly␈αa␈αslow␈αalgorithm␈αfor␈αsome␈αnew␈αconcept␈αC.
␈↓ α,␈↓The␈α∪reader␈α∪who␈α∪wishes␈α∩to␈α∪know␈α∪more␈α∪about␈α∩rules␈α∪for␈α∪creating␈α∪and␈α∪improving␈α∩LISP
␈↓ α,␈↓algorithms␈α∞is␈α∞directed␈α∞to␈α∂[Darlington␈α∞and␈α∞Burstall␈α∞73].␈α∂ A␈α∞more␈α∞general␈α∞discussion␈α∂of␈α∞the
␈↓ α,␈↓principles involved can be found in [Simon 72].


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&5.2.10. Domain/Range␈↓)αβ␈↓

␈↓ α,␈↓Another␈αfacet␈αpossessed␈αonly␈αby␈αactive␈αconcepts␈αis␈αDomain/Range.␈α The␈αsyntax␈αof␈αthis␈αfacet
␈↓ α,␈↓is␈αquite␈αsimple.␈αIt␈αis␈αa␈αlist␈αof␈αentries,␈αeach␈αof␈αthe␈αform␈α␈↓¬<␈αD␈↓#v1␈↓#␈αD␈↓#v2␈↓#...␈α→␈αR␈α>␈↓,␈αwhere␈αthere␈α
can␈αbe
␈↓ α,␈↓any␈αnumber␈αof␈αD␈↓#vi␈↓#'s␈αpreceding␈αthe␈αarrow,␈αand␈αR␈αand␈αall␈αthe␈αD␈↓#vi␈↓#'s␈αare␈αthe␈αnames␈αof␈αconcepts.
␈↓ α,␈↓Semantically,␈α∞this␈α∞entry␈α∞means␈α∞that␈α∞the␈α∞active␈α
concept␈α∞may␈α∞be␈α∞run␈α∞on␈α∞a␈α∞list␈α∞of␈α
arguments
␈↓ α,␈↓␈↓εChapter 5␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε82␈↓-

␈↓ α,␈↓where␈αthe␈α≡rst␈αone␈α
is␈αan␈αexample␈αof␈αD␈↓#v1␈↓#,␈α
the␈αsecond␈αan␈αexample␈αof␈α
D␈↓#v2␈↓#,␈αetc.,␈αand␈αin␈αthat␈α
case
␈↓ α,␈↓will␈αreturn␈αa␈αvalue␈αguaranteed␈αto␈αbe␈αan␈αexample␈αof␈αR.␈α In␈αother␈αwords,␈αthe␈αconcept␈αmay␈αbe
␈↓ α,␈↓considered␈αa␈αrelation␈αon␈αthe␈αCartesian␈αproduct␈αD␈↓#v1␈↓#xD␈↓#v2␈↓#x...xR.␈α We␈αshall␈αsay␈αthat␈αthe␈α␈↓βdomain␈↓
␈↓ α,␈↓of␈α
the␈α
concept␈α
is␈α
D␈↓#v1␈↓#xD␈↓#v2␈↓#x...,␈α
and␈α
that␈α
its␈α
␈↓βrange␈↓␈α
is␈α
R.␈α
Each␈α
D␈↓#vi␈↓#␈α
is␈α
called␈α
a␈α
␈↓βcomponent␈↓␈α
of␈α
the
␈↓ α,␈↓domain.

␈↓ α,␈↓For example, here is what the Domain/Range facet of TIMES might look like:
␈↓"␈↓ α,␈↓π␈↓ βL⊂ααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 	L⊃
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬ { ␈↓π                                            ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬    < Numbers Numbers → Numbers > ␈↓π                 ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬    < Odd-numbers Odd-numbers → Odd-numbers > ␈↓π      ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬    < Even-Numbers Even-Numbers → Even-numbers > ␈↓π    ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬    < Odd-numbers Even-Numbers → Even-Numbers > ␈↓π    ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬    < Perf-Squares Perf-Squares → Perf-Squares > ␈↓π      ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬    < Bags-of-Numbers → Numbers > ␈↓π                 ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL} ␈↓¬                                                    } ␈↓π            ␈↓ 	L}
␈↓"␈↓ α,␈↓π␈↓ βL%ααααααααααααααααααααααααααααααααααααααααααααααα␈↓ 	L$
␈↓ α,␈↓␈↓ ,-␈↓ε83␈↓-




␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬R␈↓∧Chapter 6.   Results␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓




␈↓ α,␈↓␈↓ ¬c␈↓∧␈↓&6.1. What AM Did␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|Now␈αwe␈αhave␈αseen␈αthat␈αmathematical␈αwork␈αis␈αnot␈αsimply␈αmechanical,␈αthat␈αit␈αcould
␈↓ α,␈↓β␈↓ α|not␈αbe␈αdone␈αby␈αa␈αmachine,␈αhowever␈αperfect.␈αIt␈αis␈αnot␈αmerely␈αa␈αquestion␈αof␈αapplying
␈↓ α,␈↓β␈↓ α|rules,␈α∞of␈α∞making␈α∂the␈α∞most␈α∞combinations␈α∞possible␈α∂according␈α∞to␈α∞certain␈α∂≡xed␈α∞laws.
␈↓ α,␈↓β␈↓ α|The␈α⊗combinations␈α⊗so␈α⊗obtained␈α↔would␈α⊗be␈α⊗exceedingly␈α⊗numerous,␈α↔useless,␈α⊗and
␈↓ α,␈↓β␈↓ α|cumbersome.␈α∩The␈α∩true␈α∩work␈α∩of␈α∩the␈α∩inventor␈α∩consists␈α∩in␈α∩choosing␈α∪among␈α∩these
␈↓ α,␈↓β␈↓ α|combinations␈α
so␈α
as␈α
to␈α∞eliminate␈α
the␈α
useless␈α
ones␈α
or␈α∞rather␈α
to␈α
avoid␈α
the␈α∞trouble␈α
of
␈↓ α,␈↓β␈↓ α|making␈α
them,␈α
and␈α
the␈αrules␈α
which␈α
must␈α
guide␈αthis␈α
choice␈α
are␈α
extremely␈α
≡ne␈αand
␈↓ α,␈↓β␈↓ α|delicate.␈α
It␈α
is␈α
almost␈α
impossible␈α
to␈α
state␈α
them␈α
precisely;␈α
they␈α
are␈α
felt␈α
rather␈αthan
␈↓ α,␈↓β␈↓ α|formulated.␈α⊂ Under␈α⊃these␈α⊂conditions,␈α⊃how␈α⊂imagine␈α⊂a␈α⊃sieve␈α⊂capable␈α⊃of␈α⊂applying
␈↓ α,␈↓β␈↓ α|them mechanically?

␈↓ α,␈↓¬␈↓ ε\-- Poincare'



␈↓ α,␈↓AM is both a mathematician of sorts, and a big computer program.

␈↓ α,␈↓By␈α⊃granting␈α⊃AM␈α⊃more␈α⊃anthropomorphic␈α⊃qualities␈α⊃than␈α⊃it␈α⊃deserves,␈α⊃we␈α⊃can␈α⊃describe␈α⊃its
␈↓ α,␈↓progress␈α⊃through␈α⊃elementary␈α⊃mathematics.␈α⊃ It␈α⊃rediscovered␈α⊃many␈α⊃well-known␈α∩concepts,␈α⊃a
␈↓ α,␈↓couple␈α∞interesting␈α
but␈α∞not-generally-known␈α
ones,␈α∞and␈α
several␈α∞concepts␈α
which␈α∞were␈α
hitherto
␈↓ α,␈↓unknown␈αand␈αshould␈αhave␈αstayed␈αthat␈αway.␈α Section␈α1.3,␈αon␈αpage␈α10,␈αrecaps␈αwhat␈αAM␈αdid,
␈↓ α,␈↓much as a historian might critically evaluate Euler's work.

␈↓ α,␈↓Instead␈α∂of␈α∂repeating␈α∂any␈α∞of␈α∂this␈α∂descriptive␈α∂prose␈α∂here,␈α∞Section␈α∂6.1.1␈α∂will␈α∂provide␈α∂a␈α∞very
␈↓ α,␈↓brief␈αlisting␈αof␈αwhat␈αAM␈αdid␈αin␈αa␈αsingle␈αgood␈αrun,␈αtask␈αby␈αtask.␈α These␈αtask-by-task␈αlistings
␈↓ α,␈↓are␈α∂not␈α∂complete␈α∂listings␈α∞of␈α∂every␈α∂task␈α∂AM␈α∞ever␈α∂attempted␈α∂in␈α∂any␈α∞of␈α∂its␈α∂many␈α∂runs,␈α∞but
␈↓ α,␈↓rather␈αa␈αtrace␈αof␈αa␈α
single,␈αbetter-than-average␈αrun␈αof␈αthe␈α
program.␈↓	1␈↓␈αThe␈αreader␈αmay␈αwish␈α
to
␈↓ α,␈↓consult the brief alphabetized glossary of concept names in the previous chapter (page ).

␈↓ α,␈↓Following␈α
this␈α
linear␈α
trace␈α
of␈α
AM's␈α
behavior␈α
is␈α
a␈α
more␈α
appropriate␈α
representation␈α
of␈αwhat␈α
it
␈↓ α,␈↓did:␈α⊂namely,␈α⊂a␈α⊂two-dimensional␈α⊂graph␈α⊂of␈α⊂that␈α⊂same␈α⊂behavior␈α⊂as␈α⊂seen␈α⊂in␈α∂"concept-space".
␈↓ α,␈↓This forms Section 6.1.2, and is found on page 92.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε␈αλIn␈αλfact,␈αλit␈α	is␈αλperhaps␈αλthe␈αλbest␈α	overall␈αλrun.␈αλIt␈αλoccurred␈αλin␈α	two␈αλstages␈αλ(due␈αλto␈α	space␈αλproblems;␈αλunimportant).␈αλIn␈α	this␈αλparticular
␈↓ α,␈↓ε␈↓ βLrun,␈αAM␈αmisses␈αthe␈αfew␈α"very␈αbest"␈αdiscoveries␈αit␈αever␈αmade,␈αsince␈αthe␈αruns␈αthey␈αoccurred␈αin␈αwent␈αin
␈↓ α,␈↓ε␈↓ βLsomewhat␈αλdifferent␈α	directions.␈αλIt␈α	also␈αλomits␈αλsome␈α	of␈αλthe␈α	more␈αλboring␈αλtasks:␈α	see,␈αλe.g.,␈α	the␈αλdescription␈α	of␈αλtask
␈↓ α,␈↓ε␈↓ βLnumber 69.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε84␈↓-

␈↓ α,␈↓By␈αunder-estimating␈α
AM's␈αsophistication,␈α
one␈αcan␈α
demand␈αanswers␈α
to␈αthe␈α
typical␈αquestions
␈↓ α,␈↓to␈α∂ask␈α⊂about␈α∂a␈α⊂computer␈α∂program:␈α∂how␈α⊂big␈α∂is␈α⊂it,␈α∂how␈α∂much␈α⊂cpu␈α∂time␈α⊂does␈α∂it␈α⊂use,␈α∂what
␈↓ α,␈↓language it's coded in, etc.  These are found in Section 6.1.3.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.1.1. Linear Task-by-task Summary of a Good Run␈↓)αβ␈↓


␈↓ α,␈↓␈↓ α\␈↓¬1.␈↓ Fill in examples of Compose. Failed, but suggested next task:
␈↓ α,␈↓␈↓ α\␈↓¬2.␈↓ Fill in examples of Set-union. Also failed, but suggested:
␈↓ α,␈↓␈↓ α\␈↓¬3.␈↓␈αFill␈α
in␈αexamples␈α
of␈αSets.␈α
Many␈αfound␈α(e.g.,␈α
by␈αinstantiating␈α
Set.Defn)␈αand␈α
then␈αmore
␈↓ α,␈↓␈↓ β≤derived from those examples (e.g., by running Union.Alg).
␈↓ α,␈↓␈↓ α\␈↓¬4.␈↓␈α⊂Fill␈α⊂in␈α⊂specializations␈α⊂of␈α⊂Sets␈α⊂(because␈α⊂it␈α⊂was␈α⊂very␈α⊂easy␈α⊂to␈α⊂≡nd␈α⊂examples␈α⊂of␈α∂Sets).
␈↓ α,␈↓␈↓ β≤Creation␈αof␈αnew␈α
concepts.␈αOne,␈αINT-Sets,␈αis␈α
related␈αto␈α"Singletons".␈α
 Another,␈α"BI-
␈↓ α,␈↓␈↓ β≤Sets", is all nests of braces (no atomic elements).
␈↓ α,␈↓␈↓ α\␈↓¬5.␈↓ Fill in examples of INT-Sets. This indirectly led to a rise in the worth of Equal.
␈↓ α,␈↓␈↓ α\␈↓¬6.␈↓␈α
Check␈α
all␈α
examples␈α
of␈α
INT-Sets.␈α
All␈α
were␈α
con≡rmed.␈α
 AM␈α
de≡nes␈α
the␈α
set␈α
of␈α
Nonempty
␈↓ α,␈↓␈↓ β≤INT-Sets; this is renamed "Singletons" by the user.
␈↓ α,␈↓␈↓ α\␈↓¬7.␈↓␈α∞Check␈α∞all␈α
examples␈α∞of␈α∞Sets.␈α
 To␈α∞check␈α∞a␈α
couple␈α∞conjectures,␈α∞AM␈α
will␈α∞soon␈α∞look␈α
for
␈↓ α,␈↓␈↓ β≤Bags and Osets.
␈↓ α,␈↓␈↓ α\␈↓¬8.␈↓ Fill in examples of Bags.
␈↓ α,␈↓␈↓ α\␈↓¬9.␈↓␈αFill␈αin␈αspecializations␈α
of␈αBags.␈αCreated␈αINT-Bags␈α
(contain␈αjust␈αone␈αkind␈α
of␈αelement),
␈↓ α,␈↓␈↓ β≤and BI-Bags (nests of parentheses).
␈↓ α,␈↓␈↓ α\␈↓¬10.␈↓ Fill in examples of Osets.
␈↓ α,␈↓␈↓ α\␈↓¬11.␈↓ Check examples of Osets.
␈↓ α,␈↓␈↓ α\␈↓¬12.␈↓ Fill in examples of Lists.
␈↓ α,␈↓␈↓ α\␈↓¬13.␈↓ Check examples of Lists.
␈↓ α,␈↓␈↓ α\␈↓¬14.␈↓ Fill in examples of All-but-≡rst.
␈↓ α,␈↓␈↓ α\␈↓¬15.␈↓ Fill in examples of All-but-last.
␈↓ α,␈↓␈↓ α\␈↓¬16.␈↓ Fill in specializations of All-but-last. Failed.
␈↓ α,␈↓␈↓ α\␈↓¬17.␈↓ Fill in examples of List-union.
␈↓ α,␈↓␈↓ α\␈↓¬18.␈↓ Fill in examples of Proj1.
␈↓ α,␈↓␈↓ α\␈↓¬19.␈↓ Check examples of All-but-≡rst.
␈↓ α,␈↓␈↓ α\␈↓¬20.␈↓ Check examples of All-but-last.
␈↓ α,␈↓␈↓ α\␈↓¬21.␈↓ Fill in examples of Proj2.
␈↓ α,␈↓␈↓ α\␈↓¬22.␈↓ Fill in examples of Empty-structures. 4 found.
␈↓ α,␈↓␈↓ α\␈↓¬23.␈↓ Fill in generalizations of Empty-structures. Failed.
␈↓ α,␈↓␈↓ α\␈↓¬24.␈↓ Check examples of List-union.
␈↓ α,␈↓␈↓ α\␈↓¬25.␈↓ Check examples of Bags. De≡ned Singleton-bags.
␈↓ α,␈↓␈↓ α\␈↓¬26.␈↓ Fill in examples of Bag-union.
␈↓ α,␈↓␈↓ α\␈↓¬27.␈↓ Check examples of Proj2.
␈↓ α,␈↓␈↓ α\␈↓¬28.␈↓ Fill in examples of Set-union.
␈↓ α,␈↓␈↓ α\␈↓¬29.␈↓ Check examples of Set-union. De≡ne λ (x,y) x∪y=x, later called Superset.
␈↓ α,␈↓␈↓ α\␈↓¬30.␈↓ Fill in examples of Bag-insert.
␈↓ α,␈↓␈↓ α\␈↓¬31.␈↓␈αCheck␈αexamples␈αof␈αBag-insert.␈αRange␈αis␈αreally␈αNonempty␈αbags.␈αIsolate␈αthe␈αresults␈αof
␈↓ α,␈↓␈↓ β≤insertion restricted to Singletons: call them Doubleton-bags.
␈↓ α,␈↓␈↓ α\␈↓¬32.␈↓ Fill in examples of Bag-intersect.
␈↓ α,␈↓␈↓ α\␈↓¬33.␈↓ Fill in examples of Set-insert.
␈↓ α,␈↓␈↓ α\␈↓¬34.␈↓␈α∞Check␈α∞examples␈α∞of␈α∞Set-insert.␈α∞Range␈α∞is␈α∞always␈α∞Nonempty␈α∞sets.␈α∞ De≡ne␈α∞λ␈α∞(x,S)␈α∞Set-
␈↓ α,␈↓␈↓ β≤insert(x,S)=S; i.e., set membership.  De≡ne Doubleton sets.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε85␈↓-

␈↓ α,␈↓␈↓ α\␈↓¬35.␈↓ Fill in examples of Bag-delete.
␈↓ α,␈↓␈↓ α\␈↓¬36.␈↓ Fill in examples of Bag-di≥erence.
␈↓ α,␈↓␈↓ α\␈↓¬37.␈↓ Check examples of Bag-intersect. De≡ne λ (x,y) x∩y=(); i.e. disjoint bags.
␈↓ α,␈↓␈↓ α\␈↓¬38.␈↓ Fill in examples of Set-intersect.
␈↓ α,␈↓␈↓ α\␈↓¬39.␈↓␈α∪Check␈α∪examples␈α∪of␈α∪Set-intersect.␈α∪De≡ne␈α∩λ␈α∪(x,y)␈α∪x∩y=x;␈α∪i.e.,␈α∪subset.␈α∪ Also␈α∩de≡ne
␈↓ α,␈↓␈↓ β≤disjoint sets: λ (x,y) x∩y={}.
␈↓ α,␈↓␈↓ α\␈↓¬40.␈↓ Fill in examples of List-intersect.
␈↓ α,␈↓␈↓ α\␈↓¬41.␈↓ Fill in examples of Equal. Very di≠cult to ≡nd examples; this led to:
␈↓ α,␈↓␈↓ α\␈↓¬42.␈↓ Fill in generalizations of Equal.  De≡ne "Same-size", "Equal-CARs", and some losers.
␈↓ α,␈↓␈↓ α\␈↓¬43.␈↓ Fill in examples of Same-size.
␈↓ α,␈↓␈↓ α\␈↓¬44.␈↓␈αApply␈αan␈αAlgorithm␈αfor␈αCanonize␈αto␈αthe␈αargs␈αSame-size␈αand␈αEqual.␈α AM␈α
eventually
␈↓ α,␈↓␈↓ β≤synthesizes␈α⊗the␈α∃canonizing␈α⊗function␈α∃"Size".␈α⊗ AM␈α∃de≡nes␈α⊗the␈α∃set␈α⊗of␈α∃canonical
␈↓ α,␈↓␈↓ β≤structures: bags of T's; this later gets renamed as "Numbers".
␈↓ α,␈↓␈↓ α\␈↓¬45.␈↓␈α⊃Restrict␈α⊃the␈α⊃domain/range␈α⊃of␈α∩Bag-union.␈α⊃ A␈α⊃new␈α⊃operation␈α⊃is␈α∩de≡ned,␈α⊃Number-
␈↓ α,␈↓␈↓ β≤union, with domain/range entry <Number Number → Bag>.
␈↓ α,␈↓␈↓ α\␈↓¬46.␈↓ Fill in examples of Number-union.  Many found.
␈↓ α,␈↓␈↓ α\␈↓¬47.␈↓␈α
Check␈α
the␈α
domain/range␈α
of␈α
Number-union.␈α
 Range␈α
is␈α
`Number'.␈α
 This␈α
operation␈αis
␈↓ α,␈↓␈↓ β≤renamed "Add2".
␈↓ α,␈↓␈↓ α\␈↓¬48.␈↓ Restrict the domain/range of Bag-intersect to Numbers. Renamed "Minimum".
␈↓ α,␈↓␈↓ α\␈↓¬49.␈↓ Restrict the domain/range of Bag-delete to Numbers.  Renamed "SUB1".
␈↓ α,␈↓␈↓ α\␈↓¬50.␈↓␈α∞Restrict␈α
the␈α∞domain/range␈α
of␈α∞Bag-insert␈α
to␈α∞Numbers.␈α
 AM␈α∞calls␈α
the␈α∞new␈α
operation
␈↓ α,␈↓␈↓ β≤"Number-insert". Its domain/range entry is <Anything Number → Bag>.
␈↓ α,␈↓␈↓ α\␈↓¬51.␈↓ Check the domain/range of Number-insert.  This doesn't lead anywhere.
␈↓ α,␈↓␈↓ α\␈↓¬52.␈↓ Restrict the domain/range of Bag-di≥erence to Numbers. This becomes "Subtract".
␈↓ α,␈↓␈↓ α\␈↓¬53.␈↓ Fill in examples of Subtract.  This leads to de≡ning the relation LEQ (␈↓¬≤␈↓).␈↓	2␈↓
␈↓ α,␈↓␈↓ α\␈↓¬54.␈↓ Fill in examples of LEQ. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬55.␈↓ Check examples of LEQ.
␈↓ α,␈↓␈↓ α\␈↓¬56.␈↓ Apply algorithm of Coalesce to LEQ.  LEQ(x,x) is Constant-True.
␈↓ α,␈↓␈↓ α\␈↓¬57.␈↓␈α
Fill␈α
in␈α
examples␈α
of␈α
Parallel-join2.␈α
 Included␈α
is␈α
Parallel-join2(Bags,Bags,Proj2),␈α
which
␈↓ α,␈↓␈↓ β≤is␈α
renamed␈α
"TIMES",␈α
and␈α
Parallel-join2(Structures,Structures,Proj1),␈α
a␈α
generalized
␈↓ α,␈↓␈↓ β≤Union operation renamed "G-Union", and a bunch of losers.
␈↓ α,␈↓␈↓ α\␈↓¬58.␈↓ ¬ ␈↓¬69.␈↓ Fill in and check examples of the operations just created.
␈↓ α,␈↓␈↓ α\␈↓¬70.␈↓␈α∂Fill␈α∂in␈α∂examples␈α∂of␈α∂Coalesce.␈α∂ Created:␈α∂Self-Compose,␈α∂Self-Insert,␈α∂Self-Delete,␈α∂Self-
␈↓ α,␈↓␈↓ β≤Add, Self-Times, Self-Union, etc.  Also: Coa-repeat2, Coa-join2, etc.
␈↓ α,␈↓␈↓ α\␈↓¬71.␈↓ Fill in examples of Self-Delete. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬72.␈↓ Check examples of Self-Delete.  Self-Delete is just Identity-op.
␈↓ α,␈↓␈↓ α\␈↓¬73.␈↓ Fill in examples of Self-Member. No positive examples found.
␈↓ α,␈↓␈↓ α\␈↓¬74.␈↓ Check examples of Self-Member.  Self-member is just Constant-False.
␈↓ α,␈↓␈↓ α\␈↓¬75.␈↓ Fill in examples of Self-Add. Many found. User renames this "Doubling".
␈↓ α,␈↓␈↓ α\␈↓¬76.␈↓ Check examples of Coalesce. Con≡rmed.
␈↓ α,␈↓␈↓ α\␈↓¬77.␈↓ Check examples of Add2. Con≡rmed.
␈↓ α,␈↓␈↓ α\␈↓¬78.␈↓ Fill in examples of Self-Times. Many found. Renamed "Squaring" by the user.
␈↓ α,␈↓␈↓ α\␈↓¬79.␈↓␈α∞Fill␈α∂in␈α∞examples␈α∞of␈α∂Self-Compose.␈α∞De≡ned␈α∞Squaring␈↓εo␈↓Squaring.␈α∂ Created␈α∞Add␈↓εo␈↓Add
␈↓ α,␈↓␈↓ β≤(two␈α∩versions:␈α∩Add21␈α∩which␈α∩is␈α∩λ␈α∩(x,y,z)␈α∩(x+y)+z,␈α∩and␈α∩Add22␈α∩which␈α∩is␈α⊃x+(y+z)).
␈↓ α,␈↓␈↓ β≤Similarly, two versions of Times␈↓εo␈↓Times and of Compose␈↓εo␈↓Compose.
␈↓ α,␈↓␈↓ α\␈↓¬80.␈↓ Fill in examples of Add21. (x+y)+z. Many are found.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	2␈↓ε␈α
If␈α
a␈α
larger␈α
number␈α
is␈α
"subtracted"␈αfrom␈α
a␈α
smaller,␈α
the␈α
result␈α
is␈α
zero.␈α AM␈α
explicitly␈α
defines␈α
the␈α
set␈α
of␈α
ordered␈αpairs␈α
of
␈↓ α,␈↓ε␈↓ βLnumbers having zero "difference". <x,y> is in that set iff x is less than or equal to y.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε86␈↓-

␈↓ α,␈↓␈↓ α\␈↓¬81.␈↓ Fill in examples of Add22. x+(y+z). Again many are found.
␈↓ α,␈↓␈↓ α\␈↓¬82.␈↓ Check examples of Squaring. Con≡rmed.
␈↓ α,␈↓␈↓ α\␈↓¬83.␈↓ Check examples of Add22.  Add21 and Add22 appear equivalent. But ≡rst:
␈↓ α,␈↓␈↓ α\␈↓¬84.␈↓␈α
Check␈α
examples␈αof␈α
Add21.␈α
 Add21␈α
and␈αAdd22␈α
still␈α
appear␈α
equivalent.␈α Merge␈α
them.
␈↓ α,␈↓␈↓ β≤So the proper argument for a generalized "Add" operation is a Bag.
␈↓ α,␈↓␈↓ α\␈↓¬85.␈↓␈α∞Apply␈α∞algorithm␈α∞for␈α∞Invert␈α∞to␈α∞argument␈α∞`Add'.␈α∞De≡ne␈α∞Inv-add(x)␈α∞as␈α∞the␈α∞set␈α∞of␈α∞all
␈↓ α,␈↓␈↓ β≤bags of numbers (>0) whose sum is x. Also denoted Add␈↓	-1␈↓(x).
␈↓ α,␈↓␈↓ α\␈↓¬86.␈↓ Fill in examples of TIMES21. (xy)z. Many are found.
␈↓ α,␈↓␈↓ α\␈↓¬87.␈↓ Fill in examples of TIMES22. x(yz). Again many are found.
␈↓ α,␈↓␈↓ α\␈↓¬88.␈↓ Check examples of TIMES22.  TIMES21 and TIMES22 may be equivalent.
␈↓ α,␈↓␈↓ α\␈↓¬89.␈↓␈α⊂Check␈α⊂examples␈α⊂of␈α⊂TIMES21.␈α⊂ TIMES21␈α⊂and␈α⊂TIMES22␈α⊂still␈α⊂appear␈α∂equivalent.
␈↓ α,␈↓␈↓ β≤Merge␈α∞them.␈α∞ So␈α∞the␈α∞proper␈α∞argument␈α∞for␈α∞a␈α∞generalized␈α∞"TIMES"␈α∞operation␈α∂is␈α∞a
␈↓ α,␈↓␈↓ β≤Bag. Set up an analogy between TIMES and ADD, because of this fact.
␈↓ α,␈↓␈↓ α\␈↓¬90.␈↓␈α
Apply␈αalgorithm␈α
for␈αInvert␈α
to␈αargument␈α
`TIMES'.␈α De≡ne␈α
Inv-TIMES(x)␈αas␈α
the␈αset
␈↓ α,␈↓␈↓ β≤of all bags of numbers (>1) whose product is x.  Analogic to Inv-Add.
␈↓ α,␈↓␈↓ α\␈↓¬91.␈↓␈α4Fill␈α3in␈α4examples␈α3of␈α4Parallel-replace2.␈α3 Included␈α4are␈α3Parallel-
␈↓ α,␈↓␈↓ β≤replace2(Bags,Bags,Proj2) (called MR2-BBP2), and many losers.
␈↓ α,␈↓␈↓ α\␈↓¬92.␈↓ ¬ ␈↓¬107.␈↓ Fill in and check examples of the operations just created.
␈↓ α,␈↓␈↓ α\␈↓¬108.␈↓ Fill in examples of Compose.  So easy that AM creates Int-Compose.
␈↓ α,␈↓␈↓ α\␈↓¬109.␈↓␈α∞Fill␈α∞in␈α∞examples␈α∞of␈α∞Int-Compose.␈α∂ The␈α∞two␈α∞chosen␈α∞operations␈α∞G,H␈α∞must␈α∂be␈α∞such
␈↓ α,␈↓␈↓ β≤that␈α∪ran(H)␈↓¬ε␈↓dom(G),␈α∀and␈α∪ran(G)␈↓¬ε␈↓dom(H);␈α∀both␈α∪G␈α∪and␈α∀H␈α∪must␈α∀be␈α∪interesting.
␈↓ α,␈↓␈↓ β≤Create G-Union␈↓εo␈↓MR2-BBP2,␈↓	3␈↓ Insert␈↓εo␈↓Delete, Times␈↓εo␈↓Squaring, etc.
␈↓ α,␈↓␈↓ α\␈↓¬110.␈↓␈α¬␈α␈↓¬127.␈↓␈αFill␈αin␈αand␈αcheck␈αexamples␈αof␈αthe␈αcompositions␈αjust␈αcreated.␈α Notice␈αthat␈αG-
␈↓ α,␈↓␈↓ β≤Union␈↓εo␈↓MR2-BBP2 is just TIMES.
␈↓ α,␈↓␈↓ α\␈↓¬128.␈↓␈α∪Fill␈α∪in␈α∀examples␈α∪of␈α∪Coa-repeat2.␈α∪ Among␈α∀them:␈α∪Coa-repeat2(Bags-of-Numbers,
␈↓ α,␈↓␈↓ β≤Add2)␈α<[multiplication␈α<again!],␈α=Coa-repeat2(Bags-of-Numbers,␈α<Times)
␈↓ α,␈↓␈↓ β≤[exponentiation],␈α_Coa-repeat2(Structures,␈α_Proj1)␈α_[CAR],␈α_Coa-repeat2(Structures,
␈↓ α,␈↓␈↓ β≤Proj2) [Last-element-of], etc.
␈↓ α,␈↓␈↓ α\␈↓¬129.␈↓ Check the examples of Coa-repeat2. All con≡rmed.
␈↓ α,␈↓␈↓ α\␈↓¬130.␈↓␈α∞Apply␈α
algorithms␈α∞for␈α∞Invert␈α
to␈α∞`Doubling'.␈α
 The␈α∞result␈α∞is␈α
called␈α∞"Halving"␈α∞by␈α
the
␈↓ α,␈↓␈↓ β≤user.  AM then de≡nes "Evens".
␈↓ α,␈↓␈↓ α\␈↓¬131.␈↓ Fill in examples of Self-Insert.
␈↓ α,␈↓␈↓ α\␈↓¬132.␈↓ Check examples of Self-Insert. Nothing special found.
␈↓ α,␈↓␈↓ α\␈↓¬133.␈↓ Fill in examples of Coa-repeat2-Add2.
␈↓ α,␈↓␈↓ α\␈↓¬134.␈↓ Check examples of Coa-repeat2-Add2. It's the same as TIMES.
␈↓ α,␈↓␈↓ α\␈↓¬135.␈↓ Apply algorithm for Invert to argument `Squaring'.  De≡ne "Square-root".
␈↓ α,␈↓␈↓ α\␈↓¬136.␈↓ Fill in examples of Square-root. Some found, but very ine≠ciently.
␈↓ α,␈↓␈↓ α\␈↓¬137.␈↓ Fill in new algorithms for Square-root. Had to ask user for a good one.
␈↓ α,␈↓␈↓ α\␈↓¬138.␈↓ Check examples of Square-root. De≡ne the set of numbers "Perfect-squares".
␈↓ α,␈↓␈↓ α\␈↓¬139.␈↓ Fill in examples of Coa-repeat2-Times. This is exponentiation.
␈↓ α,␈↓␈↓ α\␈↓¬140.␈↓ Check examples of Coa-repeat2-Times.  Nothing special noticed, unfortunately.
␈↓ α,␈↓␈↓ α\␈↓¬141.␈↓ Fill in examples of Inv-TIMES.  Many found, but ine≠ciently.
␈↓ α,␈↓␈↓ α\␈↓¬142.␈↓ Fill in new algorithms for Inv-TIMES. Obtained opaquely from the user.
␈↓ α,␈↓␈↓ α\␈↓¬143.␈↓ Check examples of Inv-TIMES. This task suggests the next one:
␈↓ α,␈↓␈↓ α\␈↓¬144.␈↓ Compose G-Union with Inv-TIMES. Good domain/range. Renamed "Divisors".
␈↓ α,␈↓␈↓ α\␈↓¬145.␈↓ Fill in examples of Divisors. Many found, but not very e≠ciently.
␈↓ α,␈↓␈↓ α\␈↓¬146.␈↓ Fill in new algorithms for Divisors. Obtained from the user.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	3␈↓ε an alternate derivation of the operation of multiplication.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε87␈↓-

␈↓ α,␈↓␈↓ α\␈↓¬147.␈↓ Fill in examples of Perfect-squares. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬148.␈↓␈α
Fill␈α
in␈α
specializations␈α
of␈α
TIMES.␈α
Times1(x)␈↓¬≡␈↓1␈↓π#␈↓x,␈α
Times2(x)␈↓¬≡␈↓2x,␈α
Times-sq␈αis␈α
TIMES
␈↓ α,␈↓␈↓ β≤with␈α⊂its␈α⊂domain␈α⊂restricted␈α⊂to␈α⊃bags␈α⊂of␈α⊂perfect␈α⊂squares,␈α⊂Times-ev␈α⊂takes␈α⊃only␈α⊂even
␈↓ α,␈↓␈↓ β≤arguments, Times-to-evens requires that the result be even, Times-to-sq, ...
␈↓ α,␈↓␈↓ α\␈↓¬149.␈↓␈α∂Check␈α∂examples␈α∂of␈α∂Divisors.␈α∂ De≡ne␈α∂0-Div,␈α∂1-Div,␈α∂2-Div,␈α∂and␈α∂3-Div,␈α∂the␈α∂sets␈α∞of
␈↓ α,␈↓␈↓ β≤numbers␈α∂whose␈α∂Divisors␈α∂value␈α⊂is␈α∂the␈α∂empty␈α∂set,␈α⊂a␈α∂singleton,␈α∂a␈α∂doubleton,␈α⊂and␈α∂a
␈↓ α,␈↓␈↓ β≤tripleton, respectively.
␈↓ α,␈↓␈↓ α\␈↓¬150.␈↓ Fill in examples of 1-Div. Only one example found: "1". Lower 1-Div.Worth.
␈↓ α,␈↓␈↓ α\␈↓¬151.␈↓ Fill in examples of 0-Div.  None found. Lower the worth of this concept.
␈↓ α,␈↓␈↓ α\␈↓¬152.␈↓ Fill in examples of 2-Div.  A nice number are found. Raise 2-Div.Worth.
␈↓ α,␈↓␈↓ α\␈↓¬153.␈↓ Check examples of 2-Div. All con≡rmed, but no pattern noticed.
␈↓ α,␈↓␈↓ α\␈↓¬154.␈↓ Fill in examples of 3-Div. A nice number found.
␈↓ α,␈↓␈↓ α\␈↓¬155.␈↓ Check examples of 3-Div. All con≡rmed. All are perfect squares.
␈↓ α,␈↓␈↓ α\␈↓¬156.␈↓ Restrict Square-root to numbers which are in 3-Div. Call this Root3.
␈↓ α,␈↓␈↓ α\␈↓¬157.␈↓ Fill in examples of Root3. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬158.␈↓ Check examples of Root3. All con≡rmed. All are in 2-Div. Raise their worths.
␈↓ α,␈↓␈↓ α\␈↓¬159.␈↓ Restrict Squaring to 2-divs. Call the result Square2.
␈↓ α,␈↓␈↓ α\␈↓¬160.␈↓ Fill in examples of Square2. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬161.␈↓␈α∞Check␈α
the␈α∞range␈α∞of␈α
Square2.␈α∞Always␈α∞3-Divs.␈α
 Conjecture:␈α∞x␈α∞has␈α
2␈α∞divisors␈α∞i≥␈α
x␈↓#
2␈↓#
␈↓ α,␈↓␈↓ β≤has 3 divisors.
␈↓ α,␈↓␈↓ α\␈↓¬162.␈↓ Restrict Squaring to 3-Divs. Call the result Square3.
␈↓ α,␈↓␈↓ α\␈↓¬163.␈↓ Restrict Square-rooting to 2-Divs. Call the result Root2.
␈↓ α,␈↓␈↓ α\␈↓¬164.␈↓ Fill in examples of Square3. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬165.␈↓ Compose Divisors-of and Square3. Call the result Div-Sq3.
␈↓ α,␈↓␈↓ α\␈↓¬166.␈↓ Fill in examples of Div-Sq3. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬167.␈↓ Check examples of Div-Sq3.  All such examples are Same-size.
␈↓ α,␈↓␈↓ α\␈↓¬168.␈↓␈α∞¬␈α
␈↓¬175.␈↓␈α∞More␈α∞con≡rmations␈α
and␈α∞explorations␈α∞of␈α
the␈α∞above␈α∞conjecture.␈α
 Gradually,
␈↓ α,␈↓␈↓ β≤all its rami≡cations lead to dead-ends (as far as AM is concerned).
␈↓ α,␈↓␈↓ α\␈↓¬176.␈↓ Fill in examples of Root2. None found.  Conjecture that there are none.
␈↓ α,␈↓␈↓ α\␈↓¬177.␈↓␈α
Check␈αexamples␈α
of␈α
Inv-TIMES.␈αInv-TIMES␈α
always␈α
contains␈αa␈α
singleton␈α
bag,␈αand
␈↓ α,␈↓␈↓ β≤always contains a bag of primes.
␈↓ α,␈↓␈↓ α\␈↓¬178.␈↓ Restrict the range of Inv-TIMES to bags of primes. Call this Prime-Times.
␈↓ α,␈↓␈↓ α\␈↓¬179.␈↓ Restrict the range of Inv-TIMES to singletons. Called Single-Times.
␈↓ α,␈↓␈↓ α\␈↓¬180.␈↓ Fill in examples of Prime-times. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬181.␈↓␈α⊗Check␈α⊗examples␈α⊗of␈α⊗Prime-times.␈α⊗Always␈α⊗a␈α⊗singleton␈α⊗set.␈α⊗User␈α⊗renames␈α∃this
␈↓ α,␈↓␈↓ β≤conjecture "The unique factorization theorem".
␈↓ α,␈↓␈↓ α\␈↓¬182.␈↓ Fill in examples of Single-TIMES. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬183.␈↓␈α∀Check␈α∀examples␈α∀of␈α∀Single-TIMES.␈α∀Always␈α∀a␈α∀singleton␈α∀set.␈α∀ Single-TIMES␈α∀is
␈↓ α,␈↓␈↓ β≤actually the same as Bag-insert!
␈↓ α,␈↓␈↓ α\␈↓¬184.␈↓ Fill in examples of Self-set-union. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬185.␈↓ Check examples of Self-set-union. This operation is same as Identity.
␈↓ α,␈↓␈↓ α\␈↓¬186.␈↓ Fill in examples of Self-bag-union. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬187.␈↓ Check examples of Self-bag-union. Con≡rmed. Nothing interesting noticed.
␈↓ α,␈↓␈↓ α\␈↓¬188.␈↓ Fill in examples of Inv-ADD.
␈↓ α,␈↓␈↓ α\␈↓¬189.␈↓ Check examples of Inv-ADD. Hordes of boring conjectures, so:
␈↓ α,␈↓␈↓ α\␈↓¬190.␈↓␈α
Restrict␈α∞the␈α
domain␈α∞of␈α
Inv-ADD␈α∞to␈α
primes␈α∞(Inv-Add-primes),␈α
to␈α∞evens␈α
(Inv-Add-
␈↓ α,␈↓␈↓ β≤evens), to squares, etc.
␈↓ α,␈↓␈↓ α\␈↓¬191.␈↓ Fill in examples of Inv-add-primes. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬192.␈↓ Check examples of Inv-add-primes. Con≡rmed, but nothing special noticed.
␈↓ α,␈↓␈↓ α\␈↓¬193.␈↓ Fill in examples of Inv-add-evens. Many found.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε88␈↓-

␈↓ α,␈↓␈↓ α\␈↓¬194.␈↓ Check examples of Inv-add-evens. Always contains a bag of primes.
␈↓ α,␈↓␈↓ α\␈↓¬195.␈↓ Restrict the range of Inv-Add-evens to bags of primes. Called Prime-ADD.
␈↓ α,␈↓␈↓ α\␈↓¬196.␈↓ Restrict the range of Inv-ADD to singletons. Call that new operation Single-ADD.
␈↓ α,␈↓␈↓ α\␈↓¬197.␈↓ Fill in examples of Prime-ADD. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬198.␈↓␈α
Check␈αexamples␈α
of␈αPrime-ADD.␈α
Always␈αa␈α
nonempty␈αset␈α
(of␈αbags␈α
of␈α
primes).␈αUser
␈↓ α,␈↓␈↓ β≤renames this conjecture "Goldbach's conjecture".
␈↓ α,␈↓␈↓ α\␈↓¬199.␈↓ Fill in examples of Single-ADD. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬200.␈↓␈αCheck␈αexamples␈αof␈αSingle-ADD.␈αAlways␈αa␈αsingleton␈αset.␈αThis␈αoperation␈αis␈αthe␈α
same
␈↓ α,␈↓␈↓ β≤as Bag-insert and Single-TIMES.
␈↓ α,␈↓␈↓ α\␈↓¬201.␈↓␈αRestrict␈αthe␈αrange␈αof␈αPrime-ADD␈αto␈αsingletons,␈αby␈αanalogy␈αto␈αPrime-TIMES.␈↓	4␈↓␈αCall
␈↓ α,␈↓␈↓ β≤the new operation Prime-ADD-SING.
␈↓ α,␈↓␈↓ α\␈↓¬202.␈↓ Fill in examples of Prime-ADD-SING. Many found.
␈↓ α,␈↓␈↓ α\␈↓¬203.␈↓ Check examples of Prime-ADD-SING. Nothing special noticed.
␈↓ α,␈↓␈↓ α\␈↓¬204.␈↓ Fill in examples of Times-sq.␈↓	5␈↓ Many examples found.
␈↓ α,␈↓␈↓ α\␈↓¬205.␈↓ Check domain/range of Times-sq. Is the range actually Perfect-squares?  Yes!
␈↓ α,␈↓␈↓ α\␈↓¬206.␈↓ Fill in examples of Times1. Recall that Times1(x)␈↓¬≡␈↓TIMES(1,x).
␈↓ α,␈↓␈↓ α\␈↓¬207.␈↓ Check examples of Times1. Apparently just a restriction of Identity.
␈↓ α,␈↓␈↓ α\␈↓¬208.␈↓ Check examples of Times-sq. Con≡rmed.
␈↓ α,␈↓␈↓ α\␈↓¬209.␈↓ Fill in examples of Times0.
␈↓ α,␈↓␈↓ α\␈↓¬210.␈↓ Fill in examples of Times2.
␈↓ α,␈↓␈↓ α\␈↓¬211.␈↓␈α
Check␈α
examples␈αof␈α
Times2.␈α
Apparently␈αthe␈α
same␈α
as␈αDoubling.␈α
 That␈α
is,␈αx+x=2␈↓π#␈↓x.
␈↓ α,␈↓␈↓ β≤Very important.  By analogy, de≡ne Ad2(x) as x+2.
␈↓ α,␈↓␈↓ α\␈↓¬212.␈↓ Fill in examples of Ad2.
␈↓ α,␈↓␈↓ α\␈↓¬213.␈↓ Check examples of Ad2. Nothing interesting noticed.
␈↓ α,␈↓␈↓ α\␈↓¬214.␈↓␈αFill␈αin␈αspecializations␈αof␈αAdd.␈α
Among␈αthose␈αcreated␈αare:␈αAdd0␈α(x+0),␈α
Add1,␈αAdd3,
␈↓ α,␈↓␈↓ β≤ADD-sq␈α⊂(addition␈α∂restricted␈α⊂to␈α∂perfect␈α⊂squares),␈α∂Add-ev␈α⊂(sum␈α∂of␈α⊂even␈α∂numbers),
␈↓ α,␈↓␈↓ β≤Add-pr (sum of primes), etc.
␈↓ α,␈↓␈↓ α\␈↓¬215.␈↓ Check examples of Times0. The value always seems to be 0.
␈↓ α,␈↓␈↓ α\␈↓¬216.␈↓ Fill in examples of Times-ev.␈↓	6␈↓ Many examples found.
␈↓ α,␈↓␈↓ α\␈↓¬217.␈↓ Check examples of Times-ev. Apparently all the results are Evens.
␈↓ α,␈↓␈↓ α\␈↓¬218.␈↓ Fill in examples of Times-to-ev.␈↓	7␈↓ Many found.
␈↓ α,␈↓␈↓ α\␈↓¬219.␈↓ Fill in examples of Times-to-sq. Only a few found.
␈↓ α,␈↓␈↓ α\␈↓¬220.␈↓␈αCheck␈αexamples␈αof␈αTimes-to-sq.␈αAll␈αarguments␈αalways␈αseem␈αto␈αbe␈αsquares.␈α Conjec:
␈↓ α,␈↓␈↓ β≤Times-to-sq␈α∀is␈α∀really␈α∀the␈α∀same␈α∀as␈α∀Times-sq.␈α∀Merge␈α∀the␈α∀two.␈α∀ This␈α∀is␈α∀a␈α∪false
␈↓ α,␈↓␈↓ β≤conjecture, but did AM no harm.
␈↓ α,␈↓␈↓ α\␈↓¬221.␈↓ Check examples of Times-to-ev. The domain always contains an even number.
␈↓ α,␈↓␈↓ α\␈↓¬222.␈↓ Fill in examples of Self-Union.
␈↓ α,␈↓␈↓ α\␈↓¬223.␈↓ Check examples of Self-Union.
␈↓ α,␈↓␈↓ α\␈↓¬224.␈↓ Fill in examples of SubSet.
␈↓ α,␈↓␈↓ α\␈↓¬225.␈↓ Check example of SubSet.
␈↓ α,␈↓␈↓ α\␈↓¬226.␈↓ Fill in examples of SuperSet.
␈↓ α,␈↓␈↓ α\␈↓¬227.␈↓ Check examples of SuperSet. Conjec: Subset(x,y) i≥ Superset(y,x). Important.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	4␈↓ε In this case, AM is asking which numbers are uniquely representable as the sum of two primes.
␈↓ α,␈↓ε␈↓	5␈↓ε Recall that this is just TIMES restricted to operate on perfect squares.
␈↓ α,␈↓ε␈↓	6␈↓ε Recall that Times-ev is just like TIMES restricted to operating on even numbers.
␈↓ α,␈↓ε␈↓	7␈↓ε That is, consider bags of numbers which multiply to give an even number.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε89␈↓-

␈↓ α,␈↓␈↓ α\␈↓¬228.␈↓␈α
Fill␈α
in␈α
examples␈α
of␈α
Compose␈↓εo␈↓Compose-1.␈α
AM␈α
creates␈α
some␈α
explosive␈α
combination
␈↓ α,␈↓␈↓ β≤(e.g.,␈α≤(Compose␈↓εo␈↓Compose)␈↓εo␈↓(Compose␈↓εo␈↓Compose)␈↓εo␈↓(Compose␈↓εo␈↓Compose)),␈α≥some␈α≤poor
␈↓ α,␈↓␈↓ β≤ones␈α⊃(e.g.,␈α∩Square␈↓εo␈↓Count␈↓εo␈↓ADD␈↓	-1␈↓),␈α⊃and␈α∩even␈α⊃a␈α⊃few␈α∩¬␈α⊃very␈α∩few␈α⊃¬␈α∩winners␈α⊃(e.g.,
␈↓ α,␈↓␈↓ β≤SUB1␈↓εo␈↓Count␈↓εo␈↓Self-Insert).
␈↓ α,␈↓␈↓ α\␈↓¬229.␈↓ Check examples of Compose␈↓εo␈↓Compose-1.
␈↓ α,␈↓␈↓ α\␈↓¬230.␈↓␈α⊃Fill␈α⊃in␈α⊃examples␈α⊃of␈α⊃Compose␈↓εo␈↓Compose-2.␈↓	8␈↓␈α⊃AM␈α⊃recreates␈α⊃many␈α⊃of␈α⊃the␈α⊃previous
␈↓ α,␈↓␈↓ β≤tasks' operations.
␈↓ α,␈↓␈↓ α\␈↓¬231.␈↓ Check examples of Compose␈↓εo␈↓Compose-2. Nothing noticed yet␈↓	9␈↓.
␈↓ α,␈↓␈↓ α\␈↓¬232.␈↓ ¬ ␈↓¬252.␈↓ Fill in and check examples of the losing compositions just created.
␈↓ α,␈↓␈↓ α\␈↓¬253.␈↓ Fill in examples of Add-sq (i.e., sum of squares).
␈↓ α,␈↓␈↓ α\␈↓¬254.␈↓␈α∞Check␈α∞domain/range␈α∞entries␈α∞of␈α∞Add-sq.␈α∞The␈α∞range␈α∞is␈α∞not␈α∞always␈α∞perfect␈α
squares.
␈↓ α,␈↓␈↓ β≤De≡ne␈αAdd-sq-sq(x,y),␈αwhich␈αis␈αTrue␈αi≥␈αx␈αand␈αy␈αare␈αperfect␈αsquares␈αand␈αtheir␈αsum
␈↓ α,␈↓␈↓ β≤is a perfect square as well.
␈↓ α,␈↓␈↓ α\␈↓¬255.␈↓ Fill in examples of Add-pr; i.e., addition of primes.
␈↓ α,␈↓␈↓ α\␈↓¬256.␈↓␈α⊂Check␈α⊂Domain/range␈α⊂entries␈α⊂of␈α⊂Add-pr.␈α∂AM␈α⊂de≡nes␈α⊂the␈α⊂set␈α⊂of␈α⊂pairs␈α⊂of␈α∂primes
␈↓ α,␈↓␈↓ β≤whose sum is also a prime. This is a bizarre derivation of prime pairs.





























␈↓ α,␈↓_______________________________________________________________________________
␈↓ α,␈↓ε␈↓	8␈↓ε␈αλRecall␈αλthat␈αλthe␈αλdifference␈αλbetween␈αλthis␈αλoperation␈αλand␈αλthe␈αλlast␈αλone␈αλis␈αλmerely␈αλin␈αλthe␈αλorder␈αλof␈αλthe␈αλcomposing:␈α	Fo(GoH)␈αλversus
␈↓ α,␈↓ε␈↓ βL(FoG)oH.
␈↓ α,␈↓ε␈↓	9␈↓ε Later on, AM will use these new operations to discover the associativity of Compose.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε90␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.1.2. Two-Dimensional Behavior Graph␈↓)αβ␈↓

␈↓ α,␈↓On␈αthe␈αnext␈αtwo␈αpages␈αis␈αa␈αgraph␈αof␈αthe␈αsame␈α"best␈αrun"␈αwhich␈αAM␈αexecuted.␈α The␈αnodes
␈↓ α,␈↓are␈αconcepts,␈αand␈αthe␈α
links␈αare␈αactions␈αwhich␈αAM␈α
performed.␈α Labels␈αon␈αthe␈α
links␈αindicate
␈↓ α,␈↓when␈αeach␈αaction␈αwas␈αtaken,␈αso␈αthe␈α
reader␈αmay␈αobserve␈αhow␈αAM␈αjumped␈αaround.␈αIt␈α
should
␈↓ α,␈↓also␈α⊂be␈α⊃easy␈α⊂to␈α⊃perceive␈α⊂from␈α⊃the␈α⊂graph␈α⊃which␈α⊂paths␈α⊃of␈α⊂development␈α⊃were␈α⊂abandoned,
␈↓ α,␈↓which␈αconcepts␈αignored,␈α
and␈αwhich␈αones␈α
concentrated␈αupon.␈αThese␈α
are␈αprecisely␈αthe␈α
features
␈↓ α,␈↓of␈α
AM's␈α
behavior␈α
which␈α
are␈α
awkward␈α
to␈α
infer␈α
from␈α
a␈α
simple␈α
linear␈α
trace␈α
(as␈α
in␈αthe␈α
previous
␈↓ α,␈↓section).

␈↓ α,␈↓In␈αmore␈αdetail,␈αhere␈αis␈αhow␈αto␈αread␈αthe␈αgraph:␈αEach␈αnode␈αis␈αa␈αconcept.␈αTo␈αsave␈αspace,␈αthese
␈↓ α,␈↓names are often highly abbreviated. For example, "x0" is used in place of "TIMES-0".

␈↓ α,␈↓Each concept name is surrounded by from zero to four numbers:
␈↓ α,␈↓¬␈↓ ∧l318      288
␈↓ α,␈↓¬␈↓ ∧lFROBNATION
␈↓ α,␈↓¬␈↓ ∧l310      291

␈↓ α,␈↓The␈αupper␈α
right␈αnumber␈α
indicates␈αthe␈α
task␈αnumber␈α
(see␈αlast␈α
section)␈αduring␈αwhich␈α
examples
␈↓ α,␈↓of␈α
this␈α∞concept␈α
were␈α∞≡lled␈α
in.␈α∞The␈α
lower␈α∞right␈α
number␈α∞tells␈α
when␈α∞they␈α
were␈α∞checked.␈α
The
␈↓ α,␈↓upper␈α∂left␈α∂number␈α∂indicates␈α∂when␈α⊂the␈α∂Domain/range␈α∂facet␈α∂of␈α∂that␈α∂concept␈α⊂was␈α∂modi≡ed.
␈↓ α,␈↓Finally,␈α
the␈α
lower␈α
left␈α
number␈α
is␈α
the␈α
task␈α
number␈α
during␈α
which␈α
some␈α
new␈α
Algorithms␈α
for
␈↓ α,␈↓that␈α⊂concept␈α⊂were␈α⊂obtained.␈α⊂ A␈α⊂number␈α∂in␈α⊂parentheses␈α⊂indicates␈α⊂that␈α⊂the␈α⊂task␈α⊂with␈α∂that
␈↓ α,␈↓number was a total failure.

␈↓ α,␈↓Because␈α∞of␈α∞the␈α∂limited␈α∞space,␈α∞it␈α∞was␈α∂decided␈α∞that␈α∞if␈α∞a␈α∂concept␈α∞were␈α∞ever␈α∞renamed␈α∂by␈α∞the
␈↓ α,␈↓user,␈αthen␈α
only␈αthat␈αnewer,␈α
mnemonic␈αname␈αwould␈α
be␈αgiven␈αin␈α
the␈αdiagram.␈αThus␈α
there␈αis
␈↓ α,␈↓an arrow from "Coalesce" to "␈↓βSquare␈↓", an operation originally called "Self-Times" by AM.

␈↓ α,␈↓Sometimes,␈αa␈αconcept␈αwill␈αhave␈αunder␈αit␈αa␈αnote␈αof␈αthe␈αform␈α␈↓¬≡GROK␈↓.␈α This␈αsimply␈αmeans␈αthat
␈↓ α,␈↓AM␈α∂eventually␈α∂discovered␈α∂that␈α∂the␈α∂concept␈α∂was␈α∂equivalent␈α∂to␈α∂the␈α∂already-known␈α∞concept
␈↓ α,␈↓"Grok",␈αand␈αprobably␈αforgot␈αabout␈αthis␈αone␈α(merged␈αit␈αinto␈αthe␈αone␈αit␈αalready␈αknew␈αabout).
␈↓ α,␈↓The␈α"trail"␈αof␈αdiscovery␈αmay␈αpick␈αup␈αagain␈αat␈αthat␈αpre-existing␈αconcept.␈α A␈αnode␈αwritten␈αas
␈↓ α,␈↓␈↓¬=GROK␈↓␈αmeans␈αthat␈αthe␈αconcept␈αwas␈αreally␈αthe␈αsame␈αas␈α"Grok",␈αbut␈αAM␈αnever␈αinvestigated␈αit
␈↓ α,␈↓enough to notice this.

␈↓ α,␈↓The␈α∩arrows␈α∪indicate␈α∩the␈α∩creation␈α∪of␈α∩new␈α∪concepts.␈α∩Thus␈α∩an␈α∪arrow␈α∩leading␈α∪to␈α∩concept
␈↓ α,␈↓"Frobnate"␈αindicates␈αhow␈αthat␈αconcept␈αwas␈αcreated.␈αAn␈αarrow␈αdirected␈αaway␈α
from␈αFrobnate
␈↓ α,␈↓points␈α∀to␈α∀a␈α∀concept␈α∀created␈α∀as,␈α∀e.g.,␈α∀a␈α∀specialization␈α∀or␈α∀an␈α∀example␈α∀of␈α∀Frobnate.␈α∪No
␈↓ α,␈↓arrowheads are in practice necessary: all arrows are directed ␈↓βdownwards␈↓.

␈↓ α,␈↓The␈α⊃arrows␈α⊃may␈α⊃be␈α⊃labelled,␈α⊃indicating␈α⊃precisely␈α⊃what␈α⊃they␈α⊃represent␈α⊃(e.g.,␈α⊂composition,
␈↓ α,␈↓restriction)␈α⊂and␈α⊂what␈α⊂the␈α⊃task␈α⊂number␈α⊂was␈α⊂when␈α⊃they␈α⊂occurred.␈α⊂ For␈α⊂space␈α⊃reasons,␈α⊂the
␈↓ α,␈↓following␈αconvention␈αhas␈αproven␈αnecessary:␈αif␈αan␈αarrow␈αemanating␈αfrom␈αC␈αis␈αun-numbered,
␈↓ α,␈↓it␈α
is␈α
assumed␈α
to␈α
have␈α
occurred␈α
at␈α
the␈α
same␈α
time␈α
as␈α
the␈α
arrow␈α
to␈α
its␈α
immediate␈α
left␈α
which␈α
also
␈↓ α,␈↓points␈αfrom␈αC;␈αif␈αall␈αthe␈αarrows␈αemanating␈αfrom␈αC␈αhave␈αno␈αnumber,␈αthan␈αall␈αtheir␈αtimes␈αof
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε91␈↓-

␈↓ α,␈↓occurrence␈α∞are␈α∞assumed␈α∞to␈α∞be␈α∞the␈α∞␈↓βlower␈α∞right␈↓␈↓	10␈↓␈α∞number␈α∞of␈α∞C.␈α∞ Finally,␈α∞if␈α∞C␈α∞has␈α∞no␈α∞lower
␈↓ α,␈↓right number, the arrow is assumed to have the value of the upper right number of C.

␈↓ α,␈↓An␈α∩unlabelled␈α∩arrow␈α∩is␈α∩assumed␈α∩to␈α∩be␈α∩an␈α∩act␈α∩of␈α∩Specialization␈α∩or␈α∩the␈α∩creation␈α∩of␈α∩an
␈↓ α,␈↓Example.␈↓	11␈↓␈α⊂Labels,␈α⊂when␈α⊂they␈α⊃do␈α⊂occur,␈α⊂are␈α⊂given␈α⊃in␈α⊂capitals␈α⊂and␈α⊂small␈α⊃letters;␈α⊂concept
␈↓ α,␈↓names (nodes) are by contrast in all capitals.

␈↓ α,␈↓All␈αthe␈αnumbers␈αcorrespond␈αto␈αthose␈α
given␈αto␈αthe␈αtasks␈αin␈αthe␈αtask-by-task␈α
traces␈αpresented
␈↓ α,␈↓in the last section (p. 84).

␈↓ α,␈↓The␈α⊃≡rst␈α⊃part␈α⊃of␈α⊃this␈α∩graph␈α⊃(presented␈α⊃below)␈α⊃contains␈α⊃static␈α⊃structural␈α∩(and␈α⊃ultimately
␈↓ α,␈↓numerical) concepts which were studied by AM:




















␈↓ α,␈↓The␈α∪rest␈α∩of␈α∪the␈α∩graph␈α∪(presented␈α∩on␈α∪the␈α∩next␈α∪page)␈α∩deals␈α∪with␈α∩activities␈α∪which␈α∩were
␈↓ α,␈↓investigated:












␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	10␈↓ε This is often true because many concepts are created while checking examples of some known concept.
␈↓ α,␈↓ε␈↓	11␈↓ε␈αλIt␈αλshould␈αλbe␈αλclear␈αλin␈αλeach␈αλcontext␈αλwhich␈αλis␈αλhappening.␈αλIf␈αλnot,␈αλrefer␈αλto␈αλthe␈αλshort␈αλtrace␈αλin␈αλthe␈αλpreceding␈αλsection,␈αλand␈αλlook␈αλup
␈↓ α,␈↓ε␈↓ βLthe appropriate task number.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε92␈↓-

␈↓ α,␈↓( Paste Concept Development Behavior Graph here. )
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε93␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.1.3. AM as a Computer Program␈↓)αβ␈↓

␈↓ α,␈↓When␈αviewed␈αas␈αa␈αlarge␈αLISP␈αprogram,␈αthere␈αis␈αvery␈αlittle␈αof␈αinterest␈αabout␈αAM.␈αThere␈αare
␈↓ α,␈↓the␈α
usual␈α
battery␈α
of␈α
customized␈α
functions␈α(e.g.,␈α
a␈α
conditional␈α
PRINT␈α
function),␈α
the␈αstorage
␈↓ α,␈↓hacks␈α_(special␈α↔emergency␈α_garbage␈α_collection␈α↔routines,␈α_which␈α↔know␈α_which␈α_facets␈α↔are
␈↓ α,␈↓expendible),␈α
the␈α
time␈α
hacks␈α
(omnisciently␈α
arrange␈α
clauses␈α
in␈α
a␈α
conjunction␈α
so␈α
that␈α∞the␈α
one
␈↓ α,␈↓most␈αlikely␈αto␈αfail␈αwill␈αcome␈α≡rst),␈αand␈αthe␈αbugs␈α(if␈αthe␈αuser␈αrenames␈αa␈αconcept␈αwhile␈αit's␈αthe
␈↓ α,␈↓current one being worked on, there is a 5% chance of AM entering an in≡nite loop).

␈↓ α,␈↓Below␈α⊃are␈α⊂listed␈α⊃a␈α⊃few␈α⊂parameters␈α⊃of␈α⊂the␈α⊃system,␈α⊃although␈α⊂I␈α⊃doubt␈α⊂that␈α⊃they␈α⊃hold␈α⊂any
␈↓ α,␈↓theoretical␈αsigni≡cance.␈αThe␈αreader␈αmay␈α
be␈αcurious␈αabout␈αhow␈αbig␈α
AM,␈αhow␈αlong␈αit␈αtakes␈α
to
␈↓ α,␈↓execute, etc.

␈↓ α,␈↓Machine: SUMEX, PDP-10, KI-10 uniprocessor, 256k core memory.

␈↓ α,␈↓Language:␈α
Interlisp,␈α
January␈α
'75␈α
release,␈α
which␈αoccupies␈α
140k␈α
of␈α
the␈α
total␈α
256k,␈α
but␈αwhich
␈↓ α,␈↓provides␈αa␈αsurplus␈α"shadow␈αspace"␈αof␈α256k␈αadditional␈αwords␈αavailable␈αfor␈αholding␈α
compiled
␈↓ α,␈↓code.

␈↓ α,␈↓AM␈α∞support␈α∞code:␈α∞200␈α∞compiled␈α∞(not␈α∞block-compiled)␈α∞utility␈α∞routines,␈α∞control␈α∞routines,␈α∞etc.
␈↓ α,␈↓They occupy roughly 100k, but all are pushed into the shadow space.

␈↓ α,␈↓AM␈α∂itself:␈α∂115␈α∂concepts,␈α⊂each␈α∂occupying␈α∂about␈α∂.7k␈α⊂(about␈α∂two␈α∂typed␈α∂pages,␈α⊂when␈α∂Pretty-
␈↓ α,␈↓printed␈α⊂with␈α⊃indentation).␈α⊂ Facet/entries␈α⊂stored␈α⊃as␈α⊂property/value␈α⊂on␈α⊃the␈α⊂property␈α⊃list␈α⊂of
␈↓ α,␈↓atoms whose names are concepts' names.␈↓	12␈↓ Each concept has about 8 facets ≡lled in.

␈↓ α,␈↓Heuristics␈α∂are␈α∂tacked␈α∂onto␈α⊂the␈α∂facets␈α∂of␈α∂the␈α∂concepts.␈α⊂The␈α∂more␈α∂general␈α∂the␈α⊂concept,␈α∂the
␈↓ α,␈↓more␈α∞heuristic␈α∞rules␈α∞it␈α∞has␈α∞attached␈α∞to␈α∞it.␈↓	13␈↓␈α∞"Any-concept"␈α∞has␈α∞121␈α∞rules;␈α∂"Active␈α∞concept"
␈↓ α,␈↓has␈α⊂24;␈α⊃"Coalesce"␈α⊂has␈α⊃7;␈α⊂"Set-Insertion"␈α⊃has␈α⊂none.␈α⊂There␈α⊃are␈α⊂250␈α⊃heuristic␈α⊂rules␈α⊃in␈α⊂all,
␈↓ α,␈↓divided␈αinto␈α4␈α∨avors␈α(Fillin,␈αCheck,␈αSuggest,␈αInterestingness).␈α Although␈αthe␈αmean␈αnumber
␈↓ α,␈↓of␈αrules␈αis␈αtherefore␈αonly␈αabout␈α2.2␈α(i.e.,␈α
less␈αthan␈α1␈αof␈αeach␈α∨avor)␈αper␈αconcept,␈αthe␈α
standard
␈↓ α,␈↓deviation␈αof␈αthis␈αis␈αa␈αwhopping␈α127.4.␈αThe␈αaverage␈αnumber␈αof␈αheuristics␈α(of␈αa␈αgiven␈α∨avor)
␈↓ α,␈↓encountered␈α⊃rippling␈α⊂upward␈α⊃from␈α⊃a␈α⊂randomly-chosen␈α⊃concept␈α⊃C␈α⊂(along␈α⊃the␈α⊃network␈α⊂of
␈↓ α,␈↓generalization links) is about 35, even though the mean path length is only about 4.␈↓	14␈↓

␈↓ α,␈↓The␈α
total␈α
number␈α
of␈α
jobs␈αexecuted␈α
in␈α
a␈α
typical␈α
run␈α(from␈α
scratch)␈α
is␈α
about␈α
200.␈α
 The␈αrun
␈↓ α,␈↓ends␈α∂because␈α⊂of␈α∂space␈α⊂problems,␈α∂but␈α⊂AM's␈α∂performance␈α⊂begins␈α∂to␈α⊂degrade␈α∂near␈α⊂the␈α∂end
␈↓ α,␈↓anyway.

␈↓ α,␈↓"Final"␈α∞state␈α∞of␈α∞AM:␈α
300␈α∞concepts,␈α∞each␈α∞occupying␈α∞about␈α
1k.␈α∞Many␈α∞are␈α∞swapped␈α∞out␈α
onto

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	12␈↓ε␈α	Snazzy␈α	feature:␈α	Executable␈α	entries␈α	on␈αλfacets␈α	(e.g.,␈α	an␈α	entry␈α	on␈α	Union.Alg)␈αλare␈α	stored␈α	uncompiled␈α	until␈α	the␈α	first␈α	time␈αλthey
␈↓ α,␈↓ε␈↓ βLare actually called on, at which time they are compiled and then executed.
␈↓ α,␈↓ε␈↓	13␈↓ε This was not done consciously, and may or may not hold some theoretical significance.
␈↓ α,␈↓ε␈↓	14␈↓ε␈α	If␈αλthe␈α	heuristics␈αλwere␈α	homogeneously␈αλdistributed␈α	among␈αλthe␈α	concepts,␈αλthe␈α	number␈αλof␈α	heuristics␈αλ(of␈α	a␈αλgiven␈α	type)␈α	along␈αλa
␈↓ α,␈↓ε␈↓ βLtypical␈α	path␈α	of␈α	length␈α	4␈α	would␈αλonly␈α	be␈α	about␈α	2,␈α	not␈α	35.␈α	 If␈αλall␈α	the␈α	heuristics␈α	were␈α	tacked␈α	onto␈α	Anything␈αλand
␈↓ α,␈↓ε␈↓ βLAny-concept, the number encountered in any path would be 75, not 35.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε94␈↓-

␈↓ α,␈↓disk.␈α∃Number␈α∃of␈α∀winning␈α∃concepts␈α∃discovered:␈α∀25␈α∃(estimated).␈α∃ Number␈α∃of␈α∀acceptable
␈↓ α,␈↓concepts␈αde≡ned:␈α
100␈α(est.).␈↓	15␈↓␈α
Number␈αof␈αlosing␈α
concepts␈αunfortunately␈α
worked␈αon:␈α
60␈α(est.).
␈↓ α,␈↓The␈αoriginal␈α
115␈αconcepts␈αhave␈α
grown␈αto␈αan␈α
average␈αsize␈α
of␈α2k.␈α Each␈α
concept␈αhas␈αabout␈α
11
␈↓ α,␈↓facets ≡lled in.

␈↓ α,␈↓About␈α∂30␈α∂seconds␈α∂of␈α∂cpu␈α⊂time␈α∂were␈α∂allocated␈α∂to␈α∂each␈α⊂task,␈α∂on␈α∂the␈α∂average,␈α∂but␈α⊂the␈α∂task
␈↓ α,␈↓typically␈αused␈αonly␈αabout␈α18␈αseconds␈αbefore␈αquitting.␈α Total␈αCPU␈αtime␈αfor␈αa␈αrun␈αis␈αabout␈α1
␈↓ α,␈↓hour. Total cpu time consumed by this research project was about 500 cpu hours.

␈↓ α,␈↓Real␈αtime:␈α
about␈α1␈αminute␈α
per␈αtask,␈α2␈α
hours␈αper␈αrun.␈α
The␈αidea␈αfor␈α
AM␈αwas␈α
formulated␈αin
␈↓ α,␈↓the␈αFall␈αof␈α1974,␈αand␈αAM␈αwas␈αcoded␈αin␈αthe␈αsummer␈αof␈α1975.␈α Total␈αtime␈αconsumed␈αby␈αthis
␈↓ α,␈↓project␈α
to␈α
date␈αhas␈α
been␈α
about␈α
2600␈αman-hours:␈α
700␈α
for␈α
planning,␈α500␈α
for␈α
coding,␈α
600␈αfor
␈↓ α,␈↓modifying␈α
and␈α
debugging␈α
and␈α
experimenting,␈α
700␈αfor␈α
writing␈α
the␈α
thesis,␈α
and␈α
100␈αfor␈α
editing
␈↓ α,␈↓it into book form.

␈↓ α,␈↓AM␈αwas␈αworking␈αby␈αitself:␈αit␈αreceived␈αno␈αhelp␈αfrom␈αthe␈αuser,␈αand␈αall␈αits␈αconcepts'␈αIntuitions
␈↓ α,␈↓facets had been removed.



␈↓ α,␈↓␈↓ ¬≠␈↓∧␈↓&6.2. Experiments with AM␈↓)αβ␈↓

␈↓ α,␈↓One␈αvaluable␈αaspect␈αof␈αAM␈αis␈αthat␈αit␈αis␈αamenable␈αto␈αmany␈αkind␈αof␈αinteresting␈αexperiments.
␈↓ α,␈↓Although␈αAM␈αis␈αtoo␈α␈↓βad␈αhoc␈↓␈αfor␈αnumerical␈αresults␈αto␈αhave␈αmuch␈αsigni≡cance,␈αthe␈αqualitative
␈↓ α,␈↓results␈αperhaps␈αdo␈αhave␈αsome␈αvalid␈α
things␈αto␈αsay␈αabout␈αresearch␈αin␈αelementary␈α
mathematics,
␈↓ α,␈↓about automating research, and at least about the e≠cacy of various parts of AM's design.

␈↓ α,␈↓This␈α
section␈α
will␈α
explain␈α
what␈α
it␈α
means␈α
to␈α
perform␈α
an␈α
experiment␈α
on␈α
AM,␈α
what␈α∞kinds␈α
of
␈↓ α,␈↓experiments␈αare␈αimaginable,␈αwhich␈αof␈αthose␈α
are␈αfeasible,␈αand␈α≡nally␈αwill␈αdescribe␈α
the␈αmany
␈↓ α,␈↓experiments which were performed on AM.

␈↓ α,␈↓By␈α⊃modifying␈α⊂AM␈α⊃in␈α⊃various␈α⊂ways,␈α⊃its␈α⊂behavior␈α⊃can␈α⊃be␈α⊂altered,␈α⊃and␈α⊂the␈α⊃␈↓βquality␈↓␈α⊃of␈α⊂its
␈↓ α,␈↓behavior␈αwill␈α
change␈αas␈α
well.␈α As␈α
a␈αdrastic␈α
example,␈αone␈α
experiment␈αinvolved␈α
forcing␈αAM
␈↓ α,␈↓to␈α∞select␈α∞the␈α∞next␈α∞task␈α
to␈α∞work␈α∞on␈α∞␈↓βrandomly␈↓␈α∞from␈α∞the␈α
agenda,␈α∞not␈α∞the␈α∞top␈α∞task␈α∞each␈α
time.
␈↓ α,␈↓Needless to say, the performance was very di≥erent from usual.

␈↓ α,␈↓By␈αcareful␈αplanning,␈α
each␈αexperiment␈αcan␈αtell␈α
us␈αsomething␈αnew␈α
about␈αAM:␈αhow␈αvaluable␈α
a
␈↓ α,␈↓certain␈α∩piece␈α⊃of␈α∩it␈α⊃is,␈α∩how␈α⊃robust␈α∩a␈α⊃certain␈α∩scheme␈α⊃really␈α∩is,␈α⊃etc.␈α∩ The␈α⊃results␈α∩of␈α⊃these
␈↓ α,␈↓experiments␈α⊗would␈α↔then␈α⊗have␈α↔something␈α⊗to␈α⊗contribute␈α↔to␈α⊗a␈α↔discussion␈α⊗of␈α↔the␈α⊗"real
␈↓ α,␈↓intelligence"␈α∞of␈α∞AM␈α
(e.g.,␈α∞what␈α∞features␈α
were␈α∞super∨uous),␈α∞and␈α
contribute␈α∞to␈α∞the␈α∞design␈α
of
␈↓ α,␈↓the␈α∀"next"␈α∀AM-like␈α∀system.␈α∀ Generalizing␈α∃from␈α∀those␈α∀results,␈α∀one␈α∀might␈α∃suggest␈α∀some
␈↓ α,␈↓hypotheses about the larger task of automated math research.

␈↓ α,␈↓Let's cover the di≥erent ␈↓βkinds␈↓ of experiments one could perform on AM:

␈↓ α,␈↓(i)␈α∂Remove␈α∂individual␈α∂concept␈α∂modules,␈α∂and/or␈α∂individual␈α∂heuristic␈α∂rules.␈α⊂Then␈α∂examine

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	15␈↓ε For a list of most of the `winners' and `acceptables', see the final section in Appendix 1, page 136.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε95␈↓-

␈↓ α,␈↓how␈αAM's␈αperformance␈αis␈αdegraded.␈α AM␈αshould␈αoperate␈αeven␈αif␈αmost␈αof␈αits␈αheuristic␈αrules
␈↓ α,␈↓and␈α∂most␈α∂of␈α∂its␈α∂concept␈α∂modules␈α∂were␈α∞excised.␈α∂ If␈α∂the␈α∂remaining␈α∂fragment␈α∂of␈α∂AM␈α∂is␈α∞too
␈↓ α,␈↓small,␈αhowever,␈αit␈α
may␈αnot␈αbe␈αable␈α
to␈α≡nd␈αanything␈αinteresting␈α
to␈αdo.␈αIn␈αfact,␈α
this␈αsituation
␈↓ α,␈↓was␈αactually␈αencountered␈αexperimentally,␈αwhen␈αthe␈α≡rst␈αfew␈αpartially␈αcomplete␈αconcepts␈αwere
␈↓ α,␈↓inserted.␈α∞If␈α∞only␈α∞a␈α∞little␈α∞bit␈α∞of␈α∞AM␈α∞is␈α∞removed,␈α∞the␈α∞remainder␈α∞will␈α∞in␈α∞fact␈α∞keep␈α
operating
␈↓ α,␈↓without␈αthis␈α
"uninteresting␈αcollapse".␈α
 The␈αconverse␈α
situation␈αshould␈α
also␈αhold:␈αalthough␈α
still
␈↓ α,␈↓functional␈α⊃with␈α⊂any␈α⊃concept␈α⊂module␈α⊃unplugged,␈α⊂AM's␈α⊃performance␈α⊂␈↓βshould␈↓␈α⊃be␈α⊂noticeably
␈↓ α,␈↓degraded.␈α∩ That␈α∩is,␈α∩while␈α∩not␈α∩indispensable,␈α∩each␈α∩concept␈α∩should␈α∩nontrivially␈α∪help␈α∩the
␈↓ α,␈↓others.␈α∩ The␈α∩same␈α∪holds␈α∩for␈α∩each␈α∪individual␈α∩heuristic␈α∩rule.␈α∩ When␈α∪a␈α∩piece␈α∩of␈α∪AM␈α∩is
␈↓ α,␈↓removed,␈α which␈α∨concepts␈α does␈α∨AM␈α then␈α∨"miss"␈α discovering?␈α∨ Is␈α the␈α∨removed
␈↓ α,␈↓concept/heuristic␈α⊂later␈α⊂discovered␈α⊂anyway␈α⊂by␈α⊂those␈α⊂which␈α⊂are␈α⊂left␈α⊂in␈α⊂AM?␈α⊃ This␈α⊂should
␈↓ α,␈↓indicate the importance of each kind of concept and rule which AM starts with.

␈↓ α,␈↓(ii)␈α∩Vary␈α∩the␈α∩relative␈α⊃weights␈α∩given␈α∩to␈α∩features␈α⊃by␈α∩the␈α∩criteria␈α∩which␈α∩judge␈α⊃aesthetics,
␈↓ α,␈↓interestingness,␈αworth,␈αutility,␈αetc.␈α See␈αhow␈α
important␈αeach␈αfactor␈αis␈αin␈αdirecting␈α
AM␈αalong
␈↓ α,␈↓successful␈α
routes.␈α In␈α
other␈αwords,␈α
vary␈αthe␈α
little␈αnumbers␈α
in␈αthe␈α
formulae␈α(both␈α
the␈αglobal
␈↓ α,␈↓priority-assigning␈α⊃formula␈α⊃and␈α∩the␈α⊃local␈α⊃reason-rating␈α∩ones␈α⊃inside␈α⊃heuristic␈α∩rules).␈α⊃ One
␈↓ α,␈↓important␈α
result␈α
will␈α
be␈α
some␈α
idea␈α
of␈α
the␈α
robustness␈α
or␈α
"toughness"␈α
of␈α
the␈α
numeric␈α
weighting
␈↓ α,␈↓factors. If the system easily collapses, it was too ≡nely tuned to begin with.

␈↓ α,␈↓(iii)␈αAdd␈α
several␈αnew␈α
concept␈αmodules␈α(including␈α
new␈αheuristics␈α
relevant␈αto␈α
them)␈αand␈αsee␈α
if
␈↓ α,␈↓AM␈αcan␈αwork␈αin␈αsome␈αunanticipated␈α≡eld␈αof␈αmathematics␈α(like␈αgraph␈αtheory␈αor␈α
calculus␈αor
␈↓ α,␈↓plane␈α∩geometry).␈α∩ Do␈α∪earlier␈α∩achievements␈α∩¬␈α∪concepts␈α∩and␈α∩conjectures␈α∪AM␈α∩synthesized
␈↓ α,␈↓already␈α
¬␈α
have␈α
any␈α
impact␈α
in␈α
the␈α
new␈α
domain?␈α
 Are␈α
some␈α
specialized␈α
heuristics␈α∞from␈α
the
␈↓ α,␈↓≡rst␈αdomain␈α
totally␈αwrong␈αhere?␈α
Do␈αall␈αthe␈α
old␈αgeneral␈αheuristics␈α
still␈αhold␈αhere?␈α
 Are␈αthey
␈↓ α,␈↓su≠cient,␈α
or␈α
are␈α
some␈α
"general"␈α
heuristics␈α
needed␈α
here␈α
which␈α
weren't␈α
needed␈α
before?␈α
Does
␈↓ α,␈↓AM "slow down" as more and more concepts get introduced?

␈↓ α,␈↓(iv)␈α∃Try␈α∃to␈α∃have␈α∃AM␈α∀develop␈α∃nonmathematical␈α∃theories␈α∃(like␈α∃elementary␈α∃physics,␈α∀or
␈↓ α,␈↓program␈α
veri≡cation).␈α
 This␈α
might␈αrequire␈α
limiting␈α
AM's␈α
freedom␈αto␈α
"ignore␈α
a␈α
given␈αbody
␈↓ α,␈↓of␈α∀data␈α∀and␈α∀move␈α∪on␈α∀to␈α∀something␈α∀more␈α∪interesting".␈α∀The␈α∀exploration␈α∀of␈α∀very␈α∪non-
␈↓ α,␈↓formalizable␈α∂≡elds␈α∂(e.g.,␈α∂politics)␈α∂might␈α∂require␈α∞much␈α∂more␈α∂than␈α∂a␈α∂small␈α∂augmentation␈α∞of
␈↓ α,␈↓AM's␈α∞base␈α
of␈α∞concepts.␈α∞For␈α
some␈α∞such␈α∞domains,␈α
the␈α∞"Intuitions"␈α∞scheme,␈α
which␈α∞had␈α∞to␈α
be
␈↓ α,␈↓abandoned for math, might prove valid and valuable.

␈↓ α,␈↓(v)␈α⊃Add␈α⊂several␈α⊃new␈α⊂concepts␈α⊃dealing␈α⊃with␈α⊂proof,␈α⊃and␈α⊂of␈α⊃course␈α⊂add␈α⊃all␈α⊃the␈α⊂associated
␈↓ α,␈↓heuristic␈αrules.␈αSuch␈αrules␈αwould␈αadvise␈αAM␈αon␈αthe␈α≡ne␈αpoints␈αof␈αusing␈αvarious␈αtechniques
␈↓ α,␈↓of␈αproof/disproof:␈αwhen␈αto␈αuse␈αthem,␈αwhat␈αto␈αtry␈αnext␈αbased␈αon␈αwhy␈αthe␈αlast␈αattempt␈αfailed,
␈↓ α,␈↓etc.  See if the ␈↓βkinds␈↓ of discoveries AM makes are increased.

␈↓ α,␈↓Several␈αexperiments␈α(of␈αtypes␈αi,␈αii,␈αand␈αiii␈αabove)␈αwere␈αset␈αup␈αand␈αperformed␈αon␈αAM.␈α
We're
␈↓ α,␈↓now␈α
ready␈α
to␈α
examine␈α
each␈α
of␈α
them␈α∞in␈α
detail.␈α
 The␈α
following␈α
points␈α
are␈α
covered␈α∞for␈α
each
␈↓ α,␈↓experiment:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓ How was it thought of?
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈αWhat␈αwill␈αbe␈αgained␈αby␈αit?␈α What␈αwould␈αbe␈αthe␈αimplications␈αof␈αthe␈αvarious␈αpossible
␈↓ α,␈↓␈↓ β≤outcomes?
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈αHow␈αwas␈α
the␈αexperiment␈αset␈αup?␈α
What␈αpreparations/modi≡cations␈αhad␈αto␈α
be␈αmade?
␈↓ α,␈↓␈↓ β≤How much time (man-hours) did it take?
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓ What happened?  How did AM's behavior change? Was this expected?  Analysis.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε96␈↓-

␈↓ α,␈↓␈↓ αl␈↓¬5.␈↓␈αWhat␈αwas␈αlearned␈αfrom␈αthis␈αexperiment?␈αCan␈αwe␈αconclude␈αanything␈αwhich␈αsuggests
␈↓ α,␈↓␈↓ β≤new␈α∞experiments␈α∞(e.g.,␈α
use␈α∞a␈α∞better␈α∞machine,␈α
a␈α∞new␈α∞domain)␈α
or␈α∞which␈α∞bears␈α∞on␈α
a
␈↓ α,␈↓␈↓ β≤more␈α
general␈α
problem␈α∞that␈α
AM␈α
faced␈α
(e.g.,␈α∞a␈α
new␈α
way␈α
to␈α∞teach␈α
math,␈α
a␈α∞new␈α
idea
␈↓ α,␈↓␈↓ β≤about doing math research)?


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.2.1. Must the Worth numbers be ≡nely tuned?␈↓)αβ␈↓

␈↓ α,␈↓Each␈αof␈αthe␈α115␈αinitial␈αconcepts␈αhas␈αsupplied␈αto␈αit␈α(by␈αthe␈αauthor)␈αa␈αnumber␈αbetween␈α0␈αand
␈↓ α,␈↓1000,␈α∂stored␈α⊂as␈α∂its␈α∂Worth␈α⊂facet,␈α∂which␈α∂is␈α⊂supposed␈α∂to␈α∂represent␈α⊂the␈α∂overall␈α∂value␈α⊂of␈α∂the
␈↓ α,␈↓concept.␈α⊂ "Compose"␈α⊂has␈α⊂a␈α⊂higher␈α⊂initial␈α⊂Worth␈α⊂than␈α⊂"Structure-delete",␈α⊂which␈α⊂is␈α∂higher
␈↓ α,␈↓than "Equality"␈↓	16␈↓.

␈↓ α,␈↓Frequently,␈α∞the␈α∂priority␈α∞of␈α∂a␈α∞task␈α∞involving␈α∂C␈α∞depends␈α∂on␈α∞the␈α∞overall␈α∂Worth␈α∞of␈α∂C.␈α∞How
␈↓ α,␈↓sensitive␈α∂is␈α∂AM's␈α∂behavior␈α∞to␈α∂the␈α∂initial␈α∂settings␈α∂of␈α∞the␈α∂Worth␈α∂facets?␈α∂ How␈α∂≡nely␈α∞tuned
␈↓ α,␈↓must these initial Worth values be?

␈↓ α,␈↓This␈α
experiment␈α
was␈α
thought␈α
of␈α
because␈α
of␈αthe␈α
`brittleness'␈α
of␈α
many␈α
other␈α
AI␈α
systems,␈αthe
␈↓ α,␈↓amount␈α
of␈α
≡ne␈α
tuning␈αneeded␈α
to␈α
elicit␈α
coherent␈α
behavior.␈α For␈α
example,␈α
see␈α
the␈αdiscussion␈α
of
␈↓ α,␈↓limitations␈α∞of␈α
PUP6,␈α∞in␈α
[Lenat␈α∞75b].␈α∞ The␈α
author␈α∞believed␈α
that␈α∞AM␈α
was␈α∞very␈α∞resilient␈α
in
␈↓ α,␈↓this␈αregard,␈αand␈αthat␈αa␈α
demonstration␈αof␈αthat␈αfact␈αwould␈α
increase␈αcredibility␈αin␈αthe␈αpower␈α
of
␈↓ α,␈↓the ideas which AM embodies.

␈↓ α,␈↓To␈αtest␈αthis,␈αa␈α
simple␈αexperiment␈αwas␈αperformed.␈αJust␈α
before␈αstarting␈αAM,␈αthe␈α
mean␈αvalue
␈↓ α,␈↓of␈α∞all␈α∞concepts'␈α∞Worth␈α∞values␈α∞was␈α∞computed.␈α∞It␈α∞turned␈α∞out␈α∞to␈α∞be␈α∞roughly␈α∞200.␈α∞Then␈α
each
␈↓ α,␈↓concept␈αhad␈αits␈αWorth␈αreset␈αto␈αthe␈αvalue␈α200.␈↓	17␈↓␈αThis␈αwas␈αdone␈α"by␈αhand",␈αby␈αthe␈αauthor,␈αin
␈↓ α,␈↓a␈αmatter␈αof␈αseconds.␈α AM␈αwas␈αthen␈αstarted␈αand␈αrun␈αas␈αif␈αthere␈αwere␈αnothing␈αamiss,␈αand␈αits
␈↓ α,␈↓behavior was watched carefully.

␈↓ α,␈↓What␈αhappened?␈α By␈αand␈αlarge,␈αthe␈αsame␈αmajor␈αdiscoveries␈αwere␈αmade␈α¬␈αand␈αmissed␈α¬␈αas
␈↓ α,␈↓usual,␈αin␈αthe␈αsame␈αorder␈αas␈αusual.␈α But␈αwhereas␈αAM␈αproceeded␈αfairly␈αsmoothly␈αbefore,␈αwith
␈↓ α,␈↓little␈αsuper∨uous␈αactivity,␈αit␈αnow␈αwandered␈αquite␈αblindly␈αfor␈αlong␈αperiods␈αof␈αtime,␈αespecially
␈↓ α,␈↓at␈α∩the␈α∩very␈α∩beginning.␈α∩ Once␈α∩AM␈α∩"hooked␈α∩into"␈α∩a␈α∩line␈α∩of␈α∩productive␈α∪development,␈α∩it
␈↓ α,␈↓followed␈αit␈αjust␈αas␈α
always,␈αwith␈αno␈αnoticeable␈αadditional␈α
wanderings.␈α As␈αone␈αof␈α
these␈αlines
␈↓ α,␈↓of developments died out, AM would wander around again, until the next one was begun.

␈↓ α,␈↓It␈α⊂took␈α⊂roughly␈α∂three␈α⊂times␈α⊂as␈α∂long␈α⊂for␈α⊂each␈α∂major␈α⊂discovery␈α⊂to␈α∂occur␈α⊂as␈α⊂normal.␈α∂ This
␈↓ α,␈↓"delay"␈αgot␈αshorter␈αand␈αshorter␈αas␈αAM␈αdeveloped␈αfurther.␈α In␈αeach␈αcase,␈αthe␈αtasks␈αpreceding
␈↓ α,␈↓the␈α∪discovery␈α∪and␈α∪following␈α∪it␈α∪were␈α∀pretty␈α∪much␈α∪the␈α∪same␈α∪as␈α∪normal;␈α∪only␈α∀the␈α∪tasks
␈↓ α,␈↓"between" two periods of development were di≥erent ¬ and much more numerous.

␈↓ α,␈↓The␈α
reader␈α
may␈α
be␈αinterested␈α
to␈α
learn␈α
that␈αthe␈α
Worth␈α
values␈α
of␈αmany␈α
of␈α
the␈α
concepts␈α¬␈α
and
␈↓ α,␈↓most␈α
of␈α
the␈α
new␈α
concepts␈α¬␈α
ended␈α
up␈α
very␈α
close␈α
to␈αthe␈α
same␈α
values␈α
that␈α
they␈α
achieved␈αin
␈↓ α,␈↓the␈α⊃original␈α⊃run.␈α⊃ Overrated␈α∩concepts␈α⊃were␈α⊃investigated␈α⊃and␈α⊃proved␈α∩boring;␈α⊃underrated

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	16␈↓ε␈αAs␈αAM␈αprogresses,␈αit␈αnotices␈αsomething␈αinteresting␈αabout␈αEquality␈αevery␈αnow␈αand␈αthen,␈αand␈αpushes␈αits␈αWorth␈αvalue
␈↓ α,␈↓ε␈↓ βLupwards.
␈↓ α,␈↓ε␈↓	17␈↓ε The initial spread of values had previously been from 100 to 600.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε97␈↓-

␈↓ α,␈↓concepts␈αhad␈αto␈αwait␈αlonger␈αfor␈αtheir␈αchances,␈αbut␈αthen␈αquickly␈αproved␈αinteresting␈αand␈αhad
␈↓ α,␈↓their Worth facets boosted.

␈↓ α,␈↓The␈αconclusion␈αI␈αdraw␈αfrom␈αthis␈αchange␈αin␈α
behavior␈αis␈αthat␈αthe␈αWorth␈αfacets␈αare␈αuseful␈α
for
␈↓ α,␈↓making␈αblind␈α
decisions␈α¬␈α
where␈αAM␈α
must␈αchoose␈α
based␈αonly␈α
on␈αthe␈α
overall␈αworths␈α
of␈αthe
␈↓ α,␈↓various␈α⊃concepts␈α⊃in␈α⊃its␈α⊃repertoire.␈α⊃ Whenever␈α⊂a␈α⊃speci≡c␈α⊃reason␈α⊃existed,␈α⊃it␈α⊃was␈α⊃far␈α⊂more
␈↓ α,␈↓in∨uential␈α∂than␈α∂the␈α⊂"erroneous"␈α∂Worth␈α∂values.␈α∂ The␈α⊂close,␈α∂blind,␈α∂random␈α⊂decisions␈α∂occur
␈↓ α,␈↓between long bursts of speci≡c-reason-driven periods of creative work.␈↓	18␈↓

␈↓ α,␈↓The␈α∞general␈α∂answer,␈α∞then,␈α∂is␈α∞␈↓βNo␈↓,␈α∞the␈α∂initial␈α∞settings␈α∂of␈α∞the␈α∞Worth␈α∂values␈α∞are␈α∂not␈α∞crucial.
␈↓ α,␈↓Guessing␈α
reasonable␈α∞initial␈α
values␈α∞for␈α
them␈α
is␈α∞merely␈α
a␈α∞time-saving␈α
device.␈α∞ This␈α
suggests
␈↓ α,␈↓an␈αinteresting␈αresearch␈αproblem:␈αwhat␈αimpact␈αdoes␈αthe␈α␈↓βquality␈↓␈αof␈αinitial␈αstarting␈αvalues␈α
have
␈↓ α,␈↓on␈α∞humans?␈α∞Give␈α∞several␈α∞bright␈α∞undergraduate␈α∞math␈α∞majors␈α∞the␈α∞same␈α∞set␈α∞of␈α∞objects␈α
and
␈↓ α,␈↓operators␈α
to␈αplay␈α
with,␈αbut␈α
tell␈αsome␈α
of␈αthem␈α
(i)␈α
nothing,␈αand␈α
some␈αof␈α
them␈α(ii)␈α
a␈αcertain␈α
few
␈↓ α,␈↓pieces␈αof␈αthe␈αsystem␈αare␈αvery␈αpromising,␈α(iii)␈αemphasize␈αa␈αdi≥erent␈αsubset␈αof␈αthe␈αobjects␈α
and
␈↓ α,␈↓operators.␈α∂ How␈α∂does␈α⊂"misinformation"␈α∂impede␈α∂the␈α⊂humans?␈α∂How␈α∂about␈α⊂no␈α∂information?
␈↓ α,␈↓Have them give verbal protocols about where they are focussing their attention, and why.

␈↓ α,␈↓Albeit␈αat␈αa␈αnontrivial␈α
cost,␈αthe␈αWorth␈αfacets␈α
did␈αmanage␈αto␈αcorrect␈α
themselves␈αby␈αthe␈αend␈α
of
␈↓ α,␈↓a␈αlong␈↓	19␈↓␈αrun.␈α What␈αwould␈αhappen␈αif␈αthe␈αWorth␈αfacets␈αof␈αthose␈α115␈αconcepts␈αwere␈αnot␈αonly
␈↓ α,␈↓initialized to 200, but were held ≡xed at 200 for the duration of the run?

␈↓ α,␈↓In␈αthis␈αcase,␈αthe␈αdelay␈αstill␈αsubsided␈αwith␈αtime.␈α That␈αis,␈αAM␈αstill␈αgot␈αmore␈αand␈αmore␈α"back
␈↓ α,␈↓to␈α⊂normal"␈α⊂as␈α⊂it␈α⊂progressed␈α⊂onward.␈α⊃The␈α⊂reason␈α⊂is␈α⊂because␈α⊂AM's␈α⊂later␈α⊂work␈α⊃dealt␈α⊂with
␈↓ α,␈↓concepts␈α
like␈α
Primes,␈αSquare-root,␈α
etc.,␈α
which␈α
were␈αso␈α
far␈α
removed␈αfrom␈α
the␈α
initial␈α
base␈αof
␈↓ α,␈↓concepts that the initial concepts' Worths were of little consequence.

␈↓ α,␈↓Even␈α∞more␈α∞drastically,␈α∂we␈α∞could␈α∞force␈α∞all␈α∂the␈α∞Worth␈α∞facets␈α∞of␈α∂all␈α∞concepts␈α∞¬␈α∂even␈α∞newly-
␈↓ α,␈↓created␈α∂ones␈α∂¬␈α∂to␈α∂be␈α∂kept␈α∂at␈α∂the␈α∂value␈α∂200␈α∂forever.␈α∂In␈α∂this␈α∂case,␈α∂AM's␈α∂behavior␈α∞doesn't
␈↓ α,␈↓completely␈α∞disintegrate,␈α∞but␈α∞that␈α∞delay␈α
factor␈α∞actually␈α∞increases␈α∞with␈α∞time:␈α∞apparently,␈α
AM
␈↓ α,␈↓begins␈αto␈α
su≥er␈αfrom␈α
the␈αexponential␈αgrowth␈α
of␈α"things␈α
to␈αdo"␈αas␈α
its␈αrepertoire␈α
of␈αconcepts
␈↓ α,␈↓grows␈α
linearly.␈α
 Its␈αpurposiveness,␈α
its␈α
directionality␈α
depends␈αon␈α
"focus␈α
of␈α
attention"␈αmore␈α
and
␈↓ α,␈↓more,␈αand␈αif␈αthat␈αfeature␈αis␈αremoved,␈αAM␈αloses␈αmuch␈αof␈αits␈αrationality.␈α A␈αfactor␈αof␈α5␈αdelay
␈↓ α,␈↓doesn't␈α⊂sound␈α⊃that␈α⊂bad␈α⊃"e≠ciency-wise",␈α⊂but␈α⊃the␈α⊂actual␈α⊃apparent␈α⊂behavior␈α⊃of␈α⊂AM␈α⊃is␈α⊂as
␈↓ α,␈↓staccato␈αbursts␈αof␈αdevelopment,␈α
followed␈αby␈αwild␈αleaps␈α
to␈αunrelated␈αconcepts.␈α AM␈αno␈α
longer
␈↓ α,␈↓can "permanently" record its interest in a certain concept.

␈↓ α,␈↓So␈α∂we␈α∂conclude␈α∂that␈α∂the␈α∂Worth␈α∂facets␈α∞are␈α∂(i)␈α∂not␈α∂≡nely␈α∂tuned,␈α∂yet␈α∂(ii)␈α∂provide␈α∞important
␈↓ α,␈↓global␈αinformation␈αabout␈αthe␈α
relative␈αvalues␈αof␈αconcepts.␈α
 If␈αthe␈αWorth␈αfacets␈αare␈α
completely
␈↓ α,␈↓disabled,␈α∪the␈α∪rationality␈α∀of␈α∪AM's␈α∪behavior␈α∀hangs␈α∪on␈α∪the␈α∀slender␈α∪thread␈α∪of␈α∀"focus␈α∪of
␈↓ α,␈↓attention".




␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	18␈↓ε Incidentally, GPS behaved just this same way. See, e.g., [Newell&Simon 72].
␈↓ α,␈↓ε␈↓	19␈↓ε A couple cpu hours, about a thousand tasks total selected from the agenda
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε98␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.2.2. How ≡nely tuned is the Agenda?␈↓)αβ␈↓

␈↓ α,␈↓The␈α
top␈α
few␈α
candidates␈α
on␈α
the␈α
agenda␈α
always␈αappear␈α
to␈α
be␈α
reasonable␈α
(to␈α
me).␈α
 If␈α
I␈αwork
␈↓ α,␈↓with␈α
the␈α
system,␈α
guiding␈α
it,␈α∞I␈α
can␈α
cause␈α
it␈α
to␈α∞make␈α
a␈α
few␈α
discoveries␈α
it␈α∞wouldn't␈α
otherwise
␈↓ α,␈↓make,␈α
and␈αI␈α
can␈α
cause␈αit␈α
to␈α
make␈αits␈α
typical␈αones␈α
much␈α
faster␈α(about␈α
a␈α
factor␈αof␈α
2).␈αThus␈α
the
␈↓ α,␈↓␈↓βvery␈↓ top task is not always the best.

␈↓ α,␈↓If␈α
AM␈α
randomly␈α
selects␈α
one␈α∞of␈α
the␈α
top␈α
20␈α
or␈α
so␈α∞tasks␈α
on␈α
the␈α
agenda␈α
each␈α
time,␈α∞what␈α
will
␈↓ α,␈↓happen␈α⊃to␈α∩its␈α⊃behavior?␈α∩Will␈α⊃it␈α∩disintegrate,␈α⊃slow␈α∩down␈α⊃by␈α∩a␈α⊃factor␈α∩of␈α⊃10,␈α∩slow␈α⊃down
␈↓ α,␈↓slightly,...?

␈↓ α,␈↓This␈α
experiment␈αrequired␈α
only␈α
a␈αfew␈α
seconds␈α
to␈αset␈α
up,␈α
but␈αdemanded␈α
a␈α
familiarity␈αwith␈α
the
␈↓ α,␈↓LISP␈α∞functions␈α∞which␈α∞make␈α∞up␈α∞AM's␈α∞control␈α∞structure.␈α∞ At␈α∞a␈α∞certain␈α∞point,␈α∞AM␈α∞asks␈α∞for
␈↓ α,␈↓Best-task(Agenda).␈α
 Typically,␈α
the␈α
LISP␈αfunction␈α
Best-task␈α
is␈α
de≡ned␈αas␈α
CAR␈α
¬␈α
i.e.,␈αpick␈α
the
␈↓ α,␈↓≡rst␈α∂member␈α⊂from␈α∂the␈α∂list␈α⊂of␈α∂tasks.␈α⊂ What␈α∂I␈α∂did␈α⊂was␈α∂to␈α∂rede≡ne␈α⊂Best-task␈α∂as␈α⊂a␈α∂function
␈↓ α,␈↓which␈αrandomly␈αselected␈αn␈αfrom␈αthe␈αset␈α{1,2,...,20},␈αand␈αthen␈αreturned␈αthe␈αn␈↓	th␈↓␈αmember␈αof␈αthe
␈↓ α,␈↓job-list.

␈↓ α,␈↓If␈α
you␈α∞watch␈α
the␈α
top␈α∞job␈α
on␈α∞the␈α
agenda,␈α
it␈α∞will␈α
take␈α∞about␈α
10␈α
cycles␈α∞until␈α
AM␈α∞chooses␈α
it.
␈↓ α,␈↓And␈α
yet␈αthere␈α
are␈αmany␈α
good,␈α
interesting,␈αworthwhile␈α
jobs␈αsprinkled␈α
among␈αthe␈α
top␈α
20␈αon
␈↓ α,␈↓the␈α
agenda,␈α
so␈α
AM's␈α
performance␈α
is␈α
cut␈α
by␈α
merely␈αa␈α
factor␈α
of␈α
3,␈α
as␈α
far␈α
as␈α
cpu␈α
time␈αper␈α
given
␈↓ α,␈↓major␈α
discovery.␈α
 Part␈α
of␈α
this␈α
better-than-20␈α
behavior␈α
is␈α
due␈α
to␈α
the␈α
fact␈α
that␈α
the␈α
18␈↓	th␈↓␈αbest
␈↓ α,␈↓task␈αhad␈αa␈αmuch␈αlower␈αpriority␈αrating␈αthan␈αthe␈αtop␈αfew,␈αhence␈αwas␈αallocated␈αmuch␈αless␈αcpu
␈↓ α,␈↓time␈α⊂for␈α⊂its␈α⊂quantum␈α⊂than␈α⊂the␈α⊂top␈α⊂task␈α⊂would␈α⊂have␈α⊂received.␈α⊂ Whether␈α⊂it␈α⊂succeeded␈α⊂or
␈↓ α,␈↓failed,␈αit␈αused␈αup␈αvery␈αlittle␈αtime.␈α Since␈αAM␈αwas␈αfrequently␈αworking␈αon␈αa␈αlow-value␈αtask,␈αit
␈↓ α,␈↓was␈αunwilling␈αto␈αspend␈αmuch␈αtime␈αor␈αspace␈αon␈αit.␈αSo␈αthe␈αmean␈αtime␈αallotted␈αper␈αtask␈αfell␈αto
␈↓ α,␈↓about␈α
15␈α
seconds␈α(from␈α
the␈α
typical␈α
30␈αsecs).␈α
Thus,␈α
the␈α
"losers"␈αwere␈α
dealt␈α
with␈α
quickly,␈αso␈α
the
␈↓ α,␈↓detriment to cpu-time performance was softened.

␈↓ α,␈↓To␈αcarry␈αthis␈αinvestigation␈αfurther,␈αanother␈αexperiment␈αwas␈αcarried␈αout.␈αAM␈αwas␈αforced␈αto
␈↓ α,␈↓alternate␈α∀between␈α∀choosing␈α∀the␈α∀top␈α∀task␈α∀on␈α∀the␈α∀agenda,␈α∀and␈α∀a␈α∀randomly-chosen␈α∀one.
␈↓ α,␈↓Although␈α∩its␈α∪rate␈α∩of␈α∪discovery␈α∩was␈α∩cut␈α∪by␈α∩less␈α∪than␈α∩half,␈α∩its␈α∪behavior␈α∩was␈α∪almost␈α∩as
␈↓ α,␈↓distasteful to the user as in the last (always-random) experiment.

␈↓ α,␈↓␈↓↓␈↓&Conclusion␈↓)αβ␈↓:␈αPicking␈α(on␈α
the␈αaverage)␈αthe␈α10th-best␈α
candidate␈αimpedes␈αprogress␈αby␈α
a␈αfactor
␈↓ α,␈↓less␈α
than␈α∞10␈α
(about␈α
a␈α∞factor␈α
of␈α
3),␈α∞but␈α
it␈α
dramatically␈α∞degrades␈α
the␈α
"sensibleness"␈α∞of␈α
AM's
␈↓ α,␈↓behavior,␈αthe␈αcontinuity␈αof␈αits␈αactions.␈α Humans␈αplace␈αa␈αbig␈αvalue␈αon␈α
absolute␈αsensibleness,
␈↓ α,␈↓and␈α∀believe␈α∀that␈α∪doing␈α∀something␈α∀silly␈α∪50%␈α∀of␈α∀the␈α∪time␈α∀is␈α∀␈↓β␈↓&much␈↓)αβ␈↓␈α∪worse␈α∀than␈α∀half␈α∪as
␈↓ α,␈↓productive as always doing the next most logical task.

␈↓ α,␈↓Corollary:␈α
Having␈α
20␈α
multi-processors␈α
simultaneously␈α∞execute␈α
the␈α
top␈α
20␈α
jobs␈α∞will␈α
increase
␈↓ α,␈↓the rate of "big" discoveries, but not by a full factor of 20 ¬ more like a factor of 3.

␈↓ α,␈↓Another␈α∞experiment␈α
in␈α∞this␈α
same␈α∞vein␈α
was␈α∞done,␈α
one␈α∞which␈α
was␈α∞designed␈α
to␈α∞be␈α∞far␈α
more
␈↓ α,␈↓crippling␈αto␈αAM.␈α Be-threshhold␈αwas␈αheld␈αat␈α0␈αalways,␈αso␈α␈↓βany␈↓␈αtask␈αwhich␈αever␈αgot␈αproposed
␈↓ α,␈↓was␈αkept␈α
forever␈αon␈α
the␈αagenda,␈α
no␈αmatter␈α
how␈αlow␈α
its␈αpriority.␈α
 The␈αBest-task␈αfunction␈α
was
␈↓ α,␈↓modi≡ed␈αso␈αit␈αrandomly␈α
selected␈αany␈αmember␈αof␈α
the␈αlist␈αof␈αjobs.␈α
As␈αa␈α≡nal␈αinsult,␈αthe␈α
Worth
␈↓ α,␈↓facets of all the concepts were initialized to 200 before starting AM.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ,␈↓-␈↓ε99␈↓-

␈↓ α,␈↓Result:␈α⊂Many␈α⊂"explosive"␈α⊂tasks␈α⊂were␈α⊂chosen,␈α⊂and␈α⊂the␈α⊂number␈α⊂of␈α⊂new␈α⊃concepts␈α⊂increased
␈↓ α,␈↓rapidly.␈α As␈α
expected,␈αmost␈α
of␈αthese␈αwere␈α
real␈α"losers".␈α
 There␈αseemed␈α
no␈αrationality␈αto␈α
AM's
␈↓ α,␈↓sequence␈αof␈αactions,␈αand␈αit␈αwas␈αquite␈αboring␈αto␈αwatch␈αit␈α∨oundering␈αso.␈α The␈αtypical␈αlength
␈↓ α,␈↓of␈α∂the␈α∂agenda␈α⊂was␈α∂about␈α∂500,␈α∂and␈α⊂AM's␈α∂performance␈α∂was␈α∂"slowed"␈α⊂by␈α∂at␈α∂least␈α⊂a␈α∂couple
␈↓ α,␈↓orders␈α⊂of␈α⊃magnitude.␈α⊂ A␈α⊂more␈α⊃subjective␈α⊂measure␈α⊃of␈α⊂its␈α⊂"intelligence"␈α⊃would␈α⊂say␈α⊃that␈α⊂it
␈↓ α,␈↓totally collapsed under this random scheme.

␈↓ α,␈↓␈↓&␈↓↓Conclusion:␈↓␈↓)αβ␈αHaving␈αan␈αunlimited␈αnumber␈αof␈αprocessors␈αsimultaneously␈αexecute␈αall␈αthe␈αjobs
␈↓ α,␈↓on␈α⊂the␈α⊃agenda␈α⊂would␈α⊃increase␈α⊂the␈α⊂rate␈α⊃at␈α⊂which␈α⊃AM␈α⊂made␈α⊂big␈α⊃discoveries,␈α⊂at␈α⊃an␈α⊂ever
␈↓ α,␈↓accelerating pace (since the length of the agenda would grow exponentially).

␈↓ α,␈↓Having␈α
a␈α∞uniprocessor␈α
␈↓βsimulate␈↓␈α∞such␈α
parallel␈α
processing␈α∞would␈α
be␈α∞a␈α
losing␈α∞idea,␈α
however.
␈↓ α,␈↓The truly "intelligent" behavior AM exhibits is its plausible sequencing of tasks.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.2.3. How valuable is tacking reasons onto each task?␈↓)αβ␈↓

␈↓ α,␈↓Let's␈α∩dig␈α∩inside␈α∩the␈α∩agenda␈α∩scheme␈α∩now.␈α∩ One␈α∩idea␈α∩I've␈α∩repeatedly␈α∩emphasized␈α∩is␈α∩the
␈↓ α,␈↓attaching␈α
of␈α
reasons␈αto␈α
the␈α
tasks␈αon␈α
the␈α
agenda,␈αand␈α
using␈α
those␈αreasons␈α
and␈α
their␈αratings␈α
to
␈↓ α,␈↓compute␈α⊃the␈α⊃overall␈α⊃priority␈α⊃value␈α⊃assigned␈α∩to␈α⊃each␈α⊃task.␈α⊃ An␈α⊃experiment␈α⊃was␈α∩done␈α⊃to
␈↓ α,␈↓ascertain the amount of intelligence that was emanating from that idea.

␈↓ α,␈↓The␈αglobal␈αformula␈αassigning␈αa␈αpriority␈αvalue␈αto␈αeach␈αjob␈αwas␈αmodi≡ed.␈αWe␈αlet␈αit␈αstill␈αbe␈αa
␈↓ α,␈↓function␈α∩of␈α∩the␈α∩reasons␈α⊃for␈α∩the␈α∩job,␈α∩but␈α⊃we␈α∩"trivialized"␈α∩it:␈α∩the␈α⊃priority␈α∩of␈α∩a␈α∩job␈α⊃was
␈↓ α,␈↓computed␈α
as␈α
simply␈α
the␈α
number␈α
of␈α∞reasons␈α
it␈α
has␈α
(normalized␈α
by␈α
multiplying␈α
by␈α∞100,␈α
and
␈↓ α,␈↓cut-o≥ if over 1000).

␈↓ α,␈↓This␈α
raised␈α
the␈α
new␈αquestion␈α
of␈α
what␈α
to␈αdo␈α
if␈α
several␈α
jobs␈αall␈α
have␈α
the␈α
same␈α
priority.␈α In
␈↓ α,␈↓that case, I had AM execute them in stack-order (most recent ≡rst)␈↓	20␈↓.

␈↓ α,␈↓Result:␈α
I␈α
secretly␈α∞expected␈α
that␈α
this␈α∞wouldn't␈α
make␈α
too␈α∞much␈α
di≥erence␈α
on␈α∞AM's␈α
apparent
␈↓ α,␈↓level␈α∞of␈α∂directionality,␈α∞but␈α∂such␈α∞was␈α∂de≡nitely␈α∞not␈α∞the␈α∂case.␈α∞ While␈α∂AM␈α∞opened␈α∂by␈α∞doing
␈↓ α,␈↓tasks␈α∀which␈α∪were␈α∀far␈α∀more␈α∪interesting␈α∀and␈α∪daring␈α∀than␈α∀usual␈α∪(e.g.,␈α∀≡lling␈α∀in␈α∪various
␈↓ α,␈↓Coalescings␈α∂right␈α∂away),␈α∞it␈α∂soon␈α∂became␈α∞obvious␈α∂that␈α∂AM␈α∞was␈α∂being␈α∂swayed␈α∂by␈α∞hitherto
␈↓ α,␈↓trivial␈α∂coding␈α∂decisions.␈α∂ Whole␈α∂classes␈α∂of␈α∂tasks␈α∂¬␈α∂like␈α∂Checking␈α∂Examples␈α∂of␈α∂C␈α∂¬␈α∂were
␈↓ α,␈↓never␈α
chosen,␈αbecause␈α
they␈αonly␈α
had␈αone␈α
or␈αtwo␈α
reasons␈αsupporting␈α
them.␈α
 Previously,␈αone
␈↓ α,␈↓or␈αtwo␈αgood␈αreasons␈αwere␈αsu≠cient.␈αNow,␈αtasks␈αwith␈αseveral␈αpoor␈αreasons␈αwere␈αrising␈αto␈αthe
␈↓ α,␈↓top␈αand␈αbeing␈αworked␈αon.␈αEven␈αthe␈αLIFO␈α(stack)␈αpolicy␈αfor␈αresolving␈αties␈αdidn't␈αkeep␈αAM's
␈↓ α,␈↓attention focussed.

␈↓ α,␈↓␈↓↓␈↓&Conclusion:␈↓)αβ␈↓␈α∞Unless␈α
a␈α∞conscious␈α
e≥ort␈α∞is␈α∞made␈α
to␈α∞ensure␈α
that␈α∞each␈α
reason␈α∞really␈α∞will␈α
carry
␈↓ α,␈↓roughly␈αan␈αequal␈αamount␈αof␈αsemantic␈αimpact␈α(charge,␈αweight),␈αit␈αis␈αnot␈αacceptable␈αmerely␈αto
␈↓ α,␈↓choose␈α⊂tasks␈α⊂on␈α⊂the␈α∂basis␈α⊂of␈α⊂how␈α⊂many␈α⊂reasons␈α∂they␈α⊂possess.␈α⊂ Even␈α⊂in␈α⊂those␈α∂constricted
␈↓ α,␈↓equal-weight␈α
cases,␈αthe␈α
similarities␈αbetween␈α
reasons␈α
supporting␈αa␈α
task␈αshould␈α
be␈α
taken␈αinto
␈↓ α,␈↓account.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	20␈↓ε␈α	Why?␈α	 Because␈α
(i)␈α	it␈α	sounds␈α
right␈α	intuitively␈α	to␈α
me,␈α	(ii)␈α	this␈α
is␈α	akin␈α	to␈α
human␈α	focus␈α	of␈α
attention,␈α	and␈α	mainly␈α
because␈α	(iii)
␈↓ α,␈↓ε␈↓ βLthis is what AM did anyway, with no extra modification.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε100␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.2.4. What if certain concepts are eliminated/added?␈↓)αβ␈↓

␈↓ α,␈↓Feeling␈α
in␈α
a␈α
perverse␈α
mood␈α
one␈α
day,␈α∞I␈α
eliminated␈α
the␈α
concept␈α
"Equality"␈α
from␈α
AM,␈α∞to␈α
see
␈↓ α,␈↓what␈α
it␈αwould␈α
then␈αdo.␈α
 Equality␈αwas␈α
a␈αkey␈α
concept,␈αbecause␈α
AM␈αdiscovered␈α
Numbers␈αvia
␈↓ α,␈↓the␈αtechnique␈αof␈αgeneralizing␈αthe␈αrelation␈α"Equality"␈α(exact␈αequality␈αof␈α2␈αgiven␈αstructures,␈α
at
␈↓ α,␈↓all␈α⊂internal␈α⊃levels).␈α⊂ What␈α⊂would␈α⊃happen␈α⊂if␈α⊃we␈α⊂eliminate␈α⊂this␈α⊃path?␈α⊂ Will␈α⊃AM␈α⊂rederive
␈↓ α,␈↓Equality?  Will it get to Cardinality via another route? Will it do some set-theoretic things?

␈↓ α,␈↓Result:␈α
Rather␈α
disappointing.␈α
AM␈α
never␈α
did␈α
re-derive␈α
Equality,␈α
nor␈α
Cardinality.␈α
 It␈αspent␈α
its
␈↓ α,␈↓time␈α∃thrashing␈α∀about␈α∃with␈α∀various␈α∃∨avors␈α∀of␈α∃data-structures␈α∀(unordered␈α∃vs.␈α∀ ordered,
␈↓ α,␈↓multiple-elements␈α⊃allowed␈α⊃or␈α⊃not,␈α⊃etc.),␈α∩deriving␈α⊃large␈α⊃quantities␈α⊃of␈α⊃boring␈α∩results␈α⊃about
␈↓ α,␈↓them.␈α∂Very␈α∂many␈α∞composings␈α∂and␈α∂coalescings␈α∞were␈α∂done,␈α∂but␈α∞no␈α∂exciting␈α∂new␈α∞operations
␈↓ α,␈↓were produced.

␈↓ α,␈↓A␈α∞kinder␈α
type␈α∞of␈α
experiment␈α∞would␈α
be␈α∞to␈α
␈↓βadd␈↓␈α∞a␈α
few␈α∞concepts.␈α
 One␈α∞such␈α∞experiment␈α
was
␈↓ α,␈↓done:␈αthe␈αaddition␈αof␈αCartesian-product.␈α This␈αoperation,␈αnamed␈αC-PROD,␈αaccepts␈αtwo␈αsets
␈↓ α,␈↓as arguments and returns a third set as its value: the Cartesian product of the ≡rst two.

␈↓ α,␈↓Result:␈αThe␈αonly␈αsigni≡cant␈αchange␈αin␈αAM's␈αbehavior␈αwas␈αthat␈αTIMES␈αwas␈αdiscovered␈α≡rst
␈↓ α,␈↓as␈αthe␈αrestriction␈αof␈αC-PROD␈αto␈αCanonical-Bags.␈α When␈αit␈αsoon␈αwas␈αrediscovered␈αin␈αa␈αfew
␈↓ α,␈↓other␈α
guises,␈α
its␈α
Worth␈α
was␈α
even␈α
higher␈αthan␈α
usual.␈α
 AM␈α
spent␈α
even␈α
more␈α
time␈αexploring
␈↓ α,␈↓concepts concerned with it, and deviated much less for quite a long time.

␈↓ α,␈↓Synthesis␈α
of␈α
the␈α
above␈α
experiments:␈α
It␈α
appears␈α
that␈α
AM␈α
may␈α
really␈α
be␈α
more␈α
specialized␈α
than
␈↓ α,␈↓expected;␈αAM␈αmay␈αonly␈αbe␈α
able␈αto␈αforge␈αahead␈αalong␈α
one␈αor␈αtwo␈αmain␈αlines␈αof␈α
development
␈↓ α,␈↓¬␈α
at␈α∞least␈α
if␈α
we␈α∞demand␈α
it␈α∞make␈α
very␈α
interesting,␈α∞well-known␈α
discoveries␈α∞quite␈α
frequently.
␈↓ α,␈↓Removing␈α∀certain␈α∀key␈α∀concepts␈α∀can␈α∀be␈α∀disastrous.␈α∀ On␈α∀the␈α∀other␈α∀hand,␈α∀adding␈α∪some
␈↓ α,␈↓carefully-chosen␈α∪new␈α∪ones␈α∪can␈α∪greatly␈α∪enhance␈α∪AM's␈α∪directionality␈α∪(hence␈α∪its␈α∩apparent
␈↓ α,␈↓intelligence).

␈↓ α,␈↓Conclusion:␈α∀In␈α∀its␈α∀current␈α∀state,␈α∀AM␈α∃is␈α∀thus␈α∀seen␈α∀to␈α∀be␈α∀␈↓βminimally␈α∀competent␈↓:␈α∃if␈α∀any
␈↓ α,␈↓knowledge␈α
is␈α
removed,␈α
it␈α
appears␈α
much␈α
less␈α
intelligent;␈α
if␈α
any␈α
is␈α
added,␈α
it␈α
appears␈α
slightly
␈↓ α,␈↓smarter.

␈↓ α,␈↓Suggestion␈αfor␈αfuture␈αresearch:␈αA␈αhypothesis,␈αwhich␈αshould␈αbe␈αtested␈αexperimentally,␈αis␈αthat
␈↓ α,␈↓the␈α
importance␈α
of␈α
the␈α
presence␈α
of␈α
each␈α
individual␈α
concept␈α
decreases␈α
as␈α
the␈α
number␈α
of␈α
¬␈α
and
␈↓ α,␈↓␈↓βdepth␈↓␈αof␈α¬␈αthe␈α
synthesized␈αconcepts␈αincrease.␈α That␈α
is,␈αany␈αexcision␈αwould␈α
eventually␈α"heal
␈↓ α,␈↓over",␈α
given␈α
enough␈α∞time.␈α
 The␈α
failure␈α
of␈α∞AM␈α
to␈α
verify␈α
this␈α∞may␈α
be␈α
due␈α
to␈α∞the␈α
relatively
␈↓ α,␈↓small␈αamount␈αof␈αdevelopment␈αin␈αtoto␈α(an␈αhour␈αof␈αcpu␈αtime,␈αa␈αcouple␈αhundred␈αnew␈αconcepts,
␈↓ α,␈↓a few levels deeper than the starting ones).


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&6.2.5. Can AM work in a new domain: Plane Geometry?␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|A true strategy should be domain-independent.

␈↓ α,␈↓¬␈↓ ε\-- Adams
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε101␈↓-

␈↓ α,␈↓As␈α
McDermott␈α
points␈α∞out␈α
[McDermott␈α
76],␈α
just␈α∞labelling␈α
a␈α
bunch␈α
of␈α∞heuristics␈α
`Operation
␈↓ α,␈↓heuristics'␈α∂doesn't␈α∂suddenly␈α∂make␈α∞them␈α∂relevant␈α∂to␈α∂any␈α∂operation;␈α∞all␈α∂it␈α∂does␈α∂is␈α∂give␈α∞that
␈↓ α,␈↓impression␈α∂to␈α∂a␈α∂human␈α∂who␈α∞looks␈α∂at␈α∂the␈α∂code␈α∂(or␈α∞a␈α∂description␈α∂of␈α∂it).␈α∂ Since␈α∂the␈α∞author
␈↓ α,␈↓hoped␈α⊂that␈α⊂the␈α⊂labelling␈α⊂really␈α⊂was␈α⊂fair,␈α⊂an␈α⊂experiment␈α⊂was␈α⊂done␈α⊂to␈α⊂test␈α⊂this.␈α⊂ Such␈α∂an
␈↓ α,␈↓experiment would be a key determiner of how general AM is.

␈↓ α,␈↓How␈α
might␈α
one␈α
demonstrate␈α
that␈α
the␈α
"Operation"␈α
heuristics␈α
really␈α
could␈α
be␈α
useful␈α
or␈α
dealing
␈↓ α,␈↓with any operation, not just the ones already in AM's initial base of concepts?

␈↓ α,␈↓One␈αway␈αwould␈αbe␈αto␈αpick␈αa␈αnew␈αdomain,␈αand␈αsee␈αhow␈αmany␈αold␈αheuristics␈αcontribute␈αto␈α¬
␈↓ α,␈↓and␈αhow␈αmany␈α
new␈αheuristics␈αhave␈αto␈α
be␈αadded␈αto␈αelicit␈α
¬␈αsome␈αsophisticated␈α
behavior␈αin
␈↓ α,␈↓that␈α
domain.␈α
 Of␈α
course,␈α
some␈α
new␈α
primitive␈α
concepts␈α
would␈α
have␈α
to␈α
be␈αintroduced␈α
(de≡ned)
␈↓ α,␈↓to AM.

␈↓ α,␈↓Only␈αone␈α
experiment␈αof␈α
this␈αtype␈α
was␈αattempted.␈α
 The␈αauthor␈α
added␈αa␈α
new␈αbase␈αof␈α
concepts
␈↓ α,␈↓to␈α∩the␈α∩ones␈α∩already␈α∪in␈α∩AM.␈α∩ Included␈α∩were:␈α∪Point,␈α∩Line,␈α∩Angle,␈α∩Triangle,␈α∪Equality␈α∩of
␈↓ α,␈↓points/lines/angles/triangles.␈α These␈αsimple␈αplane␈αgeometry␈αnotions␈αwere␈αsu≠ciently␈α
removed
␈↓ α,␈↓from␈α∞set-theoretic␈α
ones␈α∞that␈α∞those␈α
pre-existing␈α∞speci≡c␈α∞concepts␈α
would␈α∞be␈α∞totally␈α
irrelevant;
␈↓ α,␈↓on␈α
the␈α
other␈α
hand,␈α
the␈α
general␈α
concepts␈α¬␈α
the␈α
ones␈α
with␈α
the␈α
heuristics␈α
attached␈α
¬␈αwould␈α
still
␈↓ α,␈↓be just as relevant: Any-concept, Operation, Predicate, Structure, etc.

␈↓ α,␈↓For␈α∂each␈α∂new␈α∂geometric␈α∂concept,␈α∂the␈α∂only␈α⊂facet␈α∂≡lled␈α∂in␈α∂was␈α∂its␈α∂De≡nition.␈α∂ For␈α⊂the␈α∂new
␈↓ α,␈↓predicates␈αand␈αoperators,␈αtheir␈αDomain/range␈αentries␈αwere␈αalso␈αsupplied.␈α No␈αnew␈αheuristics
␈↓ α,␈↓were added to AM.

␈↓ α,␈↓Results:␈α∞fairly␈α∞good␈α∞behavior.␈α∞ AM␈α∞was␈α∞able␈α∞to␈α∞≡nd␈α∞examples␈α∞of␈α∞all␈α∞the␈α∞concepts␈α
de≡ned,
␈↓ α,␈↓and␈α
to␈α∞use␈α
the␈α∞character␈α
of␈α∞the␈α
results␈α
of␈α∞those␈α
examples␈α∞searches␈α
to␈α∞determine␈α
intelligent
␈↓ α,␈↓courses␈α∂of␈α∂action.␈α∂ AM␈α∂derived␈α∂congruence␈α∂and␈α∂similarity␈α∂of␈α∂triangles,␈α∂and␈α⊂several␈α∂other
␈↓ α,␈↓well-known␈α
simple␈αconcepts.␈α
 An␈αunusual␈α
result␈αwas␈α
the␈αrepeated␈α
derivation␈αof␈α
the␈α
idea␈αof
␈↓ α,␈↓"timberline".␈αThis␈αis␈αa␈αpredicate␈αon␈αtwo␈αtriangles:␈αTimberline(T1,T2)␈αi≥␈αT1␈αand␈αT2␈αhave␈αa
␈↓ α,␈↓common angle, and the side opposite that angle in the two triangles are parallel:

␈↓"␈↓ α,␈↓π                    ␈↓ ␈↓π    A
␈↓"␈↓ α,␈↓π                        /\
␈↓"␈↓ α,␈↓π                       /  \
␈↓"␈↓ α,␈↓π                      /    \
␈↓"␈↓ α,␈↓π                     /      \           ␈↓αTimberline(ABC,ADE)␈↓π
␈↓"␈↓ α,␈↓π                    /        \
␈↓"␈↓ α,␈↓π                B  /αααααααααα\  C
␈↓"␈↓ α,␈↓π                  /            \
␈↓"␈↓ α,␈↓π                 /              \
␈↓"␈↓ α,␈↓π                /                \
␈↓"␈↓ α,␈↓π               DααααααααααααααααααE


␈↓ α,␈↓Since␈α⊃AM␈α⊃kept␈α⊃rederiving␈α⊃this␈α⊃in␈α⊃new␈α⊃ways,␈α⊃it␈α⊃seems␈α⊃surprising␈α⊃that␈α⊃there␈α⊃is␈α⊃no␈α⊃very
␈↓ α,␈↓common␈α
name␈α
for␈α
the␈αconcept.␈α
It␈α
could␈α
be␈α
that␈αAM␈α
is␈α
using␈α
techniques␈α
which␈αhumans␈α
don't
␈↓ α,␈↓¬ at least, for geometry.
␈↓ α,␈↓␈↓εChapter 6␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε102␈↓-

␈↓ α,␈↓The␈α
only␈αnew␈α
bit␈α
of␈αknowledge␈α
that␈αcame␈α
out␈α
of␈αthis␈α
experiment␈α
was␈αa␈α
"use"␈αfor␈α
Goldbach's
␈↓ α,␈↓conjecture:␈αany␈αangle␈α(0-180␈αdegrees)␈αcan␈αbe␈αbuilt␈αup␈α(to␈αwithin␈α1␈αdegree)␈αas␈αthe␈αsum␈αof␈αtwo
␈↓ α,␈↓angles␈αof␈α
prime␈αdegrees␈α(<180).␈α
This␈αresult␈αis␈α
admittedly␈αesoteric␈αat␈α
best,␈αbut␈α
is␈αnonetheless
␈↓ α,␈↓worth reporting.

␈↓ α,␈↓The␈αtotal␈αe≥ort␈αexpended␈αon␈αthis␈α
experiment␈αwas:␈αa␈αfew␈αmonths␈αof␈αsubconscious␈α
processing,
␈↓ α,␈↓ten␈α∞hours␈α∂of␈α∞designing␈α∂the␈α∞base␈α∂of␈α∞concepts␈α∞to␈α∂insert,␈α∞ten␈α∂hours␈α∞inserting␈α∂and␈α∞debugging
␈↓ α,␈↓them. The whole task took about two days of real time.

␈↓ α,␈↓The␈α
conclusion␈α
to␈α
be␈α
drawn␈α
is␈α
that␈α
heuristics␈α
really␈α
can␈α
be␈α
generally␈α
useful;␈αtheir␈α
attachment
␈↓ α,␈↓to␈αgeneral-sounding␈αconcepts␈αis␈αnot␈α
an␈αillusion.␈↓	21␈↓␈αThe␈αimplication␈αof␈α
this␈αis␈αthat␈αAM␈αcan␈α
be
␈↓ α,␈↓grown␈αincrementally,␈αdomain␈αby␈αdomain.␈α Adding␈αexpertise␈αin␈αa␈αnew␈αdomain␈αrequires␈αonly
␈↓ α,␈↓the␈αintroduction␈αof␈αconcepts␈αlocal␈αto␈αthat␈αdomain;␈αall␈αthe␈αvery␈αgeneral␈αconcepts␈α¬␈αand␈αtheir
␈↓ α,␈↓heuristics ¬ already exist and can be used with no change.

␈↓ α,␈↓The␈αauthor␈αfeels␈αthat␈αthis␈αresult␈αcan␈αbe␈α
generalized:␈αAM␈αcan␈αbe␈αexpanded␈αin␈αscope,␈αeven␈α
to
␈↓ α,␈↓non-mathematical␈α∞≡elds␈α∂of␈α∞endeavor.␈α∞ In␈α∂each␈α∞≡eld,␈α∞however,␈α∂the␈α∞rankings␈α∞of␈α∂the␈α∞various
␈↓ α,␈↓heuristics␈↓	22␈↓␈αmay␈αshift␈αslightly.␈αAs␈αthe␈αdomain␈αgets␈αfurther␈αaway␈αfrom␈αmathematics,␈αvarious
␈↓ α,␈↓heuristics␈α∞are␈α∞important␈α∞which␈α
were␈α∞ignorable␈α∞before␈α∞(e.g.,␈α
those␈α∞dealing␈α∞with␈α∞ethics),␈α
and
␈↓ α,␈↓some␈αpure␈αmath␈αresearch-oriented␈αheuristics␈αbecome␈αless␈αapplicable␈α("giving␈αup␈αand␈αmoving
␈↓ α,␈↓on␈α∩to␈α∩another␈α∩topic"␈α∩is␈α∩not␈α∩an␈α∩acceptable␈α∩response␈α∩to␈α∩the␈α∩15-puzzle,␈α∩nor␈α∩to␈α∩a␈α∩hostage
␈↓ α,␈↓situation).

␈↓ α,␈↓Well,␈αit␈α
sounds␈αas␈αif␈α
we've␈αshifted␈αour␈α
orientation␈αfrom␈α`Results'␈α
to␈αa␈α
subjective␈αevaluation
␈↓ α,␈↓of those results. Let's start a new chapter to legitimize this type of commentary.




















␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	21␈↓ε␈α
Or:␈α
it's␈αa␈α
very␈α
good␈αillusion!␈α
But␈α
note:␈αif␈α
this␈α
phenomenon␈α
is␈αrepeatable␈α
and␈α
useful,␈αthen␈α
(like␈α
Newtonian␈αmechanics)␈α
it
␈↓ α,␈↓ε␈↓ βLwon't pragmatically matter whether it's only an illusion.
␈↓ α,␈↓ε␈↓	22␈↓ε the numeric values that should be returned by the local ratings formulae which are attached to the heuristic rules.
␈↓ α,␈↓␈↓ ≥-␈↓ε103␈↓-




␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬␈↓∧Chapter 7.   Evaluating AM␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓




␈↓ α,␈↓β␈↓ α|All mathematicians are wrong at times.

␈↓ α,␈↓¬␈↓ ε\-- Maxwell



␈↓ α,␈↓This chapter contains discussions "meta" to AM itself.

␈↓ α,␈↓First␈α
comes␈α
an␈α
essay␈α
about␈αjudging␈α
the␈α
performance␈α
of␈α
a␈αsystem␈α
like␈α
AM.␈α
 This␈α
is␈α
a␈αvery
␈↓ α,␈↓hard␈αtask,␈αsince␈αAM␈αhas␈αno␈α
"goal".␈αEven␈αusing␈αcurrent␈αmathematical␈αstandards,␈αshould␈α
AM
␈↓ α,␈↓be␈αjudged␈αon␈αwhat␈αit␈αproduced,␈αor␈αthe␈αquality␈αof␈αthe␈αpath␈αwhich␈αled␈αto␈αthose␈αresults,␈αor␈αthe
␈↓ α,␈↓di≥erence between what it started with and what it ≡nally derived?

␈↓ α,␈↓Section 7.2 then deals with the capabilities and limitations of AM:
␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ What concepts can be elicited from AM now? With a little tuning/tiny additions?
␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ What are some notable omissions in AM's behavior? Can the user elicit these?
␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ What could probably be done within a couple months of modi≡cations?
␈↓ α,␈↓␈↓ αl␈↓π#␈α
␈↓␈α
Aside␈α
from␈α
a␈α
total␈α
change␈α
of␈α
domain,␈α
what␈α
kinds␈α
of␈α
activities␈α
does␈α
AM␈α
lack␈α(e.g.,
␈↓ α,␈↓␈↓ β,proof␈α∪capabilities)?␈α∪Are␈α∪any␈α∪discoveries␈α∪(e.g.,␈α∪analytic␈α∪function␈α∪theory)␈α∪clearly
␈↓ α,␈↓␈↓ β,beyond its design limitations?

␈↓ α,␈↓Finally,␈αall␈αthe␈αconclusions␈αwill␈αbe␈αgathered␈αtogether,␈αand␈αa␈αshort␈αsummary␈αof␈αthis␈αproject's
␈↓ α,␈↓`contribution to knowledge' will be tolerated.


␈↓ α,␈↓␈↓ ¬'␈↓∧␈↓&7.1. Judging Performance␈↓)αβ␈↓

␈↓ α,␈↓One␈αmay␈αview␈αAM's␈αactivity␈αas␈αa␈αprogression␈αfrom␈αan␈αinitial␈αcore␈αof␈αknowledge␈αto␈α
a␈αmore
␈↓ α,␈↓sophisticated␈αbody␈αof␈αconcepts␈αand␈αtheir␈αfacets.␈α Then␈αeach␈αof␈αthe␈αfollowing␈αis␈αa␈αreasonable
␈↓ α,␈↓way to measure success, to "judge" AM:

␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α⊂By␈α⊂AM's␈α⊂ultimate␈α⊂achievements.␈α⊃Examine␈α⊂the␈α⊂list␈α⊂of␈α⊂concepts␈α⊂and␈α⊃methods␈α⊂AM
␈↓ α,␈↓␈↓ β≤developed.␈α Did␈αAM␈αever␈αdiscover␈α
anything␈αinteresting␈αyet␈αunknown␈αto␈α
the␈αuser?␈↓	1␈↓
␈↓ α,␈↓␈↓ β≤Anything new to Mankind?

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε␈α	The␈αλ"user"␈α	is␈αλa␈α	human␈αλwho␈α	works␈αλwith␈α	AM␈αλinteractively,␈α	giving␈αλit␈α	hints,␈αλcommands,␈α	questions,␈αλetc.␈α	Notice␈αλthat␈α	by␈α	"new"␈αλwe
␈↓ α,␈↓ε␈↓ βLmean␈α
new␈α
to␈α
the␈α
user,␈α	not␈α
new␈α
to␈α
Mankind.␈α
 This␈α	might␈α
occur␈α
if␈α
the␈α
user␈α	were␈α
a␈α
child,␈α
and␈α
AM␈α	discovered
␈↓ α,␈↓ε␈↓ βLsome␈αλelementary␈α	facts␈αλof␈α	arithmetic.␈αλThis␈α	is␈αλnot␈αλreally␈α	so␈αλprovincial:␈α	mathematicians␈αλtake␈α	"new"␈αλto␈α	mean␈αλnew
␈↓ α,␈↓ε␈↓ βLto Mankind, not new in the Universe.  I feel philosophy slipping in, so this footnote is terminated.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε104␈↓-

␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α
By␈α
the␈α∞character␈α
of␈α
the␈α∞di≥erence␈α
between␈α
the␈α∞initial␈α
and␈α
≡nal␈α∞states.␈α
 Progressing
␈↓ α,␈↓␈↓ β≤from␈αset␈α
theory␈αto␈αnumber␈α
theory␈αis␈αmuch␈α
more␈αimpressive␈αthan␈α
progressing␈αfrom
␈↓ α,␈↓␈↓ β≤two-dimensional geometry to three-dimensional geometry.
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈αBy␈αthe␈α
quality␈αof␈αthe␈αroute␈α
AM␈αtook␈αto␈αaccomplish␈α
these␈αadvances:␈αHow␈αclever,␈α
how
␈↓ α,␈↓␈↓ β≤circuitous,␈α→how␈α_many␈α→of␈α→the␈α_detours␈α→were␈α→quickly␈α_identi≡ed␈α→as␈α→such␈α_and
␈↓ α,␈↓␈↓ β≤abandoned?
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓␈α∩By␈α∩the␈α∩character␈α⊃of␈α∩the␈α∩User¬System␈α∩interactions:␈α⊃How␈α∩important␈α∩is␈α∩the␈α⊃user's
␈↓ α,␈↓␈↓ β≤guidance?␈α∪How␈α∀closely␈α∪must␈α∀he␈α∪guide␈α∪AM?␈α∀What␈α∪happens␈α∀if␈α∪he␈α∀doesn't␈α∪say
␈↓ α,␈↓␈↓ β≤anything␈α⊂ever?␈α⊂ When␈α⊂he␈α⊂does␈α⊂want␈α⊂to␈α⊂say␈α⊂something,␈α⊂is␈α⊂there␈α⊂an␈α⊂easy␈α⊃way␈α⊂to
␈↓ α,␈↓␈↓ β≤express␈αthat␈αto␈αAM,␈αand␈αdoes␈αAM␈αrespond␈αwell␈αto␈αit?␈α Given␈αa␈αreasonable␈αkick␈αin
␈↓ α,␈↓␈↓ β≤the␈αright␈αdirection,␈αcan␈αAM␈αdevelop␈αthe␈αmini-theories␈αwhich␈αthe␈αuser␈α
intended,␈αor
␈↓ α,␈↓␈↓ β≤at least something equally interesting?
␈↓ α,␈↓␈↓ αl␈↓¬5.␈↓␈α
By␈α
its␈α
intuitive␈α
heuristic␈αpowers:␈α
Does␈α
AM␈α
believe␈α
in␈α
"reasonable"␈αconjectures?␈α
 How
␈↓ α,␈↓␈↓ β≤accurately␈αdoes␈αAM␈αestimate␈αthe␈αdi≠culty␈α
of␈αtasks␈αit␈αis␈αconsidering?␈α Does␈α
AM␈αtie
␈↓ α,␈↓␈↓ β≤together␈α(e.g.,␈α
as␈αanalogous)␈α
concepts␈αwhich␈α
are␈αformally␈α
unrelated␈αyet␈αwhich␈α
bene≡t
␈↓ α,␈↓␈↓ β≤from such a tie?
␈↓ α,␈↓␈↓ αl␈↓¬6.␈↓␈α
By␈α
the␈α∞results␈α
of␈α
the␈α∞experiments␈α
described␈α
in␈α
Section␈α∞6.2␈α
(beginning␈α
on␈α∞page␈α
94).
␈↓ α,␈↓␈↓ β≤How␈α⊂"tuned"␈α⊂is␈α∂the␈α⊂worth␈α⊂numbering␈α⊂scheme?␈α∂The␈α⊂task␈α⊂priority␈α⊂rating␈α∂scheme?
␈↓ α,␈↓␈↓ β≤How␈αfragile␈αis␈αthe␈αset␈α
of␈αinitial␈αconcepts␈αand␈αheuristic␈αrules?␈α
 How␈αdomain-speci≡c
␈↓ α,␈↓␈↓ β≤are those heuristics really? The set of facets?
␈↓ α,␈↓␈↓ αl␈↓¬7.␈↓␈αBy␈αthe␈αvery␈αfact␈αthat␈αthe␈αkinds␈αof␈αexperiments␈αoutlined␈αin␈αSection␈α6.2␈αcan␈αeasily␈αbe
␈↓ α,␈↓␈↓ β≤"set␈α⊃up"␈α⊃and␈α⊃performed␈α⊃on␈α∩AM.␈α⊃ Regardless␈α⊃of␈α⊃the␈α⊃experiments'␈α∩outcomes,␈α⊃the
␈↓ α,␈↓␈↓ β≤features of AM which allow them to be carried out at all are worthy of note.
␈↓ α,␈↓␈↓ αl␈↓¬8.␈↓␈α∞By␈α∞the␈α∞implications␈α
of␈α∞this␈α∞project.␈α∞What␈α
can␈α∞AM␈α∞suggest␈α∞about␈α∞educating␈α
young
␈↓ α,␈↓␈↓ β≤mathematicians␈α(and␈α
scientists␈αin␈α
general)?␈α What␈α
can␈αAM␈α
say␈αabout␈α
doing␈αmath?
␈↓ α,␈↓␈↓ β≤about empirical research in general?
␈↓ α,␈↓␈↓ αl␈↓¬9.␈↓␈α
By␈α
the␈α
number␈α
of␈α
new␈αavenues␈α
for␈α
research␈α
and␈α
experimentation␈α
it␈α
opens␈αup.␈α
 What
␈↓ α,␈↓␈↓ β≤new projects can we propose?
␈↓ α,␈↓␈↓ αl␈↓¬10.␈↓ By comparisons to other, similar systems.

␈↓ α,␈↓For␈α
each␈αof␈α
these␈α
10␈αmeasuring␈α
criteria,␈α
a␈αsubsection␈α
will␈α
now␈αbe␈α
provided.␈α Other␈α
measures
␈↓ α,␈↓of judging performance exist␈↓	2␈↓, of course, but haven't been applied to AM.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.1. AM's Ultimate Discoveries␈↓)αβ␈↓

␈↓ α,␈↓Two of the ideas which AM proposed were totally new and unexpected:␈↓	3␈↓
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α
Consider␈α
numbers␈α
with␈α
an␈α∞abnormally␈α
high␈α
number␈α
of␈α
divisors.␈α
If␈α∞d(n)␈α
represents

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	2␈↓ε␈α
For␈α
example,␈α
Colby␈αsent␈α
transcripts␈α
of␈α
a␈α
session␈αwith␈α
PARRY␈α
to␈α
various␈αpsychiatrists,␈α
and␈α
had␈α
them␈α
evaluate␈αeach
␈↓ α,␈↓ε␈↓ βLinteraction␈αalong␈αseveral␈α
dimensions.␈αThe␈αsame␈α
kind␈αof␈αsurvey␈α
could␈αbe␈αdone␈α
for␈αAM.␈αA␈α
quite␈αseparate
␈↓ α,␈↓ε␈↓ βLmeasure␈α
of␈α
AM␈α
would␈α
be␈α
to␈α
wait␈α
and␈α
see␈α
how␈αmany␈α
future␈α
articles␈α
in␈α
the␈α
field␈α
refer␈α
to␈α
this␈α
work␈α(and␈α
in
␈↓ α,␈↓ε␈↓ βLwhat light!).
␈↓ α,␈↓ε␈↓	3␈↓ε␈α	Note␈α
that␈α	these␈α
are␈α	"ultimate␈α
discoveries"␈α	only␈α	in␈α
the␈α	sense␈α
of␈α	what␈α
has␈α	been␈α	done␈α
at␈α	the␈α
time␈α	of␈α
writing␈α	this␈α
book.␈α	For
␈↓ α,␈↓ε␈↓ βLone␈αλof␈αλAM's␈αλideas␈αλto␈αλbe␈αλ"new",␈αλit␈αλshould␈αλbe␈αλpreviously␈αλunknown␈αλto␈αλboth␈αλthe␈αλauthor␈αλand␈αλthe␈αλuser.␈αλWhy?␈αλIf␈αλthe
␈↓ α,␈↓ε␈↓ βLauthor␈α
knew␈αabout␈α
it,␈αthen␈α
the␈α
heuristics␈αhe␈α
provided␈αAM␈α
with␈α
might␈αunconsciously␈α
encode␈αa␈α
path␈αto␈α
that
␈↓ α,␈↓ε␈↓ βLknowledge.␈α	If␈α	the␈αλuser␈α	knew␈α	about␈α	that␈αλidea,␈α	his␈α	guidance␈αλmight␈α	unconsciously␈α	help␈α	AM␈αλto␈α	derive␈α	it.␈α	An␈αλeven
␈↓ α,␈↓ε␈↓ βLmore␈α	stringent␈α	interpretation␈α	would␈αλbe␈α	that␈α	the␈α	idea␈αλbe␈α	hitherto␈α	unknown␈α	to␈αλthe␈α	collective␈α	written␈α	record␈αλof
␈↓ α,␈↓ε␈↓ βLMathematics.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε105␈↓-

␈↓ α,␈↓␈↓ β≤the␈α⊃number␈α⊃of␈α⊂divisors␈α⊃of␈α⊃n,␈↓	4␈↓␈α⊂then␈α⊃AM␈α⊃de≡nes␈α⊂the␈α⊃set␈α⊃of␈α⊂"maximally-divisible
␈↓ α,␈↓␈↓ β≤numbers"␈α∩to␈α⊃be␈α∩␈↓¬{nεN␈α⊃|␈α∩(∀m<n)␈α∩d(m)<d(n)}␈↓.␈α⊃ By␈α∩factoring␈α⊃each␈α∩such␈α∩number␈α⊃into
␈↓ α,␈↓␈↓ β≤primes,␈α⊂AM␈α⊂noticed␈α⊃a␈α⊂regularity␈α⊂in␈α⊂them.␈α⊃The␈α⊂author␈α⊂then␈α⊂developed␈α⊃a␈α⊂"mini-
␈↓ α,␈↓␈↓ β≤theory"␈α⊂about␈α⊃these␈α⊂numbers.␈α⊃ It␈α⊂later␈α⊃turned␈α⊂out␈α⊃that␈α⊂Ramanujan␈α⊃had␈α⊂already
␈↓ α,␈↓␈↓ β≤proposed␈α
that␈αvery␈α
same␈αde≡nition␈α
(in␈α1915),␈α
and␈αhad␈α
found␈αthat␈α
same␈αregularity.
␈↓ α,␈↓␈↓ β≤His␈α∞results␈α
only␈α∞partially␈α
overlap␈α∞those␈α
of␈α∞AM␈α
and␈α∞the␈α
author,␈α∞however,␈α∞and␈α
his
␈↓ α,␈↓␈↓ β≤methods are radically di≥erent.
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈αAM␈αfound␈αa␈αcute␈αgeometric␈αapplication␈α
of␈αGoldbach's␈αconjecture.␈α Given␈αa␈αset␈αof␈α
all
␈↓ α,␈↓␈↓ β≤angles␈αof␈αprime␈αdegree,␈αfrom␈α0␈αto␈α
180␈↓#
o␈↓#,␈↓	5␈↓␈αthen␈α␈↓βany␈↓␈αangle␈αbetween␈α0␈αand␈α180␈α
degrees
␈↓ α,␈↓␈↓ β≤can␈αbe␈αapproximated␈αto␈αwithin␈α1␈↓#
o␈↓#␈αby␈αadding␈αa␈αpair␈αof␈αangles␈αfrom␈αthis␈αprime␈αset.
␈↓ α,␈↓␈↓ β≤In␈αfact,␈α
it␈αis␈α
hard␈αto␈α
≡nd␈αsmaller␈αsets␈α
than␈αthis␈α
one␈αwhich␈α
approximate␈αany␈αangle␈α
to
␈↓ α,␈↓␈↓ β≤that accuracy.

␈↓ α,␈↓By␈α
and␈α
large,␈α
the␈α
other␈α
concepts␈α
which␈α
AM␈α
developed␈α
were␈α
either␈α
already-known,␈α
or␈α
real
␈↓ α,␈↓losers.␈α
For␈α∞example,␈α
AM␈α
composed␈α∞Set-insert␈α
with␈α
the␈α∞predicate␈α
Equality.␈α
 The␈α∞result␈α
was
␈↓ α,␈↓an␈α∞operation␈α∞Insert␈↓εo␈↓Equal(x,y,z),␈α∞which␈α∞≡rst␈α∞tested␈α∞whether␈α∞x␈α∞was␈α∞Equal␈α∞to␈α∞y␈α∞or␈α∂not.␈α∞The
␈↓ α,␈↓value␈α⊂of␈α⊂this␈α⊂was␈α⊂either␈α⊂True␈α⊂or␈α⊂False.␈α∂ Next,␈α⊂this␈α⊂T/F␈α⊂value␈α⊂was␈α⊂inserted␈α⊂into␈α⊂z.␈α∂ For
␈↓ α,␈↓example,␈α
Insert␈↓εo␈↓Equal({1,2},{3,4},{5,6})␈α
=␈α
{False,5,6}.␈α
 The␈α
≡rst␈α
two␈α
arguments␈α
are␈α
not␈αequal,
␈↓ α,␈↓so␈α
the␈α
atom␈α
`False'␈α
was␈α
inserted␈αinto␈α
the␈α
third.␈α
 Although␈α
hitherto␈α
"unknown",␈αthis␈α
operation
␈↓ α,␈↓would clearly be better o≥ left in that state.

␈↓ α,␈↓Another␈αkind␈αof␈α
loser␈αoccurred␈αwhenever␈αAM␈α
entered␈αupon␈αsome␈α"regular"␈α
behavior.␈α For
␈↓ α,␈↓example,␈αif␈αit␈αdecided␈αthat␈αCompose␈αwas␈α
interesting,␈αit␈αmight␈αtry␈αto␈αcreate␈αsome␈αexamples␈α
of
␈↓ α,␈↓compositions.␈αIt␈αcould␈αdo␈αthis␈αby␈αpicking␈αtwo␈αoperations␈αand␈αcomposing␈αthem.␈α What␈αbetter
␈↓ α,␈↓operations␈α
to␈α
pick␈α
than␈α
Compose␈α
and␈α
Compose!␈α
 Thus␈α
Compose␈↓εo␈↓Compose␈α
would␈α
be␈αborn.
␈↓ α,␈↓By␈α≤composing␈α≤that␈α≤with␈α≤itself,␈α≤an␈α≤even␈α≤more␈α≤monstrous␈α≤operation␈α≤is␈α≤spawned:
␈↓ α,␈↓Compose␈↓εo␈↓Compose␈↓εo␈↓Compose␈↓εo␈↓Compose.␈α_ Since␈α_AM␈α_actually␈α_uses␈α_the␈α_word␈α_"Compose"
␈↓ α,␈↓instead␈αof␈αthat␈αlittle␈αin≡x␈αcircle,␈αthe␈αPNAME␈αof␈αthe␈αdata␈αstructure␈αit␈αcreates␈αis␈αhorrendous.
␈↓ α,␈↓Its␈α
use␈α
is␈α
almost␈αnonexistent:␈α
it␈α
must␈α
take␈α5␈α
operations␈α
as␈α
arguments,␈αand␈α
it␈α
returns␈α
a␈αnew
␈↓ α,␈↓operation␈αwhich␈αis␈αthe␈αcomposition␈αof␈α
those␈α≡ve.␈α An␈αanalogous␈αdanger␈αwhich␈αexists␈α
is␈αfor
␈↓ α,␈↓AM␈α
to␈αbe␈α
content␈αconjecturing␈α
a␈αstream␈α
of␈αvery␈α
similar␈αrelationships␈α
(e.g.,␈αthe␈α
multiplication
␈↓ α,␈↓table).␈α In␈αall␈αsuch␈αcases,␈αAM␈αmust␈αhave␈αmeta-rules␈αwhich␈αpull␈αit␈αup␈αout␈αof␈αsuch␈α
whirlpools,
␈↓ α,␈↓to perceive a higher generalization of its previous sequence of related activities.

␈↓ α,␈↓In␈αsummary,␈αthen,␈αwe␈α
may␈αsay␈αthat␈αAM␈α
produced␈αa␈αfew␈αwinning␈α
ideas␈αnew␈αto␈αthe␈αauthor,␈α
a
␈↓ α,␈↓couple␈α⊃of␈α⊂which␈α⊃were␈α⊂new␈α⊃to␈α⊃Mankind.␈α⊂Several␈α⊃additional␈α⊂"new"␈α⊃concepts␈α⊃were␈α⊂created
␈↓ α,␈↓which both AM and the user agreed were better forgotten.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.2. The Magnitude of AM's Progress␈↓)αβ␈↓





␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	4␈↓ε e.g., d(12) = Size({1,2,3,4,6,12}) = 6.
␈↓ α,␈↓ε␈↓	5␈↓ε Included are 0␈↓#
o␈↓# and 1␈↓#
o␈↓#, as well as the "typical" primes 2␈↓#
o␈↓#, 3␈↓#
o␈↓#, 5␈↓#
o␈↓#, 7␈↓#
o␈↓#, 11␈↓#
o␈↓#,..., 179␈↓#
o␈↓#.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε106␈↓-



␈↓ α,␈↓β␈↓ α|Even␈αwith␈αmen␈α
of␈αgenius,␈αwith␈αwhom␈α
the␈αbirth␈αrate␈αof␈α
hypotheses␈αis␈αvery␈α
high,␈αit
␈↓ α,␈↓β␈↓ α|only just manages to exceed the death rate.

␈↓ α,␈↓¬␈↓ ε\-- W. H. George␈↓	6␈↓¬



␈↓ α,␈↓We␈α
can␈α
ask␈α
the␈α
following␈α
kind␈α
of␈α
question:␈α
how␈α
many␈α
"levels"␈α
did␈α
AM␈α
ascend?␈α
This␈α
is␈α
a
␈↓ α,␈↓fuzzy␈α∞notion,␈α
but␈α∞basically␈α
we␈α∞shall␈α
say␈α∞that␈α∞a␈α
new␈α∞level␈α
is␈α∞reached␈α
when␈α∞a␈α∞valuable␈α
new
␈↓ α,␈↓bunch of connected concepts are de≡ned in terms of concepts at a lower level.

␈↓ α,␈↓For␈α
example,␈αAM␈α
started␈αout␈α
knowing␈αabout␈α
Sets␈αand␈α
Set-operations.␈α When␈α
it␈αprogressed
␈↓ α,␈↓to␈αnumbers␈αand␈αarithmetic,␈αthat␈α
was␈αone␈αbig␈αstep␈αup␈αto␈α
a␈αnew␈αlevel.␈αWhen␈αit␈αzeroed␈α
in␈αon
␈↓ α,␈↓primes, unique-factorization, and divisibility, it had moved up another level.

␈↓ α,␈↓When␈α∪fed␈α∀simple␈α∪geometry␈α∪concepts,␈α∀AM␈α∪moved␈α∀up␈α∪one␈α∪level␈α∀when␈α∪it␈α∀de≡ned␈α∪some
␈↓ α,␈↓generalizations␈α⊂of␈α⊂the␈α⊂equality␈α⊂of␈α⊂geometric␈α⊂≡gures␈α⊂(parallel␈α⊂lines,␈α⊂congruent␈α⊃and␈α⊂similar
␈↓ α,␈↓triangles, angles equal in measure) and their invariants (rotations, translations, re∨ections).

␈↓ α,␈↓The␈αabove␈α
few␈αexamples␈αare␈α
unfortunately␈αexhaustive:␈α
that␈αjust␈αabout␈α
sums␈αup␈α
the␈αmajor
␈↓ α,␈↓advances␈α
AM␈α
made.␈α Its␈α
progress␈α
was␈αhalted␈α
not␈α
so␈α
much␈αby␈α
cpu␈α
time␈αand␈α
space,␈α
as␈α
by␈αa
␈↓ α,␈↓paucity␈α⊂of␈α∂meta-knowledge:␈α⊂heuristic␈α∂rules␈α⊂for␈α∂≡lling␈α⊂in␈α∂new␈α⊂heuristic␈α∂rules.␈α⊂ Thus␈α∂AM's
␈↓ α,␈↓successes are ≡nite, and its failures in≡nite, along this dimension.

␈↓ α,␈↓A␈α
more␈αcharitable␈α
view␈α
might␈αcompare␈α
AM␈α
to␈αa␈α
human␈αwho␈α
was␈α
forced␈αto␈α
start␈α
from␈αset
␈↓ α,␈↓theory,␈α
with␈αAM's␈α
sparse␈α
abilities.␈α In␈α
that␈α
sense,␈αperhaps,␈α
AM␈α
would␈αrate␈α
quite␈α
well.␈αThe
␈↓ α,␈↓"unfair"␈α∩advantage␈α⊃it␈α∩had␈α⊃was␈α∩the␈α⊃presence␈α∩of␈α⊃many␈α∩heuristics␈α⊃which␈α∩themselves␈α⊃were
␈↓ α,␈↓gleaned␈α∞from␈α∞mathematicians:␈α∂i.e.,␈α∞they␈α∞are␈α∞like␈α∂compiled␈α∞hindsight.␈α∞ A␈α∞major␈α∂purpose␈α∞of
␈↓ α,␈↓mathematics␈α∞education␈α
in␈α∞the␈α
university␈α∞is␈α
to␈α∞instill␈α
these␈α∞heuristics␈α
into␈α∞the␈α
minds␈α∞of␈α
the
␈↓ α,␈↓students.

␈↓ α,␈↓AM␈α
is␈α∞thus␈α
characterized␈α∞as␈α
possessing␈α∞heuristics␈α
which␈α
are␈α∞powerful␈α
enough␈α∞to␈α
take␈α∞it␈α
a
␈↓ α,␈↓few␈α∂"levels"␈α∞away␈α∂from␈α∂the␈α∞kind␈α∂of␈α∂knowledge␈α∞it␈α∂began␈α∞with,␈α∂but␈α∂␈↓βonly␈↓␈α∞a␈α∂few␈α∂levels.␈α∞The
␈↓ α,␈↓limiting␈α∂factors␈α⊂are␈α∂(i)␈α⊂the␈α∂heuristic␈α⊂rules␈α∂AM␈α∂begins␈α⊂with,␈α∂and␈α⊂more␈α∂speci≡cally␈α⊂(ii)␈α∂the
␈↓ α,␈↓expertise␈α∂in␈α∂recognizing␈α∂and␈α∂compiling␈α∂new␈α∂heuristics,␈α∂and␈α∂more␈α∂generally␈α∂(iii)␈α∂a␈α∂lack␈α∂of
␈↓ α,␈↓real-world situations to draw upon for analogies, intuitions, and applications.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.3. The Quality of AM's Route␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|Thinking␈α∞is␈α∞not␈α∞measured␈α∞by␈α∞what␈α∞is␈α∞produced,␈α∞but␈α∞rather␈α∞is␈α∞a␈α∞property␈α∂of␈α∞the
␈↓ α,␈↓β␈↓ α|way something is done.

␈↓ α,␈↓¬␈↓ ε\-- Hamming

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	6␈↓ε Quoted from [Beveridge 50].
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε107␈↓-

␈↓ α,␈↓No␈α∞matter␈α∞what␈α∂its␈α∞achievements␈α∞were,␈α∂or␈α∞the␈α∞magnitude␈α∂of␈α∞its␈α∞advancement␈α∂from␈α∞initial
␈↓ α,␈↓knowledge,␈αAM␈αcould␈↓	7␈↓␈αstill␈αbe␈αjudged␈α
"unintelligent"␈αif,␈αe.g.,␈αit␈αwere␈αexploring␈αvast␈α
numbers
␈↓ α,␈↓of␈αabsurd␈αavenues␈α
for␈αeach␈αworthwhile␈α
one␈αit␈αfound.␈α
The␈αquality␈αof␈α
the␈αroute␈αAM␈α
followed
␈↓ α,␈↓is thus quite signi≡cant.

␈↓ α,␈↓Of␈αthe␈αtwo␈αhundred␈αnew␈αconcepts␈αit␈αde≡ned,␈αabout␈α130␈αwere␈αacceptable␈α¬␈αin␈αthe␈αsense␈αthat
␈↓ α,␈↓one␈α⊂can␈α⊂defend␈α⊂AM's␈α⊂reasoning␈α⊂in␈α⊂at␈α⊃least␈α⊂exploring␈α⊂them;␈α⊂in␈α⊂the␈α⊂sense␈α⊂that␈α⊃a␈α⊂human
␈↓ α,␈↓mathematician␈α⊃might␈α⊃have␈α⊃considered␈α⊃them.␈α⊃Of␈α⊃these␈α⊃"winners",␈α⊃about␈α⊃two␈α⊃dozen␈α⊂were
␈↓ α,␈↓signi≡cant␈α≠¬␈α≠that␈α≠is,␈α≠useful,␈α~catalytic,␈α≠well-known␈α≠by␈α≠human␈α≠mathematicians,␈α~etc.
␈↓ α,␈↓Unfortunately,␈αthe␈αsixty␈αor␈αseventy␈αconcepts␈αwhich␈αwere␈αlosers␈αwere␈α␈↓βreal␈↓␈αlosers.␈α Once␈αagain
␈↓ α,␈↓we␈αmust␈αobserve␈αthat␈αthe␈αquality␈αof␈αthe␈αroute␈αis␈αa␈αfunction␈αof␈αthe␈αquality␈αof␈α
the␈αheuristics.
␈↓ α,␈↓AM␈α
did␈α
not␈α
develop␈αthe␈α
sixty␈α
"losers"␈α
very␈αmuch:␈α
they␈α
ended␈α
up␈αwith␈α
an␈α
average␈α
of␈αonly
␈↓ α,␈↓1.5␈α
tasks␈αrelevant␈α
to␈αthem␈α
ever␈αhaving␈α
been␈α
chosen.␈α The␈α
"winners"␈αaveraged␈α
about␈αtwice␈α
as
␈↓ α,␈↓many␈αtasks␈αwhich␈αhelped␈α≡ll␈αthem␈αout␈αmore.␈α
 Also,␈αthe␈αworth␈αratings␈αof␈αthe␈αlosers␈αended␈α
up
␈↓ α,␈↓far below those of the winners in all but two cases.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.4. The Character of the User-System Interactions␈↓)αβ␈↓

␈↓ α,␈↓AM␈α
is␈α∞not␈α
a␈α∞"user-oriented"␈α
system.␈α
 There␈α∞were␈α
many␈α∞nice␈α
human-interaction␈α∞features␈α
in
␈↓ α,␈↓the␈α∂original␈α∂grandiose␈α∂proposal␈α∂for␈α∂AM␈α⊂which␈α∂never␈α∂got␈α∂o≥␈α∂the␈α∂drawing␈α∂board.␈α⊂At␈α∂the
␈↓ α,␈↓heart of these features were two assumptions:
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α
The␈α
user␈α
must␈α
understand␈α
AM,␈α
and␈α
AM␈α
must␈α
likewise␈α
have␈α
a␈α
good␈α
model␈α∞of␈α
the
␈↓ α,␈↓␈↓ β≤particular␈α⊂human␈α⊂using␈α⊂AM.␈α⊂The␈α⊂only␈α⊂time␈α⊂either␈α⊂should␈α⊂initiate␈α⊂a␈α⊃message␈α⊂is
␈↓ α,␈↓␈↓ β≤when␈αhis␈αmodel␈α
of␈αthe␈αother␈α
is␈αnot␈αwhat␈αhe␈α
wants␈αthat␈αmodel␈α
to␈αbe.␈α In␈α
that␈αcase,
␈↓ α,␈↓␈↓ β≤the message should be speci≡cally designed to ≡x that discrepancy.␈↓	8␈↓
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α∞Each␈α∞kind␈α∞of␈α∞message␈α
which␈α∞is␈α∞to␈α∞pass␈α∞between␈α
AM␈α∞and␈α∞its␈α∞user␈α∞should␈α∞have␈α
its
␈↓ α,␈↓␈↓ β≤own␈α∀appropriate␈α∪language.␈α∀ Thus␈α∪there␈α∀should␈α∪be␈α∀a␈α∪terse␈α∀comment␈α∪language,
␈↓ α,␈↓␈↓ β≤whereby␈α∞the␈α∞user␈α∞can␈α∞note␈α∞how␈α∞he␈α∞feels␈α∞about␈α∞what␈α∞AM␈α∞is␈α∞doing,␈α∞a␈α∞questioning
␈↓ α,␈↓␈↓ β≤language␈α∞for␈α∞either␈α∞party␈α∞to␈α∞get/give␈α∞reasons␈α∞to␈α∞the␈α∞other,␈α∞a␈α∞picture␈α∞language␈α
for
␈↓ α,␈↓␈↓ β≤communicating certain relationships, etc.

␈↓ α,␈↓Neither␈αof␈αthese␈αideas␈αever␈αmade␈αit␈αinto␈αthe␈αLISP␈αcode␈αthat␈αis␈αnow␈αAM,␈αalthough␈αthey␈αare
␈↓ α,␈↓certainly␈α
not␈α∞prohibited␈α
in␈α
any␈α∞way␈α
by␈α
AM's␈α∞design.␈α
As␈α
one␈α∞might␈α
expect,␈α
the␈α∞reason␈α
for
␈↓ α,␈↓this␈α∞atrophy␈α∞is␈α
simply␈α∞because␈α∞very␈α
little␈α∞guidance␈α∞from␈α
the␈α∞user␈α∞was␈α
needed␈α∞by␈α∞AM.␈α
 In
␈↓ α,␈↓fact,␈α∞all␈α
the␈α∞discoveries,␈α
cpu␈α∞time␈α
quotes,␈α∞etc.␈α
 mentioned␈α∞in␈α
this␈α∞document␈α
are␈α∞taken␈α
from
␈↓ α,␈↓totally␈αunguided␈αruns␈α
by␈αAM.␈α If␈α
the␈αuser␈αguides␈α
as␈αwell␈αas␈α
he␈αcan,␈αthen␈α
about␈αa␈αfactor␈αof␈α
2
␈↓ α,␈↓or␈α∂3␈α∞speedup␈α∂is␈α∞possible.␈α∂ Of␈α∞course,␈α∂this␈α∞assumes␈α∂that␈α∞the␈α∂user␈α∞is␈α∂dragging␈α∂AM␈α∞directly
␈↓ α,␈↓along␈α
a␈α
line␈αof␈α
development␈α
he␈α
knows␈αwill␈α
be␈α
successful.␈α
The␈αuser's␈α
"reasons"␈α
at␈α
each␈αstep
␈↓ α,␈↓are␈αbased␈αessentially␈αon␈αhindsight.␈αThus␈αthis␈αis␈αnot␈αat␈αall␈α"fair".␈αIf␈αAM␈αever␈αbecomes␈αmore
␈↓ α,␈↓user-oriented,␈α∂it␈α∂would␈α∂be␈α∂nice␈α∂to␈α∂let␈α∂children␈α∂(say␈α∂6-12␈α∂years␈α∂old)␈α∂experiment␈α∂with␈α∂it,␈α∞to

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	7␈↓ε␈α	not␈α	necessarily␈α	WOULD␈α	be␈α	so␈α
judged.␈α	Humans␈α	may␈α	very␈α	well␈α	consider␈α	an␈α
incredible␈α	number␈α	of␈α	silly␈α	ideas␈α	before␈α
the␈α	right
␈↓ α,␈↓ε␈↓ βLpair␈α	of␈αλ"hooked␈α	atoms"␈αλcollide␈α	into␈αλa␈α	sensible␈αλthought,␈α	which␈αλis␈α	then␈αλconsidered␈α	in␈αλfull␈α	consciousness.␈α	If,␈αλlike
␈↓ α,␈↓ε␈↓ βLhumans,␈αAM␈αwas␈α
capable␈αof␈αdoing␈α
this␈αprocessing␈αin␈α
a␈αsufficiently␈αbrief␈α
period␈αof␈αreal␈α
time,␈αit␈αwould␈α
not
␈↓ α,␈↓ε␈↓ βLreflect ill on its evaluation. Of course, this may simply be the DEFINITION of "sufficiently brief".
␈↓ α,␈↓ε␈↓	8␈↓ε This idea was motivated by a lecture given in 1975 by Terry Winograd
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε108␈↓-

␈↓ α,␈↓observe them working with it in domains unfamiliar to either of them.␈↓	9␈↓

␈↓ α,␈↓The␈α
user␈α
can␈α"kick"␈α
AM␈α
in␈α
one␈αdirection␈α
or␈α
another,␈α
e.g.,␈αby␈α
interrupting␈α
and␈α
telling␈αAM
␈↓ α,␈↓that␈α∂Sets␈α∂are␈α∂more␈α∂interesting␈α∂than␈α∂Numbers.␈α∂ Even␈α∂in␈α∂that␈α∂particular␈α∂case,␈α∂AM␈α∂fails␈α∂to
␈↓ α,␈↓develop␈αany␈αhigher-level␈αset␈αconcepts␈α(diagonalization,␈αin≡nite␈αsets,␈αetc.)␈αand␈αsimply␈αwallows
␈↓ α,␈↓around␈αin␈α≡nite␈αset␈αtheory␈α(de␈αMorgan's␈αlaws,␈αassociativity␈αof␈αUnion,␈αetc.).␈α When␈αgeometric
␈↓ α,␈↓concepts␈αare␈αinput,␈αand␈αAM␈αis␈αkicked␈α
in␈α␈↓βthat␈↓␈αdirection,␈αmuch␈αnicer␈αresults␈αare␈αobtained.␈α
See
␈↓ α,␈↓the report on the Geometry experiment, page 101.

␈↓ α,␈↓There␈α∂is␈α∂one␈α∂important␈α∂result␈α∂to␈α∂observe:␈α∂the␈α∂very␈α∂best␈α∂examples␈α∂of␈α∂AM␈α∂in␈α⊂action␈α∂were
␈↓ α,␈↓brought␈α
to␈αfull␈α
fruition␈α
only␈αby␈α
a␈αhuman␈α
developer.␈α
 That␈αis,␈α
AM␈α
thought␈αof␈α
a␈αcouple␈α
great
␈↓ α,␈↓concepts,␈αbut␈αcouldn't␈αdevelop␈αthem␈αwell␈αon␈αits␈αown.␈αA␈αhuman␈α(the␈αauthor)␈αthen␈αtook␈αthem
␈↓ α,␈↓and␈αworked␈αon␈αthem␈αby␈αhand,␈αand␈αinteresting␈αresults␈αwere␈αachieved.␈αThese␈αresults␈αcould␈αbe
␈↓ α,␈↓told␈α∪to␈α∪AM,␈α∀who␈α∪could␈α∪then␈α∪go␈α∀o≥␈α∪and␈α∪look␈α∪for␈α∀new␈α∪concepts␈α∪to␈α∀investigate.␈α∪ This
␈↓ α,␈↓interaction␈α~is␈α~of␈α~course␈α~at␈α~a␈α~much␈α~lower␈α~frequency␈α~than␈α~the␈α~kind␈α~of␈α~rapid≡re
␈↓ α,␈↓question/answering␈αtalked␈αabout␈αabove.␈αYet␈αit␈αseems␈αthat␈αsuch␈αsynergy␈αmay␈αbe␈αthe␈αultimate
␈↓ α,␈↓mode of AM-like systems.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.5. AM's Intuitive Powers␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|Intuitive␈αconviction␈αsurpasses␈αlogic␈αas␈αthe␈α
brilliance␈αof␈αthe␈αsun␈αsurpasses␈αthe␈α
pale
␈↓ α,␈↓β␈↓ α|light of the moon.

␈↓ α,␈↓¬␈↓ ε\-- Kline



␈↓ α,␈↓Let␈αme␈αhasten␈αto␈αmention␈αthat␈αthe␈αword␈α"intuitive"␈αin␈αthis␈αsubsection's␈αtitle␈αis␈αnot␈αrelated␈αto
␈↓ α,␈↓the␈α(currently␈αnon-existent)␈α"Intuitions"␈αfacets␈αof␈αthe␈αconcepts.␈α What␈αis␈αmeant␈αis␈αthe␈αtotality
␈↓ α,␈↓of␈α≠plausible␈α≠reasoning␈α~which␈α≠AM␈α≠engages␈α~in:␈α≠empirical␈α≠induction,␈α~generalization,
␈↓ α,␈↓specialization,␈α
maintaining␈αreasons␈α
for␈α
jobs␈αon␈α
the␈α
agenda␈αlist,␈α
creation␈α
of␈αanalogies␈α
between
␈↓ α,␈↓bunches of concepts, etc.

␈↓ α,␈↓AM␈α∂only␈α∂considers␈α∂conjectures␈α∂which␈α∂have␈α∂been␈α∂explicitly␈α∂suggested:␈α∂either␈α⊂by␈α∂empirical
␈↓ α,␈↓evidence,␈α
by␈αanalogy,␈α
or␈α(de-implemented␈α
now:)␈α
by␈αIntuition␈α
facets.␈α Once␈α
a␈α
conjecture␈αhas
␈↓ α,␈↓been␈α⊃formulated,␈α⊃it␈α⊃is␈α∩tested␈α⊃in␈α⊃all␈α⊃ways␈α∩possible:␈α⊃new␈α⊃experimental␈α⊃evidence␈α∩is␈α⊃sought
␈↓ α,␈↓(especially␈αextreme␈αcases),␈αit␈αis␈αexamined␈α
formally␈↓	10␈↓␈αto␈αsee␈αif␈αit␈αfollows␈α
from␈αalready-known
␈↓ α,␈↓conjectures, etc.

␈↓ α,␈↓Because␈α
of␈α
this␈αgrounding␈α
in␈α
plausibility,␈α
the␈αonly␈α
conjectures␈α
the␈αuser␈α
ever␈α
sees␈α
(the␈αones
␈↓ α,␈↓AM␈α∂is␈α⊂testing)␈α∂are␈α⊂quite␈α∂believable.␈α∂ If␈α⊂they␈α∂turn␈α⊂out␈α∂to␈α∂be␈α⊂false,␈α∂both␈α⊂he␈α∂and␈α⊂AM␈α∂are

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	9␈↓ε␈α
Starred␈α
(*)␈α
exercise␈α
for␈α
the␈α
reader:␈α
carry␈α
out␈α
such␈α	a␈α
project␈α
on␈α
a␈α
statistically␈α
significant␈α
sample␈α
of␈α
children,␈α
wait␈α	thirty
␈↓ α,␈↓ε␈↓ βLyears,␈α
and␈α
observe␈αthe␈α
incidence␈α
of␈α
mathematicians␈αand␈α
scientists␈α
in␈αgeneral,␈α
compared␈α
to␈α
the␈αnational
␈↓ α,␈↓ε␈↓ βLaverages. Within whatever occupation they've chosen, rate their creativity and productivity.
␈↓ α,␈↓ε␈↓	10␈↓ε␈αCurrently,␈αthis␈αis␈αdone␈αin␈αtrivial␈αways.␈αAn␈αopen␈αproblem,␈αwhich␈αis␈αunder␈αattack␈αnow,␈αis␈αto␈αadd␈αmore␈αpowerful␈αformal
␈↓ α,␈↓ε␈↓ βLreasoning abilities to AM.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε109␈↓-

␈↓ α,␈↓surprised.␈α For␈αexample,␈αboth␈αAM␈αand␈αthe␈αuser␈αwere␈αdisappointed␈αwhen␈αnothing␈αcame␈αout
␈↓ α,␈↓of␈α↔the␈α↔concept␈α↔of␈α↔Uniquely-prime-addable␈α⊗numbers␈α↔(positive␈α↔integers␈α↔which␈α↔can␈α⊗be
␈↓ α,␈↓represented␈α⊃as␈α∩the␈α⊃sum␈α∩of␈α⊃two␈α∩primes␈α⊃in␈α∩precisely␈α⊃one␈α∩way).␈α⊃ Several␈α∩conjectures␈α⊃were
␈↓ α,␈↓proposed␈α≤via␈α≠analogy␈α≤with␈α≠unique␈α≤prime␈α≠factorization,␈α≤but␈α≠none␈α≤of␈α≤them␈α≠held
␈↓ α,␈↓experimentally.␈α Each␈αof␈αthem␈αseemed␈αworth␈αinvestigating,␈αto␈αboth␈αthe␈αuser␈αand␈αthe␈αsystem.
␈↓ α,␈↓It is still not known whether there is anything interesting about that concept or not.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.6. Experiments on AM␈↓)αβ␈↓

␈↓ α,␈↓The␈α
experiments␈α
described␈αin␈α
Section␈α
6.2␈α
(page␈α94␈α
≥)␈α
provide␈αsome␈α
results␈α
relevant␈α
to␈αthe
␈↓ α,␈↓overall␈αvalue␈αof␈αthe␈αAM␈αsystem.␈α The␈αreader␈αshould␈αconsult␈αthat␈αsection␈αfor␈αdetails;␈αneither
␈↓ α,␈↓the␈α∃experiments␈α∃nor␈α∃their␈α∃results␈α∃will␈α∃be␈α∃repeated␈α∃here.␈α∃ A␈α∃few␈α∃conclusions␈α∃will␈α∃be
␈↓ α,␈↓summarized:

␈↓ α,␈↓The␈αworth-numbering␈αscheme␈αfor␈αthe␈αconcepts␈αis␈αfairly␈αrobust:␈αeven␈αwhen␈αall␈αthe␈αconcepts's
␈↓ α,␈↓worths␈αare␈αinitialized␈αat␈αthe␈αsame␈αvalue,␈αthe␈αperformance␈αof␈αAM␈αdoesn't␈αcollapse,␈αalthough
␈↓ α,␈↓it is noticeably degraded.

␈↓ α,␈↓Certain␈α
mutilations␈α
of␈α
the␈α
priority-value␈α
scheme␈α
for␈α
tasks␈α
on␈α
the␈α
agenda␈α
will␈α∞cripple␈α
AM,
␈↓ α,␈↓but it can resist most of the small changes tried in various experiments.

␈↓ α,␈↓Sometimes,␈α∞removing␈α∞just␈α∞a␈α∞single␈α∞concept␈α∂(e.g.,␈α∞Equality)␈α∞was␈α∞enough␈α∞to␈α∞block␈α∂AM␈α∞from
␈↓ α,␈↓discovering␈α∂some␈α⊂valuable␈α∂concepts␈α∂it␈α⊂otherwise␈α∂got␈α∂(in␈α⊂this␈α∂case,␈α∂Numbers).␈α⊂This␈α∂makes
␈↓ α,␈↓AM's␈αbehavior␈αsound␈αvery␈α
fragile,␈αlike␈αa␈αslender␈αchain␈α
of␈αadvancement.␈α But␈αon␈α
the␈αother
␈↓ α,␈↓hand,␈α∃many␈α∀concepts␈α∃(e.g.,␈α∀TIMES,␈α∃Timberline,␈α∀Primes␈↓	11␈↓)␈α∃were␈α∀discovered␈α∃in␈α∀several
␈↓ α,␈↓independent␈α∞ways.␈α∞ If␈α
AM's␈α∞behavior␈α∞is␈α∞a␈α
chain,␈α∞it␈α∞is␈α
multiply-stranded␈α∞(except␈α∞for␈α∞a␈α
few
␈↓ α,␈↓weak spots, like Numbers).

␈↓ α,␈↓The␈α
heuristics␈α
are␈α
speci≡c␈α
to␈α
their␈α
stated␈α
domain␈α
of␈α
applicability.␈α
 Thus␈α
when␈α∞working␈α
in
␈↓ α,␈↓geometry,␈α∞the␈α∞Operation␈α∞heuristics␈α∞were␈α∞just␈α∞as␈α
useful␈α∞as␈α∞they␈α∞were␈α∞when␈α∞AM␈α∞worked␈α
in
␈↓ α,␈↓elementary␈α∩set␈α∪theory␈α∩or␈α∩number␈α∪theory.␈α∩The␈α∪set␈α∩of␈α∩facets␈α∪seemed␈α∩adequate␈α∪for␈α∩those
␈↓ α,␈↓domains,␈α
too.␈α
The␈αIntuition␈α
facet,␈α
which␈αwas␈α
rejected␈α
as␈α
a␈αvalid␈α
source␈α
of␈αinformation␈α
about
␈↓ α,␈↓sets␈αand␈αnumbers,␈αmight␈αhave␈αbeen␈α
more␈αacceptable␈αin␈αgeometry␈α(e.g.,␈αsomething␈α
similar␈αto
␈↓ α,␈↓Gelernter's model of a geometric situation).


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.7. How to Perform Experiments on AM␈↓)αβ␈↓

␈↓ α,␈↓The␈αvery␈αfact␈αthat␈αthe␈αkinds␈αof␈αexperiments␈αmentioned␈αin␈αthe␈αlast␈αsection␈α(and␈αdescribed␈αin
␈↓ α,␈↓detail␈αin␈α
Section␈α6.2)␈αcan␈α
be␈α"set␈αup"␈α
and␈αperformed␈α
on␈αAM,␈αre∨ects␈α
an␈αimportant␈αquality␈α
of
␈↓ α,␈↓the AM program.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	11␈↓ε␈α	Primes␈α	was␈α	discovered␈α	independently␈α	as␈α
follows:␈α	all␈α	numbers␈α	(>0)␈α	were␈α	seen␈α
to␈α	be␈α	representable␈α	as␈α	the␈α	sum␈α
of␈α	smaller
␈↓ α,␈↓ε␈↓ βLnumbers;␈α	Add␈α	was␈α	known␈α	to␈α	be␈α	analogous␈α	to␈α	TIMES;␈α	But␈α	not␈α	all␈α	numbers␈α	(>1)␈α	appeared␈α	to␈α	be␈αλrepresentable
␈↓ α,␈↓ε␈↓ βLas␈αthe␈α
␈↓&product␈↓)αβ␈αof␈α
two␈αsmaller␈αones;␈α
Rule␈αnumber␈α
81␈αtriggered␈α(see␈α
Appendix␈α2,␈α
page␈α154),␈α
and␈αAM
␈↓ α,␈↓ε␈↓ βLdefined␈αthe␈αset␈αof␈αexceptions:␈αthe␈αset␈αof␈αnumbers␈αwhich␈αcould␈αnot␈αbe␈αexpressed␈αas␈αthe␈αproduct␈αof␈αtwo
␈↓ α,␈↓ε␈↓ βLsmaller ones; i.e., the primes.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε110␈↓-

␈↓ α,␈↓Most␈α∪of␈α∩those␈α∪experiments␈α∩took␈α∪only␈α∩a␈α∪matter␈α∪of␈α∩minutes␈α∪to␈α∩set␈α∪up,␈α∩only␈α∪a␈α∪few␈α∩tiny
␈↓ α,␈↓modi≡cations␈αto␈αAM.␈α For␈αexample,␈αthe␈αone␈αwhere␈αall␈αthe␈αWorth␈αratings␈αwere␈αinitialized␈αto
␈↓ α,␈↓the same value was done by evaluating the single LISP expression:

␈↓ α,␈↓¬␈↓ β,(MAPC CONCEPTS '(λ (c) (PUT c 'Worth 200)))

␈↓ α,␈↓Similarly,␈α
here␈α∞is␈α
how␈α
AM␈α∞was␈α
modi≡ed␈α
to␈α∞treat␈α
all␈α
tasks␈α∞as␈α
if␈α
they␈α∞had␈α
equal␈α∞value:␈α
the
␈↓ α,␈↓function Pick-task has a statement of the form

␈↓ α,␈↓¬␈↓ β,(SETQ Current-task (First-member-of Agenda))

␈↓ α,␈↓All␈α⊂that␈α⊂was␈α⊂necessary␈α⊃was␈α⊂to␈α⊂replace␈α⊂the␈α⊂call␈α⊃on␈α⊂the␈α⊂function␈α⊂"␈↓¬First-member-of␈↓"␈α⊃by␈α⊂the
␈↓ α,␈↓function "␈↓¬Random-member-of␈↓".

␈↓ α,␈↓Even␈α∂the␈α∂most␈α∂sophisticated␈α∂experiment,␈α∂the␈α∂introduction␈α∂of␈α∂a␈α∂new␈α∂bunch␈α∂of␈α∂concepts␈α∂¬
␈↓ α,␈↓those␈α∩dealing␈α∩with␈α∩geometric␈α⊃notions␈α∩like␈α∩Between,␈α∩Angle,␈α∩Line␈α⊃¬␈α∩took␈α∩only␈α∩a␈α∩day␈α⊃of
␈↓ α,␈↓conscious work to set up.

␈↓ α,␈↓Of course ␈↓βrunning␈↓ the experiment involves the expenditure of hours of cpu time.

␈↓ α,␈↓There␈αare␈α
certain␈αexperiments␈αone␈α
can't␈αeasily␈α
perform␈αon␈αAM:␈α
removing␈αall␈α
its␈αheuristics,
␈↓ α,␈↓for␈α∂example.␈α∞ Most␈α∂heuristic␈α∂search␈α∞programs␈α∂would␈α∂then␈α∞wallow␈α∂around,␈α∂displaying␈α∞just
␈↓ α,␈↓how␈αbig␈αtheir␈αsearch␈αspace␈αreally␈αwas.␈αBut␈αAM␈αwould␈αjust␈αsit␈αthere,␈αsince␈αit'd␈αhave␈αnothing
␈↓ α,␈↓plausible to do.

␈↓ α,␈↓Many␈α
other␈αexperiments,␈α
while␈α
cute␈αand␈α
easy␈α
to␈αset␈α
up,␈α
are␈αquite␈α
costly␈α
in␈αterms␈α
of␈αcpu␈α
time.
␈↓ α,␈↓For␈α⊂example,␈α⊂the␈α⊃class␈α⊂of␈α⊂experiments␈α⊂of␈α⊃the␈α⊂form:␈α⊂"remove␈α⊂heuristics␈α⊃x,␈α⊂y,␈α⊂and␈α⊃z,␈α⊂and
␈↓ α,␈↓observe␈α
the␈α
resultant␈α
a≥ect␈α
on␈αAM's␈α
behavior".␈α
 This␈α
observation␈α
would␈α
entail␈αrunning␈α
AM
␈↓ α,␈↓for␈α∞an␈α
hour␈α∞or␈α
two␈α∞of␈α
cpu␈α∞time!␈α
 Considering␈α∞the␈α
number␈α∞of␈α
subsets␈α∞of␈α
heuristics,␈α∞not␈α
all
␈↓ α,␈↓these␈αquestions␈αare␈αgoing␈αto␈αget␈αanswered␈αin␈αour␈αuniverse's␈αlifetime.␈α Considering␈αthe␈αsmall
␈↓ α,␈↓probable payo≥ from any one such experiment, very few should actually be attempted.

␈↓ α,␈↓One␈α
nice␈αexperiment␈α
would␈αbe␈α
to␈αmonitor␈α
the␈αcontribution␈α
each␈αheuristic␈α
is␈α
making.␈αThat
␈↓ α,␈↓is,␈αrecord␈αeach␈αtime␈αit␈αis␈αused␈αand␈αrecord␈αthe␈α≡nal␈αoutcome␈αof␈αits␈αactivation␈α(which␈αmay␈αbe
␈↓ α,␈↓several␈α∩cycles␈α∩later).␈α∩Unfortunately,␈α∩AM's␈α∩heuristics␈α∩are␈α∩not␈α∩all␈α∩coded␈α∩as␈α∪separate␈α∩Lisp
␈↓ α,␈↓entities,␈α∞which␈α∂one␈α∞could␈α∞then␈α∂"trace".␈α∞Rather,␈α∂they␈α∞are␈α∞often␈α∂interwoven␈α∞with␈α∂each␈α∞other
␈↓ α,␈↓into large program pieces. So this experiment can't be easily set up and run on AM.

␈↓ α,␈↓Even␈αthose␈αexperiments␈αwhich␈αcan␈αbe␈αrun,␈αcan␈α
be␈αset␈αup␈αonly␈αby␈αsomeone␈αfamiliar␈αwith␈α
the
␈↓ α,␈↓LISP␈αcode␈αof␈αAM.␈α It␈αwould␈αbe␈αquite␈αhard␈αto␈αmodify␈αAM␈αso␈αthat␈αthe␈αuntrained␈αuser␈αcould
␈↓ α,␈↓easily␈α⊂perform␈α∂these␈α⊂experiments.␈α∂Essentially,␈α⊂that␈α∂would␈α⊂demand␈α∂that␈α⊂AM␈α∂have␈α⊂a␈α∂deep
␈↓ α,␈↓understanding␈αof␈αits␈αown␈αstructure.␈αThis␈αis␈αof␈αcourse␈αdesirable,␈αfascinating,␈αchallenging,␈αbut
␈↓ α,␈↓wasn't part of the design of AM.␈↓	12␈↓




␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	12␈↓ε␈α
A␈α
general␈α
suggestion␈αfor␈α
future␈α
research␈α
projects␈α
in␈αthis␈α
area:␈α
such␈α
systems␈α
should␈αbe␈α
designed␈α
in␈α
a␈α
way␈αwhich
␈↓ α,␈↓ε␈↓ βLfacilitates a poorly-trained user not only ␈↓&using␈↓)αβ the system but ␈↓&experimenting␈↓)αβ on it.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε111␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.8. Future Implications of this Project␈↓)αβ␈↓

␈↓ α,␈↓One␈α∞harsh␈α∞measure␈α∂of␈α∞AM␈α∞would␈α∂be␈α∞to␈α∞demand␈α∞what␈α∂possible␈α∞applications␈α∞it␈α∂will␈α∞have.
␈↓ α,␈↓This␈αreally␈α
means␈α(i)␈αthe␈α
uses␈αfor␈αthe␈α
AM␈αsystem,␈α
(ii)␈αthe␈αuses␈α
for␈αthe␈αideas␈α
of␈αhow␈αto␈α
create
␈↓ α,␈↓such␈αsystems,␈α(iii)␈αconclusions␈αabout␈αmath␈αand␈αscience␈αone␈αcan␈αdraw␈αfrom␈αexperiments␈αwith
␈↓ α,␈↓AM.

␈↓ α,␈↓Here are some of these implications, both real and potential:

␈↓ α,␈↓␈↓¬1.␈↓␈α∞New␈α∞tools␈α∞for␈α∞computer␈α∞scientists␈α∞who␈α∞want␈α∞to␈α∞create␈α∞large␈α∞knowledge-based␈α∂systems␈α∞to
␈↓ α,␈↓␈↓ α\emulate some creative human activity.
␈↓ α,␈↓␈↓ β,␈↓¬1a.␈↓␈α∞The␈α∂modular␈α∞representation␈α∂of␈α∞knowledge␈α∞that␈α∂AM␈α∞uses␈α∂might␈α∞prove␈α∂to␈α∞be
␈↓ α,␈↓␈↓ α\e≥ective␈αin␈αany␈αknowledge-based␈αsystem.␈α Division␈αof␈αa␈αglobal␈αproblem␈αinto␈αa␈αmultitude
␈↓ α,␈↓␈↓ α\of␈α
small␈αchunks,␈α
each␈αof␈α
them␈αof␈α
the␈αform␈α
of␈αsetting␈α
up␈αone␈α
quite␈αlocal␈α
"expert"␈αon␈α
some
␈↓ α,␈↓␈↓ α\concept,␈αis␈αa␈αnice␈αway␈αto␈αmake␈αa␈αhard␈αtask␈αmore␈αmanagable.␈α Conceivably,␈αeach␈αneeded
␈↓ α,␈↓␈↓ α\expert␈α
could␈α
be␈α
≡lled␈α
in␈α
by␈α
a␈α
human␈αwho␈α
really␈α
is␈α
an␈α
expert␈α
on␈α
that␈α
topic.␈α
 Then␈αthe
␈↓ α,␈↓␈↓ α\global␈α
abilities␈αof␈α
the␈α
system␈αwould␈α
be␈α
able␈αto␈α
rely␈α
on␈αquite␈α
sophisticated␈α
local␈αcriteria.
␈↓ α,␈↓␈↓ α\Fixing a set of facets once and for all permits e≥ective inter-module communication.
␈↓ α,␈↓␈↓ β,␈↓¬1b.␈↓␈α
Some␈α
ideas␈α∞may␈α
carry␈α
over␈α
unchanged␈α∞into␈α
many␈α
≡elds␈α
of␈α∞human␈α
creativity,
␈↓ α,␈↓␈↓ α\wherever␈α⊂local␈α⊃guiding␈α⊂rules␈α⊂exist.␈α⊃ These␈α⊂include:␈α⊂(a)␈α⊃ideas␈α⊂about␈α⊃heuristics␈α⊂having
␈↓ α,␈↓␈↓ α\domains␈αof␈α
applicability,␈α(b)␈α
the␈αpolicy␈α
of␈αtacking␈α
them␈αonto␈α
the␈αmost␈αgeneral␈α
knowledge
␈↓ α,␈↓␈↓ α\source␈α
(concept,␈α
module)␈α
they␈αare␈α
relevant␈α
to,␈α
(c)␈αthe␈α
rippling␈α
scheme␈α
to␈α
locate␈αrelevant
␈↓ α,␈↓␈↓ α\knowledge, etc.,

␈↓ α,␈↓␈↓¬2.␈↓ A body of heuristics which can be built upon by others.
␈↓ α,␈↓␈↓ β,␈↓¬2a.␈↓␈αMost␈αof␈α
the␈αparticular␈αheuristic␈α
judgmental␈αcriteria␈αfor␈α
interestingness,␈αutility,
␈↓ α,␈↓␈↓ α\etc.,␈αmight␈αbe␈αvalid␈αin␈αdeveloping␈αtheorizers␈αin␈αother␈αsciences.␈α Recall␈αthat␈αeach␈αrule␈αhas
␈↓ α,␈↓␈↓ α\its domain of applicability; many of the heuristics in AM are quite general.
␈↓ α,␈↓␈↓ β,␈↓¬2b.␈↓␈α∩Just␈α∩within␈α∩the␈α∩small␈α∩domain␈α∪in␈α∩which␈α∩AM␈α∩already␈α∩works,␈α∩this␈α∪base␈α∩of
␈↓ α,␈↓␈↓ α\heuristics␈α
might␈α∞be␈α
enlarged␈α∞through␈α
contact␈α∞with␈α
various␈α∞mathematicians.␈α
If␈α∞they␈α
are
␈↓ α,␈↓␈↓ α\willing␈α⊃to␈α⊃introspect␈α⊃and␈α⊃add␈α⊃some␈α⊃of␈α⊃their␈α⊃"rules"␈α⊃to␈α⊃AM's␈α⊃existing␈α⊃base,␈α⊃it␈α⊂might
␈↓ α,␈↓␈↓ α\gradually grow more and more powerful.
␈↓ α,␈↓␈↓ β,␈↓¬2c.␈↓␈α⊃Carrying␈α⊃this␈α⊃last␈α⊃point␈α⊃to␈α⊂the␈α⊃limit␈α⊃of␈α⊃possibility,␈α⊃one␈α⊃might␈α⊃imagine␈α⊂the
␈↓ α,␈↓␈↓ α\program␈αpossessing␈αmore␈αheuristics␈αthan␈αany␈αsingle␈αhuman.␈α Of␈αcourse,␈αAM␈αas␈αit␈α
stands
␈↓ α,␈↓␈↓ α\now␈α→is␈α→missing␈α_so␈α→much␈α→of␈α_the␈α→`human␈α→element',␈α_the␈α→life␈α→experiences␈α→that␈α_a
␈↓ α,␈↓␈↓ α\mathematician␈α∀draws␈α∀upon␈α∀continually␈α∀for␈α∀inspiration,␈α∀that␈α∀merely␈α∀amassing␈α∀more
␈↓ α,␈↓␈↓ α\heuristics␈α∃won't␈α∃automatically␈α∃push␈α∃it␈α∀to␈α∃the␈α∃level␈α∃of␈α∃a␈α∃super-human␈α∀intelligence.
␈↓ α,␈↓␈↓ α\Another␈αfar-out␈αscenario␈αis␈αthat␈αof␈α
the␈αgreat␈αmathematicians␈αof␈αeach␈αgeneration␈α
pouring
␈↓ α,␈↓␈↓ α\their␈αindividual␈αheuristics␈αinto␈αan␈αAM-like␈αsystem.␈α After␈αa␈αfew␈αgenerations␈α
have␈αcome
␈↓ α,␈↓␈↓ α\and␈α
gone,␈α
running␈α
that␈α∞program␈α
could␈α
be␈α
a␈α∞valuable␈α
way␈α
to␈α
bring␈α∞about␈α
`interactions'
␈↓ α,␈↓␈↓ α\between people who were not contemporaries.

␈↓ α,␈↓␈↓¬3.␈↓ New and better strategies for math educators.
␈↓ α,␈↓␈↓ β,␈↓¬3a.␈↓␈αSince␈αthe␈αkey␈αto␈αAM's␈αsuccess␈αseems␈αto␈αbe␈αits␈αheuristics,␈αand␈αnot␈αthe␈αparticular
␈↓ α,␈↓␈↓ α\concepts␈α∞it␈α∂knows,␈α∞the␈α∂whole␈α∞orientation␈α∂of␈α∞mathematics␈α∂education␈α∞should␈α∂perhaps␈α∞be
␈↓ α,␈↓␈↓ α\modi≡ed,␈α
to␈α
provide␈α
experiences␈αfor␈α
the␈α
student␈α
which␈α
will␈αbuild␈α
up␈α
these␈α
rules␈α
in␈αhis
␈↓ α,␈↓␈↓ α\mind.␈αLearning␈αa␈αnew␈αtheorem␈αis␈αworth␈αmuch␈αless␈αthan␈αlearning␈αa␈αnew␈αheuristic␈αwhich
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε112␈↓-

␈↓ α,␈↓␈↓ α\lets␈α
you␈α∞discover␈α
new␈α∞theorems.␈↓	13␈↓␈α
I␈α∞am␈α
far␈α
from␈α∞the␈α
≡rst␈α∞to␈α
urge␈α∞such␈α
a␈α∞revision␈α
(see,
␈↓ α,␈↓␈↓ α\e.g., [Koestler 67], p.265, or see [Papert 72]).
␈↓ α,␈↓␈↓ β,␈↓¬3b.␈↓␈α
If␈αthe␈α
repertoire␈αof␈α
intuition␈α(simulated␈α
real-world␈αscenarios)␈α
were␈αsu≠cient␈α
for
␈↓ α,␈↓␈↓ α\AM␈αto␈αdevelop␈αelementary␈α
concepts␈αof␈αmath,␈αthen␈α
educators␈αshould␈αensure␈αthat␈α
children
␈↓ α,␈↓␈↓ α\(4-6␈αyears␈αold)␈αare␈αthoroughly␈αexposed␈αto␈αthose␈αscenarios.␈α Such␈αactivities␈αwould␈αinclude
␈↓ α,␈↓␈↓ α\seesaws,␈α∂slides,␈α∂piling␈α∂marbles␈α∂into␈α∂pans␈α∞of␈α∂a␈α∂balance␈α∂scale,␈α∂comparing␈α∂the␈α∂heights␈α∞of
␈↓ α,␈↓␈↓ α\towers␈α
built␈α
out␈α
of␈α
cubical␈α
blocks,␈α
solving␈α
a␈α
jigsaw␈α
puzzle,␈α
etc.␈α
 Unfortunately,␈α
AM␈α
failed
␈↓ α,␈↓␈↓ α\to␈α
show␈α
the␈α
value␈α
of␈α
these␈α∞few␈α
scenarios.␈α
 This␈α
was␈α
a␈α
potential␈α
application␈α∞which␈α
was
␈↓ α,␈↓␈↓ α\not con≡rmed.
␈↓ α,␈↓␈↓ β,␈↓¬3c.␈↓␈α
One␈αuse␈α
for␈α
AM␈αitself␈α
would␈α
be␈αas␈α
a␈α
"fun"␈αteaching␈α
tool.␈α
If␈αa␈α
very␈α
nice␈αuser
␈↓ α,␈↓␈↓ α\interface␈α
is␈α
constructed,␈α
AM␈α
could␈α
serve␈α
as␈α
a␈α
model␈α
for,␈α
say,␈α
college␈α
freshmen␈α∞with␈α
no
␈↓ α,␈↓␈↓ α\math␈α
research␈α
experience.␈α
They␈α
could␈α
watch␈α
AM,␈α
see␈α
the␈α
kinds␈α
of␈α
things␈α
it␈α
does,␈αplay
␈↓ α,␈↓␈↓ α\with␈αit,␈α
and␈αperhaps␈α
get␈αa␈αreal␈α
∨avor␈αfor␈α
(and␈αget␈αturned␈α
on␈αby)␈α
doing␈αmath␈αresearch.␈α
A
␈↓ α,␈↓␈↓ α\vast␈α∞number␈α∂of␈α∞brilliant␈α∂minds␈α∞are␈α∂too␈α∞turned␈α∂o≥␈α∞by␈α∂high-school␈α∞drilling␈α∂and␈α∞college
␈↓ α,␈↓␈↓ α\calculus␈α
to␈α
stick␈αaround␈α
long␈α
enough␈α
to␈α≡nd␈α
out␈α
how␈α
exciting␈α¬␈α
and␈α
di≥erent␈α¬␈α
research
␈↓ α,␈↓␈↓ α\math is compared to textbook math.

␈↓ α,␈↓␈↓¬4.␈↓␈α
Further␈α
experiments␈α
on␈α
AM␈α
might␈α
tell␈α
us␈α
something␈α
about␈α
how␈α
the␈α
theory␈αformation␈α
task
␈↓ α,␈↓␈↓ α\changes␈αas␈αa␈αtheory␈αgrows␈αin␈αsophistication.␈α For␈αexample,␈αcan␈αthe␈αsame␈αmethods␈αwhich
␈↓ α,␈↓␈↓ α\lead␈α∩AM␈α⊃from␈α∩premathematical␈α∩concepts␈α⊃to␈α∩arithmetic␈α∩also␈α⊃lead␈α∩AM␈α∩from␈α⊃number
␈↓ α,␈↓␈↓ α\systems␈α∞up␈α∞to␈α∞abstract␈α∞algebra?␈α∞ Or␈α∞are␈α∂a␈α∞new␈α∞set␈α∞of␈α∞heuristic␈α∞rules␈α∞or␈α∂extra␈α∞concepts
␈↓ α,␈↓␈↓ α\required?␈α
 My␈α
guess␈αis␈α
that␈α
a␈αfew␈α
of␈α
each␈α
are␈αlacking␈α
currently,␈α
but␈α␈↓βonly␈↓␈α
a␈α
few.␈α There␈α
is
␈↓ α,␈↓␈↓ α\a␈α⊃great␈α⊃deal␈α⊃of␈α⊃disagreement␈α∩about␈α⊃this␈α⊃subject␈α⊃among␈α⊃mathematicians.␈α∩ By␈α⊃tracing
␈↓ α,␈↓␈↓ α\along␈α∂the␈α∂development␈α∂of␈α∂mathematics,␈α∂one␈α∂might␈α∂categorize␈α∂discoveries␈α∂by␈α∂how␈α∂easy
␈↓ α,␈↓␈↓ α\they␈α⊃would␈α⊂be␈α⊃for␈α⊂an␈α⊃AM-like␈α⊂system␈α⊃to␈α⊂≡nd.␈α⊃ Sometimes,␈α⊂a␈α⊃discovery␈α⊃required␈α⊂the
␈↓ α,␈↓␈↓ α\invention␈α∪of␈α∀a␈α∪brand␈α∀new␈α∪heuristic␈α∀rule,␈α∪which␈α∀would␈α∪clearly␈α∀be␈α∪beyond␈α∀AM␈α∪as
␈↓ α,␈↓␈↓ α\currently␈αdesigned.␈α Sometimes,␈αdiscovery␈αis␈αbased␈αon␈αthe␈αlucky␈αrandom␈αcombination␈αof
␈↓ α,␈↓␈↓ α\existing␈α
concepts,␈α∞for␈α
no␈α∞good␈α
␈↓βa␈α
priori␈↓␈α∞reason.␈α
It␈α∞would␈α
be␈α
instructive␈α∞to␈α
≡nd␈α∞out␈α
how
␈↓ α,␈↓␈↓ α\often␈αthis␈αis␈αnecessarily␈αthe␈αcase:␈αhow␈αoften␈α␈↓βcan't␈↓␈αa␈αmathematical␈αdiscovery␈αbe␈αmotivated
␈↓ α,␈↓␈↓ α\and "explained" using heuristic rules of the kind AM possesses?

␈↓ α,␈↓␈↓¬5.␈↓␈αAn␈αunanticipated␈αresult␈αwas␈αthe␈αcreation␈αof␈αnew-to-Mankind␈αmath␈α(both␈αdirectly␈αand␈αby
␈↓ α,␈↓␈↓ α\de≡ning␈α
new,␈α
interesting␈α
concepts␈α
to␈α
investigate␈α
by␈α
hand).␈α
 The␈α
amount␈α
of␈α
new␈α
bits␈αof
␈↓ α,␈↓␈↓ α\mathematics developed to date is minuscule.
␈↓ α,␈↓␈↓ β,␈↓¬5a.␈↓␈α⊗As␈α⊗described␈α⊗in␈α⊗(2c)␈α∃above,␈α⊗AM␈α⊗might␈α⊗absorb␈α⊗heuristics␈α⊗from␈α∃several
␈↓ α,␈↓␈↓ α\individuals␈αand␈αthereby␈αintegrate␈αtheir␈αparticular␈αinsights.␈α This␈αmight␈αeventually␈αresult
␈↓ α,␈↓␈↓ α\in new mathematics being discovered.
␈↓ α,␈↓␈↓ β,␈↓¬5b.␈↓␈α∩An␈α⊃even␈α∩more␈α⊃exciting␈α∩prospect,␈α⊃which␈α∩never␈α⊃materialized,␈α∩was␈α∩that␈α⊃AM
␈↓ α,␈↓␈↓ α\would␈α⊃≡nd␈α∩a␈α⊃new␈α⊃redivision␈α∩of␈α⊃existing␈α⊃concepts,␈α∩an␈α⊃alternate␈α⊃formulation␈α∩of␈α⊃some
␈↓ α,␈↓␈↓ α\established␈α∞theory,␈α
much␈α∞like␈α
Hamiltonian␈α∞mechanics␈α
is␈α∞an␈α
alternate␈α∞uni≡cation␈α∞of␈α
the
␈↓ α,␈↓␈↓ α\data␈α∞which␈α∞led␈α∞to␈α∞Newtonian␈α
mechanics.␈α∞ The␈α∞only␈α∞rudimentary␈α∞behavior␈α∞along␈α
these
␈↓ α,␈↓␈↓ α\lines␈α∞was␈α
when␈α∞AM␈α∞occasionally␈α
derived␈α∞a␈α
familiar␈α∞concept␈α∞in␈α
an␈α∞abnormal␈α∞way␈α
(e.g.,
␈↓ α,␈↓␈↓ α\TIMES␈α
was␈α
derived␈α
in␈αfour␈α
ways;␈α
Prime␈α
pairs␈αwere␈α
noticed␈α
by␈α
restricting␈α
Addition␈αto
␈↓ α,␈↓␈↓ α\primes).



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	13␈↓ε␈αUsually.␈αOne␈α
kind␈αof␈αexception␈α
is␈αthe␈αfollowing:␈αthe␈α
ability␈αto␈αtake␈α
a␈αpowerful␈αtheorem,␈αand␈α
extract␈αfrom␈αit␈α
a␈αnew,
␈↓ α,␈↓ε␈↓ βLpowerful␈α
heuristic.␈α
AM␈α	cannot␈α
do␈α
this,␈α	but␈α
it␈α
may␈α	turn␈α
out␈α
that␈α	this␈α
mechanism␈α
is␈α	quite␈α
crucial␈α
for␈α	humans'
␈↓ α,␈↓ε␈↓ βLobtaining new heuristics. This is another open research problem.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε113␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.9. Open Problems: Suggestions for Future Research␈↓)αβ␈↓

␈↓ α,␈↓While␈αAM␈αcan␈αand␈αshould␈αstand␈αas␈αa␈αcomplete␈αresearch␈αproject,␈αpart␈αof␈αits␈αvalue␈αwill␈αstem
␈↓ α,␈↓from␈α
whatever␈α
future␈α
studies␈α∞are␈α
sparked␈α
by␈α
it.␈α
 Of␈α∞course␈α
the␈α
"evaluation"␈α
of␈α∞AM␈α
along
␈↓ α,␈↓this␈α
dimension␈αmust␈α
wait␈αfor␈α
years,␈αbut␈α
even␈αat␈α
the␈αpresent␈α
time␈αseveral␈α
such␈αopen␈α
problems
␈↓ α,␈↓come to mind:

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αDevise␈α
Meta-heuristics,␈αrules␈αcapable␈α
of␈αoperating␈αon␈α
and␈αsynthesizing␈αnew␈α
heuristic
␈↓ α,␈↓␈↓ β≤rules.␈α⊃ AM␈α⊃has␈α⊃shown␈α⊃the␈α⊃solution␈α⊃of␈α⊃this␈α⊃problem␈α⊃to␈α⊃be␈α⊃both␈α∩nontrivial␈α⊃and
␈↓ α,␈↓␈↓ β≤indispensable.␈α∞ AM's␈α
progress␈α∞ground␈α∞to␈α
a␈α∞halt␈α
because␈α∞fresh,␈α∞powerful␈α
heuristics
␈↓ α,␈↓␈↓ β≤were␈α∞never␈α∞produced.␈α
 The␈α∞next␈α∞point␈α
suggests␈α∞that␈α∞the␈α
same␈α∞need␈α∞for␈α∞new␈α
rules
␈↓ α,␈↓␈↓ β≤exists in mathematics as a whole:

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈α
Examine␈αthe␈α
history␈αof␈αmathematics,␈α
and␈αgradually␈α
build␈αup␈αa␈α
list␈αof␈α
the␈αheuristic
␈↓ α,␈↓␈↓ β≤rules␈α∪used.␈α∪ Does␈α∪the␈α∪following␈α∀thesis␈α∪have␈α∪any␈α∪validity:␈α∪"␈↓βThe␈α∀development␈α∪of
␈↓ α,␈↓β␈↓ β≤mathematics␈αis␈αessentially␈αthe␈αdevelopment␈αof␈αnew␈αheuristics␈↓."␈αThat␈αis,␈αcan␈αwe␈α`factor
␈↓ α,␈↓␈↓ β≤out'␈α
all␈α
the␈α
discoveries␈α∞reachable␈α
by␈α
the␈α
set␈α∞of␈α
heuristics␈α
available␈α
(known)␈α∞to␈α
the
␈↓ α,␈↓␈↓ β≤mathematicians␈α
at␈α
some␈α
time␈α
in␈α
history,␈αand␈α
then␈α
explain␈α
each␈α
new␈α
big␈αdiscovery
␈↓ α,␈↓␈↓ β≤as␈α⊂requiring␈α⊃the␈α⊂synthesis␈α⊃of␈α⊂a␈α⊂brand␈α⊃new␈α⊂heuristic?␈α⊃ For␈α⊂example,␈α⊃Bolyai␈α⊂and
␈↓ α,␈↓␈↓ β≤Lobachevsky␈α∪did␈α∪this␈α∪a␈α∪century␈α∪ago␈α∪when␈α∪they␈α∪decided␈α∀that␈α∪counter-intuitive
␈↓ α,␈↓␈↓ β≤systems␈α
might␈α
still␈α
be␈α∞consistent␈α
and␈α
interesting.␈α
Non-Euclidean␈α∞geometry␈α
resulted,
␈↓ α,␈↓␈↓ β≤and␈α⊃no␈α∩mathematician␈α⊃today␈α⊃would␈α∩think␈α⊃twice␈α⊃about␈α∩using␈α⊃the␈α∩heuristic␈α⊃they
␈↓ α,␈↓␈↓ β≤developed.␈α⊃ Einstein␈α⊂invented␈α⊃a␈α⊃new␈α⊂heuristic␈α⊃more␈α⊃recently,␈α⊂when␈α⊃he␈α⊃dared␈α⊂to
␈↓ α,␈↓␈↓ β≤consider␈αthat␈αcounter-intuitive␈αsystems␈αmight␈αactually␈αhave␈αphysical␈αreality.␈↓	14␈↓␈αWhat
␈↓ α,␈↓␈↓ β≤was once a bold new method is now a standard tool in theoretical physics.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αIn␈αa␈α
far␈αless␈αdramatic␈αvein,␈αa␈α
hard␈αopen␈αproblem␈αis␈αthat␈α
of␈αbuilding␈αup␈αa␈α
body␈αof
␈↓ α,␈↓␈↓ β≤rules␈α
for␈α
symbolically␈α
instantiating␈α
a␈α
de≡nition␈α
(a␈α
LISP␈α
predicate).␈α
 These␈αrules␈α
may
␈↓ α,␈↓␈↓ β≤be␈αstructured␈αhierarchically,␈αso␈αthat␈αrules␈αspeci≡c␈αto␈αoperating␈αon␈α`operations␈αwhose
␈↓ α,␈↓␈↓ β≤domain␈α
and␈α
range␈αare␈α
equal'␈α
may␈αbe␈α
gathered.␈α
 Is␈α
this␈αset␈α
≡nite␈α
and␈αmanagable;␈α
i.e.,
␈↓ α,␈↓␈↓ β≤does␈α∞some␈α∞sort␈α∞of␈α∞"closure"␈α∞occur␈α
after␈α∞a␈α∞few␈α∞hundred␈α∞(thousand?)␈α∞such␈α∞rules␈α
are
␈↓ α,␈↓␈↓ β≤assembled?

␈↓ α,␈↓␈↓ αl␈↓π#␈α⊃␈↓␈α⊃More␈α⊃generally,␈α⊃we␈α⊃can␈α⊃ask␈α⊃for␈α⊂the␈α⊃expansion␈α⊃of␈α⊃all␈α⊃the␈α⊃heuristic␈α⊃rules,␈α⊃of␈α⊂all
␈↓ α,␈↓␈↓ β≤categories.␈α~This␈α~may␈α~be␈α~done␈α~by␈α~eliciting␈α~them␈α~from␈α≠mathematicians,␈α~or
␈↓ α,␈↓␈↓ β≤automatically␈α↔by␈α⊗the␈α↔application␈α↔of␈α⊗very␈α↔sophisticated␈α↔meta-heuristics.␈α⊗ Some
␈↓ α,␈↓␈↓ β≤categories␈α∩of␈α∩rules␈α⊃include:␈α∩how␈α∩to␈α⊃generalize/specialize␈α∩de≡nitions,␈α∩how␈α∩to␈α⊃≡nd
␈↓ α,␈↓␈↓ β≤examples of a given concept, how to optimize LISP algorithms.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αExperiments␈αcan␈αbe␈αdone␈αon␈αAM.␈αA␈αfew␈αhave␈αbeen␈αperformed␈αalready,␈αmany␈αmore
␈↓ α,␈↓␈↓ β≤are␈α⊃proposed␈α∩in␈α⊃Section␈α⊃6.2,␈α∩and␈α⊃no␈α⊃doubt␈α∩some␈α⊃additional␈α⊃ones␈α∩have␈α⊃already
␈↓ α,␈↓␈↓ β≤occurred to the reader.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αExtend␈αthe␈αanalysis␈αalready␈αbegun␈α(see␈αp.␈α56)␈αof␈αthe␈αset␈αof␈αheuristics␈αAM␈αpossesses.
␈↓ α,␈↓␈↓ β≤One␈αreason␈αfor␈αsuch␈αan␈αanalysis␈αwould␈α
be␈αto␈αachieve␈αa␈αbetter␈αunderstanding␈αof␈α
the

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	14␈↓ε␈αAs␈αCourant␈αsays,␈α"When␈αEinstein␈αtried␈αto␈αreduce␈αthe␈αnotion␈αof␈α`simultaneous␈αevents␈αoccurring␈αat␈αdifferent␈αplaces'␈αto
␈↓ α,␈↓ε␈↓ βLobservable␈αphenomena,␈αwhen␈αhe␈αunmasked␈αas␈αa␈αmetaphysical␈αprejudice␈αthe␈αbelief␈αthat␈αthis␈αconcept␈αmust
␈↓ α,␈↓ε␈↓ βLhave a scientific meaning in itself, he had found the key to his theory of relativity."
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε114␈↓-

␈↓ α,␈↓␈↓ β≤contribution␈α∩of␈α∪the␈α∩heuristics.␈α∩In␈α∪some␈α∩sense,␈α∩the␈α∪heuristics␈α∩and␈α∩the␈α∪choice␈α∩of
␈↓ α,␈↓␈↓ β≤starting␈αconcepts␈α"encode"␈αthe␈αdiscoveries␈αwhich␈αAM␈αmakes,␈αand␈αthe␈αway␈α
it␈αmakes
␈↓ α,␈↓␈↓ β≤them.␈αA␈αbetter␈αunderstanding␈αof␈αthat␈αencoding␈αmay␈αlead␈αto␈αnew␈αideas␈αfor␈αAM␈αand
␈↓ α,␈↓␈↓ β≤for future AM-like systems.

␈↓ α,␈↓␈↓ αl␈↓π#␈α⊃␈↓␈α⊃Rewrite␈α⊃AM.␈α⊃In␈α⊃Chapter␈α⊃1,␈α⊃on␈α⊃page␈α⊃9,␈α⊃it␈α⊃was␈α⊃pointed␈α⊃out␈α⊃that␈α⊃there␈α∩are␈α⊃two
␈↓ α,␈↓␈↓ β≤common␈α⊗species␈α⊗of␈α↔heuristic␈α⊗search␈α⊗programs.␈α⊗ One␈α↔type␈α⊗has␈α⊗a␈α↔legal␈α⊗move
␈↓ α,␈↓␈↓ β≤generator,␈αand␈αheuristics␈αto␈αconstrain␈αit.␈α The␈αsecond␈αtype,␈αincluding␈αAM,␈αhas␈αonly
␈↓ α,␈↓␈↓ β≤a␈αset␈αof␈αheuristics,␈αand␈αthey␈αact␈αas␈αplausible␈αmove␈αgenerators.␈αSince␈αAM␈αseemed␈αto
␈↓ α,␈↓␈↓ β≤create␈α∞new␈α∞concepts,␈α∞propose␈α∞new␈α∞conjectures,␈α∞and␈α∞formulate␈α∞new␈α∞tasks␈α∞in␈α∂a␈α∞very
␈↓ α,␈↓␈↓ β≤few␈α∂distinct␈α∂ways,␈α∂it␈α∂might␈α∂very␈α∂well␈α∂be␈α∂feasible␈α∂to␈α∂≡nd␈α∂a␈α∂purely␈α∂syntactic␈α∂"legal
␈↓ α,␈↓␈↓ β≤move␈α∂generator"␈α⊂for␈α∂AM,␈α⊂and␈α∂to␈α⊂convert␈α∂each␈α⊂existing␈α∂heuristic␈α⊂into␈α∂a␈α⊂form␈α∂of
␈↓ α,␈↓␈↓ β≤constraint.␈α⊂In␈α⊂that␈α⊂case,␈α⊂one␈α⊂could,␈α⊂e.g.,␈α∂remove␈α⊂all␈α⊂the␈α⊂heuristics␈α⊂and␈α⊂still␈α⊂see␈α∂a
␈↓ α,␈↓␈↓ β≤meaningful␈α∞(if␈α∂explosive)␈α∞activity␈α∞proceed.␈α∂There␈α∞might␈α∞be␈α∂a␈α∞few␈α∂surprises␈α∞down
␈↓ α,␈↓␈↓ β≤that path.

␈↓ α,␈↓␈↓ αl␈↓π#␈α∞␈↓␈α∞A␈α∞more␈α
tractible␈α∞project,␈α∞a␈α∞subset␈α
of␈α∞the␈α∞former␈α∞one,␈α
would␈α∞be␈α∞to␈α∞recode␈α∞just␈α
the
␈↓ α,␈↓␈↓ β≤conjecture-≡nding␈α↔heuristics␈α_as␈α↔constraints␈α_on␈α↔a␈α_new,␈α↔purely␈α_syntactic␈α↔"legal
␈↓ α,␈↓␈↓ β≤conjecture␈α∩generator".␈α∩A␈α∩simple␈α⊃Generate-and-Test␈α∩paradigm␈α∩would␈α∩be␈α∩used␈α⊃to
␈↓ α,␈↓␈↓ β≤synthesize␈α∩and␈α⊃examine␈α∩large␈α∩numbers␈α⊃of␈α∩conjectures.␈α⊃Again,␈α∩removing␈α∩all␈α⊃the
␈↓ α,␈↓␈↓ β≤heuristics would be a worthwhile experiment.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αAt␈α
the␈αreaches␈αof␈α
feasability,␈αone␈αcan␈α
imagine␈αtrying␈αto␈α
extend␈αAM␈αinto␈α
more␈αand
␈↓ α,␈↓␈↓ β≤more␈α∞≡elds,␈α∞into␈α
less-formalizable␈α∞domains.␈α∞International␈α
politics␈α∞has␈α∞already␈α
been
␈↓ α,␈↓␈↓ β≤suggested as a very hard future applications area.

␈↓ α,␈↓␈↓ αl␈↓π#␈α
␈↓␈α
Abstracting␈α
that␈α
last␈α
point,␈α
try␈α
to␈αbuild␈α
up␈α
a␈α
set␈α
of␈α
criteria␈α
which␈α
make␈α
a␈αdomain
␈↓ α,␈↓␈↓ β≤ripe␈α∂for␈α∞automating␈α∂(e.g.,␈α∞it␈α∂possesses␈α∂a␈α∞strong␈α∂theory,␈α∞it␈α∂is␈α∂knowledge-rich␈α∞(many
␈↓ α,␈↓␈↓ β≤heuristics␈α∞exist),␈α
the␈α∞performance␈α∞of␈α
the␈α∞professionals/experts␈α
is␈α∞much␈α∞better␈α
than
␈↓ α,␈↓␈↓ β≤that␈α
of␈αthe␈α
typical␈αpractitioners,␈α
the␈αnew␈α
discoveries␈αin␈α
that␈α≡eld␈α
all␈αfall␈α
into␈αa␈α
small
␈↓ α,␈↓␈↓ β≤variety␈α
of␈α
syntactic␈αformats,...?).␈α
 Initially,␈α
this␈α
study␈αmight␈α
help␈α
humans␈αbuild␈α
better
␈↓ α,␈↓␈↓ β≤and␈α
more␈α
appropriate␈α
scienti≡c␈αdiscovery␈α
programs.␈α
 Someday,␈α
it␈α
might␈αeven␈α
permit
␈↓ α,␈↓␈↓ β≤the creation of an automatic-theory-formation-program-writer.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αThe␈αinteraction␈αbetween␈αAM␈αand␈αthe␈αuser␈αis␈αminimal␈αand␈αpainful.␈α Is␈αthere␈αa␈αmore
␈↓ α,␈↓␈↓ β≤e≥ective␈α∂language␈α∂for␈α∂communication?␈α∞Should␈α∂several␈α∂languages␈α∂exist,␈α∞depending
␈↓ α,␈↓␈↓ β≤on␈α
the␈α
type␈α
of␈α
message␈α
to␈α
be␈α
sent␈α
(pictures,␈α
control␈α
characters,␈α
a␈α
subset␈α
of␈αnatural
␈↓ α,␈↓␈↓ β≤language,␈α→induction␈α→from␈α→examples,␈α→etc.)?␈α_ Can␈α→AM's␈α→output␈α→be␈α→raised␈α_in
␈↓ α,␈↓␈↓ β≤sophistication␈α∀by␈α∪introducing␈α∀an␈α∪internal␈α∀model␈α∪of␈α∀the␈α∪user␈α∀and␈α∪his␈α∀state␈α∪of
␈↓ α,␈↓␈↓ β≤knowledge␈α∞at␈α∞each␈α∂moment?␈α∞ Perhaps␈α∞AM␈α∞should␈α∂also␈α∞contain␈α∞models␈α∂of␈α∞several
␈↓ α,␈↓␈↓ β≤known␈α␈↓βgroups␈↓␈αof␈αusers␈α(mathematicians,␈αcomputer␈αscientists,␈αstudents)␈α¬␈αsay,␈αstored
␈↓ α,␈↓␈↓ β≤as concepts in a specialization/generalization hierarchy.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αHuman␈αprotocol␈α
studies␈αmay␈αbe␈αappropriate,␈αto␈α
test␈αout␈αthe␈αmodel␈α
of␈αmathematical
␈↓ α,␈↓␈↓ β≤research␈α∞which␈α∂AM␈α∞puts␈α∂forward.␈α∞Are␈α∞the␈α∂sequences␈α∞of␈α∂actions␈α∞similar?␈α∂Are␈α∞the
␈↓ α,␈↓␈↓ β≤mistakes␈α~analogous?␈α~Do␈α→the␈α~pauses␈α~which␈α→the␈α~humans␈α~emit␈α→␈↓βquantitatively␈↓
␈↓ α,␈↓␈↓ β≤correspond to AM's periods of gathering and running `Suggest' heuristics?

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αCan␈αthe␈αidea␈αof␈αIntuition␈αfunctions␈αbe␈αdeveloped␈αinto␈αa␈αuseful␈αmechanism?␈α If␈αnot,
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε115␈↓-

␈↓ α,␈↓␈↓ β≤how␈α↔else␈α↔might␈α↔real-world␈α↔experiences␈α↔be␈α↔made␈α↔available␈α↔to␈α_an␈α↔automated
␈↓ α,␈↓␈↓ β≤researcher␈α⊂to␈α⊂draw␈α⊂upon␈α⊂(for␈α⊂analogies,␈α∂to␈α⊂base␈α⊂new␈α⊂theories␈α⊂upon)?␈α⊂Could␈α∂one
␈↓ α,␈↓␈↓ β≤interface␈α∞physical␈α∞e≥ectors␈α∞and␈α∞receptors␈α∂and␈α∞quite␈α∞literally␈α∞allow␈α∞the␈α∂program␈α∞to
␈↓ α,␈↓␈↓ β≤`play around in the real world' for his analogies?

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈α
Most␈αof␈αthe␈α
`future␈αimplications'␈αdiscussed␈α
in␈αthe␈αlast␈α
section␈αsuggest␈αfuture␈α
activities
␈↓ α,␈↓␈↓ β≤(e.g., new educational experiments and techniques).

␈↓ α,␈↓␈↓ αl␈↓π#␈α∂␈↓␈α∂Most␈α∂of␈α∂the␈α∂`limiting␈α∂assumptions'␈α∂discussed␈α∂in␈α∂a␈α∂later␈α∂section␈α∂(page␈α∂121)␈α∂can␈α∞be
␈↓ α,␈↓␈↓ β≤tackled␈α∂with␈α∞today's␈α∂techniques␈α∞(plus␈α∂a␈α∞great␈α∂deal␈α∞of␈α∂e≥ort).␈α∞ Thus␈α∂each␈α∂of␈α∞them
␈↓ α,␈↓␈↓ β≤counts as an open problem for research.

␈↓ α,␈↓␈↓ αl␈↓π#␈α∩␈↓␈α∩Perform␈α∩an␈α⊃information-theoretic␈α∩analysis␈α∩on␈α∩AM.␈α⊃What␈α∩is␈α∩the␈α∩value␈α∩of␈α⊃each
␈↓ α,␈↓␈↓ β≤heuristic? the new information content of each new conjecture?

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αIf␈αyou're␈αinterested␈αin␈αnatural␈α
language,␈αthe␈αvery␈αhard␈αproblem␈αexists␈αof␈αgiving␈α
AM
␈↓ α,␈↓␈↓ β≤(or␈α∞a␈α∞similar␈α∞system)␈α∞the␈α∞ability␈α∞to␈α∞really␈α∞do␈α∞inferential␈α∞processing␈α∞on␈α∞the␈α∞␈↓βreasons␈↓
␈↓ α,␈↓␈↓ β≤attached␈α
to␈α∞tasks␈α
on␈α∞the␈α
agenda.␈α
Instead␈α∞of␈α
just␈α∞being␈α
able␈α
to␈α∞test␈α
for␈α∞equality␈α
of
␈↓ α,␈↓␈↓ β≤two␈α⊂reasons,␈α⊂it␈α⊂would␈α⊂be␈α∂much␈α⊂more␈α⊂intelligent␈α⊂to␈α⊂be␈α∂able␈α⊂to␈α⊂infer␈α⊂the␈α⊂kind␈α∂of
␈↓ α,␈↓␈↓ β≤relationship␈α
between␈α
any␈αtwo␈α
reasons;␈α
if␈αthey␈α
overlap␈α
semantically,␈αwe'd␈α
like␈α
to␈αbe
␈↓ α,␈↓␈↓ β≤able␈αto␈α
compute␈αprecisely␈α
how␈αthat␈αshould␈α
that␈αe≥ect␈α
the␈αoverall␈α
rating␈αfor␈αthe␈α
task;
␈↓ α,␈↓␈↓ β≤etc.

␈↓ α,␈↓␈↓ αl␈↓π#␈α⊃␈↓␈α⊂Modify␈α⊃the␈α⊃control␈α⊂structure␈α⊃of␈α⊃AM,␈α⊂as␈α⊃follows.␈α⊃Allow␈α⊂mini-goals␈α⊃to␈α⊃exist,␈α⊂and
␈↓ α,␈↓␈↓ β≤supply␈αnew␈αrules␈αfor␈αsetting␈αthem␈αup␈α(plausible␈αgoal␈αgenerators)␈αand␈αaltering␈αthose
␈↓ α,␈↓␈↓ β≤goals,␈α
plus␈α
some␈α
new␈α
rules␈α
and␈αalgorithms␈α
for␈α
satisfying␈α
them.␈α
The␈α
modi≡cation␈αI
␈↓ α,␈↓␈↓ β≤have␈αin␈αmind␈αwould␈αresult␈αin␈αnew␈αtasks␈αbeing␈αproposed␈αbecause␈αof␈αcertain␈αcurrent
␈↓ α,␈↓␈↓ β≤goals,␈αand␈αexisting␈αtasks␈αwould␈αbe␈αreordered␈αso␈αas␈αto␈αraise␈αthe␈αchance␈αof␈αsatisfying
␈↓ α,␈↓␈↓ β≤some␈α
important␈αgoal.␈α
 Finally,␈αthe␈α
human␈α
watching␈αAM␈α
would␈αbe␈α
able␈α
to␈αobserve
␈↓ α,␈↓␈↓ β≤the␈α∪rationality␈α∪(hopefully)␈α∪of␈α∩the␈α∪goals␈α∪which␈α∪were␈α∩set.␈α∪The␈α∪simple␈α∪"Focus␈α∩of
␈↓ α,␈↓␈↓ β≤Attention"␈αmechanism␈αalready␈αin␈αAM␈α
is␈αa␈αtiny␈αstep␈αin␈αthis␈α
goal-oriented␈αdirection.
␈↓ α,␈↓␈↓ β≤Note␈αthat␈αthis␈αproposal␈αitself␈αdemonstrates␈αthat␈αAM␈αis␈αnot␈αinherently␈αopposed␈αto␈αa
␈↓ α,␈↓␈↓ β≤goal-directed␈α∂control␈α∞structure.␈α∂Rather,␈α∂AM␈α∞simply␈α∂possesses␈α∂only␈α∞a␈α∂partial␈α∂set␈α∞of
␈↓ α,␈↓␈↓ β≤mechanisms for complete reasoning about its domain.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.1.10. Comparison to Other Systems␈↓)αβ␈↓

␈↓ α,␈↓One␈α∂popular␈α∂way␈α∞to␈α∂judge␈α∂a␈α∞system␈α∂is␈α∂to␈α∞compare␈α∂it␈α∂to␈α∞other,␈α∂similar␈α∂systems,␈α∂and/or␈α∞to
␈↓ α,␈↓others'␈αproposed␈αcriteria␈α
for␈αsuch␈αsystems.␈α
There␈αis␈αno␈αother␈α
project␈α(known␈αto␈α
the␈αauthor)
␈↓ α,␈↓having␈αthe␈α
same␈αobjective:␈αautomated␈α
math␈αresearch.␈↓	15␈↓␈αMany␈α
somewhat␈αrelated␈αe≥orts␈α
have
␈↓ α,␈↓been reported in the literature and will be mentioned here.

␈↓ α,␈↓Several␈α
projects␈α
have␈αbeen␈α
undertaken␈α
which␈αoverlap␈α
small␈α
pieces␈α
of␈αthe␈α
AM␈α
system␈αand␈α
in
␈↓ α,␈↓addition␈α
concentrate␈α
deeply␈α∞upon␈α
some␈α
area␈α∞␈↓βnot␈↓␈α
present␈α
in␈α∞AM.␈α
 For␈α
example,␈α∞the␈α
CLET

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	15␈↓ε In [Atkin & Birch 1971], e.g., we find no mention of the computer except as a number cruncher.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε116␈↓-

␈↓ α,␈↓system␈α→[Badre␈α→73]␈α→worked␈α→on␈α~learning␈α→the␈α→decimal␈α→addition␈α→algorithm␈↓	16␈↓␈α~but␈α→the
␈↓ α,␈↓"␈↓βmathematics␈α→discovery␈↓"␈α→aspects␈α→of␈α→that␈α_system␈α→were␈α→neither␈α→emphasized␈α→nor␈α_worth
␈↓ α,␈↓emphasizing;␈α∀it␈α∀was␈α∀an␈α∀interesting␈α∪natural␈α∀language␈α∀communication␈α∀study.␈α∀ The␈α∪same
␈↓ α,␈↓comment applies to several related studies by IMSSS␈↓	17␈↓.

␈↓ α,␈↓Boyer␈α
and␈αMoore's␈α
theorem-prover␈α[Boyer&Moore␈α
75]␈αembodies␈α
some␈αof␈α
the␈αspirit␈α
of␈αAM
␈↓ α,␈↓(e.g.,␈αgeneralizing␈αthe␈αde≡nition␈αof␈αa␈αLISP␈αfunction),␈αbut␈αits␈αmotivations␈αare␈αquite␈αdi≥erent,
␈↓ α,␈↓its␈αknowledge␈αbase␈αis␈αminimal,␈αand␈αits␈αmethods␈αpurely␈αformal.␈↓	18␈↓␈αThe␈αsame␈αcomments␈αapply
␈↓ α,␈↓to␈α
the␈α
SAM␈α
program␈α∞[Guard␈α
69],␈α
in␈α
which␈α
a␈α∞resolution␈α
theorem-prover␈α
is␈α
set␈α
to␈α∞work␈α
on
␈↓ α,␈↓unsolved problems in lattice theory.

␈↓ α,␈↓Among␈α∞the␈α
attempts␈α∞to␈α
incorporate␈α∞heuristic␈α
knowledge␈α∞into␈α
a␈α∞theorem␈α
prover,␈α∞we␈α
should
␈↓ α,␈↓also␈α
mention␈α
[Wang␈α60],␈α
[Pitrat␈α
70],␈α
[Bledsoe␈α71],␈α
and␈α
[Brotz␈α
74].␈α How␈α
did␈α
AM␈αdi≥er␈α
from
␈↓ α,␈↓these␈α∞"heuristic␈α∞theorem-provers"?␈α∞ The␈α∞goal-driven␈α∞control␈α∞structure␈α∞of␈α∞these␈α∞systems␈α∞is␈α∞a
␈↓ α,␈↓real␈αbut␈αonly␈α
minor␈αdi≥erence␈αfrom␈αAM's␈α
control␈αstructure␈α(e.g.,␈α
AM's␈α"focus␈αof␈αattention"␈α
is
␈↓ α,␈↓a␈α⊂rudimentary␈α⊂step␈α∂in␈α⊂that␈α⊂direction;␈α∂see␈α⊂p.␈α⊂115).␈α∂ The␈α⊂fact␈α⊂that␈α∂their␈α⊂overall␈α⊂activity␈α∂is
␈↓ α,␈↓typically␈α∂labelled␈α∂as␈α∂deductive␈α∂is␈α⊂also␈α∂not␈α∂a␈α∂fundamental␈α∂distinction␈α∂(since␈α⊂constructing␈α∂a
␈↓ α,␈↓proof␈αis␈α
usually␈αin␈α
practice␈αquite␈α
␈↓&␈↓βin␈↓␈↓)αβductive).␈α Even␈α
the␈αcharacter␈α
of␈αthe␈α
inference␈αprocesses
␈↓ α,␈↓are␈α∂analogous:␈α∞The␈α∂provers␈α∞typically␈α∂contain␈α∂a␈α∞couple␈α∂binary␈α∞inference␈α∂rules,␈α∂like␈α∞Modus
␈↓ α,␈↓Ponens,␈α∂which␈α∂are␈α∂relatively␈α∂risky␈α∂to␈α∂apply␈α∂but␈α∂can␈α∂yield␈α∂big␈α∂results;␈α∂AM's␈α⊂few␈α∂"binary"
␈↓ α,␈↓operators␈αhave␈αthe␈αsame␈αcharacteristics:␈αCompose,␈αCanonize,␈αLogically-combine␈α(disjoin␈αand
␈↓ α,␈↓conjoin).␈α The␈α
main␈αdistinction␈αis␈α
that␈αthe␈α
theorem␈αprovers␈αeach␈α
incorporate␈αonly␈αa␈α
handful
␈↓ α,␈↓of␈αheuristics.␈αThe␈αreason␈αfor␈α
this,␈αin␈αturn,␈αis␈αthe␈α
paucity␈αof␈αgood␈αheuristics␈αwhich␈α
exist␈αfor
␈↓ α,␈↓the␈α
very␈α
general␈α
task␈α
environment␈α
in␈α
which␈α
they␈α
operate:␈α∞domain-independent␈α
(asemantic)
␈↓ α,␈↓predicate␈α∞calculus␈α
theorem␈α∞proving.␈α
 The␈α∞need␈α
for␈α∞additional␈α
guidance␈α∞was␈α∞recognized␈α
by
␈↓ α,␈↓these researchers.  For example, see [Wang 60], p. 3 and p. 17. Or as Bledsoe says␈↓	19␈↓:

␈↓ α,␈↓α␈↓ α|There␈α⊗is␈α⊗a␈α⊗real␈α⊗difference␈α⊗between␈α⊗␈↓βdoing␈↓α␈α⊗some␈α⊗mathematics␈α⊗and␈α↔␈↓βbeing␈↓α␈α⊗a
␈↓ α,␈↓α␈↓ α|mathematician.␈αThe␈αdifference␈αis␈αprincipally␈αone␈αof␈αjudgment:␈αin␈αthe␈αselection␈αof␈αa
␈↓ α,␈↓α␈↓ α|problem␈α(theorem␈αto␈αbe␈αproved);␈αin␈αdetermining␈αits␈αrelevance;...␈α It␈αis␈αprecisely␈αin
␈↓ α,␈↓α␈↓ α|these␈α
areas␈αthat␈α
machine␈αprovers␈α
have␈αbeen␈α
so␈αlacking.␈α
This␈αkind␈α
of␈αjudgment␈α
has
␈↓ α,␈↓α␈↓ α|to␈α∞be␈α∞supplied␈α∞by␈α∞the␈α∞user...␈α∞Thus␈α∂a␈α∞crucial␈α∞part␈α∞of␈α∞the␈α∞resolution␈α∞proof␈α∂is␈α∞the
␈↓ α,␈↓α␈↓ α|␈↓βselection␈↓α␈α
of␈α
the␈α
reference␈α
theorems␈α
by␈α
the␈α
␈↓βhuman␈↓α␈α
user;␈α
the␈α
human,␈α
by␈α
this␈αone
␈↓ α,␈↓α␈↓ α|action, usually employs more skill than that used by the computer in the proof.

␈↓ α,␈↓Many␈αresearchers␈αhave␈αconstructed␈αprograms␈αwhich␈αpioneered␈αsome␈αof␈αthe␈α
techniques␈αAM
␈↓ α,␈↓uses.␈α∞ [Gelernter␈α∞63]␈α∞reports␈α∞the␈α∞use␈α
of␈α∞prototypical␈α∞examples␈α∞as␈α∞analogic␈α∞models␈α∞to␈α
guide
␈↓ α,␈↓search␈α
in␈α∞geometry,␈α
and␈α∞[Bundy␈α
73]␈α∞employs␈α
models␈α
of␈α∞"sticks"␈α
to␈α∞help␈α
his␈α∞program␈α
work
␈↓ α,␈↓with␈α⊃natural␈α⊂numbers.␈α⊃ The␈α⊃single␈α⊂heuristic␈α⊃of␈α⊂analogy␈α⊃was␈α⊃studied␈α⊂in␈α⊃[Evans␈α⊃68]␈α⊂and
␈↓ α,␈↓[Kling 71].  Brotz's program, [Brotz 74], uses this to propose useful lemmata.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	16␈↓ε␈αGiven␈α
the␈αaddition␈α
table␈αup␈α
to␈α10␈α
+␈α10,␈α
plus␈αan␈α
English␈αtext␈α
description␈αof␈α
what␈αit␈α
means␈αto␈α
carry,␈αhow␈α
and␈αwhen␈α
to
␈↓ α,␈↓ε␈↓ βLcarry, etc., actually write a program capable of adding two 3-digit numbers
␈↓ α,␈↓ε␈↓	17␈↓ε See [Smith 74a], for example.
␈↓ α,␈↓ε␈↓	18␈↓ε␈αλThis␈αλis␈αλnot␈αλmeant␈αλas␈αλcriticism;␈αλconsidering␈αλthe␈αλgoals␈αλof␈αλthose␈αλresearchers,␈αλand␈αλthe␈αλage␈αλof␈αλthat␈αλsystem,␈αλtheir␈αλwork␈α	is␈αλquite
␈↓ α,␈↓ε␈↓ βLsignificant.
␈↓ α,␈↓ε␈↓	19␈↓ε [Bledsoe 71], p. 73
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε117␈↓-

␈↓ α,␈↓Theory␈α∀formation␈α∃systems␈α∀in␈α∀␈↓βany␈↓␈α∃≡eld␈α∀have␈α∀been␈α∃few.␈α∀Meta-Dendral␈α∃[Buchanan␈α∀74]
␈↓ α,␈↓represents␈α⊂perhaps␈α⊂the␈α⊃best␈α⊂of␈α⊂these.␈α⊃ Its␈α⊂task␈α⊂is␈α⊃to␈α⊂unify␈α⊂a␈α⊃body␈α⊂of␈α⊂mass␈α⊃spectral␈α⊂data
␈↓ α,␈↓(examples␈α⊃of␈α⊃"proper"␈α⊃identi≡cations␈α⊃of␈α⊃spectra)␈α∩into␈α⊃a␈α⊃small␈α⊃body␈α⊃of␈α⊃rules␈α∩for␈α⊃making
␈↓ α,␈↓identi≡cations.␈α⊂ Thus␈α⊂even␈α⊂this␈α⊂system␈α⊃is␈α⊂given␈α⊂a␈α⊂≡xed␈α⊂task,␈α⊃a␈α⊂≡xed␈α⊂set␈α⊂of␈α⊂data␈α⊃to␈α⊂≡nd
␈↓ α,␈↓regularities␈α∞within.␈α∞ AM,␈α∞however,␈α
must␈α∞≡nd␈α∞its␈α∞own␈α
data,␈α∞and␈α∞take␈α∞the␈α∞responsibility␈α
for
␈↓ α,␈↓managing␈αits␈αown␈αtime,␈αfor␈αnot␈αlooking␈αtoo␈αlong␈αat␈αworthless␈αdata.␈↓	20␈↓␈αThere␈αhas␈αbeen␈αmuch
␈↓ α,␈↓written␈α∞about␈α∞scienti≡c␈α∞theory␈α∞formation␈α∞(e.g.,␈α∞[Hempel␈α
52]),␈α∞but␈α∞very␈α∞little␈α∞of␈α∞it␈α∞is␈α
speci≡c
␈↓ α,␈↓enough␈αto␈αbe␈αof␈αimmediate␈αuse␈αto␈αAI␈αresearchers.␈αA␈αcouple␈αpointers␈αto␈αexcellent␈αdiscussions
␈↓ α,␈↓of␈α∩this␈α∩sort␈α∩are:␈α∪[Fogel␈α∩66],␈α∩[Simon␈α∩73],␈α∩and␈α∪[Buchanan␈α∩75].␈α∩ Also␈α∩worth␈α∩noting␈α∪is␈α∩a
␈↓ α,␈↓discussion␈αnear␈αthe␈αend␈αof␈α[Amarel␈α69],␈αin␈αwhich␈α"formation"␈αand␈α"modelling"␈αproblems␈α
are
␈↓ α,␈↓treated:
␈↓ α,␈↓α␈↓ α|The␈α
problem␈α
of␈α
model␈αfinding␈α
is␈α
related␈α
to␈αthe␈α
following␈α
general␈α
question␈αraised
␈↓ α,␈↓α␈↓ α|by␈αSchutzenberger␈α[In␈αdiscussion␈αat␈αthe␈αConference␈αon␈αIntelligence␈αand␈αIntelligent
␈↓ α,␈↓α␈↓ α|Systems,␈αAthens,␈α
Ga.,␈α1967]:␈α␈↓β`What␈α
do␈αwe␈α
want␈αto␈αdo␈α
with␈αintelligent␈αsystems␈α
that
␈↓ α,␈↓β␈↓ α|relates␈α
to␈αthe␈α
work␈α
of␈αmathematicians?'␈↓α.␈α
 So␈α
far␈αall␈α
we␈α
have␈αdone␈α
in␈α
this␈αgeneral
␈↓ α,␈↓α␈↓ α|area␈α⊂is␈α∂to␈α⊂emulate␈α⊂some␈α∂of␈α⊂the␈α⊂reasonably␈α∂simple␈α⊂activities␈α⊂of␈α∂mathematicians,
␈↓ α,␈↓α␈↓ α|which␈α_is␈α_finding␈α_consequences␈α↔from␈α_given␈α_assumptions,␈α_reasoning,␈α↔proving
␈↓ α,␈↓α␈↓ α|theorems.␈α∪A␈α∪certain␈α∪amount␈α∪of␈α∪work␈α∀of␈α∪this␈α∪type␈α∪was␈α∪already␈α∪done␈α∀in␈α∪the
␈↓ α,␈↓α␈↓ α|propositional␈α
and␈αpredicate␈α
calculi,␈α
as␈αwell␈α
as␈α
in␈αsome␈α
other␈αmathematical␈α
systems.
␈↓ α,␈↓α␈↓ α|But this is only one aspect of the work that goes on in mathematics.
␈↓ α,␈↓α␈↓ α|Another␈α∀very␈α∀important␈α∀aspect␈α∀is␈α∀the␈α∀one␈α∀of␈α∀finding␈α∀general␈α∀properties␈α∪of
␈↓ α,␈↓α␈↓ α|structures,␈α⊂finding␈α⊂analogies,␈α⊂similarities,␈α⊃isomorphisms,␈α⊂and␈α⊂so␈α⊂on.␈α⊂ This␈α⊃is␈α⊂the
␈↓ α,␈↓α␈↓ α|type␈α⊃of␈α⊃activity␈α⊃that␈α⊃is␈α∩extremely␈α⊃important␈α⊃for␈α⊃our␈α⊃understanding␈α∩of␈α⊃model-
␈↓ α,␈↓α␈↓ α|finding␈αmechanisms.␈αWork␈αin␈αthis␈αarea␈αis␈αmore␈αdifficult␈αthan␈αtheorem-proving.␈αThe
␈↓ α,␈↓α␈↓ α|problem here is that of ␈↓&theorem finding.␈↓)αβ
␈↓ α,␈↓AM␈αis␈αone␈αof␈αthe␈α≡rst␈αattempts␈αto␈αconstruct␈αa␈α"theorem-≡nding"␈αprogram.␈αAs␈αAmarel␈αnoted,
␈↓ α,␈↓it␈α
may␈α
be␈α
possible␈α
to␈α
learn␈α
from␈αsuch␈α
programs␈α
how␈α
to␈α
tackle␈α
the␈α
general␈α
task␈αof␈α
automating
␈↓ α,␈↓scienti≡c research.

␈↓ α,␈↓Besides␈α"math␈αsystems",␈αand␈α"creative␈αthinking␈αsystems",␈αand␈α"theory␈αformation␈αsystems",␈αwe
␈↓ α,␈↓should␈αat␈αleast␈αdiscuss␈αothers'␈αthoughts␈αon␈αthe␈αissue␈αof␈αalgorithmically␈αdoing␈αmath␈αresearch.
␈↓ α,␈↓Some␈αindividuals␈αfeel␈α
it␈αis␈αnot␈αso␈α
far-fetched␈αto␈αimagine␈αautomating␈α
mathematical␈αresearch
␈↓ α,␈↓(e.g.,␈α
Paul␈αCohen).␈α
 Others␈α
(e.g.,␈αPolya)␈α
would␈αprobably␈α
disagree.␈α
 The␈αpresence␈α
of␈α
a␈αhigh-
␈↓ α,␈↓speed,␈α∞general-purpose␈α∂symbol␈α∞manipulator␈α∞in␈α∂our␈α∞midst␈α∞now␈α∂makes␈α∞investigation␈α∂of␈α∞that
␈↓ α,␈↓question possible.

␈↓ α,␈↓There␈α⊃has␈α⊃been␈α⊃very␈α⊃little␈α⊃published␈α⊃thought␈α⊃about␈α⊃discovery␈α⊃in␈α⊃mathematics␈α∩from␈α⊃an
␈↓ α,␈↓algorithmic␈α
point␈α
of␈α
view;␈α
even␈α
clear␈α
thinkers␈α
like␈α
Polya␈α
and␈α
Poincare'␈α
treat␈αmathematical
␈↓ α,␈↓ability␈α∃as␈α∃a␈α∃sacred,␈α∃almost␈α∃mystic␈α∃quality,␈α∃tied␈α∃to␈α∃the␈α∃unconscious.␈α∃ The␈α∃writings␈α∀of
␈↓ α,␈↓philosophers␈α∩and␈α∩psychologists␈α∩invariably␈α⊃attempt␈α∩to␈α∩examine␈α∩human␈α∩performance␈α⊃and
␈↓ α,␈↓belief,␈αwhich␈αare␈αfar␈αmore␈αmanagable␈αthan␈αcreativity␈α␈↓βin␈αvitro␈↓.␈α Belief␈αformulae␈αin␈αinductive
␈↓ α,␈↓logic␈↓	21␈↓␈αinvariably␈αfall␈αback␈α
upon␈αhow␈αwell␈αthey␈α≡t␈α
human␈αmeasurements.␈α The␈αabilities␈αof␈α
a
␈↓ α,␈↓computer␈α⊂and␈α⊂a␈α⊂brain␈α⊂are␈α⊂too␈α⊂distinct␈α⊂to␈α⊂consider␈α⊂blindly␈α⊂working␈α⊂for␈α⊂results␈α⊂(let␈α⊂alone
␈↓ α,␈↓algorithms!) one possesses which match those of the other.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	20␈↓ε␈αIn␈αcase␈αthat␈αwasn't␈αclear:␈αMeta-Dendral␈αhas␈αa␈αfixed␈αset␈αof␈αtemplates␈αfor␈αrules␈αwhich␈αit␈αwishes␈αto␈αfind,␈αand␈αa␈αfixed
␈↓ α,␈↓ε␈↓ βLvocabulary␈α
of␈α
mass␈α
spectral␈α
concepts␈α
which␈α
can␈α
be␈α
plugged␈α
into␈α
those␈α
templates.␈α
AM␈α
also␈α
has␈α
only␈α
a␈α
few
␈↓ α,␈↓ε␈↓ βLstock formats for conjectures, but it selectively enlarges its vocabulary of math concepts.
␈↓ α,␈↓ε␈↓	21␈↓ε For example, see [Hintikka 62], [Pietarinin 72]. The latter also contains a good summary of Carnap's λ,α formalization.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε118␈↓-

␈↓ α,␈↓␈↓ ∧*␈↓∧␈↓&7.2. Capabilities and Limitations of AM␈↓)αβ␈↓


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.2.1. Current Abilities␈↓)αβ␈↓

␈↓ α,␈↓␈↓βWhat␈α≡elds␈αhas␈αAM␈αworked␈αin␈αso␈αfar?␈↓␈αAM␈αis␈αnow␈αable␈αto␈αexplore␈αa␈αsmall␈αbit␈αof␈αthe␈αtheory
␈↓ α,␈↓of␈αsets,␈αdata␈αtypes,␈αnumbers,␈αand␈αplane␈αgeometry.␈α It␈α
by␈αno␈αmeans␈αhas␈αbeen␈αfed␈α¬␈αnor␈αhas␈α
it
␈↓ α,␈↓rediscovered␈α
¬␈α
a␈α
large␈α
fraction␈α
of␈α
what␈α
is␈α
known␈α
in␈α
any␈α
of␈α
those␈α
≡elds.␈α
It␈α
might␈α∞be␈α
more
␈↓ α,␈↓accurate␈α⊂to␈α⊂be␈α⊂humble␈α⊂and␈α⊂restate␈α⊂those␈α⊂domains␈α⊂as:␈α⊂elementary␈α⊂≡nite␈α⊂set␈α⊂theory,␈α∂trivial
␈↓ α,␈↓observations␈α
about␈α∞four␈α
kinds␈α
of␈α∞data␈α
types,␈α
arithmetic␈α∞and␈α
elementary␈α∞divisibility␈α
theory,
␈↓ α,␈↓and␈αsimple␈αrelationships␈αbetween␈αlines,␈α
angles,␈αand␈αtriangles.␈α So␈αa␈αsophisticated␈α
concept␈αin
␈↓ α,␈↓each domain ¬ which was discovered by AM ¬ might be:
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ de Morgan's laws
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ the fact that Delete␈↓εo␈↓Insert␈↓	22␈↓ never alters Bags or Lists
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ unique factorization
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ similar triangles

␈↓ α,␈↓␈↓βCan␈α
AM␈α
work␈α
in␈α
a␈α
new␈α
≡eld,␈α
like␈α
politics?␈↓␈α
AM␈α
can␈α
work␈α
in␈α
a␈α
new␈α
elementary,␈α
formalized
␈↓ α,␈↓domain,␈αif␈αit␈αis␈αfed␈αa␈αsupplemental␈α
base␈αof␈αconceptual␈αprimitives␈αfor␈αthat␈αdomain.␈α To␈α
work
␈↓ α,␈↓in␈αplane␈αgeometry,␈αit␈αsu≠ced␈αto␈αgive␈αAM␈αabout␈αtwenty␈αnew␈αprimitive␈αconcepts,␈αeach␈αwith␈αa
␈↓ α,␈↓few␈αparts␈α≡lled␈αin.␈αOther␈αdomains␈αwhich␈αAM␈αcould␈αwork␈αin␈αinclude␈αelementary␈αmechanics,
␈↓ α,␈↓game-playing,␈α⊗and␈α↔programming.␈α⊗The␈α↔more␈α⊗informal␈α↔(non-internally-formalizable)␈α⊗the
␈↓ α,␈↓desired␈α≡eld,␈αthe␈αless␈αof␈αAM␈αthat␈αis␈αrelevant.␈αPerhaps␈αan␈αAM-like␈αsystem␈αcould␈αbe␈αbuilt␈αfor
␈↓ α,␈↓a␈α
constrained,␈αprecise␈α
political␈αtask.␈↓	23␈↓␈α
Disclaimer:␈αEven␈α
for␈αa␈α
very␈αsmall␈α
domain,␈αthe␈α
amount
␈↓ α,␈↓of␈α∞common-sense␈α∞knowledge␈α∞such␈α∞a␈α∞system␈α∞would␈α∞need␈α∞is␈α∞staggering.␈α∞ It␈α∞is␈α∞unfortunate␈α∞to
␈↓ α,␈↓provide␈α∞such␈α
a␈α∞trivial␈α
answer␈α∞to␈α
such␈α∞an␈α∞important␈α
question,␈α∞but␈α
there␈α∞is␈α
no␈α∞easy␈α∞way␈α
to
␈↓ α,␈↓answer it more fully until years of additional research are performed.

␈↓ α,␈↓␈↓βCan␈αAM␈αdiscover␈αX?␈αWhy␈αdidn't␈αit␈αdo␈αY?␈↓␈αIt␈αis␈αdi≠cult␈αto␈αpredict␈αwhether␈αAM␈αwill␈α(without
␈↓ α,␈↓modi≡cations)␈α
ever␈αmake␈α
a␈αspeci≡c␈α
given␈α
discovery.␈α Although␈α
its␈αcapabilities␈α
are␈α
small,␈αits
␈↓ α,␈↓limitations␈α
are␈α
hazy.␈α
What␈α
makes␈α
the␈α
matter␈α
even␈α
worse␈α
is␈α
that,␈α
given␈α
a␈α
concept␈α
C␈α
which
␈↓ α,␈↓AM␈αmissed␈αdiscovering,␈αthere␈αis␈αprobably␈αa␈αreasonable␈αheuristic␈αrule␈αwhich␈αis␈αmissing␈α
from
␈↓ α,␈↓AM,␈αwhich␈αwould␈αenable␈αthat␈αdiscovery.␈αOne␈αdanger␈αof␈αthis␈α"debugging"␈αis␈αthat␈αa␈αrule␈αwill
␈↓ α,␈↓be␈α
added␈α
which␈α
only␈α
leads␈αto␈α
that␈α
one␈α
desired␈α
discovery,␈αand␈α
isn't␈α
good␈α
for␈α
anything␈αelse.␈α
In
␈↓ α,␈↓that␈α∃case,␈α∃the␈α∃new␈α∃heuristic␈α∃rule␈α∃would␈α⊗simply␈α∃be␈α∃an␈α∃␈↓βencoding␈↓␈α∃of␈α∃a␈α∃speci≡c␈α⊗bit␈α∃of
␈↓ α,␈↓mathematics␈αwhich␈α
AM␈αwould␈α
then␈αappear␈αto␈α
discover␈αusing␈α
general␈αmethods.␈α
 This␈αmust
␈↓ α,␈↓be␈αavoided␈αat␈αall␈αcosts,␈α
␈↓βeven␈αat␈αthe␈αcost␈αof␈αintentionally␈α
giving␈αup␈αa␈αcertain␈αdiscovery.␈↓␈α
If␈αthe
␈↓ α,␈↓needed␈α
rule␈α
is␈α
general␈α
¬␈α
it␈α
has␈α∞many␈α
applications␈α
and␈α
leads␈α
to␈α
many␈α
interesting␈α∞results␈α
¬
␈↓ α,␈↓then␈αit␈αreally␈αwas␈αan␈αoversight␈αnot␈αto␈αinclude␈αit␈αin␈αAM.␈α Although␈αI␈αbelieve␈αthat␈α
there␈αare
␈↓ α,␈↓not␈αtoo␈αmany␈αsuch␈αomissions␈αstill␈αwithin␈αthe␈αsmall␈αrealm␈αAM␈αexplores,␈αthere␈αis␈αno␈αobjective
␈↓ α,␈↓way to demonstrate that, except by further long tests with AM.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	22␈↓ε Take an item x, insert it into (the front of) structure B, then delete one (the first) occurrence of x from B.
␈↓ α,␈↓ε␈↓	23␈↓ε␈αλFor␈α	example,␈αλsuch␈αλa␈α	politics-oriented␈αλAM-like␈αλsystem␈α	might␈αλconceive␈α	the␈αλnotion␈αλof␈α	a␈αλgroup␈αλof␈α	political␈αλentities␈α	which␈αλview
␈↓ α,␈↓ε␈↓ βLthemselves␈α	as␈α	quite␈α	disparate,␈α	but␈α	which␈αλare␈α	viewed␈α	from␈α	the␈α	outside␈α	as␈αλa␈α	single␈α	unit:␈α	e.g.,␈α	`the␈α	Arabs',␈αλ`the
␈↓ α,␈↓ε␈↓ βLAmerican␈αIndians'.␈αConjectures␈αabout␈αthis␈αconcept␈αmight␈αinclude␈αits␈αreputation␈αas␈αa␈αpoor␈αcombatant␈α(and
␈↓ α,␈↓ε␈↓ βLwhy). Many of the same facets AM uses would carry over to represent concepts in that new domain.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε119␈↓-

␈↓ α,␈↓␈↓βIn␈α
what␈α
ways␈α
are␈α
new␈α
concepts␈α
created?␈↓␈αAlthough␈α
the␈α
answer␈α
to␈α
this␈α
is␈α
accurately␈α
given␈αin
␈↓ α,␈↓Section␈α
4.3,␈α∞page␈α
41␈α∞(namely,␈α
this␈α∞is␈α
mainly␈α∞the␈α
jurisdiction␈α∞of␈α
the␈α∞right␈α
sides␈α∞of␈α
heuristic
␈↓ α,␈↓rules),␈α∂and␈α∞although␈α∂I␈α∞dislike␈α∂the␈α∂simple-minded␈α∞way␈α∂it␈α∞makes␈α∂AM␈α∞sound,␈α∂the␈α∂list␈α∞below
␈↓ α,␈↓does characterize the major ways in which new concepts get born:
␈↓ α,␈↓¬␈↓ αlFill in examples of a concept (e.g., by instantiating or running its definition)
␈↓ α,␈↓¬␈↓ αlCreate a generalization of a given concept (e.g., by weakening its definition)
␈↓ α,␈↓¬␈↓ αlCreate a specialization of a given concept (e.g., by restricting its domain/range)
␈↓ α,␈↓¬␈↓ αlCompose two operations f,g, thereby creating a new one h.  [Define h(x)≡f(g(x))]
␈↓ α,␈↓¬␈↓ αlCoalesce an operation f into a new one g. [Define g(x)≡f(x,x)]
␈↓ α,␈↓¬␈↓ αlPermute the order of the arguments of an operation. [Define g(x,y)≡f(y,x)]
␈↓ α,␈↓¬␈↓ αlInvert an operation [g(x)=y iff f(y)=x] (e.g., from Squaring, create Square-rooting)
␈↓ α,␈↓¬␈↓ αlCanonize␈αone␈αpredicate␈αP1␈αwith␈αrespect␈αto␈αa␈αmore␈αgeneral␈αone␈αP2␈α[create␈αa␈αnew␈αconcept␈αf,
␈↓ α,␈↓¬␈↓ β≤an operation, such that: P2(x,y) iff P1(f(x),f(y))]
␈↓ α,␈↓¬␈↓ αlCreate a new operation g, which is the repeated application of an existing operation f.
␈↓ α,␈↓¬␈↓ αlThe usual logical combinations of existing concepts x,y: x∧y, x∨y, ¬x, etc.

␈↓ α,␈↓Below is a similar list, giving the primary ways in which AM formulates new conjectures:
␈↓ α,␈↓¬␈↓ αlNotice that concept C1 is really an example of concept C2
␈↓ α,␈↓¬␈↓ αlNotice that concept C1 is really a specialization (or: generalization) of C2
␈↓ α,␈↓¬␈↓ αlNotice that C1 is equal to C2; or: ␈↓βalmost always␈↓¬ equal
␈↓ α,␈↓¬␈↓ αlNotice that C1 and C2 are related by some known concept
␈↓ α,␈↓¬␈↓ αlCheck and update the domain/range of an existing operation
␈↓ α,␈↓¬␈↓ αlIf two concepts are analogous, extend the analogy to their conjectures as well


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.2.2. Current Limitations␈↓)αβ␈↓

␈↓ α,␈↓Below␈αare␈αseveral␈αshortcomings␈αof␈αAM,␈αwhich␈αhurt␈αits␈αbehavior␈αbut␈αare␈αnot␈αbelieved␈αto␈αbe
␈↓ α,␈↓inherent limitations of its design. They are presented in order of decreasing severity.

␈↓ α,␈↓Perhaps␈α∩the␈α⊃most␈α∩serious␈α⊃limitation␈α∩on␈α⊃AM's␈α∩current␈α⊃behavior␈α∩arose␈α⊃from␈α∩the␈α∩lack␈α⊃of
␈↓ α,␈↓constraints␈αon␈αleft␈αsides␈αof␈αheuristic␈αrules.␈αIt␈αturned␈αout␈αthat␈αthis␈αexcessive␈αfreedom␈αmade␈αit
␈↓ α,␈↓di≠cult␈α
for␈α
AM␈αto␈α
inspect␈α
and␈αanalyze␈α
and␈α
synthesize␈αits␈α
own␈α
heuristics;␈αsuch␈α
a␈α
need␈αwas
␈↓ α,␈↓not␈α
foreseen␈α∞at␈α
the␈α∞time␈α
AM␈α∞was␈α
designed.␈α∞ It␈α
was␈α∞thought␈α
that␈α∞the␈α
power␈α∞to␈α
manipulate
␈↓ α,␈↓heuristic␈α
rules␈α
was␈α
an␈α
ability␈α
which␈α
the␈α
author␈α
must␈α
have,␈α
but␈α
which␈α
the␈α
system␈α
wouldn't
␈↓ α,␈↓require.␈α
 As␈αit␈α
turned␈α
out,␈αAM␈α
did␈αsuccessfully␈α
develop␈α
new␈αconcepts␈α
several␈α
levels␈αdeeper
␈↓ α,␈↓than␈α
the␈α
ones␈α
it␈α
started␈α
with.␈α
But␈α
as␈α
the␈α
new␈α
concepts␈α
got␈α
further␈α
and␈α
further␈α
away␈α
from
␈↓ α,␈↓those␈αinitial␈αones,␈αthey␈αhad␈αfewer␈αand␈αfewer␈αspeci≡c␈αheuristics␈α≡lled␈αin␈α(since␈αthey␈αhad␈αto␈αbe
␈↓ α,␈↓≡lled␈α
in␈α
by␈α
AM␈α
itself).␈α
 Gradually,␈α
AM␈α
found␈α
itself␈α
relying␈α
on␈α
heuristics␈α
which␈α
were␈α
very
␈↓ α,␈↓general␈α∞compared␈α∂to␈α∞the␈α∂concepts␈α∞it␈α∂was␈α∞dealing␈α∞with␈α∂(e.g.,␈α∞forced␈α∂to␈α∞use␈α∂heuristics␈α∞about
␈↓ α,␈↓Objects␈α
when␈α
dealing␈α
with␈α
Numbers).␈α
 Heuristics␈αfor␈α
dealing␈α
with␈α
heuristics␈α
do␈α
exist,␈αand
␈↓ α,␈↓their␈αnumber␈αcould␈αbe␈αincreased.␈α This␈αis␈αnot␈αan␈αeasy␈αjob:␈α≡nding␈αa␈αnew␈αmeta-heuristic␈αis␈αa
␈↓ α,␈↓tough␈α∂process.␈α∂ Heuristics␈α∂are␈α∂rarely␈α∂more␈α∂than␈α∂compiled␈α∂hindsight;␈α∂hence␈α∂it's␈α⊂di≠cult␈α∂to
␈↓ α,␈↓create␈α∂new␈α∂ones␈α∂"before␈α∂the␈α∂fact".␈α∂ Our␈α∂current␈α∂research␈α∂is␈α∂based␈α∂on␈α∂the␈α∂hypothesis␈α∞that
␈↓ α,␈↓heuristics␈αcan␈α
be␈αdiscovered␈α
and␈αreasoned␈α
about␈αin␈α
much␈αthe␈α
same␈αway␈α
that␈αmathematical
␈↓ α,␈↓operations␈α⊃can.␈α⊃ What␈α⊃this␈α⊂means␈α⊃is␈α⊃recoding␈α⊃each␈α⊂of␈α⊃AM's␈α⊃heuristics␈α⊃as␈α⊃a␈α⊂full-∨edged
␈↓ α,␈↓concept.␈α∞ This␈α∂would␈α∞enable␈α∞the␈α∂heuristics␈α∞to␈α∞work␈α∂on␈α∞each␈α∞other,␈α∂as␈α∞well␈α∞as␈α∂on␈α∞concepts
␈↓ α,␈↓corresponding to mathematical entities.

␈↓ α,␈↓AM␈α∀has␈α∀no␈α∀notion␈α∪of␈α∀proof,␈α∀proof␈α∀techniques,␈α∪formal␈α∀validity,␈α∀heuristics␈α∀for␈α∪≡nding
␈↓ α,␈↓counterexamples,␈αetc.␈α
 Thus␈αit␈α
never␈αreally␈α
establishes␈αany␈α
conjecture␈αformally.␈α
 This␈αcould
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε120␈↓-

␈↓ α,␈↓probably␈α⊂be␈α⊃remedied␈α⊂by␈α⊃adding␈α⊂about␈α⊃25␈α⊂new␈α⊃concepts␈α⊂(and␈α⊃their␈α⊂100␈α⊃new␈α⊂associated
␈↓ α,␈↓heuristics)␈αdealing␈αwith␈α
such␈αtopics.␈α The␈α
needed␈αconcepts␈αhave␈α
been␈αoutlined␈αon␈αpaper,␈α
but
␈↓ α,␈↓not␈α∞yet␈α
coded.␈α∞ It␈α∞would␈α
probably␈α∞require␈α
a␈α∞few␈α∞hundred␈α
hours␈α∞to␈α
code␈α∞and␈α∞debug␈α
them.
␈↓ α,␈↓[Lakatos␈α76]␈αdemonstrates␈αthe␈αvalue␈αof␈αproof␈αas␈αan␈αinstrument␈αof␈αdiscovery:␈αAs␈αone␈αspirals
␈↓ α,␈↓in␈α
from␈α
conjecture␈α
to␈αtheorem,␈α
along␈α
the␈α
path␈αof␈α
repeated␈α
criticism␈α
and␈α
improvement,␈αone
␈↓ α,␈↓often␈α∂≡nds␈α∂motivation␈α∂for␈α∂modifying␈α∞concepts'␈α∂de≡nitions,␈α∂or␈α∂for␈α∂de≡ning␈α∂new,␈α∞promising
␈↓ α,␈↓ones.

␈↓ α,␈↓The␈α
user␈α
interface␈α
is␈α
quite␈α
primitive,␈αand␈α
this␈α
again␈α
could␈α
be␈α
dramatically␈α
improved␈αwith
␈↓ α,␈↓just␈α
a␈α
couple␈α
hundred␈αhours'␈α
work.␈α
 AM's␈α
explanation␈αsystem␈α
is␈α
almost␈α
nonexistent:␈αthe␈α
user
␈↓ α,␈↓must␈αask␈αa␈αquestion␈αquickly,␈αor␈αAM␈αwill␈αhave␈αalready␈αdestroyed␈αthe␈αinformation␈αneeded␈αto
␈↓ α,␈↓construct␈αan␈αanswer.␈αA␈αclean␈αrecord␈αof␈αrecent␈αsystem␈αhistory␈αand␈αa␈αnice␈αscheme␈αfor␈αtracking
␈↓ α,␈↓down␈α
reasons␈α∞for␈α
modifying␈α
old␈α∞rules␈α
and␈α
adding␈α∞new␈α
ones␈α
dynamically␈α∞does␈α
not␈α∞exist␈α
at
␈↓ α,␈↓the␈αlevel␈α
which␈αis␈α
found,␈αe.g.,␈α
in␈αMYCIN␈α
[Davis␈α76].␈α
 There␈αis␈α
no␈αtrivial␈α
way␈αto␈α
have␈αthe
␈↓ α,␈↓system print out its heuristics in a format which is intelligible to the untrained user.

␈↓ α,␈↓An␈αimportant␈αtype␈αof␈αanalogy␈αwhich␈αwas␈αuntapped␈αby␈αAM␈αwas␈αthat␈αbetween␈αheuristics.␈α If
␈↓ α,␈↓two␈α⊃situations␈α⊃were␈α⊃similar,␈α⊃conceivably␈α∩the␈α⊃heuristics␈α⊃useful␈α⊃in␈α⊃one␈α⊃situation␈α∩might␈α⊃be
␈↓ α,␈↓useful␈α(or␈αhave␈αuseful␈α
analogues)␈αin␈αthe␈αnew␈α
situation␈α(see␈α[Koppelman␈α75]).␈α
 Perhaps␈αthis
␈↓ α,␈↓is␈αa␈αviable␈αway␈αof␈αenlarging␈αthe␈αknown␈αheuristics.␈α Such␈α"meta-level"␈αactivities␈αwere␈αkept␈αto
␈↓ α,␈↓a minimum throughout AM, and this proved to be a serious limitation.

␈↓ α,␈↓The␈α
idea␈α
of␈α
"Intuitions"␈α
facets␈α
was␈α
a␈α
∨op.␈α
 Intuitions␈α
were␈α
meant␈α
to␈α
model␈α
reality,␈α∞at␈α
least
␈↓ α,␈↓little␈α
pieces␈α
of␈α
it,␈α
so␈α
that␈α
AM␈α
could␈α
perform␈α
(simulate)␈α
physical␈α
experiments,␈α
and␈α
observe␈α
the
␈↓ α,␈↓results.␈αThe␈αmajor␈α
problem␈αhere␈αwas␈αthat␈α
␈↓βso␈↓␈αlittle␈αof␈αthe␈α
world␈αwas␈αmodelled␈αthat␈α
the␈αonly
␈↓ α,␈↓relationships␈α∩derivable␈α∩were␈α⊃those␈α∩foreseen␈α∩by␈α⊃the␈α∩author.␈α∩This␈α⊃lack␈α∩of␈α∩generality␈α⊃was
␈↓ α,␈↓unacceptable,␈α⊃and␈α∩the␈α⊃intuitions␈α∩were␈α⊃completely␈α∩excised.␈α⊃The␈α∩original␈α⊃idea␈α∩might␈α⊃lead
␈↓ α,␈↓somewhere if it were developed fully.

␈↓ α,␈↓Currently,␈α∪it␈α∪is␈α∪di≠cult␈α∀to␈α∪include␈α∪heuristics␈α∪which␈α∀interact␈α∪with␈α∪one␈α∪another␈α∀in␈α∪any
␈↓ α,␈↓signi≡cant␈α∂way.␈α∂The␈α∂whole␈α∂≡bre␈α∂of␈α∂the␈α∂Agenda␈α∂scheme␈α∂assumes␈α∂perfect␈α⊂independence␈α∂of
␈↓ α,␈↓heuristics.␈α_The␈α→global␈α_formula␈α→used␈α_to␈α_rate␈α→tasks␈α_on␈α→the␈α_agenda␈α→assumes␈α_perfect
␈↓ α,␈↓superposition␈α∪of␈α∀reasons:␈α∪there␈α∪are␈α∀no␈α∪"cross-terms".␈α∪ Is␈α∀this␈α∪assumption␈α∀always␈α∪valid?
␈↓ α,␈↓Unfortunately␈α⊃␈↓βno␈↓,␈α∩not␈α⊃even␈α⊃for␈α∩the␈α⊃limited␈α⊃domain␈α∩AM␈α⊃has␈α⊃explored.␈α∩ Sometimes,␈α⊃two
␈↓ α,␈↓reasons␈α
are␈α
very␈α
similar:␈α"Examples␈α
of␈α
Sets␈α
would␈αpermit␈α
≡nding␈α
examples␈α
of␈α
Union"␈αand
␈↓ α,␈↓"Examples␈α
of␈α∞Sets␈α
would␈α∞permit␈α
≡nding␈α∞examples␈α
of␈α∞Intersection".␈α
 In␈α∞that␈α
case,␈α∞their␈α
two
␈↓ α,␈↓ratings␈α∞shouldn't␈α
cause␈α∞such␈α∞a␈α
big␈α∞increase␈α
in␈α∞the␈α∞overall␈α
priority␈α∞value␈α
of␈α∞the␈α∞task␈α
"␈↓¬Fillin
␈↓ α,␈↓¬examples of Sets␈↓".

␈↓ α,␈↓There␈α∂are␈α∞a␈α∂few␈α∞other␈α∂minor␈α∞limitations␈α∂of␈α∞the␈α∂current␈α∞Agenda␈α∂scheme:␈α∞(i)␈α∂Sometimes,␈α∞a
␈↓ α,␈↓heuristic␈α
rule␈αwill␈α
want␈αto␈α
␈↓βdissuade␈↓␈αthe␈α
system␈αfrom␈α
some␈αactivity;␈α
so␈αoccasionally␈α
we'd␈αlike␈α
a
␈↓ α,␈↓␈↓βnegative␈↓␈α
numeric␈α
contribution␈α
to␈α
a␈α
task's␈α
priority␈α
value.␈α
 (ii)␈α
AM␈α
discards␈α
the␈α
formula␈α
which
␈↓ α,␈↓produced␈αa␈α
reason's␈αrating,␈α
and␈αkeeps␈α
around␈αonly␈αthe␈α
numeric␈αresult␈α
of␈αthat␈α
formula;␈αbut
␈↓ α,␈↓many␈αof␈αthose␈αformulae␈αinvolve␈αtime-dependent␈αquantities␈α(e.g.,␈αthe␈αnumber␈αof␈αentries␈αon␈αa
␈↓ α,␈↓particular␈α∞facet␈α∞of␈α∞a␈α∞particular␈α∞concept).␈α∞ (iii)␈α
Better␈α∞than␈α∞a␈α∞single␈α∞priority␈α∞value␈α∞for␈α
each
␈↓ α,␈↓task␈αwould␈αbe␈αfor␈αAM␈αto␈αmaintain␈αa␈αvector␈αof␈αnumbers.␈α(iv)␈αThe␈αagenda␈αlist␈αshould␈αreally
␈↓ α,␈↓be␈αan␈α
agenda␈α␈↓βtree␈↓␈α(maybe␈α
an␈αagenda␈α␈↓&Heap␈↓)αβ),␈α
since␈αthe␈α
ordering␈αof␈αtasks␈α
is␈αreally␈αjust␈α
partial,
␈↓ α,␈↓not␈α∞total.␈α
The␈α∞utility␈α
of␈α∞this␈α∞is␈α
extremely␈α∞high␈α
if␈α∞multiprocessor␈α
technology␈α∞is␈α∞available␈α
to
␈↓ α,␈↓run the program on.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε121␈↓-

␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.2.3. Limiting Assumptions␈↓)αβ␈↓

␈↓ α,␈↓AM␈α⊃only␈α⊃"got␈α⊃o≥␈α⊃the␈α⊃ground"␈α⊃because␈α⊃a␈α⊃number␈α⊃of␈α⊃sweeping␈α⊃assumptions␈α⊃were␈α⊃made,
␈↓ α,␈↓pertaining␈αto␈αwhat␈αcould␈αbe␈αignored,␈αhow␈αa␈αcomplex␈αprocess␈αcould␈αbe␈αadequately␈αsimulated,
␈↓ α,␈↓etc.␈α Now␈αthat␈αAM␈α␈↓βis␈↓␈αrunning,␈αhowever,␈αthose␈αsame␈αsimpli≡cations␈αcrop␈αup␈αas␈αlimitations␈αto
␈↓ α,␈↓the␈α
system's␈α
behavior.␈α
 Each␈αof␈α
the␈α
following␈α
points␈αis␈α
a␈α
`convenient␈α
falsehood'.␈α Although
␈↓ α,␈↓the␈α∞reader␈α∞has␈α∞already␈α∞been␈α∞told␈α∞about␈α∂some␈α∞of␈α∞these,␈α∞it's␈α∞worth␈α∞listing␈α∞them␈α∂all␈α∞together
␈↓ α,␈↓here:

␈↓ α,␈↓␈↓ αl␈↓π#␈α∩␈↓␈α∩The␈α∩only␈α∩communication␈α∩necessary␈α∪from␈α∩AM␈α∩to␈α∩the␈α∩user␈α∩is␈α∩keeping␈α∪the␈α∩user
␈↓ α,␈↓␈↓ β≤informed␈α∞of␈α∞what␈α∞AM␈α∞is␈α∞doing.␈α∞No␈α∞natural␈α∞language␈α∞ability␈α∞is␈α∞required␈α∞by␈α
AM;
␈↓ α,␈↓␈↓ β≤simple template instantiation is su≠cient.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αThe␈αonly␈αcommunication␈αfrom␈αthe␈αuser␈αto␈αAM␈αis␈αan␈αoccasional␈αinterrupt,␈αwhen␈αthe
␈↓ α,␈↓␈↓ β≤user␈α∞wishes␈α∞to␈α∞provide␈α∞some␈α∞guidance␈α∞or␈α∞to␈α∞pose␈α∞a␈α∞query.␈α∞ Both␈α∞of␈α∞these␈α∂can␈α∞be
␈↓ α,␈↓␈↓ β≤stereotyped and passed easily through a very narrow channel.␈↓	24␈↓

␈↓ α,␈↓␈↓ αl␈↓π#␈α∞␈↓␈α∂Each␈α∞heuristic␈α∂has␈α∞a␈α∂well-de≡ned␈α∞domain␈α∂of␈α∞applicability,␈α∂which␈α∞can␈α∂be␈α∞speci≡ed
␈↓ α,␈↓␈↓ β≤just by giving the name of a single concept.

␈↓ α,␈↓␈↓ αl␈↓π#␈α∩␈↓␈α∩If␈α∩concept␈α∩C1␈α∩is␈α∩more␈α∩specialized␈α∩than␈α∩C2,␈α∩then␈α∩C1's␈α∩heuristics␈α∩will␈α∪be␈α∩more
␈↓ α,␈↓␈↓ β≤powerful␈α∞and␈α∞should␈α∞be␈α∞executed␈α
before␈α∞C2's␈α∞(whenever␈α∞both␈α∞concepts'␈α
heuristics
␈↓ α,␈↓␈↓ β≤are relevant).

␈↓ α,␈↓␈↓ αl␈↓π#␈α∞␈↓␈α
If␈α∞h1␈α
and␈α∞h2␈α
are␈α∞two␈α∞heuristics␈α
attached␈α∞to␈α
concept␈α∞C,␈α
then␈α∞it␈α
is␈α∞not␈α∞necessary␈α
to
␈↓ α,␈↓␈↓ β≤spend any time ordering them.

␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ Heuristics superimpose perfectly; they never interact strongly with each other.

␈↓ α,␈↓␈↓ αl␈↓π#␈α
␈↓␈α
The␈α
reasons␈α
supporting␈αa␈α
task␈α
can␈α
be␈α
mere␈α
tokens;␈αit␈α
su≠ces␈α
to␈α
be␈α
able␈α
to␈αinspect
␈↓ α,␈↓␈↓ β≤them␈α⊃for␈α⊂equality.␈α⊃They␈α⊂need␈α⊃not␈α⊂follow␈α⊃a␈α⊂constrained␈α⊃syntax.␈α⊂The␈α⊃value␈α⊃of␈α⊂a
␈↓ α,␈↓␈↓ β≤reason is adequately characterized by a unidimensional numeric rating.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αThe␈αreasons␈αsupporting␈αa␈αtask␈αsuperimpose␈αperfectly;␈αthey␈αnever␈αinteract␈αwith␈αeach
␈↓ α,␈↓␈↓ β≤other.

␈↓ α,␈↓␈↓ αl␈↓π#␈α∪␈↓␈α∩Supporting␈α∪reasons␈α∩¬␈α∪and␈α∩their␈α∪ratings␈α∩¬␈α∪never␈α∩change␈α∪with␈α∩time,␈α∪with␈α∩one
␈↓ α,␈↓␈↓ β≤exception: the ephemeron `Focus of attention'.

␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ It doesn't matter in what order the supporting reasons for a task were added.

␈↓ α,␈↓␈↓ αl␈↓π#␈α⊂␈↓␈α⊂There␈α⊂is␈α⊂no␈α⊂need␈α⊂for␈α⊂negative␈α⊂or␈α⊂inhibitory␈α⊂reasons,␈α⊂which␈α⊂would␈α⊃decrease␈α⊂the
␈↓ α,␈↓␈↓ β≤priority value of a task.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αAt␈αany␈αmoment,␈αthe␈αtop␈αfew␈αtasks␈αon␈αthe␈αagenda␈αare␈αnot␈αcoupled␈αstrongly;␈αit␈αis␈αnot
␈↓ α,␈↓␈↓ β≤necessary to expend extra processing time to carefully order them.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	24␈↓ε␈α
E.g.,␈α	a␈α
set␈α
of␈α	escape␈α
characters,␈α	so␈α
↑W␈α
means␈α	`␈↓βWhy␈α
did␈α	you␈α
do␈α
that?␈↓ε',␈α	↑U␈α
means␈α	`␈↓βUninteresting!␈α
Go␈α
on␈α	to
␈↓ α,␈↓β␈↓ βLsomething else␈↓ε', etc.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε122␈↓-

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αThe␈αtasks␈αon␈αthe␈αagenda␈α
are␈αcompletely␈αindependent␈αof␈αeach␈αother,␈αin␈αthe␈α
sense␈αof
␈↓ α,␈↓␈↓ β≤one task `enabling' or `waking-up' another.

␈↓ α,␈↓␈↓ αl␈↓π#␈α∩␈↓␈α∪Mathematics␈α∩research␈α∪has␈α∩a␈α∪clean,␈α∩simple␈α∪model␈α∩(see␈α∪Section␈α∩7.2.5,␈α∪page␈α∩125),
␈↓ α,␈↓␈↓ β≤which␈α
indicates␈α
that␈α
it␈α
is␈α
a␈α
search␈α
process␈α
governed␈α
by␈α
a␈α
large␈α
collection␈α
of␈α
heuristic
␈↓ α,␈↓␈↓ β≤rules.

␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αElementary␈α
mathematics␈αis␈αsuch␈α
that␈αvaluable␈αnew␈α
concepts␈αwill␈αbe␈αdiscovered␈α
fairly
␈↓ α,␈↓␈↓ β≤regularly.

␈↓ α,␈↓␈↓ αl␈↓π#␈α∀␈↓␈α∃The␈α∀worth␈α∃of␈α∀each␈α∀new␈α∃concept␈α∀can␈α∃be␈α∀estimated␈α∀easily,␈α∃after␈α∀just␈α∃a␈α∀brief
␈↓ α,␈↓␈↓ β≤investigation.

␈↓ α,␈↓␈↓ αl␈↓π#␈α
␈↓␈α
Contradictions␈α
will␈α
arise␈α
very␈α
rarely,␈α
and␈α
it␈α
is␈α
not␈α
disastrous␈α
to␈α
ignore␈α∞them␈α
when
␈↓ α,␈↓␈↓ β≤they␈α∞do␈α∞occur.␈α∞ The␈α∞same␈α∞indi≥erence␈α
applies␈α∞to␈α∞the␈α∞danger␈α∞of␈α∞believing␈α∞in␈α
false
␈↓ α,␈↓␈↓ β≤conjectures.

␈↓ α,␈↓␈↓ αl␈↓π#␈α∃␈↓␈α∀When␈α∃doing␈α∃theory␈α∀formation␈α∃in␈α∃elementary␈α∀mathematics,␈α∃proof␈α∃and␈α∀formal
␈↓ α,␈↓␈↓ β≤reasoning are dispensable.

␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ Even as more knowledge is obtained, the set of facets need never change.

␈↓ α,␈↓␈↓ αl␈↓π#␈α
␈↓␈α
For␈α
any␈αpiece␈α
of␈α
knowledge␈α
sought␈αor␈α
obtained,␈α
there␈α
is␈αprecisely␈α
one␈α
facet␈α
of␈αone
␈↓ α,␈↓␈↓ β≤existing␈↓	25␈↓␈α∩concept␈α⊃where␈α∩that␈α∩knowledge␈α⊃ought␈α∩to␈α⊃be␈α∩stored,␈α∩and␈α⊃it␈α∩is␈α∩easy␈α⊃to
␈↓ α,␈↓␈↓ β≤determine that proper location.

␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ Even as more concepts are de≡ned, the body of heuristics need not grow much.

␈↓ α,␈↓␈↓ αl␈↓π#␈α
␈↓␈αAny␈α
common-sense␈αknowledge␈α
required␈αby␈α
AM␈αis␈α
automatically␈αpresent␈α
within␈αthe
␈↓ α,␈↓␈↓ β≤heuristic rules.  So, e.g., no special spatial visualization abilities are needed.

␈↓ α,␈↓It␈αis␈α
worth␈αrepeating␈αhere␈α
that␈αthe␈αabove␈α
assumptions␈αare␈α
all␈αclearly␈α␈↓βfalse␈↓.␈α
Yet␈αnone␈αof␈α
them
␈↓ α,␈↓was␈α∂too␈α∂damaging␈α∂to␈α∞AM's␈α∂behavior,␈α∂and␈α∂their␈α∞combined␈α∂presence␈α∂made␈α∂the␈α∂creation␈α∞of
␈↓ α,␈↓AM feasible.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.2.4. Choice of Domain␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|The␈αgenesis␈αof␈αmathematical␈αcreation␈αis␈αa␈αproblem␈αwhich␈αshould␈αintensely␈αinterest
␈↓ α,␈↓β␈↓ α|the␈α∞psychologist.␈α
It␈α∞is␈α∞the␈α
activity␈α∞in␈α∞which␈α
the␈α∞human␈α∞mind␈α
seems␈α∞to␈α∞take␈α
least
␈↓ α,␈↓β␈↓ α|from␈αthe␈α
outside␈αworld,␈αin␈α
which␈αit␈αacts␈α
or␈αseems␈αto␈α
act␈αonly␈αof␈α
itself␈αand␈αon␈α
itself,
␈↓ α,␈↓β␈↓ α|so␈αthat␈α
in␈αstudying␈αthe␈α
procedure␈αof␈α
mathematical␈αthought␈αwe␈α
may␈αhope␈α
to␈αreach
␈↓ α,␈↓β␈↓ α|what is most essential in man's mind.

␈↓ α,␈↓¬␈↓ ε\-- Poincare'
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	25␈↓ε␈α
The␈α	only␈α
allowable␈α	exception␈α
is␈α
that␈α	a␈α
new␈α	piece␈α
of␈α	information␈α
might␈α
require␈α	the␈α
creation␈α	of␈α
a␈α	brand␈α
new␈α
concept,␈α	and
␈↓ α,␈↓ε␈↓ βLthen require storage somewhere on that concept.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε123␈↓-

␈↓ α,␈↓Here are some questions this subsection will address:
␈↓ α,␈↓␈↓ αl␈↓π#␈α␈↓␈αWhat␈αare␈αthe␈αinherent␈αlimitations␈α¬␈αand␈αadvantages␈α¬␈αin␈α≡xing␈αa␈αdomain␈αfor␈αAM
␈↓ α,␈↓␈↓ β≤to work in?
␈↓ α,␈↓␈↓ αl␈↓π# ␈↓ What characteristics are favorable to automating research in any given domain?
␈↓ α,␈↓␈↓ αl␈↓π#␈α∂␈↓␈α∂What␈α∂are␈α∂the␈α∂speci≡c␈α∂reasons␈α∂for␈α∂and␈α∂against␈α∂elementary␈α∂≡nite␈α∂set␈α∂theory␈α∂as␈α∂the
␈↓ α,␈↓␈↓ β≤chosen starting domain?

␈↓ α,␈↓Research␈α
in␈α
various␈α
domains␈α
of␈α
science␈α
and␈α
math␈α
proceeds␈α
slightly␈α
di≥erently.␈α
 For␈α
example,
␈↓ α,␈↓psychology␈αis␈αinterested␈αin␈αexplaining␈αpeople,␈α
not␈αin␈αcreating␈αnew␈αkinds␈αof␈αpeople.␈α
Math␈αis
␈↓ α,␈↓not␈α
interested␈α
in␈αindividual␈α
entities␈α
so␈αmuch␈α
as␈α
in␈αnew␈α
kinds␈α
of␈αentities.␈α
There␈α
are␈αethical
␈↓ α,␈↓restrictions␈α⊃on␈α⊃physicians␈α⊃which␈α∩prevent␈α⊃certain␈α⊃experiments␈α⊃from␈α⊃being␈α∩done.␈α⊃Political
␈↓ α,␈↓experiments rarely permit backtracking, etc.  Each ≡eld has its own peculiarities.

␈↓ α,␈↓If␈αwe␈α
want␈αa␈α
system␈αto␈αwork␈α
in␈αmany␈α
domains,␈αwe␈αhave␈α
to␈αsacri≡ce␈α
some␈αpower.␈↓	26␈↓.␈αWithin␈α
a
␈↓ α,␈↓given␈α
≡eld␈α
of␈α
knowledge␈α
(like␈α
math),␈α
the␈α≡ner␈α
the␈α
category␈α
we␈α
limit␈α
ourselves␈α
to,␈α
the␈αmore
␈↓ α,␈↓speci≡c␈α
are␈α
the␈α∞heuristics␈α
which␈α
become␈α∞available.␈α
 So␈α
it␈α
was␈α∞reasonable␈α
to␈α
make␈α∞this␈α
≡rst
␈↓ α,␈↓attempt limited to one narrow domain.

␈↓ α,␈↓This␈α⊂brings␈α⊂up␈α⊃the␈α⊂choice␈α⊂of␈α⊃domain.␈α⊂ What␈α⊂should␈α⊂it␈α⊃be?␈α⊂As␈α⊂the␈α⊃DENDRAL␈α⊂project
␈↓ α,␈↓illustrated␈α⊂so␈α⊃clearly␈↓	27␈↓,␈α⊂choice␈α⊂of␈α⊃subject␈α⊂domain␈α⊂is␈α⊃quite␈α⊂important␈α⊂when␈α⊃studying␈α⊂how
␈↓ α,␈↓researchers␈αdiscover␈αand␈αdevelop␈αtheir␈αtheories.␈α Mathematics␈αwas␈αchosen␈αas␈αthe␈αdomain␈αof
␈↓ α,␈↓this investigation, because
␈↓ α,␈↓␈↓ αl␈↓¬1.␈↓␈α∞In␈α∞doing␈α∞math␈α∞research,␈α∞one␈α∞needn't␈α∞cope␈α∞with␈α∞the␈α∞uncertainties␈α∞and␈α∂fallability␈α∞of
␈↓ α,␈↓␈↓ β≤testing␈αequipment;␈αthat␈αis,␈αthere␈αare␈αno␈αuncertainties␈αin␈αthe␈αdata␈α(compared␈αto,␈αe.g.,
␈↓ α,␈↓␈↓ β≤molecular structure inference from mass spectrograms).
␈↓ α,␈↓␈↓ αl␈↓¬2.␈↓␈α∩Reliance␈α∪on␈α∩experts'␈α∪introspections␈α∩is␈α∩one␈α∪of␈α∩the␈α∪most␈α∩powerful␈α∪techniques␈α∩for
␈↓ α,␈↓␈↓ β≤codifying␈α∀the␈α∀judgmental␈α∀criteria␈α∀necessary␈α∀to␈α∀do␈α∀e≥ective␈α∀work␈α∀in␈α∀a␈α∀≡eld;␈α∪I
␈↓ α,␈↓␈↓ β≤personally␈α∂have␈α∞had␈α∂enough␈α∂training␈α∞in␈α∂elementary␈α∂mathematics␈α∞so␈α∂that␈α∂I␈α∞didn't
␈↓ α,␈↓␈↓ β≤have␈α∪to␈α∩rely␈α∪completely␈α∪on␈α∩external␈α∪sources␈α∪for␈α∩guidance␈α∪in␈α∪formulating␈α∩such
␈↓ α,␈↓␈↓ β≤heuristic␈α∃rules.␈α∃ Also,␈α∃several␈α∃excellent␈α∃sources␈α∃were␈α∃available␈α⊗[Polya,␈α∃Skemp,
␈↓ α,␈↓␈↓ β≤Hadamard, Kershner, etc.].
␈↓ α,␈↓␈↓ αl␈↓¬3.␈↓␈α
The␈αmore␈α
formal␈α
a␈αscience␈α
is,␈αthe␈α
easier␈α
it␈αis␈α
to␈αautomate.␈α
For␈α
a␈αmachine␈α
to␈αcarry␈α
out
␈↓ α,␈↓␈↓ β≤research␈α∞in␈α∂psychology␈α∞would␈α∂require␈α∞more␈α∂knowledge␈α∞about␈α∂human␈α∞information
␈↓ α,␈↓␈↓ β≤processing␈αthan␈α
now␈αis␈α
known,␈αbecause␈α
psychology␈αdeals␈α
with␈αentities␈α
as␈αcomplex␈α
as
␈↓ α,␈↓␈↓ β≤you␈α
and␈αI.␈α
Also,␈αin␈α
a␈αformal␈α
science,␈αthe␈α
␈↓βlanguages␈↓␈αto␈α
communicate␈αinformation␈α
can
␈↓ α,␈↓␈↓ β≤be simple even though the ␈↓βmessages␈↓ themselves be sophisticated.
␈↓ α,␈↓␈↓ αl␈↓¬4.␈↓␈α
Since␈αmathematics␈α
can␈αdeal␈α
with␈αany␈α
conceivable␈αconstructs,␈α
a␈αresearcher␈α
there␈αis␈α
not
␈↓ α,␈↓␈↓ β≤limited␈α
to␈αexplaining␈α
observed␈αdata.␈α
 Related␈αto␈α
this␈αis␈α
the␈αfreedom␈α
to␈αinvestigate␈α
¬
␈↓ α,␈↓␈↓ β≤or␈αto␈αgive␈αup␈αon␈α¬␈αwhatever␈αthe␈αresearcher␈αwants␈αto.␈αThere␈αis␈αno␈αsingle␈αdiscovery
␈↓ α,␈↓␈↓ β≤which is the "goal", no given problem to solve, no right or wrong behavior.
␈↓ α,␈↓␈↓ αl␈↓¬5.␈↓␈α
Unlike␈α
"simpler"␈α
≡elds,␈α
such␈α
as␈α
propositional␈α
logic,␈α
there␈α
is␈α
an␈α
abundance␈α
of␈α
heuristic
␈↓ α,␈↓␈↓ β≤rules available for the picking.

␈↓ α,␈↓The␈α
limitations␈α
of␈α
math␈α
as␈α
a␈α
domain␈α
are␈α
closely␈α
intertwined␈α
with␈α
its␈α∞advantages.␈α
 Having

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	26␈↓ε␈α
This␈α
is␈α
assuming␈α
a␈α
system␈α
of␈α
a␈α
given␈α	fixed␈α
size.␈α
 If␈α
this␈α
restriction␈α
isn't␈α
present,␈α
then␈α
a␈α
reasonable␈α	"general-purpose"
␈↓ α,␈↓ε␈↓ βLsystem could be built as several systems linked by one giant switch.
␈↓ α,␈↓ε␈↓	27␈↓ε see [Feigenbaum et. al. 71].  In that case, the choice of subject was enabled by [Lederberg 64].
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε124␈↓-

␈↓ α,␈↓no␈αties␈αto␈αreal-world␈αdata␈αcan␈αbe␈αviewed␈αas␈αa␈αlimitation,␈αas␈αcan␈αhaving␈αno␈αclear␈αgoal.␈αThere
␈↓ α,␈↓is␈α
always␈αthe␈α
danger␈α
that␈αAM␈α
will␈αgive␈α
up␈α
on␈αeach␈α
theory␈αas␈α
soon␈α
as␈αthe␈α
≡rst␈αtough␈α
obstacle
␈↓ α,␈↓crops up.

␈↓ α,␈↓Since␈α∂math␈α∂has␈α∂been␈α⊂worked␈α∂on␈α∂for␈α∂millenia␈α∂by␈α⊂some␈α∂of␈α∂the␈α∂greatest␈α∂minds␈α⊂from␈α∂many
␈↓ α,␈↓di≥erent␈α∞cultures,␈α
it␈α∞is␈α∞unlikely␈α
that␈α∞a␈α
small␈α∞e≥ort␈α∞like␈α
AM␈α∞would␈α
make␈α∞any␈α∞new␈α
inroads,
␈↓ α,␈↓have␈α∞any␈α∞startling␈α∞insights.␈α∞ In␈α∞that␈α∞respect,␈α∞Dendral's␈α∞space␈α∞was␈α∞much␈α∞less␈α∂explored.␈α∞ Of
␈↓ α,␈↓course␈α
math␈α
¬␈α
even␈α∞at␈α
the␈α
elementary␈α
level␈α
that␈α∞AM␈α
explored␈α
it␈α
¬␈α
still␈α∞has␈α
undiscovered
␈↓ α,␈↓gems (e.g., the recent unearthing of Conway's numbers [Knuth 74]).

␈↓ α,␈↓One␈α
point␈α
of␈α
agreement␈α
between␈α
Weizenbaum␈α
and␈α
Lederberg␈↓	28␈↓␈α
is␈α
that␈α
AI␈α
can␈α∞succeed␈α
in
␈↓ α,␈↓automating␈αan␈αactivity␈αonly␈αwhen␈αa␈α"strong␈αtheory"␈αof␈αthat␈αactivity␈αexists.␈αAM␈αis␈αbuilt␈αon␈αa
␈↓ α,␈↓detailed␈α
model␈αof␈α
how␈αhumans␈α
do␈αmath␈α
research.␈α In␈α
the␈αnext␈α
subsection,␈αwe'll␈α
discuss␈αthe
␈↓ α,␈↓model of math research that AM assumes.


␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.2.5. Limitations of the Model of Math Research␈↓)αβ␈↓


␈↓ α,␈↓β␈↓ α|Weizenbaum␈αdoes␈α
point␈αto␈αprojects␈α
in␈αmathematics␈α
and␈αchemistry␈αwhere␈α
computers
␈↓ α,␈↓β␈↓ α|have␈αshown␈αtheir␈αpotential␈αfor␈αassisting␈αhuman␈αscientists␈αin␈αsolving␈αproblems.␈αHe
␈↓ α,␈↓β␈↓ α|correctly␈α⊃points␈α⊃out␈α⊃that␈α⊃these␈α⊃successes␈α⊂are␈α⊃based␈α⊃on␈α⊃the␈α⊃existence␈α⊃of␈α⊂"strong
␈↓ α,␈↓β␈↓ α|theories" about their subject matter.

␈↓ α,␈↓¬␈↓ ε\-- Lederberg



␈↓ α,␈↓AM,␈αlike␈α
anything␈αelse␈α
in␈αthis␈α
world,␈αis␈α
constrained␈αby␈α
a␈αmass␈α
of␈αassumptions.␈α
Most␈αof␈α
these
␈↓ α,␈↓are␈α∀"compiled"␈α∀or␈α∀interwoven␈α∀into␈α∀the␈α∀very␈α∀fabric␈α∀of␈α∀AM,␈α∀hence␈α∀can't␈α∀be␈α∃tested␈α∀by
␈↓ α,␈↓experiments on AM.  Some of these were just discussed a few pages ago, in Section 7.2.3.

␈↓ α,␈↓Another␈α∩body␈α∩of␈α∩assumptions␈α∪exists.␈α∩ AM␈α∩is␈α∩built␈α∪around␈α∩a␈α∩particular␈α∩model␈α∪of␈α∩how
␈↓ α,␈↓mathematicians␈α∩actually␈α∩go␈α∩about␈α∩doing␈α∩their␈α∩research.␈α∩ This␈α∩model␈α∩was␈α∩derived␈α∩from
␈↓ α,␈↓introspection,␈α⊃but␈α⊂can␈α⊃be␈α⊂supported␈α⊃by␈α⊂quotes␈α⊃from␈α⊂Polya,␈α⊃Kershner,␈α⊃Hadamard,␈α⊂Saaty,
␈↓ α,␈↓Skemp,␈α
and␈α
many␈α
others.␈α
 No␈αattempt␈α
will␈α
be␈α
made␈α
to␈αjustify␈α
any␈α
of␈α
these␈α
premises.␈α On␈α
the
␈↓ α,␈↓next␈α∞page␈α∞is␈α∞a␈α∂simpli≡ed␈α∞summary␈α∞of␈α∞that␈α∂information␈α∞processing␈α∞model␈α∞for␈α∂math␈α∞theory
␈↓ α,␈↓formation:








␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	28␈↓ε␈α	See␈α	the␈α	quote␈α	at␈α	the␈α	front␈α	of␈α	the␈α	next␈α	subsection.␈α	It␈α	is␈α	from␈α	[Lederberg␈α	76],␈α	a␈α	review␈α	of␈α	[Weizenbaum␈α	76].␈α	This␈αλreview
␈↓ α,␈↓ε␈↓ βLalso exists as file WEIZEN.LED[pub,jmc]@SAIL.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε125␈↓-


␈↓ α,␈↓∧␈↓ ∧bMODEL OF MATH RESEARCH
␈↓ α,␈↓π⊂ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L␈↓ L⊃
␈↓"␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬1.␈↓␈αThe␈αorder␈αin␈αwhich␈αa␈αmath␈αtextbook␈αpresents␈αa␈αtheory␈αis␈αalmost␈αthe␈αexact␈αopposite␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βof␈α
the␈αorder␈α
in␈αwhich␈α
it␈αwas␈α
actually␈αdiscovered␈α
and␈αdeveloped.␈α
 In␈αa␈α
text,␈αnew␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βde≡nitions␈α
are␈α
stated␈α
with␈α
little␈α
or␈α
no␈αmotivation,␈α
and␈α
they␈α
turn␈α
out␈α
to␈α
be␈αjust␈α
the␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βones␈α
needed␈αto␈α
state␈αthe␈α
next␈α
big␈αtheorem,␈α
whose␈αproof␈α
then␈α
magically␈αappears.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βIn␈α∂contrast,␈α∂a␈α∂mathematician␈α∞doing␈α∂research␈α∂will␈α∂examine␈α∂some␈α∞already-known␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βconcepts,␈α
perhaps␈α
trying␈α
to␈α∞≡nd␈α
some␈α
regularity␈α
in␈α
experimental␈α∞data␈α
involving␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βthem.␈α∂The␈α∂patterns␈α∂he␈α∂notices␈α∞are␈α∂the␈α∂conjectures␈α∂he␈α∂must␈α∂investigate␈α∞further,␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βand these relationships directly motivate him to make new de≡nitions.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬2.␈↓␈α∞Each␈α∂step␈α∞the␈α∞researcher␈α∂takes␈α∞while␈α∞developing␈α∂a␈α∞new␈α∞theory␈α∂involves␈α∞choosing␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βfrom␈αa␈αlarge␈αset␈α
of␈α"legal"␈αalternatives␈α¬␈αthat␈α
is,␈αsearching.␈α The␈αkey␈α
to␈αkeeping␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βthis␈α⊂from␈α∂becoming␈α⊂a␈α∂blind,␈α⊂explosive␈α⊂search␈α∂is␈α⊂the␈α∂proper␈α⊂use␈α⊂of␈α∂evaluation␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βcriteria.␈α
Each␈α
mathematician␈αuses␈α
his␈α
own␈α
personal␈αheuristics␈α
to␈α
choose␈αthe␈α
"best"␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βalternative available at each moment.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬3.␈↓␈α⊂Non-formal␈α⊂criteria␈α⊂(aesthetic␈α⊂interestingness,␈α⊂inductive␈α⊂inference␈α⊃from␈α⊂empirical␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βevidence,␈α
analogy,␈α
and␈α
utility)␈α
are␈α
much␈α
more␈α
important␈α
than␈α∞formal␈α
deductive␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βmethods␈α∪in␈α∪developing␈α∪mathematically␈α∩worthwhile␈α∪theories,␈α∪and␈α∪in␈α∩avoiding␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βbarren diversions.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬4.␈↓␈αProgress␈αin␈α␈↓βany␈↓␈α≡eld␈αof␈αmathematics␈αdemands␈αmuch␈αnon-formal␈αheuristic␈αexpertise␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βin␈α∂␈↓βmany␈↓␈α∂di≥erent␈α⊂"nearby"␈α∂mathematical␈α∂≡elds.␈α∂ So␈α⊂a␈α∂broad,␈α∂universal␈α⊂core␈α∂of␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βknowledge␈α∪must␈α∪be␈α∀mastered␈α∪before␈α∪any␈α∀single␈α∪theory␈α∪can␈α∀meaningfully␈α∪be␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βdeveloped.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬5.␈↓␈αIt␈αis␈αsu≠cient␈α(and␈α
pragmatically␈αnecessary)␈αto␈αhave␈αand␈α
use␈αa␈αlarge␈αset␈αof␈α
informal␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βheuristic␈α∞rules.␈α∞These␈α∞rules␈α∞direct␈α
the␈α∞researcher's␈α∞next␈α∞activities,␈α∞depending␈α
on␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βthe␈α∩current␈α∩situation␈α∩he␈α∩is␈α∩in.␈α∩ These␈α∩rules␈α∩can␈α∩be␈α∩assumed␈α∩to␈α∩superimpose␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βideally:␈α∂the␈α⊂combined␈α∂e≥ect␈α∂of␈α⊂several␈α∂rules␈α⊂is␈α∂just␈α∂the␈α⊂sum␈α∂of␈α⊂the␈α∂individual␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βe≥ects.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬6.␈↓␈αThe␈αnecessary␈αheuristic␈α
rules␈αare␈αvirtually␈αthe␈α
same␈αin␈αall␈αbranches␈αof␈α
mathematics,␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βand␈α∞at␈α∞all␈α∞levels␈α∞of␈α∞sophistication.␈α∞ Each␈α∞specialized␈α∞≡eld␈α∞will␈α∞have␈α∞some␈α∞of␈α∞its␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βown␈αheuristics;␈α
those␈αare␈α
normally␈αmuch␈α
more␈αpowerful␈α
than␈αthe␈α
general-purpose␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βheuristics.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬7.␈↓␈α
For␈α
true␈α
understanding,␈α
the␈αresearcher␈α
should␈α
grasp␈↓	29␈↓␈α
each␈α
concept␈α
in␈αseveral␈α
ways:␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βdeclaratively,␈α⊃abstractly,␈α⊃operationally,␈α⊃knowing␈α⊃when␈α⊃it␈α⊃is␈α⊃relevant,␈α⊃and␈α⊃as␈α⊃a␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βbunch of examples.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ αL␈↓¬8.␈↓␈α∀Common␈α∀metaphysical␈α∀assumptions␈α∀about␈α∀nature␈α∀and␈α∀science:␈α∀Nature␈α∀is␈α∪fair,␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βuniform,␈αand␈αregular.␈α Coincidences␈αhave␈αmeaning.␈α Statistical␈αconsiderations␈α
are␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βvalid␈αwhen␈αlooking␈αat␈αmathematical␈αdata.␈α Simplicity␈αand␈αsymmetry␈αand␈αsynergy␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓␈↓ βare the rule, not the exception.␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓π}                                                                         ␈↓ L␈↓ L}
␈↓"␈↓ α,␈↓π%ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L␈↓ L$



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	29␈↓ε Have access to, relate to, store, be able to manipulate, be able to answer questions about
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε126␈↓-



␈↓ α,␈↓␈↓ αL␈↓↓␈↓&7.2.6. Ultimate powers and weaknesses␈↓)αβ␈↓

␈↓ α,␈↓Consider␈α∞now␈α∞␈↓βany␈↓␈α∞system␈α∞which␈α∞is␈α
consistent␈α∞with␈α∞the␈α∞preceding␈α∞model␈α∞of␈α∞math␈α
research,
␈↓ α,␈↓and␈αwhose␈αorientation␈αis␈αto␈αdiscover␈αand␈αdevelop␈αnew␈α(to␈αthe␈αsystem)␈αmathematical␈α
theories.
␈↓ α,␈↓This␈α
includes␈αAM␈α
itself,␈α
but␈αmight␈α
also␈α
include␈αa␈α
bright␈α
high-school␈αsenior␈α
who␈α
has␈αbeen
␈↓ α,␈↓taught a large body of heuristic rules.

␈↓ α,␈↓What␈α⊂can␈α⊃such␈α⊂systems␈α⊂ultimately␈α⊃achieve?␈α⊂What␈α⊂are␈α⊃their␈α⊂ultimate␈α⊂limits?␈α⊃ Answers␈α⊂to
␈↓ α,␈↓ultimate␈α⊃questions␈α⊃are␈α⊃hard␈α⊂to␈α⊃come␈α⊃by␈α⊃experimentally,␈α⊂so␈α⊃this␈α⊃discussion␈α⊃will␈α⊃be␈α⊂quite
␈↓ α,␈↓philosophical,␈α
speculative,␈α
and␈α
short.␈α The␈α
model␈α
of␈α
math␈αresearch␈α
hinges␈α
around␈α
the␈αuse␈α
of
␈↓ α,␈↓heuristic␈α
rules␈α
for␈α∞guidance␈α
at␈α
all␈α
levels␈α∞of␈α
behavior.␈α
It␈α
is␈α∞questionable␈α
whether␈α
or␈α∞not␈α
all
␈↓ α,␈↓known␈α∪mathematics␈α∩could␈α∪evolve␈α∪smoothly␈α∩in␈α∪this␈α∩way.␈α∪ As␈α∪a␈α∩≡rst␈α∪order␈α∪≡xup,␈α∩we've
␈↓ α,␈↓mentioned␈αthe␈αneed␈αto␈αprovide␈αgood␈αmeta-heuristics,␈αto␈αkeep␈αenlarging␈αthe␈αset␈αof␈αheuristics.
␈↓ α,␈↓If␈α
this␈α∞is␈α
not␈α
enough␈α∞(if␈α
meta-meta-...-heuristics␈α
are␈α∞needed),␈α
then␈α
the␈α∞model␈α
is␈α
a␈α∞poor␈α
one
␈↓ α,␈↓and␈α∞has␈α∞some␈α∞inherent␈α∞limitations.␈↓	30␈↓␈α∞If␈α∞some␈α∞discoveries␈α∞can␈α∞only␈α∞be␈α∞made␈α∞non-rationally
␈↓ α,␈↓(by␈α∞random␈α∂chance,␈α∞by␈α∂Gestalt,␈α∞etc.)␈α∂then␈α∞any␈α∞such␈α∂system␈α∞would␈α∂be␈α∞incapable␈α∂of␈α∞≡nding
␈↓ α,␈↓those concepts.

␈↓ α,␈↓Turning␈αaside␈αfrom␈αmath,␈αwhat␈αabout␈αsystems␈αwhose␈αdesign␈α¬␈αas␈αa␈αcomputer␈αprogram␈α¬␈αis
␈↓ α,␈↓similar␈α⊃to␈α⊃AM?␈↓	31␈↓␈α⊃Building␈α⊃such␈α⊃systems␈α⊃will␈α⊃be␈α⊃"fun",␈α⊃and␈α⊃perhaps␈α⊃will␈α⊃result␈α∩in␈α⊃new
␈↓ α,␈↓discoveries␈αin␈αother␈α≡elds.␈α Eventually,␈αscientists␈α
(at␈αleast␈αin␈αa␈αfew␈αvery␈αhard␈α
domains)␈αmay
␈↓ α,␈↓relegate␈α
more␈α∞and␈α
more␈α
of␈α∞their␈α
"hack"␈α∞research␈α
duties␈α
to␈α∞AM-like␈α
systems.␈α∞ The␈α
ultimate
␈↓ α,␈↓limitations␈α⊂will␈α⊂be␈α⊂those␈α⊂arising␈α⊂from␈α∂incorrect␈α⊂(e.g.,␈α⊂partial)␈α⊂models␈α⊂of␈α⊂the␈α⊂activities␈α∂the
␈↓ α,␈↓system␈αmust␈αperform.␈α The␈αsystems␈αthemselves␈αmay␈αhelp␈αimprove␈αthese␈αmodels:␈αexperiments
␈↓ α,␈↓that␈α∂are␈α∂performed␈α∂on␈α∂the␈α∂systems␈α∂are␈α∂actually␈α∂tests␈α∂of␈α∂the␈α∂underlying␈α∂model;␈α∂the␈α∂results
␈↓ α,␈↓might␈α∂cause␈α∂revisions␈α∂to␈α∂be␈α∞made␈α∂in␈α∂the␈α∂model,␈α∂then␈α∞in␈α∂the␈α∂system,␈α∂and␈α∂the␈α∂whole␈α∞cycle
␈↓ α,␈↓would begin again.


␈↓ α,␈↓␈↓ ¬F␈↓∧␈↓&7.3. Final Conclusions␈↓)αβ␈↓

␈↓ α,␈↓Before quitting, let's summarize what's worth remembering about this research project.

␈↓ α,␈↓␈↓π##␈α∂␈↓␈α∂It␈α∂is␈α∂a␈α∂demonstration␈α∂that␈α∂a␈α∂few␈α∂hundred␈α∂general␈α∂heuristic␈α∂rules␈α∂su≠ce␈α∂to␈α∂guide␈α∞an
␈↓ α,␈↓␈↓ αlautomated␈α⊗math␈α⊗researcher␈α⊗as␈α⊗it␈α⊗explores␈α⊗and␈α⊗expands␈α⊗a␈α⊗large␈α⊗but␈α∃incomplete
␈↓ α,␈↓␈↓ αlknowledge␈αbase␈αof␈αmath␈αconcepts.␈α AM␈αserves␈αas␈αa␈αliving␈αexistence␈αproof␈αthat␈αcreative
␈↓ α,␈↓␈↓ αlresearch can be e≥ectively modelled as heuristic search.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	30␈↓ε␈α	If␈α	Ptolemy␈α	had␈α
had␈α	access␈α	to␈α	a␈α	digital␈α
computer,␈α	all␈α	his␈α	data␈α
could␈α	have␈α	been␈α	made␈α	to␈α
fit␈α	(to␈α	any␈α	desired␈α
accuracy),␈α	just
␈↓ α,␈↓ε␈↓ βLby␈αcomputing␈α
epi-cycles,␈αepi-epi-cycles,...␈α
to␈αthe␈α
needed␈αnumber␈α
of␈αepi's.␈α
We␈αin␈α
AI␈αmust␈α
constantly␈αbe␈α
on
␈↓ α,␈↓ε␈↓ βLguard against that error.
␈↓ α,␈↓ε␈↓	31␈↓ε␈αHaving␈αan␈αagenda␈αof␈αtasks␈αwith␈αreasons␈αand␈αreason-ratings␈αcombining␈αto␈αform␈αa␈αglobal␈αpriority␈αfor␈αeach␈αtask,␈α
having
␈↓ α,␈↓ε␈↓ βLunits/modules/frames/Beings/Actors/scripts/concepts␈α which␈α have␈α parts/slots/aspects/facets,␈α∨etc.
␈↓ α,␈↓ε␈↓ βLHeuristic␈α
rules␈αare␈α
tacked␈αonto␈α
relevant␈α
concepts,␈αand␈α
are␈αexecuted␈α
to␈α
produce␈αnew␈α
concepts,␈αnew␈α
tasks,
␈↓ α,␈↓ε␈↓ βLnew facet entries.
␈↓ α,␈↓␈↓εChapter 7␈↓ ¬π␈↓↓AM: ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε127␈↓-



␈↓ α,␈↓␈↓π##␈α
␈↓␈α
The␈α
book␈α
also␈α
introduces␈α
a␈α
control␈α
structure␈α
based␈α
upon␈α
an␈α
agenda␈α
of␈α
small␈αresearch
␈↓ α,␈↓␈↓ αltasks, each with a list of supporting reasons attached.

␈↓ α,␈↓␈↓π##␈α␈↓␈α
The␈αmain␈α
limitation␈αof␈α
AM␈αwas␈α
its␈αinability␈α
to␈αsynthesize␈α
powerful␈αnew␈α
heuristics␈αfor
␈↓ α,␈↓␈↓ αlthe new concepts it de≡ned.

␈↓ α,␈↓␈↓π##␈α
␈↓␈α
The␈α
main␈α∞successes␈α
were␈α
the␈α
few␈α∞novel␈α
ideas␈α
it␈α
came␈α
up␈α∞with,␈α
the␈α
ease␈α
with␈α∞which␈α
a
␈↓ α,␈↓␈↓ αlnew␈αtask␈αdomain␈αwas␈αfed␈αto␈αthe␈αsystem,␈αand␈α¬␈αmost␈αimportantly␈α¬␈αthe␈αoverall␈αrational
␈↓ α,␈↓␈↓ αlsequences of behavior AM exhibited.

␈↓ α,␈↓␈↓π##␈α⊃␈↓␈α⊃The␈α⊂greatest␈α⊃long-range␈α⊃importance␈α⊂of␈α⊃AM␈α⊃may␈α⊂well␈α⊃lie␈α⊃in␈α⊂the␈α⊃body␈α⊃of␈α⊂heuristics
␈↓ α,␈↓␈↓ αlassembled,␈α∩either␈α∩as␈α⊃the␈α∩seed␈α∩for␈α⊃a␈α∩huge␈α∩base␈α⊃of␈α∩experts'␈α∩heuristics,␈α⊃or␈α∩as␈α∩a␈α⊃new
␈↓ α,␈↓␈↓ αlorientation for mathematics education.
␈↓ α,␈↓␈↓ ≥-␈↓ε128␈↓-

␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬␈↓∧Appendix 1.  AM's Concepts␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓


␈↓ α,␈↓␈↓ β-␈↓↓␈↓&Appendix 1.1. ␈↓)αβ␈↓∧␈↓& Concepts and Heuristics as coded in LISP␈↓)αβ␈↓↓

␈↓ α,␈↓Two␈α
of␈α
AM's␈α
initial␈α
concepts␈α
are␈α
presented␈α
in␈α
detail,␈α
"Compose"␈α
and␈α
"Osets"␈α
(composition␈α
of
␈↓ α,␈↓relations,␈α
and␈α
ordered␈α
sets).␈α
 The␈α
entries␈α
in␈α
each␈α
of␈α
their␈α
facets␈α
are␈α
displayed:␈α
We␈α
provide␈α
(i)
␈↓ α,␈↓the␈α⊂Lisp␈α⊂expressions␈α⊃which␈α⊂were␈α⊂actually␈α⊃stored␈α⊂there␈α⊂in␈α⊂the␈α⊃AM␈α⊂program,␈α⊂and␈α⊃(ii)␈α⊂an
␈↓ α,␈↓English,␈α∂Lambda␈α∂calculus,␈α∂and␈α∂math␈α∂notation␈α∂condensation␈α∂of␈α∂all␈α∂the␈α∂knowledge␈α∞initially
␈↓ α,␈↓supplied to AM about that facet of that concept.

␈↓ α,␈↓If␈α∂there␈α∞is␈α∂any␈α∞unmentioned␈α∂facet␈α∞for␈α∂a␈α∂concept,␈α∞then␈α∂it␈α∞started␈α∂out␈α∞blank.␈α∂ Many␈α∂of␈α∞the
␈↓ α,␈↓facets␈α
of␈α
the␈αoriginal␈α
concepts␈α
were␈αleft␈α
blank␈α
␈↓βintentionally␈↓,␈α
knowing␈αthat␈α
AM␈α
would␈αbe␈α
able
␈↓ α,␈↓to␈α≡ll␈αthem␈α
in␈αas␈αwell.␈α
After␈αall,␈αif␈αyou␈α
can␈α≡ll␈αin␈α
examples␈αof␈αany␈α
new␈αconcept,␈αyou␈αought␈α
to
␈↓ α,␈↓be able to ≡ll in examples of Compositions!


␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 1.1.1.␈↓)αβ␈↓␈↓↓␈↓& The `Compose' Concept␈↓)αβ␈↓

␈↓ α,␈↓α␈↓∧␈↓&ENGN␈↓)αβ␈↓α␈↓	1␈↓α (COMPOSE Compose Composition (Afterwards))

␈↓ α,␈↓α␈↓βAnglicised condensation:␈↓α
␈↓"␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Name(s): Compose, Composition, sometimes:  afterwards; ␈↓π                        ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$


␈↓ α,␈↓α␈↓∧␈↓&DEFN␈↓)αβ␈↓α  (TYPE NEC&SUFF PC DECLARATIVE SLOW (FOREACH X IN (DOMAIN BA2)
␈↓ α,␈↓α                    RETURN (APPLYB␈↓	2␈↓α BA1 ALGS (APPLYB BA2 ALGS X]
␈↓ α,␈↓α␈↓&DEFN-SUFF␈↓)αβ  [[TYPE SUFFICIENT NONRECURSIVE QUICK
␈↓ α,␈↓α                     (AND (ISA BA1 'ACTIVE)
␈↓ α,␈↓α                          (ISA BA2 'ACTIVE)
␈↓ α,␈↓α                          (ISA BA3 'ACTIVE)
␈↓ α,␈↓α                          (ARE-EQUIV BA3 (ALREADY-COMPOSED␈↓	3␈↓α BA1 BA2]
␈↓ α,␈↓α        [TYPE SUFFICIENT QUASIRECURSIVE SLOW (ARE-EQUIV BA3
␈↓ α,␈↓α                    (APPLYB 'COMPOSE 'ALGS BA1 BA2]␈↓	4␈↓α
␈↓ α,␈↓α        [TYPE SUFFICIENT QUASIRECURSIVE QUICK (EQUAL BA3
␈↓ α,␈↓α                    (APPLYB  'COMPOSE  'ALGS BA1 BA2]]

␈↓ α,␈↓α␈↓βAnglicised condensation:␈↓α
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε This is short for "English name", and is the facet called "Name(s)" everywhere else in this book.
␈↓ α,␈↓ε␈↓	2␈↓ε␈α	The␈α
function␈α	"APPLYB"␈α
indicates␈α	that␈α
a␈α	concept's␈α	facet␈α
is␈α	to␈α
be␈α	accessed␈α
and␈α	then␈α	executed.␈α
(APPLYB␈α	C␈α
F␈α	x␈α
y...)␈α	means:
␈↓ α,␈↓ε␈↓ βLaccess an entry on facet F of concept C, and then run it on the arguments x,y,...
␈↓ α,␈↓ε␈↓	3␈↓ε This LISP function checks to see whether the two operations have been composed before.
␈↓ α,␈↓ε␈↓	4␈↓ε␈α	The␈αλarguments␈α	to␈αλCompose.Defn␈α	(and␈α	to␈αλCompose.Algs␈α	as␈αλwell)␈α	are␈αλcalled␈α	BA1,␈α	BA2,...␈αλThus␈α	we␈αλwould␈α	write␈α	each␈αλdefinition
␈↓ α,␈↓ε␈↓ βLof Compose as "λ (BA1 BA2 BA3) ..."
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε129␈↓-

␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Definitions: ␈↓π                                                           ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Declarative slow: λ (A,B,C) ∀x, C(x)=A(B(x). ␈↓π                             ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Sufficient Nonrecursive Quick: λ (A,B,C) C has the Name `A␈↓εo␈↓¬B'. ␈↓π               ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Sufficient, Slow: Are-equivalent(C,Compose.Algs(A,B)). ␈↓π                     ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Sufficient, Quick: C=Compose.Algs(A,B). ␈↓π                                 ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$


␈↓ α,␈↓α␈↓∧␈↓&D-R␈↓)αβ␈↓α ((OPERATION ACTIVE OPERATION)
␈↓ α,␈↓α         (RELATION RELATION RELATION)
␈↓ α,␈↓α         (PREDICATE ACTIVE PREDICATE)
␈↓ α,␈↓α         (ACTIVE ACTIVE ACTIVE))
␈↓ α,␈↓α␈↓&D-R-FILLIN1␈↓)αβ  (PROGN (ARGS-ASA COMPOSE F1 F2) (CADAR (CON-MERGE-ARGS␈↓	5␈↓α  F1 F2)))
␈↓ α,␈↓α␈↓&EXS-D-R-FILLIN1␈↓)αβ  [PROGN (ARGS-ASA COMPOSE F1 F2)
␈↓ α,␈↓α           [SETQ RAN1 (LAST (ANY1OF (GETB F1 'D-R] (* RAN1 is the range of F1)
␈↓ α,␈↓α           [SETQ DOM1 (ALL-BUT-LAST (ANY1OF (GETB F1 'D-R]
␈↓ α,␈↓α           [SETQ RAN2 (LAST (ANY1OF (GETB F2 'D-R] (* RAN2 is the range of F2)
␈↓ α,␈↓α           [SETQ DOM2 (ALL-BUT-LAST (ANY1OF (GETB F2 'D-R]
␈↓ α,␈↓α           [SETQ X (MAXIMAL RAN2 DOM1 'FRAC-OVERLAP]
␈↓ α,␈↓α           (NCONC1 (LSUBST DOM2 for X in DOM1) RAN1]

␈↓ α,␈↓α␈↓βAnglicised condensation:␈↓α
␈↓"␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Domain/range: <Active Active → Active> ␈↓π                                     ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬                    <Operation Active → Operation> ␈↓π                                 ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬                    <Predicate Active → Predicate> ␈↓π                                  ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬                    <Relation Relation → Relation> ␈↓π                                   ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Fillin: 2 ␈↓β(out of a total of 9)␈↓¬ heuristics. ␈↓π                                      ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            In Appendix 2, these are heuristics numbers 175 and 176. ␈↓π                  ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$


␈↓ α,␈↓α␈↓∧␈↓&ALGS␈↓)αβ␈↓α ((TYPE QUASIRECURSIVE INDIRECT CASES [PROGN
␈↓ α,␈↓α        (COND
␈↓ α,␈↓α             ((NULL BA1)
␈↓ α,␈↓α               (APPLYB 'COMPOSE
␈↓ α,␈↓α                       'ALGS
␈↓ α,␈↓α                       (RAND-MEMB (EXS ACTIVE))
␈↓ α,␈↓α                       BA2 BA3 BA4))␈↓	6␈↓α
␈↓ α,␈↓α             &␈↓	7␈↓α
␈↓ α,␈↓α             ((ALREADY-COMPOSED BA1 BA2)   (* Note: this sets GTEMP12)   GTEMP12)
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	5␈↓ε␈αλThis␈αλis␈α	a␈αλLISP␈αλfunction,␈α	opaque␈αλto␈αλAM,␈α	which␈αλanalyzes␈αλthe␈α	Domain/range␈αλfacets␈αλof␈α	the␈αλtwo␈αλoperations␈α	F1␈αλand␈αλF2,␈α	and␈αλsees
␈↓ α,␈↓ε␈↓ βLhow␈α	(if␈αλat␈α	all)␈α	the␈αλrange␈α	of␈αλF1␈α	can␈α	be␈αλmade␈α	to␈αλoverlap␈α	the␈α	domain␈αλof␈α	F2.␈αλNote␈α	that␈α	F2␈αλis␈α	applied␈α	AFTER␈αλF1.
␈↓ α,␈↓ε␈↓ βLThe LISP code for this function is presented on page 133.
␈↓ α,␈↓ε␈↓	6␈↓ε␈α
Note␈α
what␈α
this␈α
clause␈α
says:␈α
if␈α
Compose.Algs␈α
is␈α
ever␈α	called␈α
with␈α
its␈α
first␈α
argument␈α
missing,␈α
randomly␈α
select␈α
an␈α
Active␈α	to
␈↓ α,␈↓ε␈↓ βLuse as that constituent of the composition.
␈↓ α,␈↓ε␈↓	7␈↓ε Similar to last case: takes care of missing second argument. The ampersand, "&", indicates an omission from this listing.
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε130␈↓-

␈↓ α,␈↓α             ((AND BA1 BA2 (IS-CON␈↓	8␈↓α BA1)
␈↓ α,␈↓α                   (IS-CON BA2)
␈↓ α,␈↓α                   (ISA BA1 'ACTIVE)
␈↓ α,␈↓α                   (ISA BA2 'ACTIVE)
␈↓ α,␈↓α                   (SETQ GTEMP11 (CON-MERGE-ARGS BA1 BA2 GTEMP12)))
␈↓ α,␈↓α               (* GTEMP12 is now the name of the new composition)
␈↓ α,␈↓α               (CREATEB␈↓	9␈↓α GTEMP12)
␈↓ α,␈↓α               [SETQ GUP1 (COND ((ISAG CS-B 'COMPOSE) CS-B)  (T 'COMPOSE]
␈↓ α,␈↓α               (* GUP1 is now the KIND of concept which GTEMP12 is to be an example of.
␈↓ α,␈↓α                   This will usually be "COMPOSE" or some variant of it. )
␈↓ α,␈↓α               [INCRB␈↓	10␈↓α GTEMP12 'DEFN
␈↓ α,␈↓α                (LIST 'TYPE 'APPLICATION 'OF GUP1
␈↓ α,␈↓α                 (APPEND (LIST 'APPLYB (Q␈↓	11␈↓α COMPOSE) (Q ALGS) (KWOTE BA1) (KWOTE BA2))
␈↓ α,␈↓α                         (FIRSTN (LENGTH (CAAR GTEMP11))  BA-LIST]
␈↓ α,␈↓α            (* Another way to fill in an entry for GTEMP12.Defn)
␈↓ α,␈↓α            (COND
␈↓ α,␈↓α              ([SETQ GTEMP308 (CAR (SOME (EXS COMPOSE)
␈↓ α,␈↓α                                 (FUNCTION (LAMBDA (C)
␈↓ α,␈↓α                                     (MEMBER (LASTELE (GETB GTEMP12 'DEFN))
␈↓ α,␈↓α                                             (GETB (LASTELE C)  'DEFN]
␈↓ α,␈↓α                (FORGET-CONCEPT GTEMP12)
␈↓ α,␈↓α                (CPRIN1S 8 GTEMP12 turned out to be equivalent to GTEMP308 DCR)␈↓	12␈↓α
␈↓ α,␈↓α                GTEMP308)
␈↓ α,␈↓α              (T (INCRB GUP1 'EXS (NCONC1 (GEARGS GUP1)  GTEMP12))
␈↓ α,␈↓α                 [SOME (RIPPLE GUP1 'GENL)
␈↓ α,␈↓α                       (FUNCTION (LAMBDA (G)
␈↓ α,␈↓α                           (SOME (GETB G 'D-R)
␈↓ α,␈↓α                                 (FUNCTION (LAMBDA (D)
␈↓ α,␈↓α                                     (AND (ISA BA1 (CAR D))
␈↓ α,␈↓α                                          (ISA BA2 (CADR D))
␈↓ α,␈↓α                                          (INCRB GTEMP12 'UP␈↓	13␈↓α (CADDR D))
␈↓ α,␈↓α                                          (INCRB (CADDR D) 'EXS GTEMP12]
␈↓ α,␈↓α          (* This last INCRB says that if an operation f maps onto range C,
␈↓ α,␈↓α           and we apply f and get a new Being, then that Being ISA C)␈↓	14␈↓α
␈↓ α,␈↓α                 (INCRB GTEMP12 'IN-RAN-OF GUP1)
␈↓ α,␈↓α                 (INCRB BA2 'IN-DOM-OF GUP1)
␈↓ α,␈↓α                 (INCRB BA1 'IN-DOM-OF GUP1)
␈↓ α,␈↓α                 (* Now see if the composition GTEMP12 shares any ISA's entries with
␈↓ α,␈↓α                        either constituent operation: BA1 or BA2)␈↓	15␈↓α
␈↓ α,␈↓α                 [MAPC [INTERSECTION (SET-DIFF [UNION (GETB BA1 'UP) (GETB BA2 'UP]
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	8␈↓ε An abbreviation for (APPLYB 'ANY-CONCEPT 'DEFN BA1); i.e., test whether BA1 is a bona fide concept or not.
␈↓ α,␈↓ε␈↓	9␈↓ε CREATEB is a function which sets up a new blank data structure for a new concept.
␈↓ α,␈↓ε␈↓	10␈↓ε The function call (INCRB C F X) means: add entry X to the F facet of concept C.
␈↓ α,␈↓ε␈↓	11␈↓ε␈α	The␈α	LISP␈α	function␈α	"Q"␈α	is␈α	like␈α	a␈α	double␈α	quote;␈α	after␈α	one␈α	evaluation␈α	(Q␈α	X)␈α	returns␈α	'X;␈α	after␈α	one␈α	more␈α	evaluation,␈α	'X␈α	returns
␈↓ α,␈↓ε␈↓ βLX; after a final evaluation, we get the VALUE of X.
␈↓ α,␈↓ε␈↓	12␈↓ε A conditional print statement. If the verbosity level is high enough (>8), this message is typed out to the user.
␈↓ α,␈↓ε␈↓	13␈↓ε The ISA's facet is called "UP" in the LISP program.
␈↓ α,␈↓ε␈↓	14␈↓ε This is a streamlined, specialized version of the more general heuristic rule number 154; see page 169.
␈↓ α,␈↓ε␈↓	15␈↓ε This next MAPC is thus the LISP encoding of heuristic rule number 177; see page 173.
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε131␈↓-

␈↓ α,␈↓α                                               (GETB GTEMP12 'UP]
␈↓ α,␈↓α                       (FUNCTION (LAMBDA (Z)
␈↓ α,␈↓α                           (COND
␈↓ α,␈↓α                             ((DEFN Z GTEMP12)
␈↓ α,␈↓α                               (INCRB Z 'EXS GTEMP12)
␈↓ α,␈↓α                               (INCRB GTEMP12 'UP Z]
␈↓ α,␈↓α                 (COND
␈↓ α,␈↓α                   [(GETB GTEMP12 'UP)
␈↓ α,␈↓α                     (SETB GTEMP12 'GUP (COPY (GETB GTEMP12 'UP]
␈↓ α,␈↓α                   (T (INCRB GTEMP12 'UP 'OPERATION)
␈↓ α,␈↓α                      (INCRB 'OPERATION 'EXS GTEMP12)))
␈↓ α,␈↓α                 & (* A similar search now for GENL/SPEC of the composition)
␈↓ α,␈↓α                 (SETB GTEMP12 'D-R (CAR GTEMP11))
␈↓ α,␈↓α                 (INCRB GTEMP12 'ALGS
␈↓ α,␈↓α                       (LIST 'TYPE 'NONRECURSIVE 'APPLICATION 'OF GUP1 (CADR GTEMP11)))
␈↓ α,␈↓α                 & (* Code for synthesizing a Defn entry for GTEMP12)
␈↓ α,␈↓α                 (SETB GTEMP12 'WORTH
␈↓ α,␈↓α                       (MAP2CAR (GETB BA1 'WORTH) (GETB BA2 'WORTH) 'TIMES1000))
␈↓ α,␈↓α                 (GS-CHECK␈↓	16␈↓α  GTEMP12]]))]

␈↓ α,␈↓α␈↓βAnglicised condensation:␈↓α
␈↓"␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Algorithms: ␈↓π                                                           ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Distributed: use the heuristics attached to Compose to guide the filling ␈↓π         ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬                    in of various facets of the new composition. ␈↓π                         ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬                    (The heuristics referred to are shown in Appendix 2.6, on page 173.) ␈↓π     ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Fillin: 5 ␈↓β(out of a total of 9)␈↓¬ heuristics. ␈↓π                                      ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Check: 1 heuristic ␈↓β(out of a total of 2)␈↓¬ ␈↓π                                      ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$


␈↓ α,␈↓α␈↓∧␈↓&UP␈↓)αβ␈↓α              (OPERATION)

␈↓ α,␈↓α␈↓βAnglicised condensation:␈↓α
␈↓"␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Isa's: Operation ␈↓π                                                        ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$


␈↓ α,␈↓α␈↓∧␈↓&WORTH␈↓)αβ␈↓α           (300)

␈↓ α,␈↓α␈↓βAnglicised condensation:␈↓α
␈↓"␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Worth: 300 ␈↓π                                                           ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	16␈↓ε␈α
This␈α
is␈α
a␈α
general-purpose␈α
function␈α
for␈α
testing␈α
that␈α
there␈α	is␈α
no␈α
hidden␈α
cycle␈α
in␈α
the␈α
Generalization␈α
network,␈α
that␈α
no␈α	two
␈↓ α,␈↓ε␈↓ βLconcepts␈α∞are␈α∞both␈α
generalizations␈α∞and␈α∞specializations␈α∞of␈α
each␈α∞other,␈α∞unless␈α
they␈α∞are␈α∞tagged␈α∞as␈α
being
␈↓ α,␈↓ε␈↓ βLequivalent to each other.
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε132␈↓-

␈↓ α,␈↓α␈↓∧␈↓&INT␈↓)αβ␈↓α␈↓	17␈↓α [(IMATRIX (1 2 3) (4 5))
␈↓ α,␈↓α (COND [(INTERSECTION (MAPAPPEND (GETB BA2 'D-R) 'LAST)
␈↓ α,␈↓α                      (MAPAPPEND (GETB BA1 'D-R) 'ALL-BUT-LAST))
␈↓ α,␈↓α        300
␈↓ α,␈↓α        (IDIFF 400 (ITIMES 100 (IPLUS (LENGTH (GETB BA1 'D-R))
␈↓ α,␈↓α                                      (LENGTH (GETB BA2 'D-R]
␈↓ α,␈↓α       (REASON (* In some interpretation, Range-of-op2 is 1 component of Domain-of-op1)))
␈↓ α,␈↓α (COND [[MEMB [CAR (LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α              (ALL-BUT-LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α        400
␈↓ α,␈↓α        (IDIFF 1000 (ITIMES 100 (LENGTH (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α       (REASON (* In canonical interpretation, Range-of-op2 is a component of Domain of op1)))
␈↓ α,␈↓α (COND [(INTERSECTION (GETB CS-B TIES)
␈↓ α,␈↓α                (UNION (GETB BA1 TIES)(GETB BA2 TIES)))
␈↓ α,␈↓α        100
␈↓ α,␈↓α        (ITIMES 100 [LENGTH (INTERSECTION (GETB CS-B TIES)
␈↓ α,␈↓α                                (UNION (GETB BA1 TIES)(GETB BA2 TIES])
␈↓ α,␈↓α        (REASON (* This composition preserves some good properties of its constituents))])
␈↓ α,␈↓α (COND [(SET-DIFFERENCE (GETB CS-B TIES)
␈↓ α,␈↓α                (UNION (GETB BA1 TIES)(GETB BA2 TIES)))
␈↓ α,␈↓α        100
␈↓ α,␈↓α        (ITIMES 100 [LENGTH (SET-DIFFERENCE (GETB CS-B TIES)
␈↓ α,␈↓α                                (UNION (GETB BA1 TIES)(GETB BA2 TIES])
␈↓ α,␈↓α        (REASON (* This composition has some new props, not true of either constituent))])
␈↓ α,␈↓α (COND [(OR (GREATERP (GETB BA1 'WORTH) 500))
␈↓ α,␈↓α            (GREATERP (GETB BA2 'WORTH) 500)))
␈↓ α,␈↓α        300
␈↓ α,␈↓α        (IQUOTIENT (ITIMES (GETB BA1 'WORTH)(GETB BA2 'WORTH))
␈↓ α,␈↓α                1000)
␈↓ α,␈↓α       (REASON (* Op1 and/or Op2 are very interesting themselves))])
␈↓ α,␈↓α (COND [[IS-ONE-OF [CAR (LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α                   (ALL-BUT-LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α        350
␈↓ α,␈↓α        (IDIFF [ITIMES 100 (IDIFF
␈↓ α,␈↓α                   [LENGTH (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α                   (LENGTH (RIPPLE [IS-ONE-OF
␈↓ α,␈↓α                                           [SETQ TMP4 (CAR (LAST (GETB BA2 'D-R]
␈↓ α,␈↓α                                           (ALL-BUT-LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α                                  'GENL]
␈↓ α,␈↓α               (ITIMES 50 (LENGTH (RIPPLE TMP4 'GENL]
␈↓ α,␈↓α       (REASON (* In canonical interpretation, Range-of-op2 is a specialization of a component
␈↓ α,␈↓α                  of Domain-of-op1)))
␈↓ α,␈↓α (COND [[MEMB [CAR (LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α              (ALL-BUT-LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α        450
␈↓ α,␈↓α        (IPLUS 300 (COND ([MEMB [CAR (LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α                                (ALL-BUT-LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α                          10)
␈↓ α,␈↓α                         (T 250))
␈↓ α,␈↓α               (COND ([MEMB [CAR (LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α                            (ALL-BUT-LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α                      11)
␈↓ α,␈↓α                     (T 250))
␈↓ α,␈↓α               (ITIMES 70 (LENGTH (RIPPLE [CAR (LAST (CAR (GETB BA1 'D-R] 'GENL]
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	17␈↓ε␈αNote␈α
that␈αalthough␈α
the␈αFillin␈αand␈α
Suggest␈αheuristics␈α
are␈αblended␈α
into␈αthe␈αrelevant␈α
facets␈α(e.g.,␈α
into␈αthe␈αAlgorithms␈α
for
␈↓ α,␈↓ε␈↓ βLCOMPOSE), the INTERESTINGNESS type heuristics are kept separate, in this facet.
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε133␈↓-

␈↓ α,␈↓α       (REASON (* In canonical interpretation,
␈↓ α,␈↓α                Range-of-op1 is one component of Domain-of-op2))
␈↓ α,␈↓α &
␈↓ α,␈↓α (COND [[ISA [CAR (LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α             (ALL-BUT-LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α        250
␈↓ α,␈↓α        (IPLUS 50 (COND ([ISA [CAR (LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α                              (ALL-BUT-LAST (CAR (GETB BA1 'D-R]
␈↓ α,␈↓α                         10)
␈↓ α,␈↓α                        (T 100))
␈↓ α,␈↓α               (COND ([ISA [CAR (LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α                           (ALL-BUT-LAST (CAR (GETB BA2 'D-R]
␈↓ α,␈↓α                      11)
␈↓ α,␈↓α                     (T 100))
␈↓ α,␈↓α               (ITIMES 50 (LENGTH (RIPPLE [CAR (LAST (CAR (GETB BA1 'D-R] 'GENL]
␈↓ α,␈↓α       (REASON (* Range-of-op1 is a specialization of a component of Domain-of-op2]

␈↓ α,␈↓α␈↓βAnglicised condensation:␈↓α
␈↓"␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Interest: 11 heuristics. ␈↓π                                                  ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            The heuristic rules encoded above are shown in English on page 175. ␈↓π          ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$


␈↓ α,␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓ \


␈↓ α,␈↓π␈↓βHere is the code for CON-MERGE-ARGS, the function which decides how to overlap
␈↓ α,␈↓βthe domain/range facets of its two arguments, F1 and F2:␈↓π


␈↓ α,␈↓ε(CON-MERGE-ARGS
␈↓ α,␈↓ε  [LAMBDA (F1 F2 F12 PGM1 SCHK SAPL DOM1 DOM2 RAN1 RAN2 TIL DOM3)
␈↓ α,␈↓ε    [SETQ RAN1 (LAST (CAR (GETB F1 'D-R]
␈↓ α,␈↓ε    (SETQ DOM1 (LDIFF (CAR (GETB F1 'D-R))
␈↓ α,␈↓ε                      RAN1))
␈↓ α,␈↓ε    [SETQ RAN2 (LAST (CAR (GETB F2 'D-R]
␈↓ α,␈↓ε    (SETQ DOM2 (LDIFF (CAR (GETB F2 'D-R))
␈↓ α,␈↓ε                      RAN2))
␈↓ α,␈↓ε    [SETQ DOM3 (AND (CDR DOM1)
␈↓ α,␈↓ε                        (LIST (CADR (MIN2 (APPEND RAN2 RAN2 RAN2
␈↓ α,␈↓ε                                        RAN2) DOM1 'FRAC-OVERLAP]
␈↓ α,␈↓ε    (* As DOMi and RANi are located, Switching of Args may be required, inside PGM1)
␈↓ α,␈↓ε    (AND (MEMB (CAR DOM3) DOM2) (SETQ DOM3 NIL))
␈↓ α,␈↓ε    (SETQ GTEMP20 (LENGTH DOM2))
␈↓ α,␈↓ε    [SETQ SAPL (NCONC (LIST 'APPLYB (KWOTE F1) (Q ALGS))
␈↓ α,␈↓ε                      (MAPCAR (SUB-ONCE 'X
␈↓ α,␈↓ε                                        [SETQ GTEMP19 (COND
␈↓ α,␈↓ε                                            ((IS-ONE-OF (CAR RAN2) DOM1))
␈↓ α,␈↓ε                                            [(SETQ SCHK (ONE-ISAG DOM1 (CAR RAN2]
␈↓ α,␈↓ε                                            ((SETQ SCHK (AND (SETQ TIL (EXS (CAR RAN2))
␈↓ α,␈↓ε                                                             (CAR (SOME DOM1 (FUNCTION (LAMBDA (D)
␈↓ α,␈↓ε                                                                            (INTERSECTION
␈↓ α,␈↓ε                                                                                TIL
␈↓ α,␈↓ε                                                                                 (EXS D]
␈↓ α,␈↓ε                                        DOM1)
␈↓ α,␈↓ε                              (FUNCTION (LAMBDA (Z)
␈↓ α,␈↓ε                                  (COND
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε134␈↓-

␈↓ α,␈↓ε                                    ((EQ Z 'X)
␈↓ α,␈↓ε                                      'X)
␈↓ α,␈↓ε                                    (T (SETQ GTEMP20 (ADD1 GTEMP20))
␈↓ α,␈↓ε                                       (CAR (FNTH BA-LIST GTEMP20]
␈↓ α,␈↓ε          (* SCHK is a flag which means that f2 maps us into an element of RAN2 which is not guaranteed
␈↓ α,␈↓ε          a priori to be an element of DOM1, hence a check for this applicability of f1 will then have to be made)
␈↓ α,␈↓ε    (COND
␈↓ α,␈↓ε      ((FMEMB 'X SAPL)
␈↓ α,␈↓ε        (SETQ DOM3 (REM-ONCE GTEMP19 DOM1))
␈↓ α,␈↓ε        (SETQ GTEMP7 (APPEND DOM3 DOM2))
␈↓ α,␈↓ε        [COND
␈↓ α,␈↓ε          [(NEQ (LENGTH GTEMP7)
␈↓ α,␈↓ε                (LENGTH (SELF-INT GTEMP7)))
␈↓ α,␈↓ε            (CPRIN1S 9 CRLF CRLF AM can later coalesce the D-R of F12 DCR)
␈↓ α,␈↓ε            [ADD-CANDS (LIST (LIST (LIST 'APPLYB (Q COALESCE) (Q ALGS) (KWOTE F12))
␈↓ α,␈↓ε                                   (IPLUS 100 (IQUO (DOTPROD (FIRSTN 2 (GETB F1 'WORTH))
␈↓ α,␈↓ε                                                             (GETB F2 'WORTH)) 2000))
␈↓ α,␈↓ε                                   (LIST (SPLIST There is an overlap in the new combined
␈↓ α,␈↓ε                                                 domain of the operation F12]
␈↓ α,␈↓ε            (SWHY 9 (There is an obvious overlap in (@ GTEMP7),the new combined domain of (@ F12]
␈↓ α,␈↓ε␈↓βThe next piece of this function is the heuristic rule numbered 186 in Appendix 2.␈↓ε
␈↓ α,␈↓ε          ([SOME GTEMP7 (FUNCTION (LAMBDA (X)
␈↓ α,␈↓ε                     (IS-ONE-OF X (CDR (FMEMB X GTEMP7]
␈↓ α,␈↓ε            (CPRIN1S 10 CRLF CRLF AM may later coalesce the D-R of F12 DCR)
␈↓ α,␈↓ε            [ADD-CANDS (LIST (LIST (LIST 'APPLYB (Q COALESCE) (Q ALGS) (KWOTE F12))
␈↓ α,␈↓ε                                   (IQUO (DOTPROD (FIRSTN 2 (GETB F1 'WORTH))
␈↓ α,␈↓ε                                                  (GETB F2 'WORTH))   2500))
␈↓ α,␈↓ε                                   (LIST (SPLIST There may be an overlap
␈↓ α,␈↓ε                                            in the new combined domain of the operation F12]
␈↓ α,␈↓ε            (SWHY 10 (There is a subtle overlap in (@ GTEMP7),the new combined domain of (@ F12]
␈↓ α,␈↓ε        [SETQ PGM1 (LIST 'PROG
␈↓ α,␈↓ε                         (LIST 'X)
␈↓ α,␈↓ε                         [LIST 'SETQ 'X
␈↓ α,␈↓ε                               (NCONC (LIST 'APPLYB (KWOTE F2) (Q ALGS))
␈↓ α,␈↓ε                                      (FIRSTN (LENGTH DOM2) (LIST 'BA1 'BA2 'BA3]
␈↓ α,␈↓ε                         (LIST 'RETURN
␈↓ α,␈↓ε                               (COND
␈↓ α,␈↓ε                                 (SCHK (LIST 'AND
␈↓ α,␈↓ε                                             (LIST 'APPLY* (Q DEFN) (KWOTE SCHK) 'X)
␈↓ α,␈↓ε                                             SAPL))
␈↓ α,␈↓ε                                 (T (LIST 'AND 'X SAPL]
␈↓ α,␈↓ε        (LIST (LIST (APPEND DOM2 DOM3 RAN1)) PGM1))
␈↓ α,␈↓ε      (T (* Composing is not possible)   NIL])









␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 1.1.2.␈↓)αβ␈↓␈↓↓␈↓& The `Osets' Concept␈↓)αβ␈↓

␈↓ α,␈↓Here␈αis␈αthe␈αactual␈α
property␈αlist␈αof␈αthe␈αdata-structure␈α
corresponding␈αto␈αthe␈αOsets␈α
concept,␈αat
␈↓ α,␈↓the time AM is started:
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε135␈↓-



␈↓ α,␈↓α␈↓∧␈↓&ENGN␈↓)αβ␈↓α (OSET Oset Oset-structure OSET-STRUC, Ordered-set (Set))
␈↓ α,␈↓α␈↓∧␈↓&DEFN␈↓)αβ␈↓α  (TYPE NEC&SUFF RECURSIVE TRANSPARENT [COND
␈↓ α,␈↓α                    ((EQUAL BA1 (OSET )) T)
␈↓ α,␈↓α                    (T (APPLYB 'OSET 'DEFN (APPLYB 'OSET-DELETE 'ALGS
␈↓ α,␈↓α                                                        (APPLYB 'SOME-MEMB 'ALGS BA1)
␈↓ α,␈↓α                                                        BA1])
␈↓ α,␈↓α              (TYPE NEC&SUFF RECURSIVE QUICK [COND
␈↓ α,␈↓α                    ((EQUAL BA1 '(OSET )) T)
␈↓ α,␈↓α                    ((CDDR BA1) (APPLYB 'OSET 'DEFN (RPLACD BA1 (CDDR BA1)))
␈↓ α,␈↓α                    (T NIL])
␈↓ α,␈↓α              (TYPE NEC&SUFF NONRECURSIVE QUICK (MATCH BA1 WITH ('OSET $)))
␈↓ α,␈↓α␈↓∧␈↓&GENL␈↓)αβ␈↓α    (ORD-STRUC NO-MULT-ELES-STRUC)
␈↓ α,␈↓α␈↓∧␈↓&WORTH␈↓)αβ␈↓α     (400)
␈↓ α,␈↓α␈↓∧␈↓&IN-DOM-OF␈↓)αβ␈↓α (OSET-JOIN OSET-INTERSECT OSET-DIFF OSET-INSERT OSET-DELETE)
␈↓ α,␈↓α␈↓∧␈↓&IN-RAN-OF␈↓)αβ␈↓α (OSET-JOIN OSET-INTERSECT OSET-DIFF OSET-INSERT OSET-DELETE)
␈↓ α,␈↓α␈↓∧␈↓&VIEW␈↓)αβ␈↓α    (STRUCTURE (RPLACA BA1 'OSET))



␈↓ α,␈↓Compare this with the following Anglicised condensation:

␈↓"␈↓ α,␈↓π␈↓ α<⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L⊃
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Name(s): Oset, Oset-structure, Ordered-set, sometimes: Set. ␈↓π                      ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Definitions: ␈↓π                                                           ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Recursive: λ (S) (S=[ ] or Oset.Definition(Oset-Delete.Alg(Member.Alg(S),S))) ␈↓π    ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Recursive quick: λ (S) (S=[ ] or Oset.Definition (CDR(S))) ␈↓π                   ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬            Quick: λ (S) (Match S with [...] ) ␈↓π                                      ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Generalizations: Ordered-Structure, No-multiple-elements-Structure ␈↓π                ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    Worth: 400 ␈↓π                                                           ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    In-domain-of: Oset-union, Oset-intersect, Oset-difference, Oset-insert, Oset-delete ␈↓π     ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    In-range-of: Oset-union, Oset-intersect, Oset-difference, Oset-insert, Oset-delete ␈↓π      ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<} ␈↓¬    View:  To view any structure as a Oset, do: λ (x) Enclose-in-square-brackets(x) ␈↓π       ␈↓ L}
␈↓"␈↓ α,␈↓π␈↓ α<%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ L$
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε136␈↓-

␈↓ α,␈↓␈↓ ∧D␈↓↓␈↓&Appendix 1.2. ␈↓)αβ␈↓∧␈↓& Concepts created by AM␈↓)αβ␈↓↓

␈↓ α,␈↓The␈α∞list␈α∞below␈α∞is␈α∂not␈α∞meant␈α∞as␈α∞an␈α∞exhaustive␈α∂catalogue,␈α∞but␈α∞rather␈α∞merely␈α∞to␈α∂suggest␈α∞the
␈↓ α,␈↓breadth␈α∩of␈α∩AM's␈α∩syntheses.␈α∩ The␈α∩concepts␈α∩are␈α∩listed␈α∩in␈α∩the␈α∩order␈α∩in␈α∩which␈α∩they␈α⊃were
␈↓ α,␈↓de≡ned.␈↓	20␈↓␈α
In␈α
place␈α
of␈αthe␈α
(usually-awkward)␈α
name␈α
chosen␈α
by␈αAM,␈α
I␈α
have␈α
given␈α
either␈αthe
␈↓ α,␈↓standard math/English name for the concept, or else a short description of what it is.

␈↓ α,␈↓Sets with less than 2 elements (singletons and empty sets).
␈↓ α,␈↓Sets with no atomic elements (nests of braces).
␈↓ α,␈↓Singleton sets.
␈↓ α,␈↓Bags containing (multiple occurrences of) just one kind of element.
␈↓ α,␈↓Superset (contains).
␈↓ α,␈↓Doubleton bags and sets.
␈↓ α,␈↓Set-membership.
␈↓ α,␈↓Disjoint bags.
␈↓ α,␈↓Subset.
␈↓ α,␈↓Disjoint sets.
␈↓ α,␈↓Singleton osets.
␈↓ α,␈↓Same-length (same number of elements).
␈↓ α,␈↓Same number of left parentheses, plus identical leftmost atoms.
␈↓ α,␈↓Count (≡nd the number of elements of a given structure).
␈↓ α,␈↓Numbers (unary representation).
␈↓ α,␈↓Add.
␈↓ α,␈↓Minimum.
␈↓ α,␈↓SUB1 (λ (x) x-1).
␈↓ α,␈↓Insert x into a given Bag-of-T's (almost ADD1, but not quite).
␈↓ α,␈↓Subtract (except: if x<y, then the result of x-y will be zero␈↓	21␈↓).
␈↓ α,␈↓Less than or equal to.
␈↓ α,␈↓Times.
␈↓ α,␈↓Union of a ␈↓βbag␈↓ of structures.
␈↓ α,␈↓& (the ampersand represents the creation of several real losers.)
␈↓ α,␈↓Compose a given operation F with itself (form F␈↓εo␈↓F).
␈↓ α,␈↓Insert structure S into itself.
␈↓ α,␈↓Try to delete structure S from itself (a loser).
␈↓ α,␈↓Double (add `x' to itself).
␈↓ α,␈↓Subtract `x' from itself (as an operation, this is a real zero␈↓	22␈↓).
␈↓ α,␈↓Square (TIMES(x,x)).
␈↓ α,␈↓Union structure S with itself.
␈↓ α,␈↓Coalesced-replace2: replace each element s of S by F(s,s).
␈↓ α,␈↓Coalesced-join2: append together F(s,s), for each member s␈↓¬ε␈↓S.
␈↓ α,␈↓Coa-repeat2: create a new op which takes a struc S, op F, and repeats F(s,t,S) all along S.
␈↓ α,␈↓Compose three operations: λ(F,G,H) F␈↓εo␈↓(G␈↓εo␈↓H).
␈↓ α,␈↓Compose three operations: λ(F,G,H) (F␈↓εo␈↓G)␈↓εo␈↓H.
␈↓ α,␈↓& (lots of losing compositions created, e.g. Self-insert␈↓εo␈↓Set-union.)
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	18␈↓ε See Section 6.1.84
␈↓ α,␈↓ε␈↓	19␈↓ε This is "natural-number subtract", in the same spirit of naming as we find for "Integer division".
␈↓ α,␈↓ε␈↓	20␈↓ε a Natural zero?
␈↓ α,␈↓␈↓εAppendix 1␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε137␈↓-

␈↓ α,␈↓ADD␈↓	-1␈↓(x): all ways of representing x as the sum of a bunch of nonzero numbers.
␈↓ α,␈↓G␈↓εo␈↓H, s.t. H(G(H(x))) is always de≡ned (wherever H is), and G and H are interesting.
␈↓ α,␈↓Insert␈↓εo␈↓Delete.
␈↓ α,␈↓Delete␈↓εo␈↓Insert.
␈↓ α,␈↓Size␈↓εo␈↓ADD␈↓	-1␈↓. (λ (n) The number of ways to partition n)
␈↓ α,␈↓Cubing
␈↓ α,␈↓&
␈↓ α,␈↓Exponentiation.
␈↓ α,␈↓Halving  (in natual numbers only; thus Halving(15)=7).
␈↓ α,␈↓Even numbers.
␈↓ α,␈↓Integer square-root.
␈↓ α,␈↓Perfect squares.
␈↓ α,␈↓Divisors-of.
␈↓ α,␈↓Numbers-with-0-divisors.
␈↓ α,␈↓Numbers-with-1-divisor.
␈↓ α,␈↓Primes (Numbers-with-2-divisors).
␈↓ α,␈↓Squares of primes (Numbers-with-3-divisors).
␈↓ α,␈↓Squares of squares of primes.
␈↓ α,␈↓Square-roots of primes (a loser).
␈↓ α,␈↓TIMES␈↓	-1␈↓(x): all ways of representing x as the product of a bunch of numbers (>1).
␈↓ α,␈↓All ways of representing x as the product of just one number (a trivial notion).
␈↓ α,␈↓All ways of representing x as the product of primes.
␈↓ α,␈↓All ways of representing x as the sum of primes.
␈↓ α,␈↓All ways of representing x as the sum of two primes.
␈↓ α,␈↓Numbers uniquely representable as the sum of two primes.
␈↓ α,␈↓Products of squares.
␈↓ α,␈↓Multiplication by 1.
␈↓ α,␈↓Multiplication by 0.
␈↓ α,␈↓Multiplication by 2.
␈↓ α,␈↓Addition of 0.
␈↓ α,␈↓Addition of 1.
␈↓ α,␈↓Addition of 2.
␈↓ α,␈↓Product of even numbers.
␈↓ α,␈↓Sum of squares.
␈↓ α,␈↓Sum of even numbers.
␈↓ α,␈↓& (losers: various compositions of 3 operations.)
␈↓ α,␈↓Pairs of perfect squares whose sum is also a perfect square (x␈↓#
2␈↓#+y␈↓#
2␈↓#=z␈↓#
2␈↓#).
␈↓ α,␈↓Prime pairs (p,p+2 are prime).
␈↓ α,␈↓ ␈↓π # # #␈↓
␈↓ α,␈↓␈↓ ≥-␈↓ε138␈↓-

␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬↓␈↓∧Appendix 2.  AM's Heuristics␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓



␈↓ α,␈↓β␈↓ α|Infallible␈α∞rules␈α∂of␈α∞discovery␈α∞leading␈α∂to␈α∞the␈α∞solution␈α∂of␈α∞all␈α∂possible␈α∞mathematical
␈↓ α,␈↓β␈↓ α|problems␈α∞would␈α∞be␈α∞more␈α∞desirable␈α∂than␈α∞the␈α∞philosophers'␈α∞stone,␈α∞vainly␈α∂sought␈α∞by
␈↓ α,␈↓β␈↓ α|the␈αalchemists.␈αSuch␈αrules␈αwould␈αwork␈αmagic;␈αbut␈αthere␈αis␈αno␈αsuch␈αthing␈αas␈α
magic.
␈↓ α,␈↓β␈↓ α|To␈α≡nd␈αunfailing␈αrules␈αapplicable␈αto␈αall␈αsorts␈αof␈αproblems␈αis␈αan␈αold␈αphilosophical
␈↓ α,␈↓β␈↓ α|dream; but this dream will never be more than a dream.

␈↓ α,␈↓¬␈↓ ε\-- Polya




␈↓ α,␈↓β␈↓ α|To␈αthe␈αextent␈αthat␈αa␈αprofessor␈αof␈αmusic␈αat␈αa␈αconservatoire␈αcan␈αassist␈αhis␈αstudents
␈↓ α,␈↓β␈↓ α|in␈α
becoming␈α
familiar␈α
with␈αthe␈α
patterns␈α
of␈α
harmony␈αand␈α
rhythm,␈α
and␈α
with␈αhow␈α
they
␈↓ α,␈↓β␈↓ α|combine,␈αit␈αmust␈αbe␈αpossible␈αto␈αassist␈αstudents␈αin␈αbecoming␈αsensitive␈αto␈αpatterns␈αof
␈↓ α,␈↓β␈↓ α|reasoning and how they combine. The analogy is not far-fetched at all

␈↓ α,␈↓¬␈↓ ε\-- Dijkstra



␈↓ α,␈↓This␈α∩appendix␈α∩lists␈α⊃all␈α∩the␈α∩heuristics␈α⊃with␈α∩which␈α∩AM␈α⊃is␈α∩initially␈α∩provided.␈α∩ They␈α⊃are
␈↓ α,␈↓organized␈αby␈αconcept,␈αmost␈αgeneral␈αconcepts␈α≡rst.␈α Within␈αa␈αconcept,␈αthey␈αare␈αorganized␈α
into
␈↓ α,␈↓four groups:
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ Fillin: rules for ≡lling in new entries on various facets.
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ Check: rules for patching up existing entries on various facets.
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ Suggest: rules which propose new tasks to break AM out of stagnant loops.
␈↓ α,␈↓␈↓ αl␈↓π#␈↓ Interest: criteria for estimating the interestingness of various entities.

␈↓ α,␈↓Each␈α∂heuristic␈α∞is␈α∂presented␈α∂in␈α∞English␈α∂translation.␈α∞Whenever␈α∂there␈α∂is␈α∞a␈α∂very␈α∂tricky,␈α∞non-
␈↓ α,␈↓obvious,␈α
or␈α
brilliant␈α
translation␈α
of␈α
some␈α
English␈α
clause␈α
into␈α
LISP,␈α
a␈α
brief␈α
note␈α∞will␈α
follow
␈↓ α,␈↓about␈αhow␈α
that␈αis␈α
coded.␈α Also␈α
given␈α(usually)␈α
are␈αsome␈α
example(s)␈αof␈α
its␈αuse,␈α
and␈αits␈α
overall
␈↓ α,␈↓importance.  Concepts which have no heuristics are not present in this appendix.

␈↓ α,␈↓Hundreds␈α∂of␈α∂heuristics␈α∂were␈α∂planned␈α∂on␈α∂paper␈α∂but␈α∂never␈α∂coded␈α∂(e.g.,␈α∂those␈α∂dealing␈α∞with
␈↓ α,␈↓proof␈α∃techniques,␈α∃those␈α∃dealing␈α∃with␈α∀the␈α∃drives␈α∃and␈α∃rewards␈α∃of␈α∃generalized␈α∀message
␈↓ α,␈↓senders/receivers),␈αand␈αwhole␈αclasses␈αof␈αrules␈αwere␈αcoded␈αbut␈αnever␈αused␈αby␈αAM␈αduring␈αany
␈↓ α,␈↓of␈α⊂its␈α⊂runs␈α⊂(e.g.,␈α⊂how␈α⊂to␈α⊂deal␈α⊂with␈α⊂contradictions,␈α⊂how␈α⊂to␈α⊂deal␈α⊂with␈α⊂Intu's␈α⊃facets).␈α⊂ Such
␈↓ α,␈↓super∨uous␈αrules␈αwill␈αnot␈αbe␈αincluded␈αhere.␈α They␈αwould␈αraise␈αthe␈αtotal␈αnumber␈αof␈αheuristic
␈↓ α,␈↓rules from 242 to about 500.

␈↓ α,␈↓The␈αrule␈αnumbering␈αin␈αthis␈αAppendix␈αis␈αreferred␈αto␈αoccasionally␈αin␈αother␈αappendices.␈α The
␈↓ α,␈↓total␈α∂number␈α∂of␈α∂rules␈α∂coded␈α∂in␈α∂AM␈α⊂is␈α∂actually␈α∂higher,␈α∂since␈α∂many␈α∂rules␈α∂are␈α⊂present␈α∂but
␈↓ α,␈↓never␈α∞used,␈α∞and␈α∂since␈α∞many␈α∞rules␈α∞listed␈α∂with␈α∞one␈α∞number␈α∞here␈α∂are␈α∞really␈α∞␈↓βseveral␈↓␈α∂rules␈α∞in
␈↓ α,␈↓LISP (e.g., see rules 97 and 129).
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε139␈↓-

␈↓ α,␈↓␈↓ βP␈↓↓␈↓&Appendix 2.1. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Anything␈↓)αβ␈↓↓

␈↓ α,␈↓All␈αthese␈αrules␈αdeal␈αwith␈αany␈αitem␈αX,␈αbe␈αit␈αconcept,␈αatom,␈αevent,␈αetc.␈α These␈αrules␈αare␈αabout
␈↓ α,␈↓as general ¬ and as ␈↓βweak␈↓ ¬ as one can imagine.


␈↓ α,␈↓∧␈↓ α<Anything . Suggest

␈↓ α,␈↓¬1. If AM has recently referenced entity X,
␈↓ α,␈↓¬␈↓ αLThen boost the priority of any tasks involving X.


␈↓ α,␈↓¬2. If the user has recently referred to X,
␈↓ α,␈↓¬␈↓ αLThen boost the priority of any tasks involving X.

␈↓ α,␈↓The␈α
above␈α
two␈α
rules␈α
simply␈α
rea≠rm␈α
the␈α
idea␈α
of␈α
"focus␈α
of␈α
attention".␈α
 The␈α
boost␈α
in␈αratings␈α
is
␈↓ α,␈↓only␈αslight,␈αand␈αonly␈αtemporary␈α(it␈αdecays␈αtoward␈αzero␈αexponentially␈αwith␈αtime).␈αBesides␈αthis
␈↓ α,␈↓gradual␈α∂decline␈α∂in␈α∂task␈α∂ratings,␈α∂the␈α∂rule␈α∂below␈α∂explicitly␈α∂modulates␈α∂this␈α∂boosting,␈α∂so␈α∞that
␈↓ α,␈↓in≡nite loops can be avoided.

␈↓ α,␈↓¬3. If AM has recently dealt with X with poor results,
␈↓ α,␈↓¬␈↓ αLThen lower the priority rating of all tasks involving X.


␈↓ α,␈↓¬4. If AM just referenced X and almost succeeded, but not quite,
␈↓ α,␈↓¬␈↓ αLThen look for a very similar entity Y, and retry the activity with Y in place of X.

␈↓ α,␈↓There␈α∞is␈α∞a␈α∞separate␈α∂precise␈α∞meaning␈α∞for␈α∞"almost␈α∂succeed",␈α∞"similar␈α∞entity",␈α∞and␈α∂"retry"␈α∞for
␈↓ α,␈↓each␈αkind␈αof␈αentity␈αand␈αactivity␈αthat␈αmight␈αbe␈αinvolved.␈αFor␈αexample,␈αif␈αthe␈αactivity␈αwere␈αa
␈↓ α,␈↓␈↓βtask␈↓␈α
(say␈α
to␈α
≡ll␈α
in␈α
examples␈α
of␈α
Odd-primes)␈α
and␈α
the␈α
entity␈α
X␈α
were␈α
a␈α
␈↓βconcept␈↓␈α
(in␈α
this␈αcase,
␈↓ α,␈↓Odd-primes),␈α
then␈αa␈α
`similar␈α
entity'␈αmight␈α
be␈α
the␈αconcept␈α
Odd-numbers,␈α
and␈αin␈α
that␈αcase␈α
the
␈↓ α,␈↓result␈αof␈αthis␈αrule␈αwould␈αbe␈α␈↓βa␈αnew␈αtask␈↓␈α(to␈α≡ll␈αin␈αexamples␈αof␈αOdd-numbers).␈α If␈αthe␈αfailure
␈↓ α,␈↓occurred␈α∞while␈α∞AM␈α∞was␈α
trying␈α∞to␈α∞access␈α∞the␈α
examples␈α∞facet␈α∞of␈α∞Primes,␈α∞with␈α
X=Examples,
␈↓ α,␈↓then␈α∂a␈α∂`similar␈α∂entity'␈α∂might␈α∂be␈α⊂the␈α∂Boundary-examples␈α∂facet,␈α∂and␈α∂the␈α∂above␈α⊂rule␈α∂would
␈↓ α,␈↓suggest␈α
that␈α
AM␈α
access␈αinstead␈α
the␈α
Boundary-examples␈α
facet␈αof␈α
Primes.␈α
 Of␈α
course,␈αthis␈α
rule
␈↓ α,␈↓is so weak that it is not often of much help.

␈↓ α,␈↓¬5.␈α
If␈αspace␈α
is␈α
running␈αout,␈α
and␈α
AM␈αhas␈α
not␈α
referenced␈αX␈α
for␈α
a␈αlong␈α
time,␈α
and␈αX␈α
is␈α
taking␈αup␈α
a␈αlot␈α
of
␈↓ α,␈↓¬␈↓ β,space, and no important conjectures reference X,
␈↓ α,␈↓¬␈↓ αLThen␈α
X␈αmay␈α
be␈α
forgotten␈αand␈α
its␈αspace␈α
liberated.␈α
 Probably␈αthe␈α
user␈α
should␈αbe␈α
informed␈αof␈α
this,
␈↓ α,␈↓¬␈↓ β,at least tersely.

␈↓ α,␈↓Just a general-purpose directive for emergency garbage-collection.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε140␈↓-

␈↓ α,␈↓∧␈↓ α<Anything . Interest

␈↓ α,␈↓¬6. Any entity X is interesting if it is referred to in several interesting conjectures.


␈↓ α,␈↓¬7.␈α∞Any␈α
entity␈α∞X␈α∞is␈α
interesting␈α∞if␈α∞it␈α
is␈α∞related␈α∞(via␈α
a␈α∞rare,␈α∞interesting␈α
relation)␈α∞to␈α∞another␈α
entity
␈↓ α,␈↓¬␈↓ β,which arose in a very different way and is not obviously tied to X.

␈↓ α,␈↓Unexpected␈α
connections␈α
are␈α
worth␈αcloser␈α
examination,␈α
typically.␈α
 X␈αmight␈α
be␈α
`related␈α
to'␈αY
␈↓ α,␈↓because␈α⊂F(X)=Y␈α⊂(for␈α∂some␈α⊂very␈α⊂interesting␈α∂operation␈α⊂F),␈α⊂because␈α∂Y(X)␈α⊂is␈α⊂true␈α⊂(for␈α∂some
␈↓ α,␈↓rarely-satis≡ed␈αpredicate␈α
Y),␈αbecause␈α
some␈αconjecture␈α
involving␈αX␈α
is␈αsyntactically␈αidentical␈α
to
␈↓ α,␈↓the same conjecture involving Y, etc.

␈↓ α,␈↓¬8.␈α
Entity␈αX␈α
is␈α(tentatively)␈α
interesting␈αif␈α
there␈αis␈α
an␈αanalogy␈α
in␈αwhich␈α
X␈αcorresponds␈α
to␈αY,␈α
and␈αY
␈↓ α,␈↓¬␈↓ β,has turned out to be very interesting.


␈↓ α,␈↓¬9. If entity X is an example of concept C, and X satisfies some features on C.Int,
␈↓ α,␈↓¬␈↓ αLThen X is interesting, and C's Interestingness features will indicate a numeric rating for X.

␈↓ α,␈↓This is practically the de≡niton of the Int facet. Below is a much more ususual rule:

␈↓ α,␈↓¬10.␈αIf␈α
entity␈αX␈α
is␈αan␈α
example␈αof␈α
concept␈αC,␈α
and␈αX␈α
satisfies␈αabsolutely␈α
none␈αof␈α
the␈αfeatures␈αon␈α
C.Int,
␈↓ α,␈↓¬␈↓ β,and X is just about the only C which doesn't satisfy something,
␈↓ α,␈↓¬␈↓ αLThen X is interesting because of its unusual boringness.

␈↓ α,␈↓Since␈α
most␈αsingletons␈α
are␈αinteresting␈α
because␈αall␈α
pairs␈α
of␈αtheir␈α
elements␈αare␈α
Equal,␈αthe␈α
above
␈↓ α,␈↓rule␈α∞says␈α∞it␈α∞would␈α
be␈α∞interesting␈α∞actually␈α∞to␈α
≡nd␈α∞a␈α∞singleton␈α∞for␈α
which␈α∞␈↓βnot␈↓␈α∞all␈α∞pairs␈α∞of␈α
its
␈↓ α,␈↓members␈α∞were␈α∞equal.␈α∞While␈α∞it␈α∞would␈α∞be␈α
interesting,␈α∞AM␈α∞has␈α∞very␈α∞little␈α∞chance␈α∞of␈α
≡nding
␈↓ α,␈↓such a critter.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε141␈↓-

␈↓ α,␈↓␈↓ β5␈↓↓␈↓&Appendix 2.2. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Any-concept␈↓)αβ␈↓↓

␈↓ α,␈↓This␈α∞concept␈α∞has␈α∂a␈α∞huge␈α∞number␈α∞of␈α∂heuristics.␈α∞For␈α∞that␈α∞reason,␈α∂I␈α∞have␈α∞partitioned␈α∂o≥␈α∞¬
␈↓ α,␈↓both here and in AM itself␈↓	1␈↓ ¬ the heuristics which apply to each kind of facet.


␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.1.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for any facet of Any-concept␈↓)αβ␈↓

␈↓ α,␈↓The␈α⊂≡rst␈α⊂set␈α∂of␈α⊂heuristics␈α⊂we'll␈α⊂look␈α∂at␈α⊂are␈α⊂very␈α⊂general,␈α∂applying␈α⊂to␈α⊂no␈α⊂particular␈α∂facet
␈↓ α,␈↓exactly.


␈↓ α,␈↓∧␈↓ α<Any-concept . Fillin

␈↓ α,␈↓¬11. When trying to fill in facet F of concept C, for any C and F,
␈↓ α,␈↓¬␈↓ α\If C is analogous to concept X, and X.F has some entries,
␈↓ α,␈↓¬␈↓ α\Then␈αtry␈αto␈αconstruct␈αthe␈αanalogs␈αof␈αthose␈αentries,␈αand␈αsee␈αif␈αthey␈αare␈αreally␈αvalid␈αentries␈αfor
␈↓ α,␈↓¬␈↓ β,C.F.

␈↓ α,␈↓Recall␈α
that␈α
"C.F"␈αis␈α
shorthand␈α
for␈α"facet␈α
F␈α
of␈α
concept␈αC".␈α
This␈α
rule␈αsimply␈α
says␈α
that␈α
if␈αan
␈↓ α,␈↓analogy␈αexists␈αbetween␈αtwo␈αconcepts␈αC␈αand␈αX,␈αthen␈αit␈αmay␈αbe␈αstrong␈αenough␈αto␈αmap␈αentries
␈↓ α,␈↓on␈α∂X.F␈α∂into␈α∂entries␈α∂for␈α⊂C.F.␈α∂ Note␈α∂that␈α∂F␈α∂can␈α∂be␈α⊂any␈α∂given␈α∂facet.␈α∂ There␈α∂is␈α⊂an␈α∂analogy
␈↓ α,␈↓between␈αSets␈αand␈αBags,␈αand␈α
AM␈αuses␈αthe␈αabove␈αrule␈αto␈α
turn␈αthe␈αextreme␈αexample␈αof␈αSets␈α
¬
␈↓ α,␈↓the empty set ¬ into the extreme kind of bag.


␈↓ α,␈↓∧␈↓ α<Any-concept . Suggest

␈↓ α,␈↓¬12. If the F facet of concept X is blank,
␈↓ α,␈↓¬␈↓ α\Then consider trying to fill it in.

␈↓ α,␈↓The␈αabove␈αsuper-weak␈α
rule␈αwill␈αresult␈α
in␈αa␈αnew␈α
task␈αbeing␈αadded␈α
to␈αthe␈αagenda,␈α
for␈αevery
␈↓ α,␈↓blank␈α∞facet␈α∞of␈α
every␈α∞concept.␈α∞It␈α
is␈α∞more␈α∞of␈α
a␈α∞legal␈α∞move␈α
generator␈α∞than␈α∞a␈α∞plausible␈α
move
␈↓ α,␈↓proposer.␈α
 The␈αrating␈α
of␈α
each␈αsuch␈α
task␈α
will␈αdepend␈α
on␈α
the␈αWorth␈α
of␈α
the␈αconcept␈α
X␈αand␈α
the
␈↓ α,␈↓overall␈α
worth␈α
of␈α
the␈α
type␈α
F␈αfacet,␈α
but␈α
in␈α
all␈α
cases␈α
will␈αbe␈α
␈↓βvery␈α
small.␈↓␈α
The␈α
"emptiness"␈α
of␈αa
␈↓ α,␈↓facet␈αis␈αalways␈αa␈αvalid␈αreason␈αfor␈αtrying␈αto␈α≡ll␈αit␈αin,␈αbut␈αnever␈αan␈αa␈αpriori␈αimportant␈αreason.
␈↓ α,␈↓So␈αthe␈αnet␈αe≥ect␈αof␈αthe␈αrule␈αis␈αto␈αslightly␈αbias␈αAM␈αtoward␈αworking␈αon␈αblank␈α¬␈αrather␈αthan
␈↓ α,␈↓non-blank ¬ facets.

␈↓ α,␈↓¬13.␈α
While␈α
trying␈α
to␈α
fill␈α
in␈α
facet␈α
F␈α
of␈α
concept␈α
C,␈α
for␈α
any␈α
C␈α
and␈α
F,␈α
if␈α
C␈α
is␈α
known␈α
to␈α
be␈α
similar␈α
to␈α
some
␈↓ α,␈↓¬␈↓ β,other concept D, except for difference d,
␈↓ α,␈↓¬␈↓ α\Then try to fill in C.F by selecting items from D.F for which d is nonexistent.

␈↓ α,␈↓This␈α∞rule␈α∞is␈α∞made␈α∞more␈α∞speci≡c␈α∞when␈α∞F␈α∞is␈α∞actually␈α∞known,␈α∞and␈α∞hence␈α∞the␈α∞format␈α∞of␈α∂d␈α∞is
␈↓ α,␈↓actually␈α∀determined.␈α∃For␈α∀example,␈α∃if␈α∀C=Reverse-at-all-levels,␈α∃F=examples,␈α∀then␈α∃(at␈α∀one
␈↓ α,␈↓particular␈αmoment)␈αa␈αnote␈αis␈αfound␈αon␈αthe␈αConjecs␈αfacet␈αof␈αconcept␈αC␈αwhich␈αsays␈αthat␈αC␈αis

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε␈α	Thus␈α
the␈α	LISP␈α
program␈α	has␈α
a␈α	separate␈α	concept␈α
called␈α	"Examples-of-any-concept",␈α
another␈α	concept␈α
called␈α	"Definitions-of-
␈↓ α,␈↓ε␈↓ βLany-concept", etc.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε142␈↓-

␈↓ α,␈↓just␈αlike␈αthe␈α
concept␈αD=Reverse-top-level,␈αexcept␈α
C␈αalso␈αrecurs␈α
on␈αthe␈αnonatomic␈αelements␈α
of
␈↓ α,␈↓its␈α∂arguments,␈α∂whereas␈α⊂D␈α∂doesn't.␈α∂ Thus␈α∂d␈α⊂is␈α∂made␈α∂null␈α∂by␈α⊂choosing␈α∂examples␈α∂of␈α⊂D␈α∂for
␈↓ α,␈↓which␈αthere␈αare␈α
no␈αnonatomic␈αelements.␈α
So␈αan␈αexample␈α
like␈α␈↓α`Reverse-top-level(<a␈αb␈αc>)=<c␈α
b
␈↓ α,␈↓αa>'␈↓␈αwill␈αbe␈αselected␈αand␈αwill␈αlead␈αto␈αthe␈αproposed␈αexample␈α␈↓α`Reverse-at-all-levels(<a␈αb␈αc>)=<c
␈↓ α,␈↓αb a>'␈↓, which is in fact valid.

␈↓ α,␈↓¬14. After dealing with concept C,
␈↓ α,␈↓¬␈↓ α\Slightly,␈α∞temporarily␈α
boost␈α∞the␈α∞priority␈α
value␈α∞of␈α
each␈α∞existing␈α∞task␈α
which␈α∞involves␈α∞an␈α
Active
␈↓ α,␈↓¬␈↓ β,concept whose domain or range is C.

␈↓ α,␈↓This␈αis␈αdone␈αe≠ciently␈αusing␈αthe␈αIn-dom-of␈αand␈αIn-ran-of␈αfacets␈αof␈αC.␈α A␈αtypical␈αusage␈αwas
␈↓ α,␈↓after␈α
checking␈α
the␈α
just-≡lled-in␈α
examples␈α
of␈α
Bags,␈α
when␈α
AM␈α
slightly␈α
boosted␈α
the␈α
rating␈αof
␈↓ α,␈↓≡lling␈α
in␈α
examples␈α
of␈α
Bag-union,␈α
and␈α
this␈αtask␈α
just␈α
barely␈α
squeaked␈α
through␈α
as␈α
the␈αnext␈α
one
␈↓ α,␈↓to␈αbe␈αchosen.␈α Note␈αthat␈αthe␈αrule␈αreinforced␈α
that␈αtask␈αtwice,␈αsince␈αboth␈αdomain␈αand␈αrange␈α
of
␈↓ α,␈↓Bag-union are bags.


␈↓ α,␈↓∧␈↓ α<Any-concept . Check

␈↓ α,␈↓¬15. When checking facet F of concept C, (for any F and C,)
␈↓ α,␈↓¬␈↓ α\Prune away at the entries there until the facet's size is reduced to the size which C merits.

␈↓ α,␈↓The␈α∞algorithm␈α∞for␈α∞doing␈α∞this␈α∞is␈α∞as␈α∞follows:␈α∞The␈α∞Worth␈α∞of␈α∞C␈α∞is␈α∞multiplied␈α∞by␈α∂the␈α∞overall
␈↓ α,␈↓worth␈αof␈α
facet␈αtype␈α
F.␈αThis␈α
is␈αnormalized␈α
in␈αtwo␈α
ways,␈αyielding␈α
the␈αmaximum␈α
amount␈αof␈α
list
␈↓ α,␈↓cells␈α∞that␈α∞C.F␈α∞may␈α∞occupy,␈α∞and␈α∞also␈α∞yielding␈α∞the␈α∞maximum␈α∞number␈α∞of␈α∞separate␈α∞entries␈α∞to
␈↓ α,␈↓keep␈αaround␈α
on␈αC.F.␈α
If␈αeither␈α
limit␈αis␈αbeing␈α
exceeded,␈αthen␈α
an␈αentry␈α
is␈αplucked␈α
at␈αrandom
␈↓ α,␈↓(but␈α
weighted␈α
to␈αfavor␈α
selection␈α
from␈α
the␈αrear␈α
of␈α
the␈α
facet)␈αand␈α
excised.␈α
This␈α
repeats␈αas␈α
long
␈↓ α,␈↓as␈α∪C.F␈α∪is␈α∪oversized.␈α∪As␈α∪space␈α∪grows␈α∪tight,␈α∪the␈α∪normalization␈α∪weights␈α∪decline,␈α∪so␈α∩each
␈↓ α,␈↓concept's allocation is reduced.

␈↓ α,␈↓¬16. When checking facet F of concept C,
␈↓ α,␈↓¬␈↓ α\Eliminate redundant entries.

␈↓ α,␈↓Although␈αit␈αmight␈αconceivably␈α
mean␈αsomething␈αfor␈αan␈αentry␈α
to␈αoccur␈αtwice,␈αthis␈α
was␈αnever
␈↓ α,␈↓desirable for the set of facets which each AM concept possessed.


␈↓ α,␈↓∧␈↓ α<Any-concept . Interest

␈↓ α,␈↓The␈α
interest␈αfeatures␈α
apply␈αto␈α
tell␈αhow␈α
interesting␈αa␈α
concept␈αis,␈α
and␈αare␈α
rarely␈αsubdivided␈α
by
␈↓ α,␈↓relevant␈α∞facet.␈α
That␈α∞is,␈α∞most␈α
of␈α∞the␈α
reasons␈α∞that␈α∞Any␈α
concept␈α∞might␈α
be␈α∞interesting␈α∞will␈α
be
␈↓ α,␈↓given below.

␈↓ α,␈↓¬17. A concept X is interesting if X.Conjecs contains some interesting entries.


␈↓ α,␈↓¬18.␈α∩A␈α∩concept␈α⊃is␈α∩interesting␈α∩if␈α⊃its␈α∩boundary␈α∩accidentally␈α⊃coincides␈α∩with␈α∩another,␈α⊃well-known,
␈↓ α,␈↓¬␈↓ β,interesting concept.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε143␈↓-

␈↓ α,␈↓The␈αboundary␈αof␈αa␈αconcept␈αmeans␈αthe␈αitems␈αwhich␈αjust␈αbarely␈αfall␈αinto␈α(or␈αjust␈αbarely␈αmiss
␈↓ α,␈↓satisfying)␈α
the␈α
de≡nition␈α
of␈α
that␈α
concept.␈α
Thus␈α
the␈α
boundary␈α
of␈α
Primes␈α
might␈αinclude␈α
1,2,3,4.
␈↓ α,␈↓If␈α
the␈α∞boundary␈α
of␈α∞Even␈α
numbers␈α∞includes␈α
numbers␈α∞di≥ering␈α
by␈α∞at␈α
most␈α∞1␈α
from␈α∞an␈α
even
␈↓ α,␈↓number,␈α∂then␈α∂clearly␈α∂their␈α⊂boundary␈α∂is␈α∂␈↓βall␈↓␈α∂numbers.␈α⊂Thus␈α∂it␈α∂coincides␈α∂with␈α⊂the␈α∂already-
␈↓ α,␈↓known␈α∂concept␈α∂Numbers,␈α⊂and␈α∂this␈α∂makes␈α∂Even-nos␈α⊂more␈α∂interesting.␈α∂ This␈α⊂expresses␈α∂the
␈↓ α,␈↓property we intuitively understand as: no number is very far from an even number.

␈↓ α,␈↓¬19.␈α
A␈α
concept␈α
is␈α
interesting␈α
if␈α∞its␈α
boundary␈α
accidentally␈α
coincides␈α
with␈α
the␈α
boundary␈α∞of␈α
another,
␈↓ α,␈↓¬␈↓ β,very different, interesting concept.

␈↓ α,␈↓Thus,␈αfor␈α
example,␈αPrimes␈α
and␈αNumbers␈α
are␈αboth␈α
a␈αlittle␈α
more␈αinteresting␈α
since␈αthe␈α
extreme
␈↓ α,␈↓cases␈αof␈αnumbers␈αare␈αall␈αboundary␈αcases␈αof␈αprimes.␈α Even␈αnumbers␈αand␈αOdd␈αnumbers␈αboth
␈↓ α,␈↓have␈αthe␈αsame␈αboundary,␈αnamely␈αNumbers.␈α This␈αis␈αa␈αtie␈αbetween␈αthem,␈αand␈αslightly␈αraises
␈↓ α,␈↓AM's interest in both concepts.

␈↓ α,␈↓¬20.␈α⊂A␈α∂concept␈α⊂is␈α∂interesting␈α⊂if␈α∂it␈α⊂is␈α⊂--␈α∂accidentally␈α⊂--␈α∂precisely␈α⊂the␈α∂boundary␈α⊂of␈α⊂some␈α∂other,
␈↓ α,␈↓¬␈↓ β,interesting concept.

␈↓ α,␈↓In␈αthe␈αcase␈αmentioned␈αfor␈αthe␈αabove␈αrule,␈αNumbers␈αis␈αraised␈αin␈αinterest␈αbecause␈αit␈αturns␈αout
␈↓ α,␈↓to be the boundary for even and odd numbers.

␈↓ α,␈↓¬21. A concept is boring if, after several attempts, only a couple examples are found.

␈↓ α,␈↓Another␈α
rule␈α
indicates,␈α
in␈α
such␈α
situations,␈α
that␈α
the␈α
concept␈α
may␈α
be␈α
forgotten␈α
and␈α
replaced␈α
by
␈↓ α,␈↓some conjecture.

␈↓ α,␈↓¬22.␈αConcept␈αC␈αis␈αinteresting␈αif␈αsome␈αnormally-inefficient␈αoperation␈αF␈αcan␈αbe␈αefficiently␈αperformed
␈↓ α,␈↓¬␈↓ β,on C's.

␈↓ α,␈↓Thus␈α
it␈αis␈α
very␈α
fast␈αto␈α
perform␈αInsert␈α
of␈α
items␈αinto␈α
lists␈α
because␈α(i)␈α
no␈αpre-existence␈α
checking
␈↓ α,␈↓need␈α
be␈α
done␈α
(as␈α
with␈α∞sets␈α
and␈α
osets),␈α
and␈α
(ii)␈α
no␈α∞ordered␈α
merging␈α
need␈α
be␈α
done␈α∞(as␈α
with
␈↓ α,␈↓bags).  So "Lists" is an interesting concept for that reason, according to the above rule.

␈↓ α,␈↓¬23.␈α∞Concept␈α∞C␈α∞is␈α∞interesting␈α∞if␈α∞each␈α∂example␈α∞of␈α∞C␈α∞accidentally␈α∞seems␈α∞to␈α∞satisfy␈α∂the␈α∞otherwise-
␈↓ α,␈↓¬␈↓ β,rarely␈αsatisfied␈αpredicate␈α
P,␈αor␈α(equivalently)␈α
if␈αthere␈αis␈α
an␈αunusual␈αconjecture␈α
involving
␈↓ α,␈↓¬␈↓ β,C.

␈↓ α,␈↓This is almost a primitive a≠rmation of intererestingness.

␈↓ α,␈↓¬24. Concept C is interesting if C is closely related to the very interesting concept X.

␈↓ α,␈↓This␈α∂is␈α∂intererestingness␈α∂by␈α∂association.␈α∂AM␈α∂was␈α∂interested␈α∂in␈α∂Divisors-of␈α∂because␈α∂it␈α∞was
␈↓ α,␈↓closely related to TIMES, which had proven to be a very interesting concept.

␈↓ α,␈↓¬25.␈α
Concept␈α
C␈αis␈α
interesting␈α
if␈α
there␈αis␈α
an␈α
analogy␈α
in␈αwhich␈α
C␈α
corresponds␈α
to␈αY,␈α
and␈α
the␈αanalogs␈α
of
␈↓ α,␈↓¬␈↓ β,the Interest features of Y indicate that C is interesting.

␈↓ α,␈↓This␈α∂might␈α∂have␈α∂been␈α∂a␈α∂very␈α∂useful␈α∂rule,␈α∂if␈α∂only␈α∂there␈α∂had␈α∂been␈α∂more␈α∂decent␈α∂analogies
␈↓ α,␈↓∨oating␈α
around␈α
the␈α
system.␈α∞As␈α
it␈α
was,␈α
the␈α
rule␈α∞was␈α
rarely␈α
used␈α
to␈α
advantage.␈α∞It␈α
essentially
␈↓ α,␈↓says that the analogs of Interest criteria are themselves (probably) valid criteria.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε144␈↓-

␈↓ α,␈↓¬26.␈α⊂A␈α∂concept␈α⊂C␈α⊂is␈α∂interesting␈α⊂if␈α⊂one␈α∂of␈α⊂its␈α∂generalizations␈α⊂or␈α⊂specializations␈α∂turns␈α⊂out␈α⊂to␈α∂be
␈↓ α,␈↓¬␈↓ β,unexpectedly very interesting.

␈↓ α,␈↓"Unexpected"␈αmeans␈αthat␈αthe␈αinteresting␈αproperty␈αhadn't␈αalready␈αbeen␈αobserved␈αfor␈αC.␈αIf␈αC
␈↓ α,␈↓is␈α∞interesting␈α
in␈α∞some␈α
way,␈α∞and␈α
then␈α∞one␈α
of␈α∞its␈α
generalizations␈α∞is␈α
seen␈α∞to␈α
be␈α∞interesting␈α
in
␈↓ α,␈↓exactly␈α⊂the␈α⊂same␈α⊂way,␈α⊂then␈α⊂that␈α⊂is␈α⊂"expected".␈α⊂It's␈α⊂almost␈α⊂more␈α⊂interesting␈α⊂if␈α⊂the␈α⊂second
␈↓ α,␈↓concept␈α
unexpectedly␈α
␈↓βlacks␈↓␈α
some␈α
fundamental␈α
property␈α
about␈α
C.␈α
At␈α
least␈α
in␈α
that␈α
case␈α
AM
␈↓ α,␈↓might learn something about what gives C that property. In fact, AM has this rule:

␈↓ α,␈↓¬27. If concept C possesses some very interesting property lacked by one of its specializations S,
␈↓ α,␈↓¬␈↓ α\Then both C and S become slightly more interesting.

␈↓ α,␈↓In the LISP program, ths is closely linked with rule 104.

␈↓ α,␈↓¬28. If a concept C is re-derived in a new way, that makes it more interesting.
␈↓ α,␈↓¬␈↓ α\If␈α
concepts␈α
C1␈α
and␈α
C2␈α
turn␈α
out␈α∞to␈α
be␈α
equivalent␈α
concepts,␈α
then␈α
merge␈α
them.␈α∞ The␈α
combined
␈↓ α,␈↓¬␈↓ β,concept is now more interesting than either of its predecessors.

␈↓ α,␈↓The␈αtwo␈αconditionals␈αabove␈αare␈αreally␈αthe␈αsame␈αrule,␈αso␈αthey␈αaren't␈αgiven␈αseparate␈αnumbers.
␈↓ α,␈↓C1␈α⊂and␈α⊂C2␈α⊂might␈α⊂be␈α⊂conjectured␈α⊂equivalent␈α⊂because␈α⊂their␈α⊂examples␈α⊂coincide,␈α⊂each␈α⊃is␈α⊂a
␈↓ α,␈↓generalization␈α∂of␈α∞the␈α∂other,␈α∂their␈α∞de≡nitions␈α∂can␈α∞be␈α∂formally␈α∂shown␈α∞to␈α∂be␈α∂equivalent,␈α∞etc.
␈↓ α,␈↓This rule is similar in spirit to rule number 114.


␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.2.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for the Examples facets of Any-concept␈↓)αβ␈↓

␈↓ α,␈↓The␈α
following␈αheuristics␈α
are␈αused␈α
for␈αdealing␈α
with␈α
the␈αmany␈α
kinds␈αof␈α
examples␈αfacets␈α
which
␈↓ α,␈↓a concept can possess: non-examples, boundary examples, Isa links, etc.


␈↓ α,␈↓∧␈↓ α<Any-concept . Examples . Fillin

␈↓ α,␈↓¬29. To fill in examples of X, where X is a kind of Y (for some more general concept Y),
␈↓ α,␈↓¬␈↓ α\Inspect the examples of Y; some of them may be examples of X as well.
␈↓ α,␈↓¬␈↓ α\The further removed Y is from X, the less cost-effective this rule is.

␈↓ α,␈↓For␈α∂the␈α∂task␈α∞of␈α∂≡lling␈α∂in␈α∂Empty-structures,␈α∞AM␈α∂knows␈α∂that␈α∞concept␈α∂is␈α∂a␈α∂specialization␈α∞of
␈↓ α,␈↓Structures,␈α
so␈α
it␈α
looks␈α
over␈α
all␈α
the␈α
then-known␈α
examples␈α
of␈α
Structures.␈α
Sure␈α
enough,␈α
a␈αfew␈α
of
␈↓ α,␈↓them␈αare␈α
empty␈α(satisfy␈α
Empty-structures.Defn).␈α Similarly,␈α
for␈αthe␈α
task␈αof␈α
≡lling␈αin␈α
examples
␈↓ α,␈↓of␈αPrimes,␈αthis␈αrule␈αwould␈αhave␈αAM␈αnotice␈αthat␈αPrimes␈αis␈αa␈αkind␈αof␈αNumber,␈αand␈α
therefore
␈↓ α,␈↓look␈α∂over␈α∂all␈α∞the␈α∂known␈α∂examples␈α∞of␈α∂Number.␈α∂ It␈α∞would␈α∂not␈α∂be␈α∞cost-e≥ective␈α∂to␈α∂look␈α∞for
␈↓ α,␈↓primes␈α∞by␈α
testing␈α∞each␈α
example␈α∞of␈α
Anything,␈α∞and␈α
the␈α∞third␈α
and␈α∞≡nal␈α
clause␈α∞in␈α∞the␈α
above
␈↓ α,␈↓rule recognizes that fact.

␈↓ α,␈↓¬30. To fill in non-examples of concept X,
␈↓ α,␈↓¬␈↓ α\Search␈αthe␈αspecializations␈αof␈α
X.␈αLook␈αat␈αall␈α
their␈αnon-examples.␈α Some␈αof␈α
them␈αmay␈αturn␈αout␈α
to
␈↓ α,␈↓¬␈↓ β,be non-examples of X as well.

␈↓ α,␈↓This␈α
rule␈α
is␈α
the␈α
counterpart␈α
of␈α
the␈αlast␈α
one,␈α
but␈α
for␈α
␈↓βnon␈↓-examples.␈α
 As␈α
expected,␈α
this␈αwas␈α
less
␈↓ α,␈↓useful than the preceding positive rule.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε145␈↓-

␈↓ α,␈↓¬31. If the current task is to fill in examples of any concept X,
␈↓ α,␈↓¬␈↓ α\Then one way to get them is to symbolically instantiate a definition of X.

␈↓ α,␈↓That␈α∂rule␈α∂simply␈α∂says␈α⊂to␈α∂use␈α∂some␈α∂known␈α∂tricks,␈α⊂some␈α∂hacks,␈α∂to␈α∂wring␈α∂examples␈α⊂from␈α∂a
␈↓ α,␈↓declarative␈αde≡nition.␈αOne␈αtrick␈αAM␈αknows␈αabout␈αis␈αto␈αplug␈αalready-known␈αexamples␈αof␈αX
␈↓ α,␈↓into␈αthe␈αrecursive␈αstep␈αof␈α
a␈αde≡nition.␈αAnother␈αtrick␈αis␈α
simply␈αto␈αtry␈αto␈αinstantiate␈α
the␈αbase
␈↓ α,␈↓step␈αof␈αa␈αrecursive␈αde≡nition.␈α Another␈αtrick␈αis␈αto␈αtake␈αa␈αde≡nition␈αof␈αthe␈αform␈α"λ␈α(x)␈αx␈αisa
␈↓ α,␈↓P,␈αand␈α
<␈↓βsub-expression␈↓>",␈αwork␈αon␈α
instantiating␈αjust␈αthe␈α
␈↓βsub-expression␈↓,␈αand␈αthen␈α
pop␈αback
␈↓ α,␈↓up and see which of those items are P's.

␈↓ α,␈↓¬32. If the current task is to fill in non-examples of concept X,
␈↓ α,␈↓¬␈↓ α\Then␈α
one␈α
fast␈α
way␈α
to␈α
get␈α
them␈α
is␈α
to␈α
pick␈α
any␈α
random␈α
item,␈α
any␈α
example␈α
of␈α
Anything,␈α
and␈α
check
␈↓ α,␈↓¬␈↓ β,that it fails X.Defn.

␈↓ α,␈↓This␈αis␈α
an␈αa≠rmation␈αthat␈α
for␈αany␈αconcept␈α
X,␈αmost␈α
things␈αin␈αthe␈α
universe␈αwill␈αprobably␈α
not
␈↓ α,␈↓be␈α
X's.␈α
This␈αrule␈α
was␈α
almost␈αnever␈α
used␈α
to␈αgood␈α
advantage:␈α
non-examples␈αof␈α
a␈α
concept␈αX
␈↓ α,␈↓were␈α
never␈α
sought␈αunless␈α
there␈α
was␈α
some␈αreason␈α
to␈α
expect␈α
that␈αthey␈α
might␈α
not␈α
exist.␈αIn␈α
those
␈↓ α,␈↓cases,␈α∩the␈α∩presumption␈α∩of␈α∩the␈α∩above␈α∩rule␈α⊃was␈α∩wrong,␈α∩and␈α∩it␈α∩failed.␈α∩ That␈α∩is,␈α∩the␈α⊃rule
␈↓ α,␈↓succeeded i≥ it was not needed.␈↓	2␈↓

␈↓ α,␈↓¬33. To fill in examples of concept X,
␈↓ α,␈↓¬␈↓ α\If X.View tells how to view a Z as if it were an X, and some examples of Z are known,
␈↓ α,␈↓¬␈↓ α\Then just run X.View on those examples, and check that the results really are X's.

␈↓ α,␈↓Thus␈α∞examples␈α∞of␈α∞osets␈α∂were␈α∞found␈α∞by␈α∞viewing␈α∂other␈α∞known␈α∞examples␈α∞of␈α∂structures␈α∞(e.g.,
␈↓ α,␈↓examples of sets) as if they were osets.

␈↓ α,␈↓¬34. To fill in examples of concept X,
␈↓ α,␈↓¬␈↓ α\Find an operation whose range is X,␈↓	3␈↓¬ and find examples of that operation being applied.

␈↓ α,␈↓To␈α∞≡ll␈α∂in␈α∞examples␈α∞of␈α∂Even-nos,␈α∞this␈α∞rule␈α∂might␈α∞have␈α∞AM␈α∂notice␈α∞the␈α∂operation␈α∞`Double'.
␈↓ α,␈↓Any␈α
example␈α
of␈αDouble␈α
will␈α
contain␈αan␈α
example␈α
of␈αan␈α
even␈α
number␈αas␈α
its␈α
value:␈αe.g.,␈α
<3→6>
␈↓ α,␈↓contains the even number 6.

␈↓ α,␈↓¬35. If the current task is to fill in examples of concept X,
␈↓ α,␈↓¬␈↓ α\One␈α∞bizarre␈α∞way␈α∞is␈α∞to␈α∞specialize␈α∞X,␈α∞adding␈α∞a␈α∞strong␈α∞constraint␈α∞to␈α∞X.Defn,␈α∞and␈α∞then␈α∂look␈α∞for
␈↓ α,␈↓¬␈↓ β,examples of that new specialization.

␈↓ α,␈↓Like␈α
the␈αclassical␈α
"insane␈α
heuristic"␈↓	4␈↓,␈αthis␈α
sounds␈αcrazy␈α
but␈α
works␈αembarassingly␈α
often.␈α
If␈αI
␈↓ α,␈↓ask␈α∞you␈α∞to␈α∂≡nd␈α∞numbers␈α∞having␈α∞a␈α∂prime␈α∞number␈α∞of␈α∞divisors,␈α∂the␈α∞rate␈α∞at␈α∞which␈α∂you␈α∞≡nd
␈↓ α,␈↓them␈αwill␈αprobably␈αbe␈αlower␈α
than␈αif␈αI'd␈αasked␈αyou␈α
to␈α≡nd␈αnumbers␈αwith␈αprecisely␈α2␈α
divisors.
␈↓ α,␈↓The␈α∞␈↓βvariety␈↓␈α∞of␈α∞examples␈α∞will␈α∞su≥er,␈α∞of␈α
course.␈α∞ The␈α∞converse␈α∞of␈α∞this␈α∞heuristic␈α∞¬␈α∞for␈α
non-
␈↓ α,␈↓examples ¬ was deemed too unaesthetic to feed to AM.

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	2␈↓ε Catch-22?
␈↓ α,␈↓ε␈↓	3␈↓ε or at least INTERSECTS X. Use the In-ran-of facets and the rippling mechanism to find such an operation.
␈↓ α,␈↓ε␈↓	4␈↓ε␈α	A␈αλharder␈α	task␈αλmight␈α	be␈α	easier␈αλto␈α	do.␈αλA␈α	stronger␈α	theorem␈αλmight␈α	be␈αλeasier␈α	to␈α	prove.␈αλThis␈α	is␈αλcalled␈α	"The␈α	Inventor's␈αλParadox",
␈↓ α,␈↓ε␈↓ βLon page 121 of [Polya 57].
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε146␈↓-

␈↓ α,␈↓¬36. To fill in examples of X,
␈↓ α,␈↓¬␈↓ α\One␈αinefficient␈αmethod␈αis␈αto␈αexamine␈αrandom␈αexamples␈αof␈αAnything,␈αchecking␈αeach␈αby␈αrunning
␈↓ α,␈↓¬␈↓ β,X.Defn␈αto␈αsee␈αif␈αit␈αis␈αan␈αX.␈α Slightly␈αbetter␈αis␈αto␈αripple␈αoutward␈αfrom␈αX␈αin␈αall␈αdirections,
␈↓ α,␈↓¬␈↓ β,testing all the examples of the concepts encountered.

␈↓ α,␈↓This is blind generate-and-test, and was (luckily) not needed much by AM.

␈↓ α,␈↓¬37.␈α
To␈αfind␈α
more␈αexamples␈α
of␈αX␈α
(or:␈αto␈α
find␈α
an␈αextreme␈α
example␈αof␈α
X),␈αwhen␈α
a␈αnice␈α
big␈αexample␈α
is
␈↓ α,␈↓¬␈↓ β,known, and X has a recursive definition,
␈↓ α,␈↓¬␈↓ α\Try␈αto␈α
plug␈αthe␈αknown␈α
example␈αinto␈αthe␈α
definition␈αand␈α
produce␈αa␈αsimpler␈α
one.␈αRepeat␈αthis␈α
until
␈↓ α,␈↓¬␈↓ β,an␈α
example␈α
is␈α
produced␈α
which␈α∞satisfies␈α
the␈α
base-step␈α
predicate␈α
of␈α
the␈α∞definition.␈α
That
␈↓ α,␈↓¬␈↓ β,entity is then an extreme (boundary) example of X.

␈↓ α,␈↓For example, AM had a de≡nition of a set as

␈↓ α,␈↓"Set(S)␈α∂if␈α⊂S={}␈α∂or␈α∂if␈α⊂Set(Remove-random-element(S))."␈α∂When␈α∂AM␈α⊂found␈α∂the␈α⊂big␈α∂example
␈↓ α,␈↓{A,B,{{C},D},{{{E}}},F}␈α∂by␈α∂some␈α∂other␈α∂means,␈α∂it␈α⊂used␈α∂the␈α∂above␈α∂rule␈α∂and␈α∂on␈α⊂he␈α∂recursive
␈↓ α,␈↓de≡nition␈α
to␈αturn␈α
this␈α
into␈α{A,B,{{{E}}},F}␈α
by␈α
removing␈αthe␈α
randomly-chosen␈α
third␈αelement.
␈↓ α,␈↓{A,B,F}␈αwas␈αproduced␈αnext,␈αfollowed␈αby␈α{B,F}␈αand␈α{F}.␈αAfter␈αthat,␈α{}␈αwas␈αproduced␈αand␈αthe
␈↓ α,␈↓rule relinquished control.

␈↓ α,␈↓¬38. To find examples of X, when X has a recursive definition,
␈↓ α,␈↓¬␈↓ α\One␈αmethod␈α
with␈αlow␈α
success␈αrate␈α
but␈αhigh␈αpayoff␈α
is␈αto␈α
try␈αto␈α
invert␈αthat␈α
definition,␈αthereby
␈↓ α,␈↓¬␈↓ β,creating a procedure for generating new examples.

␈↓ α,␈↓Using␈αthe␈αprevious␈αexample,␈αAM␈αwas␈αable␈αto␈αturn␈αthe␈αrecursive␈αde≡nition␈αof␈αa␈αset␈αinto␈αthe
␈↓ α,␈↓program␈α∩"Insert-any-random-item(S)",␈α∩which␈α∩turns␈α∪any␈α∩set␈α∩into␈α∩a␈α∩(usually␈α∪di≥erent␈α∩and
␈↓ α,␈↓larger)␈α
new␈α
set.␈α
 Since␈α
the␈α
rules␈α
which␈αAM␈α
uses␈α
to␈α
do␈α
these␈α
transformations␈α
are␈αvery␈α
special-
␈↓ α,␈↓purpose,␈α⊂they␈α⊃are␈α⊂not␈α⊂worth␈α⊃detailing␈α⊂here.␈α⊃This␈α⊂is␈α⊂one␈α⊃very␈α⊂managable␈α⊃open␈α⊂problem,
␈↓ α,␈↓where␈α
someone␈αmight␈α
spend␈αsome␈α
months␈αand␈α
create␈αa␈α
decent␈αbody␈α
of␈αde≡nition-inversion
␈↓ α,␈↓rules.  A typical rule AM has says:
␈↓ α,␈↓"Any␈αphrase␈αmatching␈α`␈↓βRemoving␈αan␈αx␈αand␈αensuring␈αthat␈αP(x)␈↓'␈αcan␈αbe␈αinverted␈αand␈αturned
␈↓ α,␈↓into␈α
this␈α
one:␈α
`␈↓βFinding␈α
any␈α
random␈α
x␈α
for␈α
which␈α
P(x)␈α
holds,␈α
then␈α
inserting␈α
x'␈↓."␈α
The␈α
class␈α
of
␈↓ α,␈↓de≡nitions␈α
which␈α∞can␈α
be␈α∞inverted␈α
using␈α∞AM's␈α
existing␈α∞rules␈α
is␈α∞quite␈α
small;␈α∞whenever␈α
AM
␈↓ α,␈↓needed␈α∪to␈α∀be␈α∪able␈α∪to␈α∀invert␈α∪another␈α∪particular␈α∀de≡nition,␈α∪the␈α∪author␈α∀simply␈α∪supplied
␈↓ α,␈↓whatever rules would be required.

␈↓ α,␈↓¬39. While filling in examples of C,
␈↓ α,␈↓¬␈↓ α\if␈α
two␈α
constructs␈α
x␈α
and␈α
y␈α
are␈α
found␈α
which␈αare␈α
very␈α
similar␈α
yet␈α
only␈α
one␈α
of␈α
which␈α
is␈αan␈α
example
␈↓ α,␈↓¬␈↓ β,of the concept C,
␈↓ α,␈↓¬␈↓ α\Then one is a boundary example of C, and the other is a boundary non-example,
␈↓ α,␈↓¬␈↓ α\and␈α∪it's␈α∪worth␈α∩creating␈α∪more␈α∪boundary␈α∪examples␈α∩and␈α∪boundary␈α∪non-examples␈α∪by␈α∩slowly
␈↓ α,␈↓¬␈↓ β,transforming x and y into each other.

␈↓ α,␈↓Thus␈α∞when␈α∞AM␈α∞notices␈α∞that␈α∞{a}␈α∞and␈α∞{a,b,a}␈α∞are␈α∞similar␈α∞yet␈α∞not␈α∞both␈α∞sets,␈α∞it␈α∂creates␈α∞{a,b},
␈↓ α,␈↓{b,a},␈α∞{a,a}␈α∞and␈α∞sees␈α∞which␈α
are␈α∞and␈α∞are␈α∞not␈α∞examples␈α
of␈α∞sets.␈α∞In␈α∞this␈α∞way,␈α∞some␈α
boundary
␈↓ α,␈↓items␈α(both␈αexamples␈αand␈αnon-examples)␈αare␈αcreated.␈αThe␈αrules␈αfor␈αthis␈αslow␈α
transformation
␈↓ α,␈↓are␈α∞again␈α∂special␈α∞purpose.␈α∂ They␈α∞examine␈α∂the␈α∞di≥erence␈α∞between␈α∂the␈α∞items␈α∂x␈α∞and␈α∂y,␈α∞and
␈↓ α,␈↓suggest␈α
operators␈α
(e.g.,␈α
Deletion)␈α
which␈α∞will␈α
reduce␈α
that␈α
di≥erence.␈α
 This␈α∞GPS-like␈α
strategy
␈↓ α,␈↓has␈α⊂been␈α∂well␈α⊂studied␈α∂by␈α⊂others,␈α∂and␈α⊂its␈α∂inferior␈α⊂implementation␈α∂inside␈α⊂AM␈α∂will␈α⊂not␈α∂be
␈↓ α,␈↓detailed.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε147␈↓-

␈↓ α,␈↓¬40. If the main task now is to fill in examples of concept C,
␈↓ α,␈↓¬␈↓ α\Consider␈α
all␈α∞the␈α
examples␈α
of␈α∞"first␈α
cousins"␈α
of␈α∞C.␈α
Some␈α
of␈α∞them␈α
might␈α
be␈α∞examples␈α
of␈α∞C␈α
as
␈↓ α,␈↓¬␈↓ β,well.

␈↓ α,␈↓By␈α
"≡rst␈αcousins",␈α
we␈αmean␈α
all␈αdirect␈α
specializations␈α
of␈αall␈α
direct␈αgeneralizations␈α
of␈αa␈α
concept,
␈↓ α,␈↓or␈αvice␈αversa.␈α That␈αis,␈αgoing␈αup␈αonce␈αalong␈αa␈αGenl␈αlink,␈αand␈αthen␈αdown␈αonce␈αalong␈αa␈αSpec
␈↓ α,␈↓link (or going down one link and then up one link).

␈↓ α,␈↓¬41. If the main task now is to fill in boundary (non-)examples of concept C,
␈↓ α,␈↓¬␈↓ α\Consider␈αall␈αthe␈αboundary␈α(non-)examples␈αof␈α"first␈αcousins"␈αof␈αC.␈α Some␈αof␈αthem␈αmight␈αlie␈αon
␈↓ α,␈↓¬␈↓ β,the boundary of C as well.

␈↓ α,␈↓If␈α⊂they␈α⊂turn␈α⊂out␈α⊂not␈α⊂to␈α⊂be␈α∂boundary␈α⊂examples,␈α⊂they␈α⊂can␈α⊂be␈α⊂recorded␈α⊂as␈α⊂boundary␈α∂non-
␈↓ α,␈↓examples, and vice versa.

␈↓ α,␈↓¬42. To fill in Isa links of concept X, (that is, to find a list of concepts of which X is an example),
␈↓ α,␈↓¬␈↓ α\Just␈α∂ripple␈α∂down␈α∂the␈α∞tree␈α∂of␈α∂concepts,␈α∂applying␈α∂a␈α∞definition␈α∂of␈α∂each␈α∂concept.␈α∂ Whenever␈α∞a
␈↓ α,␈↓¬␈↓ β,definition␈α
fails,␈α
don't␈α
waste␈α
time␈α
trying␈α
any␈αof␈α
its␈α
specializations.␈α
 The␈α
Isa's␈α
of␈α
X␈αare␈α
then
␈↓ α,␈↓¬␈↓ β,all the concepts tried whose definitions passed X.

␈↓ α,␈↓When␈α∂a␈α∞new␈α∂concept␈α∂is␈α∞created,␈α∂e.g.,␈α∞a␈α∂new␈α∂composition,␈α∞this␈α∂rule␈α∞can␈α∂ascertain␈α∂the␈α∞most
␈↓ α,␈↓speci≡c␈α
Isa␈αlinks␈α
that␈αcan␈α
be␈αattached␈α
to␈αit.␈α
 Another␈αuse␈α
for␈αthis␈α
rule␈αwould␈α
be:␈αIf␈α
the␈αIsa
␈↓ α,␈↓link␈αnetwork␈αever␈αgot␈αfouled␈αup␈α(contained␈α
paradoxes),␈αthis␈αrule␈αcould␈αbe␈αused␈αto␈α
straighten
␈↓ α,␈↓everything out (with a logarithmic expenditure of time).


␈↓ α,␈↓∧␈↓ α<Any-concept . Examples . Suggest

␈↓ α,␈↓¬43. If some (but not most) examples of X are also examples of Y (for some concept Y),
␈↓ α,␈↓¬␈↓ α\and some (but not most) examples of Y are also examples of X,
␈↓ α,␈↓¬␈↓ α\Create␈α
a␈α
new␈αconcept␈α
defined␈α
as␈αthe␈α
intersection␈α
of␈α
those␈αtwo␈α
concepts␈α
(X␈αand␈α
Y).␈α
This␈αwill␈α
be
␈↓ α,␈↓¬␈↓ β,a specialization of both concepts.

␈↓ α,␈↓If␈αyou␈αhapen␈αto␈αnotice␈αthat␈αsome␈αprimes␈αare␈αpalindromic,␈αthis␈αrule␈αwould␈αsuggest␈αcreating␈αa
␈↓ α,␈↓brand␈α
new␈α
concept,␈α
de≡ned␈α
as␈α
the␈α
set␈αof␈α
numbers␈α
which␈α
are␈α
both␈α
palindromic␈α
and␈αprime.
␈↓ α,␈↓AM␈αnever␈αactually␈α
noticed␈αthis,␈αsince␈α
it␈αrepresented␈αall␈αnumbers␈α
in␈αunary.␈α If␈α
pushed,␈αAM
␈↓ α,␈↓will␈α∩de≡ne␈α∩Palindrome(n)␈α∩to␈α∩mean␈α∩that␈α∪the␈α∩sequence␈α∩of␈α∩exponents␈α∩of␈α∩prime␈α∪factors␈α∩is
␈↓ α,␈↓symmetric;␈α∞thus␈α∞2␈↓	3␈↓3␈↓	8␈↓5␈↓	1␈↓7␈↓	1␈↓11␈↓	8␈↓13␈↓	3␈↓␈α∂is␈α∞palindromic␈α∞in␈α∞AM's␈α∂sense␈α∞because␈α∞the␈α∞sequence␈α∂of␈α∞its
␈↓ α,␈↓exponents␈α∪(3␈α∀8␈α∪1␈α∀1␈α∪8␈α∀3)␈α∪is␈α∪unchanged␈α∀upon␈α∪reversal.␈α∀In␈α∪this␈α∀sense,␈α∪the␈α∀only␈α∪Prime
␈↓ α,␈↓palindromes are the primes themselves (or: just `2', depending upon the precise de≡nition).

␈↓ α,␈↓¬44. If very few examples of X are found,
␈↓ α,␈↓¬␈↓ α\Then␈α⊂add␈α⊂the␈α∂following␈α⊂task␈α⊂to␈α⊂the␈α∂agenda:␈α⊂"Generalize␈α⊂the␈α∂concept␈α⊂X",␈α⊂for␈α⊂the␈α∂following
␈↓ α,␈↓¬␈↓ β,reason:␈α⊗"X's␈α∃are␈α⊗quite␈α∃rare;␈α⊗a␈α⊗slightly␈α∃less␈α⊗restrictive␈α∃concept␈α⊗might␈α⊗be␈α∃more
␈↓ α,␈↓¬␈↓ β,interesting".

␈↓ α,␈↓Of␈αcourse,␈αAM␈αcontains␈αa␈αprecise␈αmeaning␈αfor␈αthe␈αphrase␈α"very␈αfew".␈α When␈αAM␈αlooks␈αfor
␈↓ α,␈↓primes␈α∂among␈α∂examples␈α⊂of␈α∂already-known␈α∂kinds␈α⊂of␈α∂numbers,␈α∂it␈α⊂will␈α∂≡nd␈α∂dozens␈α⊂of␈α∂non-
␈↓ α,␈↓examples␈α⊗for␈α⊗every␈α⊗example␈α∃of␈α⊗a␈α⊗prime␈α⊗it␈α∃uncovers.␈α⊗ "Very␈α⊗few"␈α⊗is␈α⊗thus␈α∃naturally
␈↓ α,␈↓implemented␈αas␈αa␈αstatistical␈αcon≡dence␈αlevel.␈α AM␈αuses␈αthis␈αrule␈αwhen␈αvery␈αfew␈αexamples␈αof
␈↓ α,␈↓Equality are found readily.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε148␈↓-

␈↓ α,␈↓¬45. If very many examples of X are found in a short period of time,
␈↓ α,␈↓¬␈↓ α\Then try to create a new, specialized version of X.

␈↓ α,␈↓This␈αis␈αsimilar␈αto␈α
the␈αpreceding␈αrule.␈α Since␈α
numbers␈αare␈αeasy␈αto␈α
≡nd,␈αthis␈αmight␈αcause␈αus␈α
to
␈↓ α,␈↓look for certain more interesting subclasses of numbers to study.

␈↓ α,␈↓¬46. If there are no known examples for the interesting concept X,
␈↓ α,␈↓¬␈↓ α\Then consider spending some time looking for such examples.

␈↓ α,␈↓I've␈α⊂heard␈α∂of␈α⊂a␈α∂math␈α⊂student␈α∂who␈α⊂de≡ned␈α∂a␈α⊂set␈α∂of␈α⊂number␈α∂which␈α⊂had␈α⊂quite␈α∂marvelous
␈↓ α,␈↓properties.␈αAfter␈αthe␈α20␈↓#
t␈↓#␈↓#
h␈↓#␈αincredible␈αtheorem␈α
about␈αthem␈αhe'd␈αproved,␈αsomeone␈αnoticed␈α
that
␈↓ α,␈↓the␈αset␈αwas␈αempty.␈αThe␈αdanger␈αof␈αunwittingly␈αdealing␈αwith␈αa␈αvacuous␈αconcept␈αis␈αeven␈αworse
␈↓ α,␈↓for a machine than for a human mathematician.  The above rule explicitly prevents that.

␈↓ α,␈↓¬47. If the totality of examples of concept C is too small to be interesting,
␈↓ α,␈↓¬␈↓ α\Then␈α
consider␈α
these␈α
reactions:␈α
(i)␈α
generalize␈α
C;␈α
(ii)␈α
forget␈α
C␈α
completely;␈α
(iii)␈α
replace␈α
C␈α
by␈α
one
␈↓ α,␈↓¬␈↓ β,conjecture.

␈↓ α,␈↓This␈α
is␈α
a␈α
good␈α
example␈α
of␈α
when␈α
a␈α
task␈α
like␈α
␈↓¬"Fill␈α
in␈α
generalizations␈α
of␈α
Numbers-with-1-divisors"␈↓
␈↓ α,␈↓might␈αget␈αproposed␈αwith␈αa␈αhigh-priority␈αreason.␈α The␈αclass␈αof␈αentities␈αwhich␈αC␈αencompasses
␈↓ α,␈↓is␈α⊂simply␈α⊂too␈α⊂small,␈α⊂too␈α⊂trivial␈α⊂to␈α⊂be␈α⊂worth␈α⊂maintaining␈α⊂a␈α⊂separate␈α⊂concept.␈α⊂When␈α⊃C␈α⊂is
␈↓ α,␈↓numbers-with-1-divisor,␈α
C␈αis␈α
really␈α
just␈αanother␈α
disguise␈αfor␈α
the␈α
singleton␈αset␈α
{1}.␈αThe␈α
above
␈↓ α,␈↓rule␈α∞might␈α
cause␈α∞a␈α
new␈α∞task␈α
to␈α∞be␈α∞added␈α
to␈α∞the␈α
agenda,␈α∞␈↓¬Fill␈α
in␈α∞generalizations␈α∞of␈α
Numbers-
␈↓ α,␈↓¬with-1-divisor␈↓.␈α∂ When␈α∂that␈α∂task␈α∂is␈α∂executed,␈α∂AM␈α∂might␈α∂create␈α∂the␈α∂concept␈α∞Numbers-with-
␈↓ α,␈↓odd-no-of-divisors,␈α⊂Numbers-with-prime-number-of-divisors,␈α∂etc.␈α⊂ Besides␈α⊂generalizing␈α∂that
␈↓ α,␈↓concept,␈α∂the␈α∞above␈α∂rule␈α∞gives␈α∂AM␈α∞two␈α∂other␈α∞alternatives.␈α∂ AM␈α∞may␈α∂simply␈α∂obliterate␈α∞the
␈↓ α,␈↓nearly-vacuous␈α
concept,␈α
perhaps␈α
leaving␈α
around␈α
just␈α
the␈α
statement␈α
"␈↓β1␈α
is␈α
the␈α
only␈αnumber␈α
with
␈↓ α,␈↓βone␈α∂divisor␈↓".␈α∞That␈α∂conjecture␈α∂might␈α∞be␈α∂tacked␈α∞onto␈α∂the␈α∂Conjecs␈α∞facet␈α∂of␈α∂Divisors-of.␈α∞The
␈↓ α,␈↓actual␈α
rule␈α∞will␈α
specify␈α
criteria␈α∞for␈α
deciding␈α
which␈α∞of␈α
the␈α
three␈α∞alternatives␈α
to␈α
try.␈α∞In␈α
fact,
␈↓ α,␈↓AM␈αreally␈αstarts␈αall␈αthree␈αactivities:␈αa␈α
task␈αwill␈αalways␈αbe␈αcreated␈αand␈αadded␈αto␈α
the␈αagenda
␈↓ α,␈↓(to␈αgeneralize␈αC),␈αthe␈αvacuous␈αconcept␈αwill␈αbe␈αtagged␈αas␈α"forgettable",␈αand␈αAM␈αwill␈αattempt
␈↓ α,␈↓to formulate a conjecture (the only items satisfying C.Defn are C.Exs).

␈↓ α,␈↓¬48. If the totality of examples of concept C is too large to be interesting,
␈↓ α,␈↓¬␈↓ α\Then␈α∞consider␈α
these␈α∞three␈α∞possible␈α
reactions:␈α∞(i)␈α
specialize␈α∞C;␈α∞(ii)␈α
forget␈α∞C␈α∞completely;␈α
(iii)
␈↓ α,␈↓¬␈↓ β,replace C by one conjecture.

␈↓ α,␈↓This␈αis␈αanalogous␈αto␈αthe␈α
preceding␈αrule,␈αbut␈αis␈αused␈α
far␈αless␈αfrequently.␈α One␈αcommon␈αuse␈α
is
␈↓ α,␈↓when␈αa␈αdisjunction␈αof␈αtwo␈αconcepts␈αhas␈αbeen␈αformed␈αwhich␈αis␈αaccidentally␈αlarge␈αor␈αalready-
␈↓ α,␈↓known (e.g., "Evens ∪ Odds" would be replaced by a conjecture).

␈↓ α,␈↓¬49. After filling in examples of C, if some examples were found,
␈↓ α,␈↓¬␈↓ α\Look␈αat␈αall␈αthe␈αoperations␈αwhich␈αcan␈αbe␈αapplied␈αto␈αC's␈α(that␈αis,␈αaccess␈αC.In-dom-of),␈αfind␈αthose
␈↓ α,␈↓¬␈↓ β,which␈α
are␈α
interesting␈α∞but␈α
which␈α
have␈α∞no␈α
known␈α
examples,␈α
and␈α∞suggest␈α
that␈α
AM␈α∞fill␈α
in
␈↓ α,␈↓¬␈↓ β,examples␈α⊂for␈α⊂them,␈α⊂because␈α∂some␈α⊂items␈α⊂are␈α⊂now␈α∂known␈α⊂which␈α⊂are␈α⊂in␈α⊂their␈α∂domain,
␈↓ α,␈↓¬␈↓ β,namely C.Exs.

␈↓ α,␈↓This␈α
rule␈αhad␈α
AM␈α
≡ll␈αin␈α
examples␈α
of␈αSet-insertion,␈α
as␈αsoon␈α
as␈α
some␈αexamples␈α
of␈α
Sets␈αhad
␈↓ α,␈↓been found.

␈↓ α,␈↓¬50. After filling in examples of C, if some examples were found,
␈↓ α,␈↓¬␈↓ α\Consider the task of Checking the examples facet of concept C.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε149␈↓-

␈↓ α,␈↓This was very frequently used during AM's runs.

␈↓ α,␈↓¬51. After checking examples of C, if many examples remain,
␈↓ α,␈↓¬␈↓ α\Consider the task of `Filling in some Conjecs for C'.

␈↓ α,␈↓This␈α∀was␈α∀used␈α∪often␈α∀by␈α∀AM.␈α∀After␈α∪checking␈α∀the␈α∀examples␈α∀of␈α∪C,␈α∀AM␈α∀would␈α∀try␈α∪to
␈↓ α,␈↓empirically formulate some interesting conjecture about C.

␈↓ α,␈↓¬52. After successfully filling in non-examples of X, if no examples exist,
␈↓ α,␈↓¬␈↓ α\If AM has not recently tried to find examples of X, then it should do so.
␈↓ α,␈↓¬␈↓ α\If␈α
AM␈αhas␈α
recently␈αtried␈α
and␈αfailed␈α
to␈α
find␈αexamples,␈α
consider␈αthe␈α
conjecture␈αthat␈α
X␈αis␈α
vacuous,
␈↓ α,␈↓¬␈↓ β,empty, null, always-False.  Consider generalizing X.


␈↓ α,␈↓¬53. After trying in vain to find some non-examples of X, if many examples exist,
␈↓ α,␈↓¬␈↓ α\Consider the conjecture that X is universal, always-True.  Consider specializing X.


␈↓ α,␈↓¬54. After successfully filling in examples of X, if no non-examples exist,
␈↓ α,␈↓¬␈↓ α\If AM has not recently tried to find non-examples of X, then it should consider doing so.
␈↓ α,␈↓¬␈↓ α\If␈α∞AM␈α∞has␈α∞recently␈α∞tried␈α∞and␈α∞failed␈α∂to␈α∞find␈α∞non-examples,␈α∞consider␈α∞the␈α∞conjecture␈α∞that␈α∂X␈α∞is
␈↓ α,␈↓¬␈↓ β,universal, always-True.  Consider specializing X.


␈↓ α,␈↓¬55. After trying in vain to find some examples of X,
␈↓ α,␈↓¬␈↓ α\If many non-examples exist,
␈↓ α,␈↓¬␈↓ α\Consider the conjecture that X is vacuous, null, empty, always-False. Consider generalizing X.



␈↓ α,␈↓∧␈↓ α<Any-concept . Examples . Check

␈↓ α,␈↓¬56. If the current task is to Check Examples of concept X,
␈↓ α,␈↓¬␈↓ α\and (Forsome Y) Y is a generalization of X with many examples,
␈↓ α,␈↓¬␈↓ α\and all examples of Y (ignoring boundary cases) are also examples of X,
␈↓ α,␈↓¬␈↓ α\Then conjecture that X is really no more specialized than Y,
␈↓ α,␈↓¬␈↓ α\and Check the truth of this conjecture on boundary examples of Y,
␈↓ α,␈↓¬␈↓ α\and␈α∀see␈α∀whether␈α∃Y␈α∀might␈α∀itself␈α∀turn␈α∃out␈α∀to␈α∀be␈α∀no␈α∃more␈α∀specialized␈α∀than␈α∀one␈α∃of␈α∀␈↓βits␈↓¬
␈↓ α,␈↓¬␈↓ β,generalizations.

␈↓ α,␈↓This␈αrule␈αcaused␈αAM,␈αwhile␈αchecking␈αexamples␈αof␈αodd-primes,␈αto␈αconjecture␈αthat␈α␈↓βall␈↓␈αprimes
␈↓ α,␈↓were odd-primes.

␈↓ α,␈↓¬57. If the current task is to Check Examples of concept X,
␈↓ α,␈↓¬␈↓ α\and (Forsome Y) Y is a specialization of X,
␈↓ α,␈↓¬␈↓ α\and all examples of X (ignoring boundary cases) are also examples of Y,
␈↓ α,␈↓¬␈↓ α\Then conjecture that X is really no more general than Y,
␈↓ α,␈↓¬␈↓ α\and Check the truth of this conjecture on boundary examples of X,
␈↓ α,␈↓¬␈↓ α\and see whether Y might itself turn out to be no more general than one of ␈↓βits␈↓¬ specializations.

␈↓ α,␈↓This rule is analogous to the preceding one for generalizations.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε150␈↓-


␈↓ α,␈↓¬58. When checking boundary examples of a concept C,
␈↓ α,␈↓¬␈↓ α\ensure that every scrap of C.Defn has been used.

␈↓ α,␈↓It␈α
is␈αoften␈α
the␈α
tiny␈αdetails␈α
in␈αthe␈α
de≡nition␈α
that␈αdetermine␈α
the␈αprecise␈α
boundary.␈α
 Thus␈αwe
␈↓ α,␈↓must␈αlook␈αcarefully␈αto␈αsee␈αwhether␈αPrimes␈αallows␈α1␈αas␈αan␈αexample␈αor␈αnot.␈α A␈αde≡nition␈αlike
␈↓ α,␈↓"numbers␈α∀divisible␈α∀only␈α∀by␈α∀1␈α∀and␈α∀themselves"␈α∀includes␈α∀1,␈α∀but␈α∀this␈α∃de≡nition␈α∀doesn't:
␈↓ α,␈↓"numbers␈α⊂having␈α⊂precisely␈α⊂2␈α⊂divisors".␈α⊂ In␈α⊂the␈α⊂LISP␈α⊂program,␈α⊂this␈α⊂rule␈α⊂contains␈α⊂several
␈↓ α,␈↓hacks␈α
(tricks)␈αfor␈α
checking␈αthat␈α
the␈αde≡nition␈α
has␈α
been␈αstretched␈α
to␈αthe␈α
fullest.␈α For␈α
example:
␈↓ α,␈↓if␈α∞the␈α∞de≡nition␈α∞is␈α∞of␈α∞the␈α
form␈α∞"all␈α∞x␈α∞in␈α∞X␈α∞such␈α
that...",␈α∞then␈α∞pay␈α∞careful␈α∞attention␈α∞to␈α
the
␈↓ α,␈↓boundary␈αof␈αX.␈α That␈αis,␈αtake␈αthe␈αtime␈αto␈αaccess␈αX.Boundary-exs␈αand␈αX.Boundary-non-exs,
␈↓ α,␈↓and check them against C.Defn.

␈↓ α,␈↓¬59. When checking examples of C,
␈↓ α,␈↓¬␈↓ α\Ensure␈α
that␈αeach␈α
example␈α
satisfies␈αC.Defn,␈α
and␈αeach␈α
non-example␈α
fails␈αit.␈α
 The␈αprecise␈α
member
␈↓ α,␈↓¬␈↓ β,of C.Defn to use can be chosen depending on the example.

␈↓ α,␈↓As␈αdescribed␈αearlier␈αin␈αthe␈αtext,␈αde≡nitions␈αcan␈αhave␈αdescriptors␈αwhich␈αindicate␈αwhat␈αkinds
␈↓ α,␈↓of arguments they might be best for, their overall speed, etc.

␈↓ α,␈↓¬60. When checking examples of C,
␈↓ α,␈↓¬␈↓ α\If␈αan␈αentry␈αe␈αis␈αrejected␈α(i.e.,␈αit␈αis␈αseen␈αto␈αbe␈αnot␈αan␈αexample␈αof␈αC␈αafter␈αall),␈αthen␈α
remove␈αe
␈↓ α,␈↓¬␈↓ β,from C.Exs and consider inserting it on the Boundary non-examples facet of C.

␈↓ α,␈↓There␈α∞is␈α∞a␈α∞complicated␈↓	5␈↓␈α∞algorithm␈α∞for␈α∞deciding␈α∞whether␈α∞to␈α∞forget␈α∞e␈α∞entirely␈α∞or␈α∞to␈α∂keep␈α∞it
␈↓ α,␈↓around as a close but not close enough kind of example.

␈↓ α,␈↓¬61. When checking examples of C,
␈↓ α,␈↓¬␈↓ α\After an entry e has been verified as a bone fide example of C,
␈↓ α,␈↓¬␈↓ α\Check whether e is also a valid example of some direct specialization of C.
␈↓ α,␈↓¬␈↓ α\If␈α⊂it␈α⊃is,␈α⊂then␈α⊃remove␈α⊂it␈α⊃from␈α⊂C.Exs,␈α⊃and␈α⊂consider␈α⊃adding␈α⊂it␈α⊃to␈α⊂the␈α⊃examples␈α⊂facet␈α⊃of␈α⊂that
␈↓ α,␈↓¬␈↓ β,specialization, and suggest the task of Checking examples of that specialization.


␈↓ α,␈↓¬62. When checking examples of C,
␈↓ α,␈↓¬␈↓ α\If an entry e is rejected,
␈↓ α,␈↓¬␈↓ α\Then check whether e is nevertheless a valid example of some generalization of C.
␈↓ α,␈↓¬␈↓ α\If␈αit␈αis,␈αconsider␈αadding␈αit␈αto␈αthat␈αconcept's␈αboundary-examples␈αfacet,␈αand␈αconsider␈αadding␈αit␈αto
␈↓ α,␈↓¬␈↓ β,the boundary non-examples facet of C.

␈↓ α,␈↓This is similar to the preceding rule.

␈↓ α,␈↓¬63. When checking non-examples of C, including boundary non-examples,
␈↓ α,␈↓¬␈↓ α\Ensure␈α
that␈αeach␈α
one␈α
fails␈αa␈α
definition␈αof␈α
C.␈α
Otherwise,␈αtransfer␈α
it␈αto␈α
the␈α
boundary␈αexamples
␈↓ α,␈↓¬␈↓ β,facet of C.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	5␈↓ε␈αNot␈αnecessarily␈αsophisticated.␈α First,␈αAM␈αaccesses␈αthe␈αWorth␈αof␈αC.␈α From␈αthis␈αit␈αdetermines␈αhow␈αmany␈αboundary␈αnon-
␈↓ α,␈↓ε␈↓ βLexamples␈α
C␈α
deserves␈α
to␈α
keep␈αaround␈α
(and␈α
how␈α
many␈α
total␈α
list␈αcells␈α
it␈α
merits).␈α
AM␈α
compares␈αthese␈α
quotas
␈↓ α,␈↓ε␈↓ βLwith␈α
the␈α
current␈α
number␈α
of␈α
(and␈α
size␈α
of)␈α	entries␈α
already␈α
listed␈α
on␈α
C.bdy-non-exs.␈α
 The␈α
degree␈α
of␈α
need␈α	of
␈↓ α,␈↓ε␈↓ βLanother␈α
entry␈α
there␈αthen␈α
sets␈α
the␈α"odds"␈α
for␈α
insertion␈αversus␈α
forgetting.␈α
 Finally␈αa␈α
random␈α
number␈αis
␈↓ α,␈↓ε␈↓ βLcomputed, and the odds determine what range it must lie in for e to be remembered.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε151␈↓-

␈↓ α,␈↓¬64. When checking non-examples of C, including boundary non-examples,
␈↓ α,␈↓¬␈↓ α\After an entry e has been verified as a bone fide non-example of C,
␈↓ α,␈↓¬␈↓ α\Check whether e is also a non-example of some direct generalization of C.
␈↓ α,␈↓¬␈↓ α\If␈α
it␈α
is,␈α∞then␈α
remove␈α
it␈α
from␈α∞C.Non-Exs,␈α
and␈α
consider␈α
adding␈α∞it␈α
to␈α
the␈α
non-examples␈α∞facet␈α
of
␈↓ α,␈↓¬␈↓ β,that generalization, and suggest the task of Checking examples of that generalization.


␈↓ α,␈↓¬65. When checking (boundary) non-examples of C,
␈↓ α,␈↓¬␈↓ α\If an entry e is rejected, that is if it turns out to be an example of C after all,
␈↓ α,␈↓¬␈↓ α\Then check whether e is nevertheless a non-example of some specialization of C.
␈↓ α,␈↓¬␈↓ α\If it is, consider adding it to that concept's boundary non-examples facet.

␈↓ α,␈↓This is similar to the preceding rule.


␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.3.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for the Conjecs facet of Any-concept␈↓)αβ␈↓


␈↓ α,␈↓∧␈↓ α<Any-concept . Conjecs . Fillin

␈↓ α,␈↓When␈αthe␈αtask␈αis␈αto␈αlook␈αaround␈αand␈α≡nd␈αconjectures␈αdealing␈αwith␈αconcept␈αC,␈αthe␈αfollowing
␈↓ α,␈↓general rules may be useful.

␈↓ α,␈↓¬66.␈α∂If␈α⊂there␈α∂is␈α⊂an␈α∂analogy␈α∂from␈α⊂X␈α∂to␈α⊂C,␈α∂and␈α⊂a␈α∂nice␈α∂item␈α⊂in␈α∂X.Conjecs,␈α⊂formulate␈α∂and␈α⊂test␈α∂the
␈↓ α,␈↓¬␈↓ β,analogous conjecture for C.

␈↓ α,␈↓Since␈α
an␈α
analogy␈α
is␈α
not␈α
much␈α
more␈αthan␈α
a␈α
set␈α
of␈α
substitutions,␈α
formulating␈α
the␈α`analogous
␈↓ α,␈↓conjecture' is almost a purely syntactic transformation.

␈↓ α,␈↓¬67. Examine C.Exs for regularities.

␈↓ α,␈↓What␈α∞mysteries␈α∞are␈α∞lurking␈α∞in␈α∂the␈α∞LISP␈α∞code␈α∞for␈α∞␈↓βthis␈↓␈α∂rule,␈α∞you␈α∞ask?␈α∞ Nothing␈α∞but␈α∂a␈α∞few
␈↓ α,␈↓special-purpose␈α
hacks␈α
and␈α
a␈α
few␈α
ultra-general␈α
hacks.␈α
 Here␈α
is␈α
a␈α
slightly␈α
more␈α
speci≡c␈αrule␈α
for
␈↓ α,␈↓you seekers:

␈↓ α,␈↓¬68.␈αLook␈αat␈αC.Exs.␈αPick␈αone␈αelement␈αat␈αrandom.␈αWrite␈αdown␈αstatements␈αtrue␈αabout␈αthat␈αexample␈αe.
␈↓ α,␈↓¬␈↓ β,Include␈αa␈αlist␈αof␈αall␈αconcepts␈αof␈αwhich␈αit␈αis␈αan␈αexample,␈αall␈αInterests␈αfeatures␈αit␈αsatisfies,
␈↓ α,␈↓¬␈↓ β,etc.
␈↓ α,␈↓¬␈↓ α\Then␈αcheck␈αeach␈αconjecture␈α
on␈αthis␈αlist␈αagainst␈α
all␈αother␈αknown␈αexamples␈α
of␈αC.␈α If␈αany␈α
example
␈↓ α,␈↓¬␈↓ β,(except a boundary example) of C violates a conjecture, discard it.
␈↓ α,␈↓¬␈↓ α\Take all the surviving conjectures, and eliminate any which trivally follow from other ones.

␈↓ α,␈↓This␈α
is␈α
a␈α
common␈α
way␈α
AM␈α
uses:␈α
induce␈αa␈α
conjecture␈α
from␈α
one␈α
example␈α
and␈α
test␈α
it␈α
on␈αall
␈↓ α,␈↓the␈α∞rest.␈α∂ A␈α∞more␈α∞sophisticated␈α∂approach␈α∞might␈α∞be␈α∂to␈α∞induce␈α∞it␈α∂by␈α∞using␈α∞a␈α∂few␈α∞examples
␈↓ α,␈↓simultaneously,␈αbut␈αI␈αhaven't␈αthought␈αof␈αany␈αnontrivial␈αway␈αto␈αdo␈αthat.␈α The␈αcareful␈αreader
␈↓ α,␈↓will␈αperceive␈αthat␈αmost␈αof␈αthe␈αconjectures␈αAM␈αwill␈αderive␈αusing␈αthis␈αheuristic␈αwill␈αbe␈αof␈αthe
␈↓ α,␈↓form␈α"X␈α
is␈αunexpectedly␈α
a␈αspecialization␈α
of␈αY",␈αor␈α
"X␈αis␈α
unexpectedly␈αan␈α
example␈αof␈αY",␈α
etc.
␈↓ α,␈↓Indeed, most of AM's conjectures are really that simple syntactically.

␈↓ α,␈↓¬69.␈α∃Formulate␈α∀a␈α∃parameterized␈α∀conjecture,␈α∃a␈α∀"template",␈α∃which␈α∀gets␈α∃slowly␈α∃specialized␈α∀or
␈↓ α,␈↓¬␈↓ β,instantiated into a definite conjecture.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε152␈↓-

␈↓ α,␈↓AM␈α
has␈α∞only␈α
a␈α∞few␈α
trivial␈α
methods␈α∞for␈α
doing␈α∞this␈α
(e.g.,␈α
introduce␈α∞a␈α
variable␈α∞initially␈α
and
␈↓ α,␈↓≡nd␈αthe␈αconstant␈αvalue␈αto␈αplug␈αin␈αthere␈αlater).␈α As␈αusual,␈αthey␈αwill␈αbe␈αomitted␈αhere,␈αand␈αthe
␈↓ α,␈↓author␈α
encourages␈α
some␈α
research␈αin␈α
this␈α
area,␈α
to␈αturn␈α
out␈α
a␈α
decent␈αset␈α
of␈α
general␈α
rules␈αfor
␈↓ α,␈↓accomplishing␈α⊂this␈α⊂hypothesis␈α⊂template␈α⊂instantiation.␈α⊃The␈α⊂best␈α⊂e≥ort␈α⊂to␈α⊂date␈α⊃along␈α⊂these
␈↓ α,␈↓lines, in one speci≡c sophisticated scienti≡c ≡eld, is that of META-DENDRAL [Buchanan].


␈↓ α,␈↓∧␈↓ α<Any-concept . Conjecs . Check

␈↓ α,␈↓¬70.␈α∞If␈α∞a␈α∂universal␈α∞conjecture␈α∞(For␈α∞all␈α∂X's,␈α∞...)␈α∞is␈α∞contradicted␈α∂by␈α∞empirical␈α∞data,␈α∞gather␈α∂the␈α∞data
␈↓ α,␈↓¬␈↓ β,together and try to find a regularity in those exceptions.
␈↓ α,␈↓¬␈↓ α\If␈α
this␈αsucceeds,␈α
give␈α
the␈αexceptions␈α
a␈αname␈α
N␈α
(if␈αthey␈α
aren't␈α
already␈αa␈α
concept),␈αand␈α
rephrase
␈↓ α,␈↓¬␈↓ β,the conjecture (For all X's which are not N's...).  Consider making X-N a new concept.

␈↓ α,␈↓Again␈α∩note␈α∩how␈α∩"active"␈α∩this␈α∩little␈α∩checking␈α∩rule␈α∩can␈α∩be.␈α∩ It␈α∩can␈α∩patch␈α∩up␈α∩nearly-true
␈↓ α,␈↓conjectures, examine data, de≡ne new concepts, etc.

␈↓ α,␈↓¬71. After verifying a conjecture for concept C,
␈↓ α,␈↓¬␈↓ α\See if it also holds for related concepts (e.g., a generalization of C).


␈↓ α,␈↓There␈αare␈α
of␈αcourse␈α
bookeeping␈αdetails␈αnot␈α
explicitly␈αshown␈α
above,␈αwhich␈α
are␈αpresent␈αin␈α
the
␈↓ α,␈↓LISP␈αprogram.␈α
For␈αexample,␈α
if␈αconjecture␈α
X␈αis␈αtrue␈α
for␈αall␈α
specializations␈αof␈α
C,␈αthen␈αit␈α
must
␈↓ α,␈↓be added to C.Conjecs and removed from the Conjecs facets of each specialization of C.


␈↓ α,␈↓∧␈↓ α<Any-concept . Conjecs . Suggest

␈↓ α,␈↓¬72. If X is probably related to Y, but no definite connection is known,
␈↓ α,␈↓¬␈↓ α\It's worthwhile looking for a specific conjecture tying X and Y together.

␈↓ α,␈↓How␈α
might␈α
AM␈α
know␈αthat␈α
X␈α
and␈α
Y␈αare␈α
only␈α
␈↓βprobably␈↓␈α
related?␈α
 X␈αand␈α
Y␈α
may␈α
play␈αthe␈α
same
␈↓ α,␈↓role␈αin␈αan␈αanalogy␈α(e.g.,␈αthe␈αsingleton␈αbag␈α"(T)"␈αand␈α"any␈αtypical␈αsingleton␈αbag"␈αshare␈αmany
␈↓ α,␈↓properties),␈α∞or␈α∞they␈α∞may␈α∂both␈α∞be␈α∞specializations␈α∞of␈α∞the␈α∂same␈α∞concept␈α∞Z␈α∞(e.g.,␈α∞two␈α∂kinds␈α∞of
␈↓ α,␈↓numbers),␈α∂or␈α∂they␈α∂may␈α∂both␈α∂have␈α∂been␈α∂created␈α∂in␈α∂the␈α∂same␈α∂unusual␈α∂way␈α∂(e.g.,␈α⊂Plus␈α∂and
␈↓ α,␈↓Times and Exponentiation are all creatable by ␈↓βrepeating␈↓ another operation).


␈↓ α,␈↓∧␈↓ α<Any-concept . Conjecs . Interest

␈↓ α,␈↓¬73. A conjecture about X is interesting if X is very interesting.


␈↓ α,␈↓¬74. A nonconstructive existence conjecture is interesting.

␈↓ α,␈↓Thus␈α∀the␈α∀unique␈α∀factorization␈α∀theorem␈α∀is␈α∪judged␈α∀to␈α∀be␈α∀interesting␈α∀because␈α∀it␈α∪merely
␈↓ α,␈↓guarantees␈α⊃that␈α⊃some␈α⊃factoring␈α⊃will␈α⊃be␈α⊃into␈α⊃primes.␈α⊃ If␈α⊃you␈α⊃give␈α⊃an␈α⊃algorithm␈α⊃for␈α⊃that
␈↓ α,␈↓factoring,␈α
then␈α
the␈αtheorem␈α
actually␈α
loses␈α
its␈αmystique␈α
and␈α
(according␈αto␈α
this␈α
rule)␈α
some␈αof
␈↓ α,␈↓its value. But it increases in value due to the next rule.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε153␈↓-

␈↓ α,␈↓¬75. A constructive existence conjecture is interesting if it is frequently used.


␈↓ α,␈↓¬76.␈α∩A␈α∩conjecture␈α∩C␈α∩about␈α∩X␈α∩is␈α∩interesting␈α∩if␈α∩the␈α∩origin␈α∩and␈α∩the␈α∩verification␈α∩of␈α∩C␈α∪for␈α∩each
␈↓ α,␈↓¬␈↓ β,specialization␈αof␈α
X␈αwas␈α
quite␈αindependent␈α
of␈αeach␈α
other,␈αand␈α
preceded␈αC's␈αbeing␈α
noticed
␈↓ α,␈↓¬␈↓ β,applicable to all X's.

␈↓ α,␈↓This␈α
would␈α
be␈α
even␈α
more␈α
striking␈α
if␈α
␈↓βproof␈↓␈α
techniques␈α
were␈α
known,␈α
and␈α
each␈αspecialized␈α
case
␈↓ α,␈↓had␈αa␈αseparate␈αkind␈αof␈αproof.␈α Many␈αnumber␈αtheory␈αresults␈αare␈αlike␈αthis,␈αwhere␈αthere␈αexists
␈↓ α,␈↓a␈αgeneral␈αproof␈αonly␈αfor␈αnumbers␈αbigger␈αthan␈α317,␈αsay,␈αand␈αall␈αsmaller␈αnumbers␈αare␈αsimply
␈↓ α,␈↓checked␈α
individually␈α
to␈αmake␈α
sure␈α
they␈αsatisfy␈α
the␈α
conjecture.␈α Category␈α
theory␈α
is␈αbuilt␈α
upon
␈↓ α,␈↓practically nothing but this heuristic.


␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.4.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for the Analogies facet of Any-concept␈↓)αβ␈↓


␈↓ α,␈↓∧␈↓ α<Any-concept . Analogies . Fillin

␈↓ α,␈↓¬77. To fill in conjectures involving concept C, where C is analogous to D,
␈↓ α,␈↓¬␈↓ α\Consider the analogue of each conjecture involving D.


␈↓ α,␈↓¬78. If the current task involves a specific analogy, and the request is to find more conjectures,
␈↓ α,␈↓¬␈↓ α\Then␈αconsider␈αthe␈α
analog␈αof␈αeach␈αinteresting␈α
conjecture␈αabout␈αany␈αconcept␈α
involved␈αcentrally
␈↓ α,␈↓¬␈↓ β,in the analogy.

␈↓ α,␈↓That␈α
is,␈α
this␈α
rule␈α∞suggests␈α
applying␈α
the␈α
preceding␈α
rule␈α∞to␈α
each␈α
concept␈α
which␈α
is␈α∞central␈α
to
␈↓ α,␈↓the␈α
given␈α
analogy.␈αThe␈α
result␈α
is␈αa␈α
∨ood␈α
of␈αnew␈α
conjectures.␈α
 There␈αis␈α
a␈α
tradeo≥␈α(explicitly
␈↓ α,␈↓taken␈α
into␈α
account␈α∞in␈α
the␈α
LISP␈α∞version␈α
of␈α
this␈α
rule)␈α∞between␈α
how␈α
interesting␈α∞a␈α
conjecture
␈↓ α,␈↓has␈α∞to␈α∞be,␈α∞and␈α∂how␈α∞centrally␈α∞a␈α∞concept␈α∂has␈α∞to␈α∞≡t␈α∞into␈α∂the␈α∞analogy,␈α∞in␈α∞order␈α∂to␈α∞determine
␈↓ α,␈↓what␈αresources␈α
AM␈αshould␈α
be␈αwilling␈α
to␈αexpend␈α
to␈α≡nd␈α
the␈αanalogous␈α
conjecture.␈α Note␈α
that
␈↓ α,␈↓this␈α⊂is␈α⊂not␈α⊂a␈α⊂general␈α⊂suggestion␈α⊂of␈α⊂what␈α⊂to␈α⊂do,␈α⊂but␈α⊂a␈α⊂speci≡c␈α⊂strategy␈α⊂for␈α⊃enlarging␈α⊂the
␈↓ α,␈↓analogy␈α∂itself.␈α∂If␈α∞the␈α∂new␈α∂conjecture␈α∞is␈α∂veri≡ed,␈α∂then␈α∞not␈α∂only␈α∂would␈α∞it␈α∂be␈α∂entered␈α∞under
␈↓ α,␈↓some␈α
Conjecs␈α
facet,␈α
but␈αit␈α
would␈α
also␈α
go␈α
to␈αenlarging␈α
the␈α
data␈α
structure␈α
which␈αrepresents␈α
the
␈↓ α,␈↓analogy.

␈↓ α,␈↓¬79.␈α
Let␈α
the␈α
analogy␈α
suggest␈α
how␈α
to␈α
specialize␈α
and␈α
generalize␈α
each␈α
concept␈α
into␈α
what␈α
is␈α
at␈αleast
␈↓ α,␈↓¬␈↓ β,the analog of a known, very interesting concept.

␈↓ α,␈↓Like␈αthe␈α
last␈αrule,␈α
this␈αone␈αsimply␈α
says␈αto␈α
use␈αthe␈αanalogy␈α
itself␈αas␈α
the␈α"reason"␈αfor␈α
exploring
␈↓ α,␈↓certain␈αnew␈αentities,␈αin␈αthis␈αcase␈αnew␈αconcepts.␈α When␈αthe␈αBags↔Numbers␈αanalogy␈αis␈αmade,
␈↓ α,␈↓AM␈αnotices␈αthat␈αSingleton␈αbags␈αand␈αEmpty␈αbags␈αare␈αtwo␈αinteresting,␈αextreme␈αspecializations
␈↓ α,␈↓of␈α
Bags.␈α The␈α
above␈αrule␈α
then␈α
allows␈αAM␈α
to␈αconstruct␈α
and␈α
study␈αwhat␈α
we␈αknow␈α
and␈αlove␈α
as
␈↓ α,␈↓the␈αnumbers␈αone␈αand␈αzero.␈αThe␈αanalogy␈αis␈α∨awed␈αbecause␈αthere␈αis␈αonly␈αone␈α"one",␈αalthough
␈↓ α,␈↓there␈αare␈αmany␈α
di≥erent␈αsingleton␈αbags.␈α But␈α
just␈αas␈αsingletons␈α
and␈αempty␈αbags␈αhave␈α
special
␈↓ α,␈↓properties␈α∞under␈α∞bag␈α∞operations,␈α∞so␈α∞do␈α∞0,1␈α∞under␈α∞numeric␈α∞operations.␈α∞ This␈α∞was␈α∂one␈α∞case
␈↓ α,␈↓where an analogy paid o≥ handsomely.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε154␈↓-


␈↓ α,␈↓¬80.␈α
If␈α
it␈α
is␈α
desired␈αto␈α
have␈α
an␈α
analogy␈α
between␈α
concepts␈αX␈α
and␈α
Y,␈α
then␈α
look␈α
for␈αtwo␈α
already-known
␈↓ α,␈↓¬␈↓ β,analogies between X↔Z and Z↔Y, for any Z.
␈↓ α,␈↓¬␈↓ α\If found, compose the two analogies and see if the resultant analogy makes sense.


␈↓ α,␈↓Since␈α⊂the␈α⊂analogies␈α⊃are␈α⊂really␈α⊂just␈α⊂substitutions,␈α⊃composing␈α⊂them␈α⊂has␈α⊂a␈α⊃familiar,␈α⊂precise
␈↓ α,␈↓meaning.␈αThis␈αrule␈αwas␈αnever␈αreally␈αused␈αby␈αAM,␈αdue␈αto␈αthe␈αpaucity␈αof␈αanalogies.␈αThe␈α
user
␈↓ α,␈↓can␈α∂push␈α∂AM␈α∂into␈α∂creating␈α∂more␈α∂of␈α∂them,␈α∂and␈α∂ultimately␈α∂using␈α∂this␈α∂rule.␈α∂A␈α∂chain␈α∞from
␈↓ α,␈↓X↔Z↔Y↔X can be found which presents a new, bizarre analogy from X to itself.


␈↓ α,␈↓∧␈↓ α<Any-concept . Analogies . Suggest

␈↓ α,␈↓¬81.␈αIf␈αan␈αanalogy␈αis␈αstrong,␈αand␈αone␈α
concept␈αhas␈αa␈αvery␈αinteresting␈αuniversal␈αconjecture␈αC␈α(For␈α
all
␈↓ α,␈↓¬␈↓ β,x in B...), but the analog conjecture C' is false,
␈↓ α,␈↓¬␈↓ α\Then␈α
it's␈αworth␈α
constructing␈α
the␈αset␈α
of␈αitems␈α
in␈α
B'␈αfor␈α
which␈α
the␈αconjecture␈α
holds.␈α It's␈α
perhaps
␈↓ α,␈↓¬␈↓ β,even more interesting to isolate the set of exceptional elements.

␈↓ α,␈↓With␈αthe␈αAdd↔Times␈αanalogy,␈αit's␈αtrue␈αthat␈αall␈αnumbers␈αn>1␈αcan␈αbe␈αrepresented␈αas␈αthe␈α
sum
␈↓ α,␈↓of␈α
two␈α
other␈α
numbers␈α
(each␈α
of␈α
them␈α
smaller␈αthan␈α
n),␈α
but␈α
it␈α
is␈α
␈↓βnot␈↓␈α
true␈α
that␈α
all␈αnumbers␈α
(with
␈↓ α,␈↓just␈α∪a␈α∀couple␈α∪exceptions)␈α∪can␈α∀be␈α∪represented␈α∀as␈α∪the␈α∪product␈α∀of␈α∪other␈α∀(hence␈α∪smaller)
␈↓ α,␈↓numbers.␈α∪ The␈α∪above␈α∪rule␈α∩has␈α∪AM␈α∪de≡ne␈α∪the␈α∪set␈α∩of␈α∪numbers␈α∪which␈α∪can/can't␈α∪be␈α∩so
␈↓ α,␈↓represented.␈α These␈α
are␈αjust␈αthe␈α
composite␈αnumbers␈αand␈α
the␈αset␈αof␈α
primes.␈α This␈αsecond␈α
way
␈↓ α,␈↓of␈αencountering␈αprimes␈α
was␈αvery␈αunexpected␈α
¬␈αboth␈αby␈α
AM␈αand␈αby␈α
the␈αauthor.␈αIt␈α
expresses
␈↓ α,␈↓the␈αdeep␈αfact␈α
that␈αone␈αdi≥erence␈α
between␈αAdd␈αand␈α
Times␈αis␈αthe␈α
presence␈αof␈αprimes␈αonly␈α
for
␈↓ α,␈↓multiplication.  At the time of its discovery, AM didn't appreciate this fully of course.

␈↓ α,␈↓¬82.␈α
If␈α
space␈α
is␈α
tight,␈α
and␈α
no␈α
use␈α
of␈α
the␈α
analogy␈α
has␈α
ever␈α
been␈α
made,␈α
and␈α
it␈α
is␈α
very␈α
old,␈α
and␈αit␈α
takes
␈↓ α,␈↓¬␈↓ β,up a lot of space,
␈↓ α,␈↓¬␈↓ α\Then it is permissable to forget it without a trace.


␈↓ α,␈↓¬83.␈α
If␈αtwo␈α
valuable␈αconjectures␈α
are␈αsyntactically␈α
identical,␈αand␈α
can␈αbe␈α
made␈αidentical␈α
by␈α
a␈αsimple
␈↓ α,␈↓¬␈↓ β,substitution, then tentatively consider the analogy which is that very substitution.

␈↓ α,␈↓Thus␈α∩the␈α∩associative/commutative␈α∩property␈α∩of␈α∩Add␈α⊃and␈α∩Times␈α∩causes␈α∩them␈α∩to␈α∩be␈α⊃tied
␈↓ α,␈↓together in an analogy, because of this rule.

␈↓ α,␈↓¬84. If an analogy is very interesting and very complete,
␈↓ α,␈↓¬␈↓ α\Then␈α
spend␈α
some␈α
time␈α
refining␈α
it,␈α
looking␈α
for␈α
small␈α
exceptions.␈α
 If␈α
none␈α
are␈α
found,␈α
see␈α
whether
␈↓ α,␈↓¬␈↓ β,the two situations are genuinely isomorphic.


␈↓ α,␈↓¬85.␈α∩If␈α∩concepts␈α∩X␈α∩and␈α∩Y␈α∩are␈α⊃analogous,␈α∩look␈α∩for␈α∩analogies␈α∩between␈α∩their␈α∩specializations,␈α⊃or
␈↓ α,␈↓¬␈↓ β,between their generalizations.

␈↓ α,␈↓This rule is not used much by AM, although the author thought it would be.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε155␈↓-

␈↓ α,␈↓∧␈↓ α<Any-concept . Analogies . Interest

␈↓ α,␈↓¬86.␈α
An␈α
analogy␈α
which␈α
has␈α
no␈α
discrepancies␈α
whatsoever␈α
is␈α
not␈α
as␈α
interesting␈α
as␈α
a␈α
slightly␈αflawed
␈↓ α,␈↓¬␈↓ β,analogy.


␈↓ α,␈↓¬87.␈α∂An␈α∂analogy␈α∂is␈α∂interesting␈α∂if␈α∂it␈α∂associates␈α∂(for␈α∂the␈α∂first␈α∂time)␈α∂two␈α∂concepts␈α∂which␈α∂are␈α∞each
␈↓ α,␈↓¬␈↓ β,unusally␈α∀fully␈α∀filled␈α∀out␈α∀(having␈α∀many␈α∀conjectures,␈α∀many␈α∀examples,␈α∀many␈α∀interest
␈↓ α,␈↓¬␈↓ β,features, etc.).



␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.5.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for the Genl/Spec facets of Any-concept␈↓)αβ␈↓


␈↓ α,␈↓∧␈↓ α<Any-concept . Genl/Spec . Fillin

␈↓ α,␈↓¬88. To fill in specializations of X, if it was very easy to find examples of X,
␈↓ α,␈↓¬␈↓ α\Grab␈α∩some␈α∩features␈α⊃which␈α∩would␈α∩indicate␈α⊃than␈α∩an␈α∩X␈α⊃was␈α∩interesting␈α∩(some␈α∩entries␈α⊃from
␈↓ α,␈↓¬␈↓ β,X.Interest,␈αor␈αmore␈αremote␈αInterest␈αpredicates␈αgarnered␈αby␈αrippling),␈αand␈αconjoin␈αthem
␈↓ α,␈↓¬␈↓ β,onto the definition of X, thereby creating a new concept.

␈↓ α,␈↓Here's␈α⊂one␈α⊂instance␈α⊂where␈α⊂the␈α∂above␈α⊂rule␈α⊂was␈α⊂used:␈α⊂It␈α∂was␈α⊂so␈α⊂easy␈α⊂for␈α⊂AM␈α⊂to␈α∂produce
␈↓ α,␈↓examples␈αof␈αsets␈αthat␈αit␈αdecided␈αto␈αspecialize␈αthat␈αconcept.␈αThe␈αabove␈αrule␈αthen␈αplucked␈αthe
␈↓ α,␈↓interestingness␈αfeature␈α"all␈αpairs␈αof␈αmembers␈αsatisfy␈αthe␈αsame␈αrare␈αpredicate"␈αand␈αconjoined
␈↓ α,␈↓it␈α∂to␈α∂the␈α∂old␈α⊂de≡nition␈α∂of␈α∂Sets.␈α∂ The␈α⊂new␈α∂concept,␈α∂Interesting-sets,␈α∂included␈α⊂all␈α∂singletons
␈↓ α,␈↓(because␈α⊂all␈α⊃pairs␈α⊂of␈α⊂members␈α⊃drawn␈α⊂from␈α⊃a␈α⊂singleton␈α⊂satisfy␈α⊃the␈α⊂predicate␈α⊃Equal)␈α⊂and
␈↓ α,␈↓empty sets.

␈↓ α,␈↓¬89. To fill in generalizations of concept X,
␈↓ α,␈↓¬␈↓ α\Take␈αthe␈αdefinition␈αe,␈αand␈αreplace␈αit␈αby␈αa␈αgeneralization␈αof␈αe.␈α If␈αe␈αis␈αa␈αconcept,␈αuse␈αe.Genl;␈αif
␈↓ α,␈↓¬␈↓ β,e␈α∪is␈α∪a␈α∪conjunction,␈α∪then␈α∪remove␈α∪a␈α∪conjunct␈α∪or␈α∪generalize␈↓	6␈↓¬␈α∪a␈α∪conjunct;␈α∪if␈α∪e␈α∪is␈α∩a
␈↓ α,␈↓¬␈↓ β,disjunction,␈αthen␈αadd␈αa␈αdisjunct␈αor␈αgeneralize␈αa␈αdisjunct;␈αif␈αe␈αis␈αnegated,␈αthen␈αspecialize
␈↓ α,␈↓¬␈↓ β,the␈α
negate;␈αif␈α
e␈αis␈α
an␈α
example␈αof␈α
E,␈αthen␈α
replace␈αe␈α
by␈α
"any␈αexample␈α
of␈αE";␈α
if␈αe␈α
satisfies
␈↓ α,␈↓¬␈↓ β,any␈α∞property␈α∂P,␈α∞then␈α∂replace␈α∞e␈α∂by␈α∞"anything␈α∂satisfying␈α∞P";␈α∂if␈α∞e␈α∂is␈α∞a␈α∂constant␈↓	7␈↓¬,␈α∞then
␈↓ α,␈↓¬␈↓ β,replace␈αe␈αby␈αa␈αnew␈αvariable␈α(or␈αan␈αexisting␈αone)␈αwhich␈αcould␈αassume␈αvalue␈αe;␈αif␈αe␈αis␈αa
␈↓ α,␈↓¬␈↓ β,variable, then enlarge its scope of possible bindings.

␈↓ α,␈↓This␈αrule␈αcontains␈αa␈αbag␈αof␈αtricks␈α
for␈αgeneralizing␈αany␈αLISP␈αpredicate,␈αthe␈αde≡nition␈αof␈α
any
␈↓ α,␈↓concept. They are all ␈↓βsyntactic␈↓ tricks, however.

␈↓ α,␈↓¬90.␈αTo␈αfill␈αin␈αgeneralizations␈αof␈αconcept␈αX,␈αIf␈αsome␈αconjecture␈αexists␈αabout␈α"all␈αX's␈αand␈αY's"␈αor␈α"in
␈↓ α,␈↓¬␈↓ β,X or Y", for some other concept Y,
␈↓ α,␈↓¬␈↓ α\Create a new concept, a generalization of both X and Y, defined as their disjunction.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	6␈↓ε i.e., recur.
␈↓ α,␈↓ε␈↓	7␈↓ε␈α
Of␈α
course␈αit's␈α
unlikely␈α
that␈αa␈α
concept␈α
is␈α
defined␈αsimply␈α
as␈α
a␈αconstant,␈α
but␈α
the␈α
preceding␈αfootnote␈α
shows␈α
that␈αthis␈α
little
␈↓ α,␈↓ε␈↓ βLprogram can be entered recursively, being fed a sub-expression of the definition.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε156␈↓-

␈↓ α,␈↓This␈α⊗rule␈α↔contains␈α⊗another␈α↔trick␈α⊗for␈α⊗generalizing␈α↔any␈α⊗concept,␈α↔although␈α⊗it␈α↔is␈α⊗more
␈↓ α,␈↓meaningful,␈α
more␈α
semantic␈α
than␈α
the␈α∞previous␈α
rule's␈α
tricks.␈α
 Many␈α
theorems␈α
are␈α∞true␈α
about
␈↓ α,␈↓numbers␈αwith␈α1␈αor␈α2␈αdivisors,␈αso␈αthis␈αmight␈αbe␈αone␈αreasonable␈αway␈αto␈αgeneralize␈αNumbers-
␈↓ α,␈↓with-1-divisor into a new useful␈↓	8␈↓ concept.

␈↓ α,␈↓¬91. To fill in generalizations of concept X,
␈↓ α,␈↓¬␈↓ α\If other generalizations G1, G2 of X exist but are TOO general,
␈↓ α,␈↓¬␈↓ α\Create␈α
a␈α
new␈αconcept,␈α
a␈α
generalization␈αof␈α
X␈α
and␈α
a␈αspecialization␈α
of␈α
both␈αG1␈α
and␈α
G2,␈αdefined␈α
as
␈↓ α,␈↓¬␈↓ β,the conjunction of G1 and G2's definitions.

␈↓ α,␈↓Thus␈α∃when␈α∃AM␈α∃generalizes␈α∀Reverse-all-levels␈α∃into␈α∃Reverse-top-level␈α∃and␈α∀Reverse-≡rst-
␈↓ α,␈↓element,␈αthe␈α
above␈αrule␈α
causes␈αAM␈αto␈α
create␈αa␈α
new␈αoperation,␈αwhich␈α
reverses␈αthe␈α
top␈αlevel
␈↓ α,␈↓and␈αwhich␈αreverses␈αthe␈αCAR␈↓	9␈↓␈αof␈αthe␈αoriginal␈αlist.␈α While␈αnot␈αparticularly␈αuseful,␈αthe␈αreader
␈↓ α,␈↓should␈αobserve␈αthat␈αit␈αis␈αin␈αfact␈α
midway␈αin␈αgenerality␈αbetween␈αthe␈αoriginal␈αReverse␈α
function
␈↓ α,␈↓and the ≡rst two generalizations.

␈↓ α,␈↓¬92. To fill in specializations of concept X,
␈↓ α,␈↓¬␈↓ α\Take␈α
the␈αdefinition␈α
e,␈αand␈α
replace␈α
it␈αby␈α
a␈αspecialization␈α
of␈α
e.␈α If␈α
e␈αis␈α
a␈α
concept,␈αuse␈α
e.Genl;␈αif␈α
e
␈↓ α,␈↓¬␈↓ β,is␈α∞a␈α
disjunction,␈α∞then␈α
remove␈α∞a␈α
disjunct␈α∞or␈α
specialize␈α∞a␈α
disjunct;␈α∞if␈α
e␈α∞is␈α∞a␈α
conjunction,
␈↓ α,␈↓¬␈↓ β,then␈α⊃add␈α⊃a␈α⊃conjunct␈α⊃or␈α⊃specialize␈α⊃a␈α⊃conjunct;␈α⊃if␈α⊃e␈α⊃is␈α⊃negated,␈α⊃then␈α⊃generalize␈α⊃the
␈↓ α,␈↓¬␈↓ β,negate;␈αif␈αe␈αis␈α"any␈αexample␈αof␈αE",␈αthen␈αreplace␈αe␈αby␈αa␈αparticular␈αexample␈αof␈αE;␈αif␈αe␈αis
␈↓ α,␈↓¬␈↓ β,"anything␈αsatisfying␈αP",␈αthen␈αreplace␈αe␈αby␈αa␈αparticular␈αsatisfier␈αof␈αP;␈αif␈αe␈αis␈αa␈αvariable,
␈↓ α,␈↓¬␈↓ β,then replace it by a well-chosen constant or restrict its scope.

␈↓ α,␈↓This␈αrule␈αcontains␈αa␈αbag␈αof␈αtricks␈αfor␈αspecializing␈αany␈αLISP␈αpredicate,␈αthe␈αde≡nition␈αof␈αany
␈↓ α,␈↓concept.␈α∞ They␈α∞are␈α
all␈α∞␈↓βsyntactic␈↓␈α∞tricks,␈α
however.␈α∞Note␈α∞that␈α
the␈α∞Lisp␈α∞code␈α
for␈α∞this␈α∞rule␈α
will
␈↓ α,␈↓typically call itself (recur) in order to specialize small pieces of the original de≡nition.

␈↓ α,␈↓¬93.␈αTo␈α
fill␈αin␈α
specializations␈αof␈αconcept␈α
X,␈αIf␈α
some␈αconjecture␈αexists␈α
about␈α"all␈α
X's␈αwhich␈α
are␈αalso
␈↓ α,␈↓¬␈↓ β,Y's" or "in X and Y", for some other concept Y,
␈↓ α,␈↓¬␈↓ α\Create a new concept, a specialization of both X and Y, defined as their conjunction.

␈↓ α,␈↓This␈α↔rule␈α↔contains␈α↔another␈α↔trick␈α↔for␈α↔specializing␈α↔any␈α↔concept,␈α↔although␈α↔it␈α_is␈α↔more
␈↓ α,␈↓meaningful,␈α∂more␈α∂semantic␈α∂than␈α∂the␈α∞previous␈α∂rule's␈α∂tricks.␈α∂ Many␈α∂theorems␈α∂about␈α∞primes
␈↓ α,␈↓contain␈α∞the␈α
condition␈α∞"p>2";␈α∞i.e.,␈α
they␈α∞are␈α
really␈α∞true␈α∞about␈α
primes␈α∞which␈α
are␈α∞odd.␈α∞So␈α
this
␈↓ α,␈↓might␈α∂be␈α∂one␈α∞reasonable␈α∂way␈α∂to␈α∂specialize␈α∞Primes␈α∂into␈α∂a␈α∞new␈α∂concept:␈α∂by␈α∂conjoining␈α∞the
␈↓ α,␈↓de≡nitions␈α
of␈α
Primes␈α
and␈α
Odd-numbers,␈α
into␈α
the␈α
new␈α
concept␈α
Odd-primes.␈α
Here's␈αanother
␈↓ α,␈↓usage␈αof␈αthis␈αrule:␈αIf␈αAM␈αhad␈αoriginally␈α
de≡ned␈αPrimes␈αto␈αinclude␈α`1',␈αthen␈αthe␈αfrequency␈α
of
␈↓ α,␈↓conjectures␈α
where␈α
1␈α
was␈α
an␈α
exception␈α
would␈α
trigger␈α
this␈α
rule␈α
to␈α
de≡ne␈α
Primes␈α
more␈α
normally
␈↓ α,␈↓(p␈↓¬≥␈↓2).

␈↓ α,␈↓¬94. To fill in specializations of concept X,
␈↓ α,␈↓¬␈↓ α\If other specializations S1, S2 of X exist but are TOO restrictive to be interesting,
␈↓ α,␈↓¬␈↓ α\Create␈αa␈α
new␈αconcept,␈α
a␈αspecialization␈α
of␈αX␈α
and␈αa␈α
generalization␈αof␈α
both␈αS1␈α
and␈αS2,␈αdefined␈α
as
␈↓ α,␈↓¬␈↓ β,the disjunction of S1 and S2's definitions.



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	8␈↓ε at least, several theorems will be stated more concisely using this new concept: Numbers with 1 or 2 divisors.
␈↓ α,␈↓ε␈↓	9␈↓ε also the CAR of the CAR, etc., until a non-list is encountered.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε157␈↓-


␈↓ α,␈↓¬95. To fill in generalizations of concept X, when a recursive definition of X exists,
␈↓ α,␈↓¬␈↓ α\If␈α⊃the␈α⊃definition␈α⊃contains␈α∩two␈α⊃conjoined␈α⊃recursive␈α⊃calls,␈α∩replace␈α⊃them␈α⊃by␈α⊃a␈α∩disjunction␈α⊃or
␈↓ α,␈↓¬␈↓ β,eliminate one call entirely.
␈↓ α,␈↓¬␈↓ α\If␈α∞there␈α∞is␈α∞only␈α
one␈α∞recursive␈α∞call,␈α∞disjoin␈α
a␈α∞second␈α∞call,␈α∞this␈α
one␈α∞on␈α∞a␈α∞different␈α
destructive
␈↓ α,␈↓¬␈↓ β,function␈αapplied␈α
to␈αthe␈αoriginal␈α
argument.␈α If␈α
the␈αoriginal␈αdestructive␈α
function␈αis␈α
one␈αof
␈↓ α,␈↓¬␈↓ β,{CAR,CDR}, then let the new one be the other member of that pair.

␈↓ α,␈↓AM␈α
uses␈α
the␈α
≡rst␈α
part␈α
of␈α
this␈α
rule␈α
to␈α
turn␈α
Equal-lists␈α
into␈α
two␈α
variants␈α
of␈αSame-length-as.
␈↓ α,␈↓The␈α∞second␈α∞part,␈α∂while␈α∞surprisingly␈α∞unused,␈α∂could␈α∞work␈α∞on␈α∂this␈α∞de≡nition␈α∞of␈α∂␈↓αMEMBER:␈α∞"λ
␈↓ α,␈↓α(x,L)␈α∂LISTP(L)␈α∞and:␈α∂[x=CAR(L)␈α∞or␈α∂MEMBER(x,CDR(L))]"␈↓,␈α∂which␈α∞is␈α∂just␈α∞"membership␈α∂at␈α∂the␈α∞top
␈↓ α,␈↓level␈αof",␈αor␈α
␈↓¬ε␈↓,␈αand␈αtransform␈αit␈α
into␈αthis␈αone␈αof␈α
␈↓αMEM␈↓,␈αwhich␈αrepresents␈αmembership␈α
at␈α␈↓βany␈↓
␈↓ α,␈↓depth:␈α∪"␈↓αλ(x,L)␈α∪LISTP␈↓	10␈↓α(L)␈α∪and:␈α∀[x=CAR(L)␈α∪or␈α∪MEM(x,CDR(L))␈α∪or␈α∪MEM(x,CAR(L))]␈↓".␈α∀ The␈α∪rule
␈↓ α,␈↓noticed a recursive call on CDR(L), and simply disjoined a recursive call on CAR(L).

␈↓ α,␈↓¬96. To fill in specializations of concept X, when a recursive definition of C exists,
␈↓ α,␈↓¬␈↓ α\If␈α⊃the␈α⊃definition␈α⊃contains␈α∩two␈α⊃disjoined␈α⊃recursive␈α⊃calls,␈α∩replace␈α⊃them␈α⊃by␈α⊃a␈α∩conjunction␈α⊃or
␈↓ α,␈↓¬␈↓ β,eliminate one call entirely.
␈↓ α,␈↓¬␈↓ α\If␈αthere␈αis␈αonly␈αone␈αrecursive␈αcall,␈αconjoin␈αa␈αsecond␈αon␈αanother␈αdestructive␈αfunction␈αapplied␈αto
␈↓ α,␈↓¬␈↓ β,the␈αoriginal␈αargument.␈αOften␈αthe␈αtwo␈αrecursions␈αwill␈αbe␈αon␈αthe␈αCAR␈αand␈αthe␈αCDR␈αof␈αthe
␈↓ α,␈↓¬␈↓ β,original argument to the predicate which is the definition for X.

␈↓ α,␈↓This␈α
is␈α
closely␈α
related␈α
to␈α
the␈α
preceding␈α
rule.␈α
 Just␈α
as␈α
it␈α
turned␈α
the␈α
concept␈α
of␈α
`element␈α
of'␈α
into
␈↓ α,␈↓the␈α⊃more␈α⊃general␈α∩one␈α⊃of␈α⊃`membership␈α∩at␈α⊃any␈α⊃depth',␈α⊃the␈α∩above␈α⊃rule␈α⊃can␈α∩specialize␈α⊃the
␈↓ α,␈↓de≡nition␈α∀of␈α∀␈↓αMEMBER␈↓␈α∀into␈α∀this␈α∀one,␈α∀called␈α∀␈↓αAMEM:␈α∀"λ␈α∀(x,L)␈α∀LISTP(L)␈α∀and:␈α∀[x=CAR(L)␈α∪or:
␈↓ α,␈↓α[AMEM(x,CDR(L)) and AMEM(x,CAR(L))]]"␈↓.␈↓	11␈↓

␈↓ α,␈↓¬97. To fill in specializations of concept X,
␈↓ α,␈↓¬␈↓ α\Find,,␈α
within␈αa␈α
definition␈α
of␈αX␈α
(at␈α
even␈αparity␈α
of␈α
NOT's),␈αan␈α
expression␈α
of␈αthe␈α
form␈α
"For␈αsome␈α
x
␈↓ α,␈↓¬␈↓ β,in X, P(x)", and replace it either by "For all x in X, P(x)", or by P(x␈↓ε␈↓#vo␈↓#␈↓¬).

␈↓ α,␈↓Thus␈α"sets,␈αall␈αpairs␈αof␈αwhose␈αmembers␈αsatisfy␈αSOME␈αinteresting␈αpredicate"␈αgets␈αspecialized
␈↓ α,␈↓into␈α"sets,␈αall␈α
pairs␈αof␈αwhose␈α
members␈αsatisfy␈αEquality".␈α The␈α
same␈αrule,␈αwith␈α
"even␈αparity"
␈↓ α,␈↓replaced␈α∩by␈α∩"odd␈α∩parity",␈α⊃is␈α∩useful␈α∩for␈α∩␈↓βgeneralizing␈↓␈α∩a␈α⊃de≡nition.␈α∩ This␈α∩rule␈α∩is␈α∩really␈α⊃4
␈↓ α,␈↓separate␈αrules,␈αin␈αthe␈αLISP␈αprogram.␈α The␈αsame␈αrule,␈αwith␈αthe␈αtransformations␈αgoing␈αin␈αthe
␈↓ α,␈↓opposite␈α∞direction,␈α∞is␈α∞also␈α∞used␈α∞for␈α
generalizing.␈α∞ The␈α∞same␈α∞rule,␈α∞with␈α∞the␈α
transformations
␈↓ α,␈↓reversed␈αand␈αthe␈αparity␈αreversed,␈αis␈αused␈αfor␈αspecializing␈αa␈αde≡nition.␈α Here␈αis␈αthat␈αdoubly-
␈↓ α,␈↓switched rule:

␈↓ α,␈↓¬98. To fill in specializations of concept X,
␈↓ α,␈↓¬␈↓ α\Find␈αwithin␈α
a␈αdefinition␈αof␈α
X␈α(at␈αodd␈α
parity␈αof␈αNOT's)␈α
an␈αexpression␈αof␈α
the␈αform␈α"For␈α
all␈αx␈αin␈α
X,
␈↓ α,␈↓¬␈↓ β,P(x)",␈α
and␈α
replace␈α
it␈α
either␈αby␈α
"For␈α
some␈α
x␈α
in␈α
X,␈αP(x)",␈α
or␈α
by␈α
P(x␈↓ε␈↓#vo␈↓#␈↓¬).␈α
 Or␈αreplace␈α
"P(α)",
␈↓ α,␈↓¬␈↓ β,where␈αα␈αis␈αa␈αconstant,␈αby␈α"For␈αsome␈αx␈αin␈αA,␈αP(x)"␈αwhere␈αA␈αis␈αa␈αconcept␈αof␈αwhich␈αα␈αis
␈↓ α,␈↓¬␈↓ β,one example.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	10␈↓ε The Interlisp function LISTP(L) tests whether or not L is a (nonnull) list.
␈↓ α,␈↓ε␈↓	11␈↓ε␈α	This␈α	operation␈α	is␈α	almost␈α	impossible␈α	to␈αλexplain␈α	verbally.␈α	AMEM(x,L)␈α	means␈α	that␈α	x␈α	is␈αλan␈α	element␈α	of␈α	L,␈α	and␈α	for␈α	each␈αλmember
␈↓ α,␈↓ε␈↓ βLM␈αof␈α
L␈αbefore␈α
the␈αx,␈α
M␈αis␈α
an␈αordered␈α
structure␈αand␈αx␈α
is␈αan␈α
element␈αof␈α
M,␈αand␈α
for␈αeach␈α
member␈αN␈αof␈α
M
␈↓ α,␈↓ε␈↓ βLbefore the x which is inside M,... etc. E.g., <[x] [ ␈↓¬<␈↓ε<x a b> <x> x d e␈↓¬>␈↓ε <x f> x g h ] [<x i> x j] x k [l] m>.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε158␈↓-

␈↓ α,␈↓¬99. When creating in a specialization S of concept C,
␈↓ α,␈↓¬␈↓ α\Note that S.Genl should contain C, and that C.Spec should contain S.

␈↓ α,␈↓The analogous rule exists, in which all spec and genl are switched.


␈↓ α,␈↓∧␈↓ α<Any-concept . Genl/Spec . Suggest

␈↓ α,␈↓¬100. After creating a new specialization S of concept C,
␈↓ α,␈↓¬␈↓ α\Explicitly look for ties between S and other known specializations of C.

␈↓ α,␈↓For␈αexample,␈αafter␈αAM␈αde≡nes␈αthe␈αnew␈αconcept␈αof␈αNumbers-with-3-divisors,␈αit␈αlooks␈α
around
␈↓ α,␈↓for ties between that kind of number and other kinds of numbers.

␈↓ α,␈↓¬101. After creating a new generalization G of concept C,
␈↓ α,␈↓¬␈↓ αlExplicitly look for ties between G and other close generalizations of C.

␈↓ α,␈↓For␈α∂example,␈α⊂AM␈α∂de≡ned␈α⊂the␈α∂new␈α∂predicates␈α⊂Same-size-CARs␈α∂and␈α⊂Same-size-CDRs␈↓	12␈↓␈α∂as
␈↓ α,␈↓two␈α∞generalizations␈α∞of␈α∞Equality.␈α∂The␈α∞above␈α∞rule␈α∞then␈α∂suggested␈α∞that␈α∞AM␈α∞explicitly␈α∂try␈α∞to
␈↓ α,␈↓≡nd␈α
some␈αconnection␈α
between␈αthese␈α
two␈α
new␈αpredicates.␈α
 Although␈α␈↓βAM␈↓␈α
failed,␈α
Don␈αKnuth
␈↓ α,␈↓(using␈αa␈αsimilar␈α
heuristic,␈αperhaps)␈αalso␈αlooked␈α
for␈αa␈αconnection,␈αand␈α
found␈αone:␈αit␈αturns␈α
out
␈↓ α,␈↓that␈αthe␈α
two␈αpredicates␈αare␈α
both␈αways␈αof␈α
de≡ning␈αthe␈αrelation␈α
we␈αintuitively␈α
understand␈αas
␈↓ α,␈↓"having the same length as".

␈↓ α,␈↓¬102. After creating a new specialization S of concept C,
␈↓ α,␈↓¬␈↓ αlConsider looking for examples of S.

␈↓ α,␈↓This has to be said explicitly, because all too often a concept is specialized into vacuousness.

␈↓ α,␈↓¬103. After creating a new generalization G of concept C,
␈↓ α,␈↓¬␈↓ αlConsider looking for non-examples of G.

␈↓ α,␈↓This␈α∞has␈α∞to␈α
be␈α∞said␈α∞explicitly,␈α
because␈α∞all␈α∞too␈α
often␈α∞a␈α∞concept␈α
is␈α∞generalized␈α∞into␈α
vacuous
␈↓ α,␈↓universality. This rule is less useful to AM than the preceding one.

␈↓ α,␈↓¬104. If concept C possesses some very interesting property lacked by one of its specializations S,
␈↓ α,␈↓¬␈↓ αlThen␈αconsidering␈αcreating␈αa␈αconcept␈αintermediate␈αin␈αspecialization␈αbetween␈αC␈αand␈αS,␈αand␈αsee
␈↓ α,␈↓¬␈↓ β,whether that possesses the property.

␈↓ α,␈↓This rule will trigger whenever a new generalization or specialization is created.

␈↓ α,␈↓¬105.␈αIf␈α
concept␈αS␈α
is␈αnow␈αvery␈α
interesting,␈αand␈α
S␈αwas␈αcreated␈α
as␈αa␈α
specialization␈αof␈α
some␈αearlier
␈↓ α,␈↓¬␈↓ β,concept C,
␈↓ α,␈↓¬␈↓ αlGive␈α∩extra␈α∪consideration␈α∩to␈α∪specializing␈α∩S,␈α∪and␈α∩to␈α∪specializing␈α∩concept␈α∪C␈α∩again␈α∪(but␈α∩in
␈↓ α,␈↓¬␈↓ β,different ways than ever before).


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	12␈↓ε␈αTwo␈α
lists␈αsatisfy␈α
Same-size-CDRs␈αiff␈αthey␈α
have␈αthe␈α
same␈αnumber␈αof␈α
members.␈α Two␈α
lists␈αsatisfy␈αSame-size-CARs␈α
iff
␈↓ α,␈↓ε␈↓ βL(when␈α	written␈α	out␈α	in␈αλstandard␈α	LISP␈α	notation)␈α	they␈αλhave␈α	the␈α	same␈α	number␈αλof␈α	initial␈α	left␈α	parentheses␈α	and␈αλalso
␈↓ α,␈↓ε␈↓ βLhave the same first identifier following that last initial left parenthesis.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε159␈↓-

␈↓ α,␈↓The␈α∩next␈α∩rule␈α∩is␈α∩the␈α∩analog␈α∩of␈α⊃the␈α∩preceding␈α∩one.␈α∩ They␈α∩incorporate␈α∩tiny␈α∩bits␈α∩of␈α⊃the
␈↓ α,␈↓strategies of hill-climbing and learning from one's successes.

␈↓ α,␈↓¬106.␈αIf␈αconcept␈αG␈αis␈αnow␈αvery␈αinteresting,␈αand␈αG␈αwas␈αcreated␈αas␈αa␈αgeneralization␈αof␈αsome␈αearlier
␈↓ α,␈↓¬␈↓ β,concept C,
␈↓ α,␈↓¬␈↓ αlGive extra consideration to generalizing G, and to generalizing C in other ways.

␈↓ α,␈↓The␈α∩analogous␈α∩rules␈α∩exist,␈α⊃for␈α∩concepts␈α∩that␈α∩have␈α⊃become␈α∩so␈α∩boring␈α∩they've␈α∩just␈α⊃been
␈↓ α,␈↓discarded:

␈↓ α,␈↓¬107.␈α∃If␈α∃concept␈α∃X␈α∃proved␈α∀to␈α∃be␈α∃a␈α∃dead-end,␈α∃and␈α∀X␈α∃was␈α∃created␈α∃as␈α∃a␈α∃generalization␈α∀of
␈↓ α,␈↓¬␈↓ β,(specialization of) some earlier concept C,
␈↓ α,␈↓¬␈↓ αlGive␈α
less␈α
consideration␈αto␈α
generalizing␈α
(specializing)␈α
X,␈αand␈α
to␈α
generalizing␈α
(specializing)␈αC␈α
in
␈↓ α,␈↓¬␈↓ β,other ways in the future.



␈↓ α,␈↓∧␈↓ α<Any-concept . Genl/Spec . Check

␈↓ α,␈↓¬108. When checking a generalization G of concept C,
␈↓ α,␈↓¬␈↓ αlSpecifically test to ensure that G is not equivalent to C.
␈↓ α,␈↓¬␈↓ αlThe␈α
easiest␈α
way␈α
is␈α
to␈α∞examine␈α
the␈α
non-examples␈α
(especially␈α
boundary␈α
non-examples)␈α∞of␈α
C,
␈↓ α,␈↓¬␈↓ β,and␈αlook␈α
for␈αone␈αsatisfying␈α
G;␈αor␈αexamine␈α
the␈αexamples␈αof␈α
G␈α(esp.␈αboundary)␈α
and␈αlook
␈↓ α,␈↓¬␈↓ β,for one failing to satisfy C.
␈↓ α,␈↓¬␈↓ αlIf␈α
they␈α
appear␈α
to␈α
be␈αthe␈α
same␈α
concept,␈α
look␈α
carefully␈α
at␈αG.␈α
Are␈α
there␈α
any␈α
specializations␈αof␈α
G
␈↓ α,␈↓¬␈↓ β,whose␈αexamples␈αhave␈αnever␈αbeen␈αfilled␈αin?␈αIf␈αso,␈αthen␈αby␈αall␈αmeans␈αsuggest␈αlooking␈αfor
␈↓ α,␈↓¬␈↓ β,such concepts' examples before concluding that G and C are really equivalent.
␈↓ α,␈↓¬␈↓ β,If they are the same, then replace one by a conjecture.
␈↓ α,␈↓¬␈↓ β,If␈α∂they␈α∞are␈α∂different,␈α∂make␈α∞sure␈α∂that␈α∂some␈α∞boundary␈α∂non-example␈α∂of␈α∞C␈α∂(which␈α∂is␈α∞an
␈↓ α,␈↓¬␈↓ ∧,example of G) is explicitly stored for C.

␈↓ α,␈↓This␈αrule␈α
makes␈αsure␈αthat␈α
AM␈αis␈α
not␈αdeluding␈αitself.␈α
 When␈αAM␈αgeneralizes␈α
Numbers-with-
␈↓ α,␈↓1-divisor␈α↔into␈α⊗Numbers-which-equal-their-no-of-divisors,␈α↔it␈α↔still␈α⊗hasn't␈α↔gotten␈α↔past␈α⊗the
␈↓ α,␈↓singleton␈α
set␈α
{1}.␈αThe␈α
conjecture␈α
in␈α
this␈αcase␈α
would␈α
be␈α
"␈↓βThe␈αonly␈α
number␈α
which␈α
equals␈αits␈α
own
␈↓ α,␈↓βnumber␈αof␈α
divisors␈αis␈α1␈↓".␈α
 Typically,␈αwhen␈αa␈α
generalization␈αG␈α
of␈αC␈αturns␈α
out␈αto␈αbe␈α
equivalent
␈↓ α,␈↓to␈α∂C,␈α∂there␈α∂is␈α∂theorem␈α∂lurking␈α∂around,␈α∂of␈α∂the␈α∂form␈α∂"All␈α∂G's␈α∂also␈α∂satisfy␈α⊂this␈α∂property...",
␈↓ α,␈↓where␈α
the␈α∞property␈α
is␈α∞the␈α
"extra"␈α∞constraint␈α
present␈α
in␈α∞C's␈α
de≡nition␈α∞but␈α
absent␈α∞from␈α
G's.
␈↓ α,␈↓This␈α⊂rule␈α⊂also␈α⊂was␈α⊂used␈α⊂when␈α⊂AM␈α⊂had␈α⊂just␈α⊂found␈α⊂some␈α⊂examples␈α⊂of␈α⊂Sets.␈α⊂AM␈α⊂almost
␈↓ α,␈↓believed␈α
that␈α
all␈α∞Unordered-Structures␈α
were␈α
also␈α∞Sets,␈α
but␈α
the␈α
last␈α∞main␈α
clause␈α
of␈α∞the␈α
rule
␈↓ α,␈↓had␈α∂AM␈α∂notice␈α∞that␈α∂Bags␈α∂is␈α∞a␈α∂specialization␈α∂of␈α∞Unordered-structures,␈α∂and␈α∂that␈α∂the␈α∞latter
␈↓ α,␈↓concept␈α⊂had␈α⊂never␈α⊃had␈α⊂any␈α⊂of␈α⊂its␈α⊃examples␈α⊂≡lled␈α⊂in.␈α⊂As␈α⊃a␈α⊂result,␈α⊂AM␈α⊂printed␈α⊃out␈α⊂this
␈↓ α,␈↓message:␈α∞"Almost␈α∞concluded␈α∞that␈α∞Unordered-structures␈α
are␈α∞also␈α∞always␈α∞Sets.␈α∞ But␈α∞will␈α
wait
␈↓ α,␈↓until␈αexamples␈α
of␈αBags␈αare␈α
found.␈α Perhaps␈αsome␈α
Bags␈αwill␈αnot␈α
be␈αSets."␈αIn␈α
fact,␈αexamples
␈↓ α,␈↓of Bags are soon found, and they aren't sets.

␈↓ α,␈↓¬109. When checking a specialization S of concept C,
␈↓ α,␈↓¬␈↓ αlSpecifically test to ensure that S is not equivalent to C.
␈↓ α,␈↓¬␈↓ β,If they are the same, then replace one by a conjecture.
␈↓ α,␈↓¬␈↓ β,If␈α∂they␈α∂are␈α∂different,␈α∂make␈α∂sure␈α∂that␈α∞some␈α∂boundary␈α∂example␈α∂of␈α∂C␈α∂(which␈α∂is␈α∂not␈α∞an
␈↓ α,␈↓¬␈↓ ∧,example of S) is explicitly stored for C.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε160␈↓-

␈↓ α,␈↓This␈α∂rule␈α∞is␈α∂similar␈α∞to␈α∂the␈α∞preceding␈α∂one.␈α∞If␈α∂adding␈α∞a␈α∂new␈α∞constraint␈α∂P␈α∞to␈α∂the␈α∞de≡nition
␈↓ α,␈↓doesn't␈α
change␈αthe␈α
concept␈αC,␈α
then␈αthere␈α
is␈αprobably␈α
a␈αtheorem␈α
there␈αof␈α
the␈αform␈α
"All␈αC's
␈↓ α,␈↓also satisfy constraint P".

␈↓ α,␈↓¬110. When checking a specialization S of a specialization X of a concept C,
␈↓ α,␈↓¬␈↓ αlif there exist other specs. of specs. of C,
␈↓ α,␈↓¬␈↓ αlthen␈α⊂ensure␈α⊂that␈α⊂none␈α⊂of␈α∂them␈α⊂are␈α⊂the␈α⊂same␈α⊂as␈α∂S.␈α⊂ This␈α⊂is␈α⊂especially␈α⊂worthwhile␈α⊂if␈α∂the
␈↓ α,␈↓¬␈↓ β,specializing operators in each case were the same but reversed in order.

␈↓ α,␈↓Thus␈αwe␈αcan␈αadd␈αa␈αconstraint␈αto␈αC␈αand␈αcollapse␈αthe␈α≡rst␈αtwo␈αarguments,␈αor␈αwe␈αcan␈αcollapse
␈↓ α,␈↓the␈αarguments␈α
and␈αadd␈αthe␈α
constraint;␈αeither␈αway,␈α
we␈αget␈αto␈α
the␈αsame␈αvery␈α
specialized␈αnew
␈↓ α,␈↓concept.␈α
 The␈α
above␈α
rule␈αhelps␈α
detect␈α
those␈α
accidental␈α
duplicates.␈α E.g.,␈α
Coalesced-Dom=Ran-
␈↓ α,␈↓Compositions␈αare␈αreally␈α
the␈αsame␈αas␈α
Dom=Ran-Coalesced-Compositions,␈αand␈αthis␈αrule␈α
would
␈↓ α,␈↓suspect that they might be.

␈↓ α,␈↓¬111. When checking the Genl or Spec facet entries for concept C,
␈↓ α,␈↓¬␈↓ αlensure␈αthat␈α
C.Genl␈αand␈αC.Spec␈α
have␈αno␈αcommon␈α
member␈αZ.␈α
If␈αthey␈αdo,␈α
then␈αconjecture␈αthat␈α
C
␈↓ α,␈↓¬␈↓ β,and Z are actually equivalent.

␈↓ α,␈↓In␈α"fact,␈α"this␈α!rule␈α"actually␈α"ensures␈α!that␈α"Generalizations(C)␈α"does␈α"not␈α!intersect
␈↓ α,␈↓Specializations(C).␈α
If␈α
it␈α
does,␈α
a␈α
whole␈α
`cycle'␈α
of␈α
concepts␈α
exists␈α
which␈α
can␈α
be␈α∞collapsed␈α
into
␈↓ α,␈↓one single concept. See also rule 114, below.


␈↓ α,␈↓∧␈↓ α<Any-concept . Genl/Spec . Interest

␈↓ α,␈↓¬112.␈αA␈α
generalization␈αof␈α
X␈αis␈α
interesting␈αif␈α
all␈αthe␈α
previously-known␈αboundary␈α
non-examples␈αare
␈↓ α,␈↓¬␈↓ β,now boundary examples of the concept.

␈↓ α,␈↓A␈α∂check␈α⊂is␈α∂included␈α∂here␈α⊂to␈α∂ensure␈α⊂that␈α∂the␈α∂new␈α⊂concept␈α∂was␈α∂not␈α⊂simply␈α∂de≡ned␈α⊂as␈α∂the
␈↓ α,␈↓closure of the old one.

␈↓ α,␈↓¬113.␈α
A␈α
specialization␈α
of␈αX␈α
is␈α
interesting␈α
if␈α
all␈αthe␈α
previously-known␈α
boundary␈α
examples␈α
are␈αnow
␈↓ α,␈↓¬␈↓ β,boundary non-examples of the new specialized concept.

␈↓ α,␈↓A␈α∂check␈α⊂is␈α∂included␈α∂here␈α⊂to␈α∂ensure␈α⊂that␈α∂the␈α∂new␈α⊂concept␈α∂was␈α∂not␈α⊂simply␈α∂de≡ned␈α⊂as␈α∂the
␈↓ α,␈↓interior of the old one.

␈↓ α,␈↓¬114.␈αIf␈α
C1␈αis␈α
a␈αgeneralization␈α
of␈αC2,␈α
which␈αis␈α
a␈αgeneralization␈α
of␈αC3,...,␈α
which␈αis␈α
a␈αgeneralization␈α
of
␈↓ α,␈↓¬␈↓ β,Cj, and it has just been learned that Cj is a generalization of C1,
␈↓ α,␈↓¬␈↓ αlThen␈αall␈αthe␈αconcepts␈αC1,...,Cj␈αare␈αequivalent,␈αand␈αcan␈αbe␈αmerged,␈αand␈αthe␈αcombined␈αconcept
␈↓ α,␈↓¬␈↓ β,will␈αbe␈αmuch␈αmore␈αinteresting␈αthan␈αany␈αsingle␈αone,␈αand␈αthe␈αinterestingness␈αof␈αthe␈αnew
␈↓ α,␈↓¬␈↓ β,composite concept increases rapidly with j.

␈↓ α,␈↓The␈α
Lisp␈α
code␈α
has␈α
the␈α
new␈α
interest␈α
value␈α
be␈α
computed␈α
as␈α
the␈α
maximum␈α
value␈α
of␈α
the␈α
old
␈↓ α,␈↓concepts,␈αplus␈αa␈αbonus␈αwhich␈αincreases␈αlike␈αthe␈αsquare␈αof␈αj.␈α This␈αis␈αsimilar␈αto␈αrule␈αnumber
␈↓ α,␈↓28.␈α
 A␈α
rule␈α
just␈α
like␈α
the␈α
preceding␈α
one␈α
exists,␈α
with␈α
`Specialization'␈α
substituted␈αeverywhere␈α
for
␈↓ α,␈↓`Generalization'.␈α
 Thus␈α
a␈α
closed␈α
loop␈α
of␈α∞Spec␈α
links␈α
constitutes␈α
a␈α
demonstration␈α
that␈α∞all␈α
the
␈↓ α,␈↓concepts in that loop are equivalent. These rules were used more frequently than expected.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε161␈↓-

␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.6.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for the View facet of Any-concept␈↓)αβ␈↓


␈↓ α,␈↓∧␈↓ α<Any-concept . View . Fillin

␈↓ α,␈↓¬115. To fill in View facet entries for X,
␈↓ α,␈↓¬␈↓ αlFind an interesting operation F whose range is X,
␈↓ α,␈↓¬␈↓ αland indicate that any member of Domain(F) can be viewed as an X just by running F on it.

␈↓ α,␈↓While␈α∩trying␈α⊃to␈α∩≡ll␈α∩in␈α⊃the␈α∩View␈α∩facet␈α⊃of␈α∩Even-nos,␈α⊃AM␈α∩used␈α∩this␈α⊃rule.␈α∩It␈α∩located␈α⊃the
␈↓ α,␈↓operation␈α
Doubling,␈α
whose␈α
domain␈αis␈α
Numbers␈α
and␈α
whose␈αrange␈α
is␈α
Even-nos.␈α
Then␈αthe␈α
rule
␈↓ α,␈↓created␈α
a␈α
new␈α
entry:␈α
"to␈α
view␈α
any␈α
number␈α
as␈αif␈α
it␈α
were␈α
an␈α
even␈α
number,␈α
double␈α
it".␈α
This␈αis␈α
a
␈↓ α,␈↓twisted␈α∩a≠rmation␈α∩of␈α∩the␈α∩standard␈α∩correspondence␈α∩between␈α∩natural␈α∩numbers␈α∩and␈α⊃even
␈↓ α,␈↓natural numbers.


␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.7.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for the In-dom/ran-of facets of Any-concept␈↓)αβ␈↓


␈↓ α,␈↓∧␈↓ α<Any-concept . In-dom-of/In-ran-of . Fillin

␈↓ α,␈↓¬116. To fill in entries for the In-dom-of facet of concept X,
␈↓ α,␈↓¬␈↓ αlRipple␈α
down␈α
the␈α
tree␈α
of␈α
concepts,␈α
starting␈α
at␈α
Active,␈α
to␈α
empirically␈α
determine␈α∞which␈α
active
␈↓ α,␈↓¬␈↓ β,concepts can be run on X's.

␈↓ α,␈↓This␈αcan␈αusually␈αbe␈α
decided␈αby␈αinspecting␈αthe␈α
Domain/range␈αfacets␈αof␈αthe␈α
Active␈αconcepts.
␈↓ α,␈↓Occasionally,␈α
AM␈αmust␈α
actually␈α
try␈αto␈α
run␈α
an␈αactive␈α
on␈α
sample␈αX's,␈α
to␈α
see␈αwhether␈α
it␈αfails␈α
or
␈↓ α,␈↓returns a value.␈↓	13␈↓

␈↓ α,␈↓¬117. To fill in the In-ran-of facet of concept X,
␈↓ α,␈↓¬␈↓ αlRipple␈α
down␈α
the␈α
tree␈α
of␈α
concepts,␈α
starting␈α
at␈α
Active,␈α
to␈α
empirically␈α
determine␈α∞which␈α
active
␈↓ α,␈↓¬␈↓ β,concepts can be run to yield X's.

␈↓ α,␈↓This␈αcan␈αusually␈αbe␈α
decided␈αby␈αinspecting␈αthe␈α
Domain/range␈αfacets␈αof␈αthe␈α
Active␈αconcepts.
␈↓ α,␈↓Occasionally,␈α∂AM␈α∂inspects␈α∂known␈α∂examples␈α∂of␈α∂some␈α∂Active␈α∂concept,␈α∂to␈α∂see␈α∂if␈α∂any␈α∂of␈α∞the
␈↓ α,␈↓results are X's.

␈↓ α,␈↓¬118. While filling in entries for the In-dom-of facet of X,
␈↓ α,␈↓¬␈↓ αlLook␈α∂especially␈α⊂carefully␈α∂for␈α⊂Operations␈α∂which␈α∂transform␈α⊂examples␈α∂and␈α⊂non-examples␈α∂into
␈↓ α,␈↓¬␈↓ β,each other;
␈↓ α,␈↓¬␈↓ αlThis is even better if the operation pushes boundary exs/non-exs `across the boundary'.

␈↓ α,␈↓This was used to note that Insert and Delete had a lot to do with the concept of Singleton.





␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	13␈↓ε One key feature of Lisp which permits this to be done is the Errorset feature.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε162␈↓-

␈↓ α,␈↓␈↓ αL␈↓α␈↓&Appendix 2.2.8.␈↓)αβ␈↓␈↓↓␈↓& Heuristics for the De≡nition facet of Any-concept␈↓)αβ␈↓


␈↓ α,␈↓∧␈↓ α<Any-concept . Defn . Suggest

␈↓ α,␈↓¬119. If there are no known definitions for concept X,
␈↓ α,␈↓¬␈↓ αlThen it is crucial that AM spend some time looking for such definitions.

␈↓ α,␈↓This␈αsituation␈α
might␈αoccur␈αif␈α
only␈αan␈αAlgorithm␈α
is␈αpresent␈αfor␈α
some␈αconcept.␈α In␈α
that␈αcase,
␈↓ α,␈↓the␈α⊃above␈α⊂rule␈α⊃would␈α⊃suggest␈α⊂a␈α⊃new,␈α⊃high-priority␈α⊂task,␈α⊃and␈α⊃AM␈α⊂would␈α⊃then␈α⊃twist␈α⊂the
␈↓ α,␈↓algorithm␈α
into␈αa␈α
(probably␈α
very␈αine≠cient)␈α
de≡nition.␈α
 A␈αmuch␈α
more␈α
serious␈αsituation␈α
would
␈↓ α,␈↓occur␈α∂if␈α∂a␈α∂concept␈α∂were␈α∂speci≡ed␈α∂only␈α∂by␈α∂its␈α∂Intuition␈α∂entries␈α∂(created,␈α∂e.g.,␈α⊂by␈α∂modifying
␈↓ α,␈↓another␈αconcept's␈αintuitions).␈α In␈αthat␈αcase,␈αrapidly␈αformulating␈αa␈αprecise␈αde≡nition␈αwould␈α
be
␈↓ α,␈↓a necessity.  Of course, this need never arose, since all the intuitions were deleted.


␈↓ α,␈↓∧␈↓ α<Any-concept . Defn . Check

␈↓ α,␈↓¬120. When checking the Definition facet of concept C,
␈↓ α,␈↓¬␈↓ αlEnsure␈α
that␈α
each␈α
member␈α
of␈α
C.Exs␈αsatisfies␈α
all␈α
definitions␈α
present,␈α
and␈α
each␈αnon-example␈α
fails
␈↓ α,␈↓¬␈↓ β,all␈α
definitions.␈α
 If␈α∞there␈α
is␈α
one␈α∞dissenting␈α
definition,␈α
modify␈α∞it,␈α
and␈α
move␈α∞the␈α
offending
␈↓ α,␈↓¬␈↓ β,example to the boundary.

␈↓ α,␈↓There␈α∀is␈α∀little␈α∀real␈α∃"checking"␈α∀that␈α∀can␈α∀be␈α∀done␈α∃to␈α∀a␈α∀de≡nition,␈α∀aside␈α∃from␈α∀internal
␈↓ α,␈↓consistency:␈α⊂If␈α⊃there␈α⊂exist␈α⊂several␈α⊃suposedly-equivalent␈α⊂de≡nitions,␈α⊂then␈α⊃AM␈α⊂can␈α⊃at␈α⊂least
␈↓ α,␈↓ensure␈α
they␈α
agree␈α
on␈α
the␈α
known␈α
examples␈α
and␈α
non-examples␈α
of␈α
the␈α
concept.␈α
If␈αthe␈α
Intuitions
␈↓ α,␈↓facets were permitted, then each de≡nition could be checked for intuitive appeal.

␈↓ α,␈↓¬121. When checking the Definition facet of concept C,
␈↓ α,␈↓¬␈↓ αlTry␈αto␈αfind␈αand␈αeliminate␈αany␈αredundant␈αconstraints,␈αtry␈αto␈αfind␈αand␈αeliminate␈αany␈αcircularity,
␈↓ α,␈↓¬␈↓ β,check that any recursion will terminate.

␈↓ α,␈↓Here␈αare␈αthe␈αother␈αfew␈α
tricks␈αthat␈αAM␈αknows␈αfor␈α
"checking"␈αa␈αde≡nition.␈αFor␈αeach␈αclause␈α
in
␈↓ α,␈↓the␈α
rule␈α
above,␈α∞AM␈α
has␈α
a␈α∞very␈α
limited␈α
ability␈α
to␈α∞detect␈α
and␈α
patch␈α∞up␈α
"bugs"␈α
of␈α∞that␈α
sort.
␈↓ α,␈↓Checking␈αthat␈αrecursion␈αwill␈αterminate,␈αfor␈αexample,␈αis␈αdone␈αby␈αexamining␈αthe␈αargument␈αto
␈↓ α,␈↓the␈αrecursive␈α
call,␈αand␈α
verifying␈αthat␈αit␈α
contains␈α(at␈α
some␈αlevel␈α
before␈αthe␈αoriginal␈α
argument)
␈↓ α,␈↓an␈αapplication␈αof␈α
a␈αLISP␈αfunction␈αon␈α
Destructive-LISP-functions-list.␈αThere␈αis␈αno␈α
intelligent
␈↓ α,␈↓inference␈α⊂that␈α⊂is␈α⊂going␈α⊂on␈α⊂here,␈α⊂and␈α⊂for␈α⊂that␈α⊂reason␈α⊂the␈α⊂process␈α⊂is␈α⊂not␈α⊂even␈α⊂mentioned
␈↓ α,␈↓within the body of this document.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε163␈↓-

␈↓ α,␈↓␈↓ αy␈↓↓␈↓&Appendix 2.3. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with any Active concept␈↓)αβ␈↓↓

␈↓ α,␈↓All␈α∞the␈α∞rules␈α∞below␈α∞are␈α∞applicable␈α
to␈α∞tasks␈α∞which␈α∞involve␈α∞operations,␈α∞predicates,␈α
relations,
␈↓ α,␈↓functions,␈α
etc.␈α
 In␈α
short,␈α
they␈α
apply␈α
to␈α
all␈α
the␈α
concepts␈α
AM␈α
knows␈α
about␈α
which␈αinvolve␈α
␈↓βdoing␈↓
␈↓ α,␈↓something, which involve action.


␈↓ α,␈↓∧␈↓ α<Active . Fillin

␈↓ α,␈↓¬122. If the current task is to fill in examples of the activity F,
␈↓ α,␈↓¬␈↓ αlOne way to get them is to run F on randomly chosen examples of the domain of F.

␈↓ α,␈↓Thus,␈α⊃to␈α⊂≡nd␈α⊃examples␈α⊂of␈α⊃Equality,␈α⊂AM␈α⊃repeatedly␈α⊂executed␈α⊃Equality.Alg␈α⊃on␈α⊂randomly
␈↓ α,␈↓chosen␈αpairs␈αof␈αobjects.␈α AM␈αfound␈αexamples␈αof␈αCompositions␈αby␈αactually␈αpicking␈αa␈αpair␈αof
␈↓ α,␈↓operations␈α⊂at␈α⊂random␈α⊂and␈α⊂trying␈α⊂to␈α⊂compose␈α⊂them.␈α⊂Of␈α⊂course,␈α⊂most␈α⊃such␈α⊂"unmotivated"
␈↓ α,␈↓compositions turned out to be uninteresting.

␈↓ α,␈↓¬123.␈α∂While␈α∂filling␈α∂in␈α∂examples␈α∂of␈α∂the␈α∂activity␈α∂F,␈α∂by␈α∂running␈α∂F.Algs␈α∂on␈α∂random␈α⊂arguments␈α∂from
␈↓ α,␈↓¬␈↓ β,F.Domain,
␈↓ α,␈↓¬␈↓ αlIt␈αis␈αworth␈αthe␈αeffort␈αto␈αspecifically␈αinclude␈αextreme␈αor␈αboundary␈αexamples␈αof␈αthe␈αdomain␈αof
␈↓ α,␈↓¬␈↓ β,F, among the arguments on which F.Algs is run.


␈↓ α,␈↓¬124. To fill in a Domain entry for active concept F,
␈↓ α,␈↓¬␈↓ αlRun␈α
F␈α
on␈α
various␈α
entities,␈α
rippling␈αdown␈α
the␈α
tree␈α
of␈α
concepts,␈α
to␈α
determine␈αempirically␈α
where
␈↓ α,␈↓¬␈↓ β,F seems to be defined.

␈↓ α,␈↓This␈αmay␈α
shock␈αthe␈αreader,␈α
as␈αit␈α
sounds␈αdumb␈αand␈α
explosive,␈αbut␈α
the␈αconcepts␈αare␈α
arranged
␈↓ α,␈↓in␈α
a␈α
tree␈α
(using␈αGenl␈α
links),␈α
so␈α
the␈αsearch␈α
is␈α
really␈α
quite␈αfast.␈α
 Although␈α
this␈α
rule␈α
is␈αrarely
␈↓ α,␈↓used, it always seems to give surprisingly good results.

␈↓ α,␈↓¬125. To fill in generalizations of active F,
␈↓ α,␈↓¬␈↓ αlConsider just extending F, by enlarging its domain. Revise F.Defn as little as possible.

␈↓ α,␈↓Although␈αEquality␈αis␈αinitially␈αonly␈αfor␈αstructures,␈αAM␈αextends␈αit␈α(using␈αthe␈αsame␈αde≡nition,
␈↓ α,␈↓actually) to a predicate over all pairs of entities.

␈↓ α,␈↓¬126. To fill in specializations of active F,
␈↓ α,␈↓¬␈↓ αlConsider␈αjust␈αrestricting␈αF,␈αby␈αshrinking␈αits␈αdomain.␈αCheck␈αF.Defn␈αto␈αsee␈αif␈αsome␈αoptimization
␈↓ α,␈↓¬␈↓ β,is possible.


␈↓ α,␈↓¬127. After an algorithm is known for F, if AM wants a better one,
␈↓ α,␈↓¬␈↓ αlAM is permitted to ask the user to provide a fast but opaque algorithm for F.

␈↓ α,␈↓This␈α∪was␈α∪used␈α∪a␈α∪few␈α∪times,␈α∪especially␈α∪for␈α∪inverse␈α∪functions.␈α∪A␈α∪nontrivial␈α∪open-ended
␈↓ α,␈↓research␈αproblem␈α(*)␈↓	14␈↓␈αis␈αto␈αcollect␈αa␈αbody␈αof␈αrules␈αwhich␈αtransform␈αan␈αine≠cient␈αalgorithm
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	14␈↓ε␈α
Following␈α
Knuth,␈α	we␈α
shall␈α
reserve␈α	a␈α
star␈α
␈↓(*)␈↓ε␈α
for␈α	those␈α
problems␈α
which␈α	are␈α
quite␈α
difficult,␈α	which␈α
should␈α
take␈α
the␈α	reader
␈↓ α,␈↓ε␈↓ βLroughly␈α∂3␈α∞full␈α∂lifetimes␈α∞to␈α∂master.␈α∞Readers␈α∂not␈α∞believing␈α∂in␈α∞reincarnation␈α∂should␈α∞therefore␈α∂skip␈α∞such
␈↓ α,␈↓ε␈↓ βLproblems.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε164␈↓-

␈↓ α,␈↓into a computationally acceptable one.

␈↓ α,␈↓¬128. If the current task is to fill in boundary (non-)examples of the activity F,
␈↓ α,␈↓¬␈↓ αlOne␈α
way␈αto␈α
get␈α
them␈αis␈α
to␈α
run␈αF␈α
on␈αrandomly␈α
chosen␈α
boundary␈αexamples␈α
and␈α
(with␈αproper
␈↓ α,␈↓¬␈↓ β,safeguards) boundary non-examples of the domain of F.

␈↓ α,␈↓Proper␈α
safeguards␈α
are␈α
required␈α
to␈α
ensure␈α
that␈α
F.Algs␈α
doesn't␈α
loop␈α
or␈α
cause␈α
an␈α
error␈αwhen
␈↓ α,␈↓fed␈α∩a␈α⊃slightly-wrong␈α∩(slightly-illegal)␈α⊃argument.␈α∩ In␈α⊃LISP,␈α∩a␈α⊃timer␈α∩and␈α∩an␈α⊃ERRORSET
␈↓ α,␈↓su≠ce as crude safeguards.

␈↓ α,␈↓¬129. If the current task is to fill in (boundary) non-examples of the activity F,
␈↓ α,␈↓¬␈↓ αlOne␈α
low-interest␈α
way␈α
to␈α
get␈α
them␈α
is␈α∞to␈α
run␈α
F␈α
on␈α
randomly␈α
chosen␈α
examples␈α
of␈α∞its␈α
domain,
␈↓ α,␈↓¬␈↓ β,and␈αthen␈αreplace␈αthe␈αvalue␈αobtained␈αby␈αsome␈αother␈α(very␈αsimilar)␈αvalue.␈α Also,␈αbe␈αsure
␈↓ α,␈↓¬␈↓ β,to check that the resultant i/o pair doesn't accidentally satisfy F.Defn.

␈↓ α,␈↓The␈α⊃parentheses␈α⊃in␈α⊃the␈α⊃above␈α⊃rule␈α⊃mean␈α⊃that␈α⊃it␈α⊃is␈α⊃really␈α⊃two␈α⊃rules:␈α⊃for␈α⊃␈↓βboundary␈↓␈α⊂non-
␈↓ α,␈↓examples,␈α
just␈α
change␈α
the␈α≡nal␈α
value␈α
slightly.␈α
 For␈α␈↓βtypical␈↓␈α
non-examples,␈α
change␈α
the␈αresult
␈↓ α,␈↓signi≡cantly.␈α If␈αyou␈αread␈αthe␈αwords␈αinside␈αin␈αthe␈αparentheses␈αin␈αthe␈αIF␈αpart,␈αthen␈α
read␈αthe
␈↓ α,␈↓words inside the parentheses in the THEN part as well, ␈↓βor␈↓ omit them in both cases.


␈↓ α,␈↓∧␈↓ α<Active . Check

␈↓ α,␈↓¬130. When checking an algorithm for active F,
␈↓ α,␈↓¬␈↓ αlrun that algorithm and ensure that the input/output satisfy F.Defn.


␈↓ α,␈↓¬131. When checking a definition d for active concept F,
␈↓ α,␈↓¬␈↓ αlRun one of its algorithms and ensure that the input/output satisfy d.

␈↓ α,␈↓This␈α∪is␈α∩the␈α∪converse␈α∩of␈α∪the␈α∪preceding␈α∩rule.␈α∪ They␈α∩simply␈α∪say␈α∩that␈α∪the␈α∪de≡nition␈α∩and
␈↓ α,␈↓algorithm facets must be mutually consistent.

␈↓ α,␈↓¬132. While checking examples or boundary examples of the active concept F,
␈↓ α,␈↓¬␈↓ αlEnsure that each input/output pair is consistent with F.Dom/range.

␈↓ α,␈↓If␈αthe␈αdomain/range␈αentry␈αis␈α<D1␈αD2...␈α Dk␈α→␈αR>,␈αand␈αthe␈αi/o␈αpair␈αis␈α<d␈↓#v1␈↓#␈αd␈↓#v2␈↓#...␈αd␈↓#vk␈↓#␈α,␈αr>,␈αthen
␈↓ α,␈↓each␈α⊂component␈α⊂d␈↓#vi␈↓#␈α⊂of␈α⊂the␈α∂input␈α⊂must␈α⊂be␈α⊂an␈α⊂example␈α∂of␈α⊂the␈α⊂corresponding␈α⊂Di,␈α⊂and␈α∂the
␈↓ α,␈↓output r must be an example of R.

␈↓ α,␈↓¬133. When checking examples of the active concept F,
␈↓ α,␈↓¬␈↓ αlIf any argument(s) to F were concepts, tag their In-domain-of facets with `F'.
␈↓ α,␈↓¬␈↓ αlIf any values produced by F are concepts, tag their In-range-of facets with `F'.

␈↓ α,␈↓For␈αexample,␈αRestrict(Union)␈αproduced␈α
Add,␈αat␈αone␈αtime␈α
in␈αAM's␈αhistory.␈αThen␈α
the␈αabove
␈↓ α,␈↓rule␈αcaused␈α`Restrict'␈αto␈αbe␈αinserted␈αas␈αa␈αnew␈αentry␈αon␈αUnion.In-dom-of␈αand␈αalso␈αon␈αAdd.In-
␈↓ α,␈↓ran-of.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε165␈↓-

␈↓ α,␈↓∧␈↓ α<Active . Suggest

␈↓ α,␈↓¬134. If there are no known algorithms for active concept F,
␈↓ α,␈↓¬␈↓ αlThen AM should spend some time looking for such algorithms.

␈↓ α,␈↓This␈αsituation␈αmight␈αoccur␈αif␈αonly␈αa␈α
De≡nition␈αis␈αpresent␈αfor␈αsome␈αoperation.␈α In␈α
that␈αcase,
␈↓ α,␈↓the␈α⊃above␈α⊂rule␈α⊃would␈α⊃suggest␈α⊂a␈α⊃new,␈α⊃high-priority␈α⊂task,␈α⊃and␈α⊃AM␈α⊂would␈α⊃then␈α⊃twist␈α⊂the
␈↓ α,␈↓de≡nition␈α∂into␈α∞a␈α∂(probably␈α∂very␈α∞ine≠cient)␈α∂algorithm.␈α∞ The␈α∂rule␈α∂below␈α∞is␈α∂similar,␈α∂for␈α∞the
␈↓ α,␈↓Domain/range facet:

␈↓ α,␈↓¬135. If the Domain/range facet of active concept F is blank,
␈↓ α,␈↓¬␈↓ αlThen AM should spend some time looking for specifications of F's domain and range.


␈↓ α,␈↓¬136.␈α
If␈α
a␈αDomain␈α
of␈α
active␈αconcept␈α
F␈α
is␈αencountered␈α
frequently,␈α
either␈αwithin␈α
conjectures␈α
or␈αas␈α
the
␈↓ α,␈↓¬␈↓ β,domain or range of other operations and predicates,
␈↓ α,␈↓¬␈↓ αlThen define that Domain as a separate concept, and raise the Worth of F slightly.

␈↓ α,␈↓The␈α`Domain'␈αhere␈αrefers␈αto␈αthe␈αsequence␈α
of␈αcomponents,␈αwhose␈αcartesian␈αproduct␈αis␈αwhat␈α
is
␈↓ α,␈↓normally␈α∩referred␈α⊃to␈α∩in␈α⊃mathematics␈α∩as␈α⊃the␈α∩domain␈α⊃of␈α∩the␈α⊃operation.␈α∩ This␈α⊃led␈α∩to␈α⊃the
␈↓ α,␈↓de≡nition␈αof␈α"Anything␈↓ε␈αx␈α␈↓Structures",␈αwhich␈αis␈αthe␈αdomain␈αof␈αseveral␈αInsertion␈αand␈α
Deletion
␈↓ α,␈↓operations, Membership testing predicates, etc.

␈↓ α,␈↓¬137.␈αIt␈α
is␈αworthwhile␈α
to␈αexplicitly␈α
calculate␈αthe␈α
value␈αof␈α
F␈αfor␈α
all␈αdistinguished␈α(extreme,␈α
boundary,
␈↓ α,␈↓¬␈↓ β,interesting) members of and subsets of its domain.


␈↓ α,␈↓¬138. If some domain component of F has a very interesting specialization,
␈↓ α,␈↓¬␈↓ αlThen consider restricting F (along that component) to that smaller domain.

␈↓ α,␈↓Note␈αthat␈αthese␈αlast␈αcouple␈αrules␈αdeal␈αwith␈αthe␈αimage␈αof␈αinteresting␈αdomain␈αitems.␈αThe␈αnext
␈↓ α,␈↓rule␈αdeals␈α
with␈αthe␈α
inverse␈αimage␈α
(pre-image)␈αof␈α
unusual␈αrange␈α
items.␈αWe␈α
saw␈αearlier␈αin␈α
this
␈↓ α,␈↓document (Chapter 2) how this rule led to the de≡nition of Prime numbers.

␈↓ α,␈↓¬139. If the range of F contains interesting items, or an interesting specialization,
␈↓ α,␈↓¬␈↓ αlThen it is worthwhile to consider their inverse image under F.


␈↓ α,␈↓¬140. When trying to fill in new Algorithms for Active concept F,
␈↓ α,␈↓¬␈↓ αlTry to transform any conjectures about F into (pieces of) new algorithms.

␈↓ α,␈↓This␈α∂is␈α∂one␈α⊂place␈α∂where␈α∂a␈α⊂sophisticated␈α∂body␈α∂of␈α⊂transformation␈α∂rules␈α∂might␈α⊂be␈α∂inserted.
␈↓ α,␈↓Others␈α
are␈α
working␈α∞on␈α
this␈α
problem␈α∞[Burstall␈α
&␈α
Darlington␈α
75],␈α∞and␈α
AM␈α
only␈α∞contains␈α
a
␈↓ α,␈↓few␈α
simple␈α
tricks␈α
for␈αturning␈α
conjectures␈α
into␈α
procedures.␈α For␈α
example,␈α
"All␈α
primes␈αare␈α
odd,
␈↓ α,␈↓except␈α∂`2'",␈α∞is␈α∂transformed␈α∞into␈α∂a␈α∞more␈α∂e≡cient␈α∞search␈α∂for␈α∞primes:␈α∂a␈α∞separate␈α∂test␈α∂for␈α∞x=2,
␈↓ α,␈↓followed by a search through only Odd-numbers.

␈↓ α,␈↓¬141. After trying in vain to fill in examples of active concept F,
␈↓ α,␈↓¬␈↓ αlLocate␈α
the␈α
domain␈α
of␈αF,␈α
and␈α
suggest␈α
that␈αAM␈α
try␈α
to␈α
fill␈αin␈α
examples␈α
for␈α
each␈α
component␈αof
␈↓ α,␈↓¬␈↓ β,that domain.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε166␈↓-

␈↓ α,␈↓Thus␈α∞after␈α∞failing␈α∞to␈α∞≡nd␈α
examples␈α∞for␈α∞Set-union,␈α∞AM␈α∞was␈α
told␈α∞to␈α∞≡nd␈α∞examples␈α∞of␈α
Sets,
␈↓ α,␈↓because␈αthat␈αcould␈αhave␈αlet␈αthe␈αprevious␈αtask␈αsucceed.␈αThere␈αis␈αno␈αrecursion␈αhere:␈αafter␈αthe
␈↓ α,␈↓sets␈α∂are␈α∂found,␈α∂AM␈α∂will␈α∂not␈α∂automatically␈α∞go␈α∂back␈α∂to␈α∂≡nding␈α∂examples␈α∂of␈α∂Set-union.␈α∞ In
␈↓ α,␈↓practice, that task was eventually proposed and chosen again, and succeeded this time.

␈↓ α,␈↓¬142. After working on an Active concept F,
␈↓ α,␈↓¬␈↓ αlGive␈α
a␈α
slight,␈α
ephemeral␈α
boost␈α
to␈α
tasks␈α
involving␈α
Domain(F):␈α
give␈α
a␈α
moderate␈α
size␈α
boost␈αto
␈↓ α,␈↓¬␈↓ β,each␈α∞task␈α∞which␈α∞asks␈α∞to␈α∞fill␈α∂in␈α∞examples␈α∞of␈α∞that␈α∞domain/range␈α∞component,␈α∞and␈α∂give␈α∞a
␈↓ α,␈↓¬␈↓ β,very tiny boost to each other task mentioning such a concept.

␈↓ α,␈↓This␈α∞is␈α
both␈α∞a␈α
supplement␈α∞to␈α
the␈α∞more␈α
general␈α∞"focus␈α
of␈α∞attention"␈α
rule,␈α∞and␈α∞a␈α
nontrivial
␈↓ α,␈↓heuristic for ≡nding valuable new tasks.  It is the partial converse of rule 14.


␈↓ α,␈↓∧␈↓ α<Active . Interest

␈↓ α,␈↓¬143.␈αAn␈αactive␈αconcept␈αF␈αis␈αinteresting␈αif␈αthere␈αare␈αother␈αoperations␈αwith␈αthe␈αsame␈αdomain␈αas␈αF,
␈↓ α,␈↓¬␈↓ β,and␈α
if␈α
they␈α
are␈α
(on␈α
the␈α
average)␈αfairly␈α
interesting.␈α
 If␈α
the␈α
other␈α
operations'␈α
domain␈αis
␈↓ α,␈↓¬␈↓ β,only␈α∞similar,␈α∞then␈α∞they␈α∞must␈α∞be␈α∞very␈α∞interesting␈α∞and␈α∞have␈α∞some␈α∞valuable␈α∞conjectures
␈↓ α,␈↓¬␈↓ β,tied to them, if they are to be allowed to push up F's interestingness rating.

␈↓ α,␈↓The␈α∞value␈α∞of␈α
having␈α∞the␈α∞same␈α∞domain/range␈α
is␈α∞the␈α∞ability␈α∞to␈α
compose␈α∞with␈α∞them.␈α∞ If␈α
the
␈↓ α,␈↓domain/range is only similar, then AM can hope for analogies or for partial compositions.

␈↓ α,␈↓¬144. An active concept is interesting if it was recently created.

␈↓ α,␈↓This␈αis␈αa␈αslight␈αextra␈αboost␈αgiven␈αto␈αeach␈αnew␈αoperation,␈αpredicate,␈αetc.␈α This␈αbonus␈αdecays
␈↓ α,␈↓rapidly␈αwith␈αtime,␈αand␈αthus␈αso␈αwill␈αthe␈αoverall␈αworth␈αof␈αthe␈αconcept,␈αunless␈αsome␈αinteresting
␈↓ α,␈↓property is encountered quickly.

␈↓ α,␈↓¬145. An active concept is interesting if its domain is very interesting.

␈↓ α,␈↓An␈α∞important␈α∞common␈α∞case␈α∂of␈α∞this␈α∞rule␈α∞is␈α∞when␈α∂the␈α∞domain␈α∞is␈α∞interesting␈α∞because␈α∂all␈α∞its
␈↓ α,␈↓members␈α
are␈α
equal␈αto␈α
each␈α
other.␈α The␈α
corresponding␈α
statement␈αabout␈α
␈↓βranges␈↓␈α
does␈αexist,␈α
but
␈↓ α,␈↓only␈α
operations␈α
can␈α∞be␈α
said␈α
to␈α∞have␈α
a␈α
speci≡c␈α
range␈α∞(not,␈α
e.g.␈α
 Predicates).␈α∞ Therefore,␈α
the
␈↓ α,␈↓`range' rule is listed under Operation.Interest, as rule number 165.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε167␈↓-

␈↓ α,␈↓␈↓ β)␈↓↓␈↓&Appendix 2.4. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with any Predicate␈↓)αβ␈↓↓

␈↓ α,␈↓Each␈α∂of␈α∂these␈α∂heuristics␈α∂can␈α∂be␈α∂assumed␈α∂to␈α∂be␈α∂prefaced␈α∂by␈α∂a␈α∂clause␈α∂of␈α∂the␈α∂form␈α∂"If␈α∂the
␈↓ α,␈↓current␈α⊂task␈α∂is␈α⊂to␈α∂deal␈α⊂with␈α∂concept␈α⊂X,␈α∂where␈α⊂X␈α∂isa␈α⊂Predicate,...".␈α∂This␈α⊂will␈α⊂be␈α∂repeated
␈↓ α,␈↓below, for each rule.


␈↓ α,␈↓∧␈↓ α<Predicate . Fillin

␈↓ α,␈↓¬146. If the current task was (Fill-in examples of X),
␈↓ α,␈↓¬␈↓ αland X is a predicate,
␈↓ α,␈↓¬␈↓ αland more than 100 items are known in the domain of X,
␈↓ α,␈↓¬␈↓ αland at least 10 cpu seconds were spent trying to randomly instantiate X,
␈↓ α,␈↓¬␈↓ αland the ratio of successes/failures is both >0 and less than .05
␈↓ α,␈↓¬Then add the following task to the agenda: (Fill-in generalizations of X), for the following reason:
␈↓ α,␈↓¬␈↓ αl"X is rarely satisfied; a slightly less restrictive concept might be more interesting".
␈↓ α,␈↓¬␈↓ αlThis reason's rating is computed as three times the ratio of nonexamples/examples found.

␈↓ α,␈↓This␈α
rule␈α
says␈α
to␈α
generalize␈α
a␈α
predicate␈α
if␈αit␈α
rarely␈α
succeeds␈α
(returns␈α
T).␈α
 One␈α
use␈α
for␈αthis
␈↓ α,␈↓was␈αwhen␈α
Equality␈αwas␈α
found␈αto␈αbe␈α
quite␈αrare;␈α
the␈αresultant␈α
generalizations␈αdid␈αindeed␈α
turn
␈↓ α,␈↓out␈α
to␈αbe␈α
more␈α
valuable␈α(numbers).␈α
 A␈α
similar␈αuse␈α
was␈α
found␈αfor␈α
predicates␈α
which␈αtested␈α
for
␈↓ α,␈↓identical␈α∞equality␈α∂of␈α∞two␈α∞angles,␈α∂of␈α∞two␈α∂triangles,␈α∞and␈α∞of␈α∂two␈α∞lines.␈α∂ Their␈α∞generalizations
␈↓ α,␈↓were␈α∪also␈α∪valuable␈α∪(congruence,␈α∪similarity,␈α∪parallel,␈α∪equal-measure).␈α∪ Most␈α∪rules␈α∪in␈α∩this
␈↓ α,␈↓appendix␈αare␈α
not␈αpresented␈α
with␈αthe␈α
same␈αlevel␈αof␈α
detail␈αas␈α
the␈αpreceding␈α
one,␈αas␈αthe␈α
reader
␈↓ α,␈↓has no doubt observed.

␈↓ α,␈↓¬147. To fill in Domain/range entries for predicate P,
␈↓ α,␈↓¬␈↓ αlP can operate on the domain of any specialization of P,
␈↓ α,␈↓¬␈↓ αlP can operate on any specialization of the domain of P,
␈↓ α,␈↓¬␈↓ αlP can operate on some restriction of the domain of any generalization of P,
␈↓ α,␈↓¬␈↓ αlP may be able to operate on some enlargement of its current domain,
␈↓ α,␈↓¬␈↓ αlThe range of P will necessarily be the doubleton set {T,F},
␈↓ α,␈↓¬␈↓ αlP is guaranteed return T if any of its specializations do, and F if any of its generalizations do.

␈↓ α,␈↓This␈α∞contains␈α∞a␈α∞compiled␈α∞version␈α∂of␈α∞what␈α∞we␈α∞mean␈α∞when␈α∂we␈α∞say␈α∞that␈α∞one␈α∞predicate␈α∂is␈α∞a
␈↓ α,␈↓generalization␈α∞or␈α
specialization␈α∞of␈α
another.␈α∞Viewed␈α∞as␈α
relations,␈α∞as␈α
subsets␈α∞of␈α∞a␈α
Cartesian-
␈↓ α,␈↓product␈αof␈αspaces,␈αthis␈α
notion␈αof␈αgeneral/special␈αis␈αjust␈α
that␈αof␈αsuperset/subset.␈α The␈αlast␈α
line
␈↓ α,␈↓of␈αthe␈αrule␈αis␈αmeant␈αto␈αindicate␈αthat␈αadding␈αnew␈αconstraints␈αonto␈αP␈αcan␈αonly␈αmake␈αit␈αreturn
␈↓ α,␈↓True less frequently, while relaxing P's de≡nition can only make it return True more often.


␈↓ α,␈↓∧␈↓ α<Predicate . Suggest

␈↓ α,␈↓¬148.␈αIf␈αall␈αthe␈αvalues␈αof␈αActive␈αconcept␈αF␈αhappen␈αto␈αbe␈αTruth-values,␈αand␈αF␈αis␈αnot␈αknown␈αto␈αbe␈αa
␈↓ α,␈↓¬␈↓ β,predicate,
␈↓ α,␈↓¬␈↓ αlThen conjecture that F is in fact a predicate.

␈↓ α,␈↓This␈αrule␈αis␈αplaced␈αon␈αthe␈αSuggest␈αfacet␈αbecause,␈αif␈αplaced␈αanywhere␈αelse␈αon␈αthis␈αconcept,␈αit
␈↓ α,␈↓could␈α
only␈αbe␈α
seen␈αas␈α
relevant␈αby␈α
AM␈αif␈α
AM␈αalready␈α
knew␈αthat␈α
F␈αwere␈α
a␈α
predeicate.␈α On
␈↓ α,␈↓the␈αother␈αhand,␈αthe␈αrule␈αcan't␈αbe␈αplaced,␈αe.g.,␈αon␈αActive.Fillin,␈αsince␈αjust␈αforgetting␈α(deleting)
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε168␈↓-

␈↓ α,␈↓this␈α"Predicate"␈αconcept␈αshould␈αbe␈αenough␈αto␈αdelete␈αall␈αreferences␈αto␈αpredicates␈αanywhere␈αin
␈↓ α,␈↓the system.


␈↓ α,␈↓∧␈↓ α<Predicate . Interest

␈↓ α,␈↓¬149.␈αA␈αpredicate␈αP␈αis␈α
interesting␈αif␈αits␈αdomain␈αis␈α
Any-concept␈α(the␈αspace␈αof␈αall␈α
known␈αconcepts).
␈↓ α,␈↓¬␈↓ β,This␈α∩is␈α∪especially␈α∩true␈α∩if␈α∪there␈α∩is␈α∩a␈α∪significant␈α∩positive␈α∩correlation␈α∪(theoretical␈α∩or
␈↓ α,␈↓¬␈↓ β,empirical) between concepts' worths and their P-values.

␈↓ α,␈↓This very high level heuristic wasn't really used by AM during its runs.

␈↓ α,␈↓␈↓ β$␈↓↓␈↓&Appendix 2.5. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with any Operation␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Operation . Fillin

␈↓ α,␈↓¬150. To fill in examples of operation F (with domain A and range B),
␈↓ α,␈↓¬␈↓ αlwhen many examples α of A are already known,
␈↓ α,␈↓¬␈↓ αland F maps some of those examples α into distinguished members (esp: extrema) b of B,
␈↓ α,␈↓¬␈↓ αlThen␈α
(for␈α∞each␈α
such␈α
distinguished␈α∞member␈α
"b"εB)␈α
study␈α∞F␈↓	-1␈↓¬(b)␈α
as␈α
a␈α∞new␈α
concept.␈α∞ That␈α
is,
␈↓ α,␈↓¬␈↓ β,isolate those members of A whose F-value is the unusual item bεB.

␈↓ α,␈↓This␈αrule␈αsays␈αto␈αinvestigate␈αthe␈αinverse␈αimage␈αof␈αan␈αunusual␈αitem␈αb,␈αunder␈α
the␈αinteresting
␈↓ α,␈↓operation␈α⊂f.␈α⊂ When␈α⊂b=2␈α⊂and␈α⊂f=number-of-divisors-of,␈α∂this␈α⊂rule␈α⊂leads␈α⊂to␈α⊂the␈α⊂de≡nition␈α∂of
␈↓ α,␈↓prime␈α⊂numbers.␈α⊂ When␈α⊂b=Phi␈↓	15␈↓␈α∂and␈α⊂f=Intersection,␈α⊂the␈α⊂rule␈α∂led␈α⊂to␈α⊂the␈α⊂discovery␈α⊂of␈α∂the
␈↓ α,␈↓concept of disjointness of sets.

␈↓ α,␈↓¬151. To fill in Domain/range entries for operation F,
␈↓ α,␈↓¬␈↓ αlF can operate on the domain of any specialization of F,
␈↓ α,␈↓¬␈↓ αlF␈α∂can␈α⊂operate␈α∂on␈α⊂the␈α∂specialization␈α⊂of␈α∂the␈α⊂domain␈α∂of␈α⊂any␈α∂specialization␈α⊂of␈α∂F␈α⊂(including␈α∂F
␈↓ α,␈↓¬␈↓ β,itself),
␈↓ α,␈↓¬␈↓ αlF␈α
can␈α
operate␈α
on␈α
some␈α
restriction␈α
of␈α
the␈α
domain␈α
of␈α
any␈α
generalization␈α
of␈α
F,␈α
at␈α
least␈α∞on␈α
its
␈↓ α,␈↓¬␈↓ β,current domain and perhaps even on a bigger space,
␈↓ α,␈↓¬␈↓ αlF␈α⊂may␈α∂be␈α⊂able␈α∂to␈α⊂operate␈α⊂on␈α∂some␈α⊂generalization␈α∂of␈α⊂(some␈α∂component(s)␈α⊂of)␈α⊂its␈α∂current
␈↓ α,␈↓¬␈↓ β,domain,
␈↓ α,␈↓¬␈↓ αlF can only (and will always) produce values lying in the range of each generalization of F,
␈↓ α,␈↓¬␈↓ αlF␈αcan␈α
--␈αwith␈α
the␈αproper␈α
arguments␈α--␈αproduce␈α
values␈αlying␈α
in␈αthe␈α
range␈αof␈α
any␈αparticular
␈↓ α,␈↓¬␈↓ β,specialization of F.

␈↓ α,␈↓There␈α
are␈αonly␈α
a␈αfew␈α
changes␈αbetween␈α
this␈αrule␈α
and␈αthe␈α
corresponding␈αone␈α
for␈αPredicates.
␈↓ α,␈↓Recall␈α⊂that␈α⊂Operations␈α⊂can␈α∂be␈α⊂multi-valued,␈α⊂and␈α⊂those␈α⊂values␈α∂are␈α⊂not␈α⊂limited␈α⊂to␈α⊂the␈α∂set
␈↓ α,␈↓{T,F}.

␈↓ α,␈↓¬152. To fill in Domain/range entries for operation F, when some exist already,
␈↓ α,␈↓¬␈↓ αlTake␈α∂an␈α∂entry␈α∂of␈α∂the␈α∂form␈α∂<D1␈α∂D2...␈α∂ Dn␈α∞→␈α∂R>␈α∂and␈α∂see␈α∂if␈α∂DixR␈α∂is␈α∂meaningful␈α∂for␈α∂some␈α∞i
␈↓ α,␈↓¬␈↓ β,(especially: i=n).
␈↓ α,␈↓¬␈↓ αlIf␈αso,␈αthen␈αremove␈αDi␈αfrom␈αthe␈αleft␈αside␈αof␈αthe␈αentry,␈αand␈αreplace␈αR␈αby␈αDixR,␈αand␈αmodify␈αthe
␈↓ α,␈↓¬␈↓ β,definition of F.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	15␈↓ε the empty set, NIL, {}
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε169␈↓-

␈↓ α,␈↓In␈α
LISP,␈α
"meaningful"␈α
is␈αcoded␈α
as:␈α
either␈α
D␈↓εi␈↓¬x␈↓R␈αis␈α
equivalent␈α
to␈α
an␈α
already-known␈αconcept,
␈↓ α,␈↓or␈α
else␈α
it␈α∞is␈α
found␈α
in␈α∞at␈α
least␈α
two␈α∞interesting␈α
conjectures.␈α
 This␈α∞is␈α
probably␈α
an␈α∞instance␈α
of
␈↓ α,␈↓what␈α
McDermott␈α
calls␈α
natural␈α
stupidity␈↓	16␈↓.␈α
 This␈αrule␈α
is␈α
tagged␈α
as␈α
being␈α
explosive,␈α
and␈αis␈α
not
␈↓ α,␈↓used very often by AM.

␈↓ α,␈↓¬153. To fill in a Range entry for operation F,
␈↓ α,␈↓¬␈↓ αlRun␈αF␈αon␈αvarious␈αdomain␈αexamples,␈αespecially␈αboundary␈αexamples,␈αto␈αcollect␈αexamples␈αof␈αthe
␈↓ α,␈↓¬␈↓ β,range.␈α Then␈αripple␈αdown␈αthe␈αtree␈αof␈αconcepts␈αto␈αdetermine␈αempirically␈αwhere␈αF␈α
seems
␈↓ α,␈↓¬␈↓ β,to be sending its values.

␈↓ α,␈↓This␈αmay␈α
shock␈αthe␈αreader,␈α
as␈αit␈α
sounds␈αdumb␈αand␈α
explosive,␈αbut␈α
the␈αconcepts␈αare␈α
arranged
␈↓ α,␈↓in␈α
a␈α
tree␈α
(using␈αGenl␈α
links),␈α
so␈α
the␈αsearch␈α
is␈α
really␈α
quite␈αfast.␈α
 Although␈α
this␈α
rule␈α
is␈αrarely
␈↓ α,␈↓used, it always seems to give surprisingly good results.

␈↓ α,␈↓¬154. If operation F has just been applied, and has yielded a new concept C as its result,
␈↓ α,␈↓¬␈↓ αlThen␈αcarefully␈αexamine␈αF.Dom/range␈αto␈αtry␈αto␈αfind␈α
out␈αwhat␈αC.Isa␈αshould␈αbe.␈α C.Isa␈αwill␈αbe␈α
all
␈↓ α,␈↓¬␈↓ β,legal entries listed as values of the range of F.

␈↓ α,␈↓When␈αF=Compose,␈αsay␈αAM␈αhas␈αjust␈αcreated␈αC=Empty␈↓εo␈↓Insert.␈↓	17␈↓␈αWhat␈αis␈αC?␈αIt␈αis␈αa␈αconcept,
␈↓ α,␈↓of␈αcourse,␈αbut␈αwhat␈αelse?␈αBy␈αexamining␈αthe␈αDomain/range␈αfacet␈αof␈αCompose,␈αAM␈α≡nds␈αthe
␈↓ α,␈↓entry␈α<Active␈αActive␈α
→␈αActive>.␈αAha!␈α
So␈αC␈αmust␈αbe␈α
an␈αActive.␈αBut␈α
AM␈αalso␈α≡nds␈αthe␈α
entry
␈↓ α,␈↓<Predicate␈αActive␈α→␈αPredicate>.␈α Since␈α"Empty"␈αis␈αa␈αpredicate,␈αthe␈α≡nal␈αcomposition␈αC␈αmust
␈↓ α,␈↓also␈αbe␈αa␈αpredicate.␈α So␈αC.Isa␈αwould␈αbe␈α≡lled␈αin␈αwith␈α"Predicate".␈αAM␈αthus␈αused␈α
the␈αabove
␈↓ α,␈↓rule␈α∞to␈α∂determine␈α∞that␈α∞Empty␈↓εo␈↓Insert␈α∂was␈α∞a␈α∞predicate.␈α∂Even␈α∞if␈α∞this␈α∂rule␈α∞were␈α∂excised,␈α∞AM
␈↓ α,␈↓could still determine that fact, painfully, by noticing that all the values were truth-values.

␈↓ α,␈↓¬155.␈α∞If␈α∞operation␈α∞F␈α∞has␈α∞just␈α∞been␈α∞applied␈α∞to␈α∞A1,A2,...,␈α∞and␈α∞has␈α∞yielded␈α∞a␈α∞new␈α∞concept␈α∞C␈α∞as␈α∞its
␈↓ α,␈↓¬␈↓ β,result,
␈↓ α,␈↓¬␈↓ αlThen␈α
add␈αF␈α
to␈αC.In-ran-of;␈α
add␈αF␈α
to␈αthe␈α
In-dom-of␈αfacet␈α
of␈αall␈α
the␈αAi's␈α
which␈α
are␈αconcepts;
␈↓ α,␈↓¬␈↓ β,add <A1... → C> to F.Exs.

␈↓ α,␈↓There␈α∩is␈α∩some␈α∩overlap␈α∩here␈α∩with␈α∩earlier␈α⊃rules,␈α∩but␈α∩there␈α∩is␈α∩no␈α∩theoretical␈α∩or␈α⊃practical
␈↓ α,␈↓di≠culty with such redundancy.

␈↓ α,␈↓¬156.␈α∂When␈α∞filling␈α∂in␈α∞examples␈α∂of␈α∞operation␈α∂F,␈α∂if␈α∞F␈α∂takes␈α∞some␈α∂existing␈α∞concepts␈α∂A1,␈α∂A2,...␈α∞and
␈↓ α,␈↓¬␈↓ β,(may) produce a new concept,
␈↓ α,␈↓¬␈↓ αlThen␈α∂only␈α⊂consider,␈α∂as␈α⊂potential␈α∂A␈↓#vi␈↓#'s,␈α∂those␈α⊂concepts␈α∂which␈α⊂already␈α∂have␈α⊂some␈α∂examples.
␈↓ α,␈↓¬␈↓ β,Prefer␈αthe␈αA␈↓#vi␈↓#'s␈αto␈αbe␈αinteresting,␈αto␈αhave␈αa␈αhigh␈αworth␈αrating,␈αto␈αhave␈αsome␈α
interesting
␈↓ α,␈↓¬␈↓ β,conjectures about them, to have several examples and several non-examples, etc.

␈↓ α,␈↓The␈αdanger␈αhere␈αis␈αof,␈αe.g.,␈αComposing␈αtwo␈αoperations␈αwhich␈αturn␈αout␈αto␈αbe␈αvacuous,␈αor␈αof
␈↓ α,␈↓Conjoining␈αan␈αempty␈αconcept␈αonto␈αanother,␈α
or␈αof␈αproliferating␈αvariants␈αof␈αa␈αboring␈α
concept,
␈↓ α,␈↓etc.



␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	16␈↓ε␈α	See␈α	[McDermott␈α	76]␈αλfor␈α	natural␈α	stupidity.␈α	 He␈αλcriticizes␈α	the␈α	use␈α	of␈αλvery␈α	intelligent-sounding␈α	names␈α	for␈αλotherwise-simple
␈↓ α,␈↓ε␈↓ βLprogram modules. But consider "Homo sapiens", which means "wise man". Now ␈↓&there's␈↓)αβ a misleading label...
␈↓ α,␈↓ε␈↓	17␈↓ε␈α	i.e.,␈α	insert␈α	x␈α	into␈α	a␈α	structure␈α	S␈αλand␈α	then␈α	see␈α	if␈α	S␈α	is␈α	empty.␈α	This␈α	leads␈αλAM␈α	to␈α	realize␈α	that␈α	inserting␈α	can␈α	never␈α	result␈α	in␈αλan
␈↓ α,␈↓ε␈↓ βLempty structure.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε170␈↓-

␈↓ α,␈↓∧␈↓ α<Operation . Check

␈↓ α,␈↓Below are rules used to check existing entries on various facets of operations.

␈↓ α,␈↓¬157. To check the domain/range entries on the operation F,
␈↓ α,␈↓¬␈↓ αlIF a domain/range entry has the form (D D D... → R),
␈↓ α,␈↓¬␈↓ αland␈αall␈αthe␈αD's␈αare␈αequal,␈αand␈αR␈αis␈αa␈αgeneralization␈αof␈αD␈α(or,␈αwith␈αless␈αenthusiasm:␈αif␈αR␈αand␈α
D
␈↓ α,␈↓¬␈↓ β,have a significant overlap),
␈↓ α,␈↓¬␈↓ αlTHEN␈αit's␈αworth␈αseeing␈αwhether␈α(D␈αD␈αD...␈α→␈αD)␈αis␈αconsistent␈αwith␈αall␈αknown␈αexamples␈αof␈αthe
␈↓ α,␈↓¬␈↓ β,operation:
␈↓ α,␈↓¬␈↓ βlIf␈α
there␈α
are␈α
no␈α
known␈α
examples,␈α
add␈αa␈α
task␈α
to␈α
the␈α
agenda␈α
requesting␈α
they␈αbe␈α
filled
␈↓ α,␈↓¬␈↓ ∧,in.
␈↓ α,␈↓¬␈↓ βlIf␈αthere␈αare␈α
examples,␈αand␈α(D␈α
D␈αD...␈α→␈α
D)␈αis␈αconsistent,␈α
add␈αit␈αto␈αthe␈α
Domain/range
␈↓ α,␈↓¬␈↓ ∧,facet of this operation.
␈↓ α,␈↓¬␈↓ βlIf␈α
there␈α
are␈α
some␈α
contradicting␈α
examples,␈αcreate␈α
a␈α
new␈α
concept␈α
which␈α
is␈αdefined
␈↓ α,␈↓¬␈↓ ∧,as this operation restricted to (D D D... → D).

␈↓ α,␈↓When␈α∃AM␈α⊗restricts␈α∃Bag-union␈α⊗to␈α∃numbers␈α⊗(bags␈α∃of␈α⊗T's),␈α∃the␈α⊗new␈α∃operation␈α⊗has␈α∃a
␈↓ α,␈↓Domain/range␈α⊂entry␈α⊂of␈α∂the␈α⊂form␈α⊂(Numbers␈α∂Numbers␈α⊂→␈α⊂Bag).␈α∂ The␈α⊂above␈α⊂rule␈α⊂has␈α∂AM
␈↓ α,␈↓investigate␈α
whether␈αthe␈α
range␈αspeci≡cation␈α
mightn't␈αalso␈α
be␈αnarrowed␈α
down␈αto␈α
Number.␈αIn
␈↓ α,␈↓this␈αcase␈αit␈αis␈αa␈αgreat␈αhelp.␈αThe␈αrule␈αoften␈αfails,␈αof␈αcourse:␈αthe␈αsum␈αof␈αtwo␈αprimes␈αis␈αrarely␈αa
␈↓ α,␈↓prime,␈α∞the␈α∂cross-product␈α∞of␈α∞two␈α∂lists-of-atoms␈α∞is␈α∞not␈α∂a␈α∞list-of-atoms,␈α∞etc.␈α∂ Since␈α∞this␈α∂rule␈α∞is
␈↓ α,␈↓almost instantaneous to execute, it's cost-e≥ective overall.

␈↓ α,␈↓¬158. When checking the domain/range entries on the operation F,
␈↓ α,␈↓¬␈↓ αlIF a domain/range entry has the form (D D D... → R),
␈↓ α,␈↓¬␈↓ αland all the D's are equal, and R is a specialization of D,
␈↓ α,␈↓¬␈↓ αlTHEN␈αit's␈α
worth␈αinserting␈α(D␈α
D␈αD...␈α→␈α
D)␈αas␈αa␈α
new␈αentry␈αon␈α
F.Dom/ran,␈αeven␈αthough␈α
that␈αis
␈↓ α,␈↓¬␈↓ β,redundant.

␈↓ α,␈↓This␈αshows␈αthat␈αsymmetry␈αand␈αaesthetics␈αare␈αsometimes␈αpreferable␈αto␈αabsolute␈αoptimization.
␈↓ α,␈↓That's␈α
why␈αwe␈α
program␈αin␈α
Lisp,␈α
instead␈αof␈α
machine␈αlanguage.␈α
 On␈α
the␈αother␈α
hand,␈αthis␈α
rule
␈↓ α,␈↓wasn't really that useful to AM. Now, by analogy,...?

␈↓ α,␈↓¬159. When checking the Domain/range entries for operation F,
␈↓ α,␈↓¬␈↓ αlEnsure␈α⊂that␈α⊂the␈α⊂boundary␈α∂items␈α⊂in␈α⊂the␈α⊂range␈α∂can␈α⊂actually␈α⊂be␈α⊂reached␈α∂by␈α⊂F.␈α⊂ If␈α⊂not,␈α∂see
␈↓ α,␈↓¬␈↓ β,whether the range is really just some known specialization of F.

␈↓ α,␈↓This␈α∞rule␈α∞is␈α∞a␈α∞typical␈α∂checking␈α∞rule.␈α∞Note␈α∞that␈α∞it␈α∞is␈α∂active,␈α∞not␈α∞passive:␈α∞it␈α∞might␈α∂alter␈α∞the
␈↓ α,␈↓Domain/range facet of F, it it ≡nds an error there.

␈↓ α,␈↓¬160. When checking examples of the operation F, for each such example,
␈↓ α,␈↓¬␈↓ αlIf the value returned by F is a concept C, add `F' to C.In-range-of.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε171␈↓-

␈↓ α,␈↓∧␈↓ α<Operation . Suggest

␈↓ α,␈↓¬161. Whenever the domain of operation F has changed,
␈↓ α,␈↓¬␈↓ αlcheck␈α
whether␈α
the␈α
range␈α
has␈α
also␈α
changed.␈α
Often␈α
the␈α
range␈α
will␈α
change␈α
analogously␈α
to␈α
the
␈↓ α,␈↓¬␈↓ β,domain, where the operation itself is the Analogy.

␈↓ α,␈↓¬162. After working on Operation F,
␈↓ α,␈↓¬␈↓ αlGive a slight, ephemeral boost to tasks involving Range(F).

␈↓ α,␈↓This␈αwll␈αbe␈αa␈αmoderate␈αsize␈αboost␈αfor␈αeach␈αtask␈αwhich␈αasks␈αto␈α≡ll␈αin␈αexamples␈αof␈αthat␈αrange
␈↓ α,␈↓concept,␈αand␈αa␈αvery␈αtiny␈αboost␈αfor␈αeach␈αother␈αtask␈αmentioning␈αsuch␈αa␈αconcept.␈α This␈αis␈αboth
␈↓ α,␈↓a␈α∞supplement␈α∞to␈α∞the␈α∞more␈α∞general␈α∞"focus␈α∞of␈α∞attention"␈α∞rule,␈α∞and␈α∞a␈α∞nontrivial␈α∞heuristic␈α
for
␈↓ α,␈↓≡nding␈αvaluable␈α
new␈αtasks.␈α
 It␈αis␈α
an␈αextension␈α
of␈αrule␈α
number␈α142,␈α
and␈αa␈α
partial␈αconverse␈α
to
␈↓ α,␈↓rule 14.


␈↓ α,␈↓∧␈↓ α<Operation . Interest

␈↓ α,␈↓¬163.␈αAn␈αoperation␈αF␈αis␈αinteresting␈αif␈αthere␈αare␈αother␈αoperations␈αwith␈αthe␈αsame␈αdomain␈αand␈αrange,
␈↓ α,␈↓¬␈↓ β,and if they are (on the average) fairly interesting.


␈↓ α,␈↓¬164.␈αAn␈αoperation␈αF␈α
is␈αinteresting␈αif␈αit␈α
is␈αthe␈αfirst␈αoperation␈α
connecting␈αits␈αdomain␈αconcept␈α
to␈αits
␈↓ α,␈↓¬␈↓ β,range␈α↔concept,␈α↔and␈α↔if␈α_those␈α↔domain/range␈α↔components␈α↔are␈α_themselves␈α↔valuable
␈↓ α,␈↓¬␈↓ β,concepts,␈α∩and␈α∩there␈α∩is␈α∩no␈α∩analogy␈α⊃between␈α∩them,␈α∩and␈α∩there␈α∩are␈α∩some␈α⊃interesting
␈↓ α,␈↓¬␈↓ β,conjectures involving the domain of F.

␈↓ α,␈↓The␈αabove␈α
two␈αrules␈α
say␈αthat␈αF␈α
can␈αbe␈α
valuable␈αbecuase␈αit's␈α
similar␈αto␈α
other,␈αalready-liked
␈↓ α,␈↓operations,␈αor␈αbecause␈αit␈αis␈αtotally␈αdi≥erent␈αfrom␈αany␈αknown␈αoperation.␈αAlthough␈α
these␈αtwo
␈↓ α,␈↓criteria␈α∞are␈α∞nonintersecting,␈α∞their␈α∞union␈α∂represents␈α∞only␈α∞a␈α∞small␈α∞fraction␈α∞of␈α∂the␈α∞operations
␈↓ α,␈↓that get created: typically, ␈↓βneither␈↓ rule will trigger.

␈↓ α,␈↓¬165. An operation F is interesting if its range is very interesting.

␈↓ α,␈↓Range␈α⊂here␈α∂refers␈α⊂to␈α∂the␈α⊂concept␈α⊂in␈α∂which␈α⊂all␈α∂results␈α⊂of␈α∂F␈α⊂must␈α⊂lie.␈α∂ It␈α⊂is␈α∂the␈α⊂R␈α⊂in␈α∂the
␈↓ α,␈↓domain/range␈αfacet␈αentry␈α<D␈α
→␈αR>␈αfor␈αconcept␈αF.␈α
 The␈αcorresponding␈αrule␈αfor␈α
`domains'␈αis
␈↓ α,␈↓applicable␈α
to␈α
any␈α
Active,␈αnot␈α
just␈α
to␈α
Operations,␈α
hence␈αis␈α
listed␈α
under␈α
Active.Interest,␈αas␈α
rule
␈↓ α,␈↓number 145.

␈↓ α,␈↓¬166.␈α
An␈α
operation␈α
F␈α
is␈αinteresting␈α
if␈α
the␈α
values␈α
of␈α
F␈αsatisfy␈α
some␈α
unusual␈α
property␈α
which␈α
is␈αnot␈α
(in
␈↓ α,␈↓¬␈↓ β,general) satisfied by the arguments to F.

␈↓ α,␈↓Thus␈α
doubling␈α
is␈α∞interesting␈α
because␈α
it␈α∞always␈α
returns␈α
an␈α
even␈α∞number.␈α
 This␈α
is␈α∞one␈α
case
␈↓ α,␈↓where␈α
the␈αinteresting␈α
property␈αcan␈α
be␈αdeduced␈α
trivially␈αjust␈α
by␈αlooking␈α
at␈αthe␈α
domain␈αand
␈↓ α,␈↓range of the operation: Numbers→Even-nos.

␈↓ α,␈↓¬167. An operation is interesting if its values are interesting.

␈↓ α,␈↓This␈α⊃can␈α⊃mean␈α⊃that␈α⊃each␈α⊃value␈α⊃is␈α⊃interesting␈α⊃(e.g.,␈α⊃Compose␈α⊃is␈α⊃well-received␈α⊃because␈α⊂it
␈↓ α,␈↓produces␈α
many␈αnew,␈α
valuable␈α
concepts␈αas␈α
its␈α
values).␈α Or,␈α
it␈α
can␈αmean␈α
that␈α
the␈αoperations'
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε172␈↓-

␈↓ α,␈↓values,␈α
gathered␈α
together␈α
into␈α
one␈α
big␈α
set,␈α
are␈α
interesting␈α
as␈α
a␈α
set.␈α
 Unlike␈α
the␈α
preceding␈α
rule,
␈↓ α,␈↓this␈α
one␈α
has␈α
no␈α
mention␈α
whatsoever␈α∞of␈α
the␈α
domain␈α
items,␈α
the␈α
arguments␈α
to␈α∞the␈α
operation.
␈↓ α,␈↓This␈α⊃rule␈α⊃was␈α⊂used␈α⊃to␈α⊃good␈α⊂advantage␈α⊃frequently␈α⊃by␈α⊂AM.␈α⊃ For␈α⊃example,␈α⊃Factorings␈α⊂of
␈↓ α,␈↓numbers␈α∂are␈α∂interesting␈α∂because␈α∂(using␈α∂rule␈α∂232)␈α∂for␈α∂all␈α∂x,␈α∂Factorings(x)␈α∂is␈α∂interesting␈α∂in
␈↓ α,␈↓exactly␈αthe␈αsame␈αway.␈α Namely,␈αFactorings(x),␈αviewed␈αas␈αa␈αset,␈αalways␈αcontains␈αprecisely␈αone
␈↓ α,␈↓item␈α∞which␈α∞has␈α∞a␈α∞certain␈α∞interesting␈α∞property␈α
(see␈α∞rule␈α∞233).␈α∞ Namely,␈α∞all␈α∞its␈α∞members␈α
are
␈↓ α,␈↓primes␈α(see␈αrule␈α232␈αagain).␈α This␈αexplains␈αone␈αway␈αin␈αwhich␈αAM␈αnoticed␈αthat␈αall␈αnumbers
␈↓ α,␈↓seem to factor uniquely into primes.

␈↓ α,␈↓¬168.␈α∂An␈α∞operation␈α∂is␈α∂interesting␈α∞if␈α∂its␈α∂values␈α∞are␈α∂interesting,␈α∂ignoring␈α∞the␈α∂images␈α∂of␈α∞boundary
␈↓ α,␈↓¬␈↓ β,items from the domain.

␈↓ α,␈↓That is, if the image of the domain ¬ minus its boundary ¬ is interesting.

␈↓ α,␈↓¬169.␈α∞An␈α∞operation␈α∞is␈α∞interesting␈α∞if␈α∞its␈α∞values␈α
on␈α∞the␈α∞boundary␈α∞items␈α∞from␈α∞the␈α∞domain␈α∞are␈α
very
␈↓ α,␈↓¬␈↓ β,interesting.  Ignore the non-boundary parts of the domain.

␈↓ α,␈↓That is, if the image of the boundary of the domain is interesting.

␈↓ α,␈↓¬170.␈α
An␈α
operation␈α
is␈α
interesting␈α
if␈α
it␈α
leaves␈α
intact␈α
any␈α
interesting␈α
properties␈α
of␈α
its␈α
argument(s).
␈↓ α,␈↓¬␈↓ β,This␈α∂is␈α∂even␈α∂better␈α∂if␈α∂it␈α∞eliminates␈α∂some␈α∂undesirable␈α∂properties,␈α∂or␈α∂adds␈α∂some␈α∞new,
␈↓ α,␈↓¬␈↓ β,desirable ones.

␈↓ α,␈↓Thus␈α
a␈α
new,␈α
specialized␈α∞kind␈α
of␈α
Insertion␈α
operation␈α
is␈α∞interesting␈α
if,␈α
even␈α
though␈α∞it␈α
stu≥s
␈↓ α,␈↓more␈α⊂items␈α⊂into␈α⊂a␈α⊂structure,␈α⊂the␈α⊂nice␈α⊂properties␈α⊂of␈α⊂the␈α⊂structure␈α⊂remain.␈α⊂ The␈α⊂operation
␈↓ α,␈↓"Merge"␈α∞is␈α
interesting␈α∞for␈α
this␈α∞very␈α
reason:␈α∞it␈α
inserts␈α∞items␈α
into␈α∞an␈α
alphabetized␈α∞list,␈α∞yet␈α
it
␈↓ α,␈↓doesn't destroy that interesting property of the list.

␈↓ α,␈↓¬171.␈α∂An␈α∂operation␈α∂is␈α∂interesting␈α∂if␈α∂its␈α∂domain␈α∞and␈α∂range␈α∂are␈α∂equal.␈α∂If␈α∂there␈α∂is␈α∂more␈α∂than␈α∞one
␈↓ α,␈↓¬␈↓ β,domain␈α∪component,␈α∪then␈α∪at␈α∪least␈α∪one␈α∪of␈α∪them␈α∪should␈α∪equal␈α∪the␈α∪range.␈α∪The␈α∩more
␈↓ α,␈↓¬␈↓ β,components which are equal to the range, the better.

␈↓ α,␈↓Thus␈α⊃"Insertion"␈α⊃quali≡es␈α⊃here,␈α⊃since␈α⊃its␈α⊃domain/range␈α⊃entry␈α⊃is␈α⊃<Anything␈α⊃Structures␈α⊃→
␈↓ α,␈↓Structures>.␈α∞ But␈α∂"Union"␈α∞is␈α∞even␈α∂better,␈α∞since␈α∞both␈α∂domain␈α∞components␈α∞equal␈α∂the␈α∞range,
␈↓ α,␈↓namely Structures.

␈↓ α,␈↓¬172.␈α
An␈α
operation␈αis␈α
mildly␈α
interesting␈αif␈α
its␈α
range␈αis␈α
related␈α
somehow␈α(e.g.␈α
specialization␈α
of)␈αto
␈↓ α,␈↓¬␈↓ β,one or more components of its range. The more the better.

␈↓ α,␈↓A weakened form of the preceding rule.

␈↓ α,␈↓¬173. If the result of applying operation F is a new concept C,
␈↓ α,␈↓¬␈↓ αlThen the interestingness of F is weakly tied to that of C.

␈↓ α,␈↓If␈αthe␈αnew␈αconcept␈αC␈αbecomes␈αvery␈αvaluable,␈αthen␈αF␈αwill␈αrise␈αslightly␈αin␈αinterest.␈α If␈αC␈αis␈αso
␈↓ α,␈↓bad␈αit␈αgets␈αforgotten,␈αF␈αwill␈αnot␈αbe␈αregarded␈αquite␈αas␈αhighly.␈α When␈αCanonize␈αscores␈αbig␈αits
␈↓ α,␈↓≡rst␈αtime␈αused,␈αit␈αrises␈αin␈αinterest.␈αThis␈αcaused␈αAM␈αto␈αform␈αpoorly-motivated␈αcanonizations,
␈↓ α,␈↓which␈αled␈αto␈α
dismal␈αresults,␈αwhich␈αgradually␈α
lowered␈αthe␈αrating␈αof␈α
Canonize␈αto␈αwhere␈αit␈α
was
␈↓ α,␈↓originally.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε173␈↓-

␈↓ α,␈↓␈↓ β⊂␈↓↓␈↓&Appendix 2.6. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with any Composition␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Composition . Fillin

␈↓ α,␈↓¬174. To fill in algorithms for operation F, where F is a composition G␈↓εo␈↓¬H,
␈↓ α,␈↓¬␈↓ αlOne algorithm is: apply H and then apply G to the result.

␈↓ α,␈↓Of␈αcourse␈αthis␈αrule␈αis␈α
not␈αmuch␈αmore␈αthan␈αthe␈αde≡nition␈α
of␈αwhat␈αit␈αmeans␈αto␈α
compose␈αtwo
␈↓ α,␈↓operations.

␈↓ α,␈↓¬175. To fill in Domain/range entries for operation F, where F is a composition G␈↓εo␈↓¬H,
␈↓ α,␈↓¬␈↓ αlTentatively␈α
assume␈α
that␈α
the␈α
domain␈α
is␈αDomain(H),␈α
and␈α
range␈α
is␈α
Range(G).␈α
 More␈αprecisely,␈α
the
␈↓ α,␈↓¬␈↓ β,domain␈α∂will␈α∞be␈α∂the␈α∂result␈α∞of␈α∂substituting␈α∂Domain(H)␈α∞for␈α∂Range(H)␈α∂wherever␈α∞Range(H)
␈↓ α,␈↓¬␈↓ β,appears (or: just once) in Domain(G).

␈↓ α,␈↓Thus␈α
for␈α
F=Divides␈↓εo␈↓Count,␈α
where␈α
Divides:<Number,Number␈α
→␈α
{T,F}>,␈α
and␈α
Count:<Bag␈α→
␈↓ α,␈↓Number>,␈α⊂the␈α⊂above␈α⊂rule␈α⊃would␈α⊂say␈α⊂that␈α⊂the␈α⊂domain/range␈α⊃entries␈α⊂for␈α⊂F␈α⊂are␈α⊃gotten␈α⊂by
␈↓ α,␈↓substituting␈α
`Bag'␈α
for␈α
`Number'␈α
once␈α
or␈αtwice␈α
in␈α
Domain(Divides).␈α
The␈α
possible␈α
entries␈αfor
␈↓ α,␈↓F.Dom/range␈α∂are␈α∞thus:␈α∂<Bag,Bag␈α∞→␈α∂{T,F}>,␈α∞<Number,Bag␈α∂→␈α∞{T,F}>,␈α∂and␈α∂<Bag,Number␈α∞→
␈↓ α,␈↓{T,F}>.

␈↓ α,␈↓¬176.␈αTo␈αfill␈αin␈αDomain/range␈αentries␈αfor␈αoperation␈αF,␈αwhere␈αF␈αis␈αa␈αcomposition␈αG␈↓εo␈↓¬H,␈αBut␈αRange(H)
␈↓ α,␈↓¬␈↓ β,does not occur as a component of Domain(G),
␈↓ α,␈↓¬␈↓ αlThe␈αrange␈α
of␈αF␈αis␈α
still␈αRange(G),␈αbut␈α
the␈αdomain␈αof␈α
F␈αis␈αcomputed␈α
as␈αfollows:␈α
Ascertain␈αthe
␈↓ α,␈↓¬␈↓ β,component␈αX␈α
of␈αDomain(G)␈α
having␈αthe␈α
biggest␈α(fractional)␈α
overlap␈αwith␈α
Range(H).␈αThen
␈↓ α,␈↓¬␈↓ β,substitute␈α∩Domain(H)␈α∪for␈α∩X␈α∪in␈α∩Domain(G).␈α∪The␈α∩result␈α∩is␈α∪the␈α∩value␈α∪to␈α∩be␈α∪used␈α∩for
␈↓ α,␈↓¬␈↓ β,Domain(F).

␈↓ α,␈↓This␈αrule␈α
is␈αa␈α
second-order␈αcorrection␈α
to␈αthe␈α
previous␈αone.␈α
If␈αthere␈α
is␈αno␈α
absolute␈αequality,
␈↓ α,␈↓then␈α∞a␈α∞large␈α
intersection␈α∞will␈α∞su≠ce.␈α
Notice␈α∞that␈α∞F␈α∞may␈α
no␈α∞longer␈α∞be␈α
de≡ned␈α∞on␈α∞all␈α∞of␈α
its
␈↓ α,␈↓domain,␈α⊂even␈α⊂if␈α⊂G␈α⊃and␈α⊂H␈α⊂are.␈α⊂ If␈α⊂identical␈α⊃equality␈α⊂is␈α⊂taken␈α⊂as␈α⊂the␈α⊃maximum␈α⊂possible
␈↓ α,␈↓overlap␈α∩betwen␈α⊃two␈α∩concepts,␈α⊃then␈α∩this␈α∩rule␈α⊃can␈α∩be␈α⊃used␈α∩to␈α⊃replace␈α∩the␈α∩preceding␈α⊃one
␈↓ α,␈↓completely.

␈↓ α,␈↓¬177. When trying to fill in the Isa entries for a composition F=G␈↓εo␈↓¬H,
␈↓ α,␈↓¬␈↓ αlExamine␈α
G.Isa␈α
and␈α
H.Isa,␈α
and␈α
especially␈α
their␈α
intersection.␈α
Some␈α
of␈α
those␈α
concepts␈α∞may␈α
also
␈↓ α,␈↓¬␈↓ β,claim F as an example. Run their definition facet to see.

␈↓ α,␈↓To see how this is encoded into LISP, turn to page 130.

␈↓ α,␈↓¬178. When trying to fill in the Genl or Spec entries for a composition F=G␈↓εo␈↓¬H,
␈↓ α,␈↓¬␈↓ αlExamine the corresponding facet on G and on H.

␈↓ α,␈↓This rule is similar to the preceding one, but wasn't as useful or as reliable.

␈↓ α,␈↓¬179.␈α∂A␈α⊂satisfactory␈α∂initial␈α⊂guess␈α∂at␈α⊂the␈α∂Worth␈α∂value␈α⊂of␈α∂composition␈α⊂F=G␈↓εo␈↓¬H␈α∂is␈α⊂the␈α∂root-sum-of-
␈↓ α,␈↓¬␈↓ β,squares of G.Worth and H.Worth.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε174␈↓-


␈↓ α,␈↓¬180.␈α∞To␈α∞fill␈α∞in␈α∞examples␈α∂of␈α∞F,␈α∞where␈α∞F=G␈↓εo␈↓¬H,␈α∞and␈α∞both␈α∂G␈α∞and␈α∞H␈α∞are␈α∞time-consuming,␈α∂but␈α∞where
␈↓ α,␈↓¬␈↓ β,many examples of both G and H exist,
␈↓ α,␈↓¬␈↓ αlSeek␈α
an␈α
example␈α
x→y␈α
of␈α
H,␈αand␈α
an␈α
example␈α
y→z␈α
of␈α
G,␈αand␈α
then␈α
return␈α
x→z␈α
as␈α
a␈αprobable
␈↓ α,␈↓¬␈↓ β,example of F.

␈↓ α,␈↓Above,␈α
`seek'␈α
is␈α
done␈α
in␈α∞a␈α
tight,␈α
e≠cent␈α
manner.␈α
The␈α∞examples␈α
are␈α
H␈α
are␈α
hashed␈α∞into␈α
an
␈↓ α,␈↓array,␈α∞based␈α∞on␈α
the␈α∞values␈α∞y␈α
of␈α∞each␈α∞one.␈α∞Then␈α
the␈α∞arguments␈α∞of␈α
the␈α∞examples␈α∞of␈α∞G␈α
are
␈↓ α,␈↓hashed␈α
to␈αsee␈α
if␈α
they␈αoccur␈α
in␈α
this␈αarray.␈α
Those␈α
that␈αdo␈α
will␈α
generate␈αan␈α
example␈α
of␈αthe␈α
new
␈↓ α,␈↓composition.

␈↓ α,␈↓¬181.␈α
To␈α
fill␈α
in␈αexamples␈α
of␈α
F,␈α
where␈αF=G␈↓εo␈↓¬H,␈α
and␈α
G␈α
is␈α
timeconsuming,␈αbut␈α
many␈α
examples␈α
of␈αG␈α
exist,
␈↓ α,␈↓¬␈↓ β,and it is not known whether H is time-consuming or not,
␈↓ α,␈↓¬␈↓ αlSpend a moment trying to access or trivially fill in examples of H.
␈↓ α,␈↓¬␈↓ αlIf this succeeds, apply the preceding rule.
␈↓ α,␈↓¬␈↓ αlIf␈αthis␈α
fails,␈αthen␈α
formally␈αpropose␈α
that␈αAM␈αfill␈α
in␈αexamples␈α
of␈αH,␈α
with␈αpriority␈α
equal␈αto␈αthat␈α
of
␈↓ α,␈↓¬␈↓ β,the␈α∞current␈α∞task,␈α∞for␈α∞these␈α∞two␈α∞reasons:␈α∞(i)␈α∞if␈α∞examples␈α∞of␈α∞H␈α∞existed,␈α∞then␈α∂AM␈α∞could
␈↓ α,␈↓¬␈↓ β,have␈α⊃used␈α⊃the␈α⊃heuristic␈α⊃preceding␈α⊃this␈α⊃one,␈α⊃to␈α⊃fill␈α⊃in␈α⊃examples␈α⊃of␈α⊃F,␈α⊃and␈α⊃(ii)␈α⊃it␈α⊂is
␈↓ α,␈↓¬␈↓ β,dangerous␈α
to␈αspend␈α
a␈α
long␈αtime␈α
dealing␈α
with␈αG␈↓εo␈↓¬H␈α
before␈αany␈α
examples␈α
at␈αall␈α
of␈α
H␈αare
␈↓ α,␈↓¬␈↓ β,known.

␈↓ α,␈↓This␈α∞rule␈α
is␈α∞of␈α
course␈α∞tightly␈α
coupled␈α∞to␈α∞the␈α
preceding␈α∞one.␈α
 The␈α∞same␈α
rule␈α∞exists␈α∞for␈α
the
␈↓ α,␈↓case where just H is time-consuming, instead of G.

␈↓ α,␈↓¬182. When trying to fill in Conjecs about a composition F=G␈↓εo␈↓¬H,
␈↓ α,␈↓¬␈↓ αlConsider that F may be the same as G (or the same as H).

␈↓ α,␈↓It␈α
was␈α∞somewhat␈α
depressing␈α
that␈α∞this␈α
`stupid'␈α∞heuristic␈α
turned␈α
out␈α∞to␈α
be␈α∞valuable,␈α
perhaps
␈↓ α,␈↓even necessary for AM's top performance.


␈↓ α,␈↓∧␈↓ α<Composition . Check

␈↓ α,␈↓¬183.␈α
Check␈α
that␈α
F␈↓εo␈↓¬G␈α
is␈α
really␈α
not␈α
the␈α
same␈αas␈α
F,␈α
or␈α
the␈α
same␈α
as␈α
G.␈α
 Spend␈α
some␈α
time␈αchecking
␈↓ α,␈↓¬␈↓ β,whether F␈↓εo␈↓¬G is equivalent to any already-known active concept.

␈↓ α,␈↓This␈α
happens␈α
often␈α
enough␈α
to␈α
make␈α
it␈α
worth␈α
stating␈α
explicitly.␈α
Often,␈α
for␈α
example,␈α
F␈α
will
␈↓ α,␈↓not even bother looking at the result of G! For example,
␈↓ α,␈↓Proj2␈↓εo␈↓Square(x,y)  =  Proj2(Square(x),y)  =  y  =  Proj2(x,y).

␈↓ α,␈↓¬184. When checking the Algorithms entries for a composition F=G␈↓εo␈↓¬H,
␈↓ α,␈↓¬␈↓ αlIf range(H) is not wholly contained in the domain of G,
␈↓ α,␈↓¬␈↓ αlthen␈α
the␈α
algorithm␈α
must␈α
contain␈α
a␈α
"legality"␈αcheck,␈α
ensuring␈α
that␈α
H(x)␈α
is␈α
a␈α
valid␈α
member␈αof
␈↓ α,␈↓¬␈↓ β,the domain of G.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε175␈↓-

␈↓ α,␈↓∧␈↓ α<Composition . Suggest

␈↓ α,␈↓¬185. Given an interesting operation F:A␈↓#
n␈↓#→A,
␈↓ α,␈↓¬␈↓ αlconsider composing F with itself.

␈↓ α,␈↓This␈αmay␈αresult␈αin␈αmore␈αthan␈αone␈αnew␈αoperation.␈αFrom␈αF=division,␈αfor␈αexample,␈αwe␈αget␈αthe
␈↓ α,␈↓two␈α∪operations␈α∪(x/y)/z␈α∪and␈α∪x/(y/z).␈α∪ AM␈α∪quickly␈α∪realizes␈α∪that␈α∪such␈α∪variants␈α∀are␈α∪really
␈↓ α,␈↓equivalent,␈α⊃and␈α∩(if␈α⊃prodded)␈α∩eventually␈α⊃realizes␈α⊃that␈α∩F(F(x,y),z)=F(x,F(y,z))␈α⊃is␈α∩a␈α⊃common
␈↓ α,␈↓situation (which we call associativity of F).

␈↓ α,␈↓¬186.␈αIf␈αthe␈αnewly-formed␈αdomain␈αof␈αthe␈αcomposition␈αF=G␈↓εo␈↓¬H␈αcontains␈αmore␈αthan␈αone␈αoccurrence␈αof
␈↓ α,␈↓¬␈↓ β,the concept D, and this isn't true of G or H,
␈↓ α,␈↓¬␈↓ αlThen␈α_consider␈α_creating␈α_a␈α_new␈α_operation,␈α_a␈α_specialization␈α_of␈α_F,␈α_by␈α→Coalescing␈α_the
␈↓ α,␈↓¬␈↓ β,domain/range of F, by eliminating one of the D components.

␈↓ α,␈↓Thus␈α∞when␈α∞Insert␈↓εo␈↓Delete␈α∂is␈α∞formed,␈α∞the␈α∂old␈α∞Domain/range␈α∞entries␈α∂were␈α∞both␈α∞of␈α∂the␈α∞form
␈↓ α,␈↓<Anything␈αStructure␈α
→␈αStructure>.␈αThe␈α
newly-created␈αentry␈α
for␈αInsert␈↓εo␈↓Delete␈αwas␈α
<Anything
␈↓ α,␈↓Anything␈α∂Structure␈α∂→␈α∂Structure>;␈α∂i.e.,␈α∂take␈α∂x,␈α∂delete␈α∂it␈α∂from␈α∂S,␈α∂then␈α∂insert␈α∂y␈α∂into␈α∂S.␈α∂The
␈↓ α,␈↓above␈α
rule␈α
had␈α
AM␈α
turn␈α
this␈α
into␈α
a␈α
new␈α
operation,␈α
with␈α
domain/range␈α<Anything␈α
Structure
␈↓ α,␈↓→ Structure>, which deleted x from S and the inserted the very same x back into S.


␈↓ α,␈↓∧␈↓ α<Composition . Interest

␈↓ α,␈↓¬187. A composition F=G␈↓εo␈↓¬H is interesting if G and H are very interesting.


␈↓ α,␈↓¬188.␈αA␈αcomposition␈αF=G␈↓εo␈↓¬H␈αis␈αinteresting␈αif␈αF␈αhas␈αan␈αinteresting␈αproperty␈αnot␈αpossessed␈αby␈αeither
␈↓ α,␈↓¬␈↓ β,G or H.


␈↓ α,␈↓¬189.␈α∂A␈α∂composition␈α∞F=G␈↓εo␈↓¬H␈α∂is␈α∂interesting␈α∂if␈α∞F␈α∂has␈α∂most␈α∂of␈α∞the␈α∂interesting␈α∂properties␈α∂which␈α∞are
␈↓ α,␈↓¬␈↓ β,possessed␈α⊂by␈α⊂either␈α∂G␈α⊂or␈α⊂H.␈α∂ This␈α⊂is␈α⊂slightly␈α∂reduced␈α⊂if␈α⊂both␈α∂G␈α⊂and␈α⊂H␈α⊂possess␈α∂the
␈↓ α,␈↓¬␈↓ β,property.


␈↓ α,␈↓¬190.␈α
A␈αcomposition␈α
F=G␈↓εo␈↓¬H␈αis␈α
interesting␈αif␈α
F␈αlacks␈α
any␈αundesirable␈α
properties␈αtrue␈α
of␈αG␈α
or␈αH.␈α
 This
␈↓ α,␈↓¬␈↓ β,is␈αgreatly␈α
increased␈αif␈α
both␈αG␈α
and␈αH␈α
possess␈αthe␈α
bad␈αproperty,␈α
unless␈αG␈α
and␈αH␈αare␈α
very
␈↓ α,␈↓¬␈↓ β,closely related to each other (e.g., H=G,or H=G␈↓	-1␈↓¬).

␈↓ α,␈↓The␈α
numeric␈α
impact␈α
of␈α
each␈α
of␈α
these␈α
rules␈α
was␈α
guessed␈α
at␈α
initially,␈α
and␈α
has␈α
never␈αneeded
␈↓ α,␈↓tuning. Here is an area where experimentation might prove interesting.

␈↓ α,␈↓¬191.␈α
A␈αcomposition␈α
F=G␈↓εo␈↓¬H␈αis␈α
interesting␈αif␈α
F␈α
maps␈αinteresting␈α
subsets␈αof␈α
domain(H)␈αinto␈α
interesting
␈↓ α,␈↓¬␈↓ β,subsets of range(G).
␈↓ α,␈↓¬␈↓ αlF␈αis␈αto␈αbe␈αjudged␈αeven␈αmore␈αinteresting␈αif␈αthe␈αimage␈αwas␈αnot␈αthought␈αto␈αbe␈αinteresting␈αuntil
␈↓ α,␈↓¬␈↓ β,after it was explicitly isolated and studied because of part 1 of this very rule.

␈↓ α,␈↓Here,␈α∂an␈α∂"interesting"␈α⊂subset␈α∂of␈α∂domain(H)␈α∂is␈α⊂one␈α∂so␈α∂judged␈α∂by␈α⊂Interests(domain(H)).␈α∂ A
␈↓ α,␈↓completely␈α
di≥erent␈αset␈α
of␈αcriteria␈α
will␈αbe␈α
used␈αto␈α
judge␈αthe␈α
interestingness␈αof␈α
the␈αresultant
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε176␈↓-

␈↓ α,␈↓image␈α∂under␈α∂F.␈α∂Namely,␈α⊂for␈α∂that␈α∂purpose,␈α∂AM␈α⊂will␈α∂ask␈α∂for␈α∂range(G).Interest,␈α⊂and␈α∂ripple
␈↓ α,␈↓outwards to look for related interest features.

␈↓ α,␈↓¬192.␈α∀A␈α∀composition␈α∀F=G␈↓εo␈↓¬H␈α∃is␈α∀interesting␈α∀if␈α∀F␈↓	-1␈↓¬␈α∃maps␈α∀interesting␈α∀subsets␈α∀of␈α∃range(G)␈α∀into
␈↓ α,␈↓¬␈↓ β,interesting subsets of domain(F).
␈↓ α,␈↓¬␈↓ αlThis is even better if the preimage wasn't hitherto realized as interesting.

␈↓ α,␈↓This␈α
is␈α
the␈α∞converse␈α
of␈α
the␈α∞preceding␈α
rule.␈α
Again,␈α
"interesting"␈α∞is␈α
judged␈α
by␈α∞two␈α
di≥erent
␈↓ α,␈↓sets of criteria.

␈↓ α,␈↓¬193.␈α∀A␈α∀composition␈α∀F=G␈↓εo␈↓¬H␈α∪is␈α∀interesting␈α∀if␈α∀F␈α∪maps␈α∀interesting␈α∀elements␈α∀of␈α∀domain(H)␈α∪into
␈↓ α,␈↓¬␈↓ β,interesting subsets of range(G).


␈↓ α,␈↓¬194.␈α∪A␈α∪composition␈α∪F=G␈↓εo␈↓¬H␈α∪is␈α∪interesting␈α∩if␈α∪F␈↓	-1␈↓¬␈α∪maps␈α∪interesting␈α∪elements␈α∪of␈α∪range(G)␈α∩into
␈↓ α,␈↓¬␈↓ β,interesting subsets of domain(F).
␈↓ α,␈↓¬␈↓ αlThis is even better if the subset is only now seen to be interesting.

␈↓ α,␈↓This␈α⊂is␈α⊂the␈α∂analogue␈α⊂of␈α⊂an␈α∂earlier␈α⊂rule,␈α⊂but␈α∂for␈α⊂individual␈α⊂items␈α∂rather␈α⊂than␈α⊂for␈α∂whole
␈↓ α,␈↓subsets of the domain and range of F.

␈↓ α,␈↓¬195.␈αA␈αcomposition␈α
F=G␈↓εo␈↓¬H␈αis␈αinteresting␈αif␈α
range(H)␈αis␈αequal␈αto,␈α
not␈αjust␈αintersects,␈αone␈α
component
␈↓ α,␈↓¬␈↓ β,of domain(G).


␈↓ α,␈↓¬196.␈αA␈αcomposition␈αF=G␈↓εo␈↓¬H␈αis␈αmildly␈αinteresting␈αif␈αrange(H)␈αis␈αa␈αspecialization␈αof␈αone␈αcomponent␈αof
␈↓ α,␈↓¬␈↓ β,domain(G).

␈↓ α,␈↓This␈α⊂is␈α⊂a␈α⊃weakened␈α⊂version␈α⊂of␈α⊃the␈α⊂preceding␈α⊂feature.␈α⊃Such␈α⊂a␈α⊂composition␈α⊃is␈α⊂interesting
␈↓ α,␈↓because␈α∞it␈α∂is␈α∞guaranteed␈α∞to␈α∂always␈α∞be␈α∂applicable.␈α∞If␈α∞Range(H)␈α∂merely␈α∞intersects␈α∂a␈α∞domain
␈↓ α,␈↓component␈αof␈αG,␈αthen␈αthere␈αmust␈αbe␈αan␈αextra␈αcheck,␈αafter␈αcomputing␈αH(x),␈αto␈αensure␈αit␈αlies
␈↓ α,␈↓within the legal domain of G, before trying to run G on that new entity H(x).

␈↓ α,␈↓¬197. A composition F=G␈↓εo␈↓¬H is more interesting if range(G) is equal to a domain component of H.

␈↓ α,␈↓This␈αis␈αover␈αand␈αabove␈αthe␈αslight␈αboost␈αgiven␈αto␈αthe␈αcomposition␈αbecause␈αit␈αis␈αan␈α␈↓βoperation␈↓
␈↓ α,␈↓whose domain and range coincide (see rule 171).

␈↓ α,␈↓␈↓ β$␈↓↓␈↓&Appendix 2.7. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with any Insertions␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Insertion . Check

␈↓ α,␈↓¬198. When checking an example of any kind of insertion of x into S,
␈↓ α,␈↓¬␈↓ αlEnsure that x is a member of S.

␈↓ α,␈↓The␈αonly␈αtypes␈α
of␈αinsertions␈αknown␈αto␈α
AM␈αare␈α␈↓βunconditional␈↓␈αinsertions,␈α
so␈αthis␈αrule␈αis␈α
valid.
␈↓ α,␈↓It␈α
is␈α
useful␈αfor␈α
ensuring␈α
that␈αa␈α
particular␈α
new␈α
operation␈αreally␈α
is␈α
an␈αinsertion-operation␈α
after
␈↓ α,␈↓all!
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε177␈↓-

␈↓ α,␈↓␈↓ α\␈↓↓␈↓&Appendix 2.8. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with the operation Coalesce␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Coalesce . Fillin

␈↓ α,␈↓¬199. When coalescing F(a,b,c,...), whose domain/range is <A B C... → R>,
␈↓ α,␈↓¬␈↓ αlA␈α∂good␈α⊂choice␈α∂of␈α∂two␈α⊂domain␈α∂components␈α∂to␈α⊂coalesce␈α∂is␈α∂a␈α⊂pair␈α∂of␈α∂identically␈α⊂equal␈α∂ones.
␈↓ α,␈↓¬␈↓ β,Barring␈αthat,␈α
choose␈αa␈α
pair␈αrelated␈αby␈α
specialization␈α(eliminate␈α
the␈αmore␈α
general␈αone).
␈↓ α,␈↓¬␈↓ β,Barring that, choose a pair with a common specialization S, and replace both by S.

␈↓ α,␈↓Thus␈α
to␈α
coalesce␈α
the␈α
operation␈α
"Insert␈↓εo␈↓Delete"␈α[which␈α
takes␈α
two␈α
items␈α
and␈α
a␈αstructure,␈α
deletes
␈↓ α,␈↓the␈α≡rst␈αargument␈αfrom␈αthe␈αstructure␈αand␈αthen␈αinserts␈αthe␈αsecond␈αargument],␈αAM␈αexamines
␈↓ α,␈↓its␈αDomain/range␈α
entry:␈α<Anything␈αAnything␈α
Structure␈α→␈αStructure>.␈α
 Although␈αit␈αwould␈α
be
␈↓ α,␈↓legal␈αto␈α
collapse␈αthe␈αsecond␈α
and␈αthird␈αarguments,␈α
the␈αabove␈α
rule␈αsays␈αit␈α
makes␈αmore␈αsense␈α
in
␈↓ α,␈↓general␈αto␈α
collapse␈αthe␈α
≡rst␈αand␈α
second.␈α In␈αfact,␈α
in␈αthat␈α
case,␈αAM␈α
gets␈αan␈α
operation␈αwhich
␈↓ α,␈↓tells it something about multiple elements structures.

␈↓ α,␈↓¬200. When filling in Algorithms for a coalesced version G of active concept F,
␈↓ α,␈↓¬␈↓ αlOne natural algorithm is simply to call on F.Algs, with two arguments the same.

␈↓ α,␈↓Of␈α∞course␈α∞the␈α∞two␈α∞identical␈α∞arguments␈α∞are␈α∞those␈α∞which␈α∞have␈α∞been␈α∞decided␈α∞to␈α∂be␈α∞merged.
␈↓ α,␈↓This␈αwill␈α
be␈αdecided␈α
before␈αthe␈α
de≡nition␈αand␈α
algorithm␈αfacets␈α
are␈α≡lled␈α
in.␈α Thus␈αa␈α
natural
␈↓ α,␈↓algorithm for Square is to call on TIMES.Alg(x,x).  The following rule is similar:

␈↓ α,␈↓¬201. When filling in Definitions for a coalesced version G of active concept F,
␈↓ α,␈↓¬␈↓ αlOne natural Definition is simply to call on F.Defn, with two arguments the same.


␈↓ α,␈↓¬202. When filling in the Worth of a new coalesced version of F,
␈↓ α,␈↓¬␈↓ αlA suitable value is 0.9x(Worth of F) + 0.1x(Worth of Coalesce).

␈↓ α,␈↓This␈α∞is␈α∞a␈α
compromise␈α∞between␈α∞(i)␈α∞the␈α
knowledge␈α∞that␈α∞the␈α
new␈α∞operation␈α∞will␈α∞probably␈α
be
␈↓ α,␈↓less␈αinteresting␈αthan␈α
F,␈αand␈α(ii)␈αthe␈α
knowledge␈αthat␈αit␈αmay␈α
lead␈αto␈αeven␈αmore␈α
valuable␈αnew
␈↓ α,␈↓concepts␈α(e.g.,␈α␈↓βits␈↓␈αinverse␈αmay␈αbe␈αmore␈αinteresting␈αthan␈αF's).␈α The␈αformula␈αalso␈αincorporates
␈↓ α,␈↓a␈αsmall␈α
factor␈αwhich␈α
is␈αbased␈α
on␈αthe␈α
overall␈αvalue␈α
of␈αcoalescings␈α
which␈αAM␈α
has␈αdone␈αso␈α
far
␈↓ α,␈↓in the run.


␈↓ α,␈↓∧␈↓ α<Coalesce . Check

␈↓ α,␈↓¬203. If G and H are each two coalescings away from F, for any F,
␈↓ α,␈↓¬␈↓ αlThen␈αcheck␈αthat␈αG␈αand␈αH␈αaren't␈αreally␈α
the␈αsame,␈αby␈αwriting␈αtheir␈αdefinitions␈αout␈αin␈α
terms␈αof
␈↓ α,␈↓¬␈↓ β,F.Defn.

␈↓ α,␈↓Thus␈αif␈α
R(a,b,c)␈αis␈αreally␈α
F(a,b,a,c),␈αand␈αS(a,b,c)␈α
is␈αreally␈α
F(a,b,c,c),␈αand␈αR␈α
and␈αS␈αget␈α
coalesced
␈↓ α,␈↓again,␈α
into␈α
G(a,b)␈α
whch␈α
is␈αR(a,b,a)␈α
and␈α
into␈α
H(a,b)␈α
which␈α
is␈αS(a,b,a),␈α
then␈α
both␈α
G␈α
and␈αH␈α
are
␈↓ α,␈↓really␈α⊃F(a,b,a,a).␈α⊃ The␈α⊂order␈α⊃of␈α⊃coalescing␈α⊃is␈α⊂unimportant.␈α⊃ This␈α⊃is␈α⊂a␈α⊃boost␈α⊃to␈α⊃the␈α⊂more
␈↓ α,␈↓general␈α
impetus␈α
for␈α
checking␈α
this␈α
sort␈α
of␈α
thing,␈α
rule␈α
110.␈α
 This␈α
rule␈α
is␈α
faster,␈α
containing␈αa
␈↓ α,␈↓special-purpose␈α
program␈α
for␈α
untangling␈α
argument-calls␈αrapidly.␈α
 If␈α
the␈α
concept␈α
of␈αCoalesce␈α
is
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε178␈↓-

␈↓ α,␈↓excised␈α⊃from␈α⊂the␈α⊃system,␈α⊃one␈α⊂can␈α⊃easily␈α⊃imagine␈α⊂it␈α⊃being␈α⊃re-derived␈α⊂by␈α⊃a␈α⊃more␈α⊂general
␈↓ α,␈↓`coincidence'␈αstrategy,␈αbut␈αhow␈αwill␈αthese␈αspeci≡c,␈αhigh-powered,␈αtightly-coded␈αheuristics␈αever
␈↓ α,␈↓get␈αdiscovered␈αand␈αtacked␈αonto␈αthe␈αCoalesce␈αconcept?␈αThis␈αis␈αan␈αinstance␈αof␈αthe␈αmain␈αmeta-
␈↓ α,␈↓level research problem proposed earlier in the book (Chapter 7).


␈↓ α,␈↓∧␈↓ α<Coalesce . Suggest

␈↓ α,␈↓¬204. If a newly-interesting active concept F(x,y) takes a pair of N's as arguments,
␈↓ α,␈↓¬␈↓ αlThen␈α∩create␈α⊃a␈α∩new␈α∩concept,␈α⊃a␈α∩specialization␈α⊃of␈α∩F,␈α∩called␈α⊃F-Itself,␈α∩taking␈α⊃just␈α∩one␈α∩N␈α⊃as
␈↓ α,␈↓¬␈↓ β,argument, defined as F(x,x), with initial worth Worth(F).
␈↓ α,␈↓¬␈↓ αlIf AM has never coalesced F before, this gets a slight bonus value.
␈↓ α,␈↓¬␈↓ αlIf␈α
AM␈α
has␈α
coalesced␈α
F␈α
before,␈α
say␈αinto␈α
S,␈α
then␈α
modify␈α
this␈α
suggestion's␈α
value␈α
according␈αto␈α
the
␈↓ α,␈↓¬␈↓ β,current worth of S.
␈↓ α,␈↓¬␈↓ αlThe lower the system's interest-threshhold is, the more attactive this suggestion becomes.

␈↓ α,␈↓AM␈α∩used␈α∩this␈α∩rule␈α∩to␈α∩coalesce␈α∩many␈α∩active␈α∩concepts.␈α∩ Times(x,x)␈α∩is␈α∩what␈α∩we␈α∩know␈α∩as
␈↓ α,␈↓squaring;␈α∞Equality(x,x)␈α∞turned␈α
out␈α∞to␈α∞be␈α∞the␈α
constant␈α∞predicate␈α∞True;␈α∞Intersect(x,x)␈α
turned
␈↓ α,␈↓out␈α∞to␈α∞be␈α∞the␈α∞identity␈α∞operator;␈α∞Compose(f,f)␈α∞was␈α∞an␈α∞interesting␈α∞"iteration"␈α∞operator␈↓	18␈↓;␈α∞etc.
␈↓ α,␈↓This␈α
rule␈αis␈α
really␈α
a␈αbundle␈α
of␈α
little␈αmeta-rules␈α
modifying␈α
one␈αsuggestion:␈α
the␈αsuggestion␈α
that
␈↓ α,␈↓AM␈α∩coalesce␈α∩F.␈α∩ The␈α∩very␈α∩last␈α∩part␈α∪of␈α∩the␈α∩above␈α∩rule␈α∩indicates␈α∩that␈α∩if␈α∩the␈α∪system␈α∩is
␈↓ α,␈↓desparate,␈α∂this␈α⊂is␈α∂the␈α⊂least␈α∂distasteful␈α⊂way␈α∂to␈α∂"take␈α⊂a␈α∂chance"␈α⊂on␈α∂a␈α⊂high-payo≥␈α∂high-risk
␈↓ α,␈↓course␈αof␈αaction.␈αIt␈αis␈αmore␈αsane␈αthan,␈αe.g.,␈αrandomly␈αcomposing␈αtwo␈αoperations␈αuntil␈αa␈αnice
␈↓ α,␈↓new one is created.

␈↓ α,␈↓¬205. If concept F takes only one argument,
␈↓ α,␈↓¬␈↓ αlThen it is not worthwhile to try to coalesce it.

␈↓ α,␈↓This␈αrule␈αwas␈αof␈αlittle␈αhelp␈αcpu-timewise,␈αsince␈αeven␈αif␈αAM␈αtried␈αto␈αcoalesce␈αsuch␈αan␈αactive
␈↓ α,␈↓concept,it␈α
would␈αfail␈α
almost␈αinstantaneously.␈α
 The␈αrule␈α
did␈αhelp␈α
make␈αAM␈α
appear␈αsmarter,
␈↓ α,␈↓however.

␈↓ α,␈↓␈↓ αV␈↓↓␈↓&Appendix 2.9. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with the operation Canonize␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Canonize . Fillin

␈↓ α,␈↓¬206.␈αIf␈αthe␈αtask␈αis␈αto␈αCanonize␈αpredicates␈αP1␈αand␈αP2␈α(over␈αAxA)␈↓	19␈↓¬,␈αand␈αthe␈αdifference␈αbetween␈αa
␈↓ α,␈↓¬␈↓ β,definition␈αof␈αP1␈αand␈αdefinition␈αof␈αP2␈αis␈αjust␈αthat␈αP2␈αperforms␈αsome␈αextra␈αcheck␈αthat␈αP1
␈↓ α,␈↓¬␈↓ β,doesn't,
␈↓ α,␈↓¬␈↓ αlThen␈α
F␈α
should␈α
convert␈α
any␈α
aεA␈α
into␈αa␈α
member␈α
of␈α
A␈α
which␈α
automatically␈α
satisfies␈α
that␈αextra
␈↓ α,␈↓¬␈↓ β,constraint.

␈↓ α,␈↓Thus␈α
when␈α
P1=Same-length,␈α
P2=Equality,␈α
A=Lists,␈α
the␈α
extra␈α
constraint␈α
that␈α
P2␈α
satis≡es␈α
is
␈↓ α,␈↓just␈α
that␈αit␈α
recurs␈αin␈α
the␈αCAR␈α
direction:␈αthe␈α
CARs␈αof␈α
the␈αtwo␈α
arguments␈αmust␈α
also␈αsatisfy

␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	18␈↓ε␈αe.g.,␈α
Compose(Compose,Compose)␈αis␈α
an␈αoperator␈α
which␈αtakes␈α
3␈αoperations␈α
f,g,h␈αand␈α
forms␈α"f␈α
o␈αg␈α
o␈αh";␈α
i.e.,␈αtheir␈α
joint
␈↓ α,␈↓ε␈↓ βLcomposition.
␈↓ α,␈↓ε␈↓	19␈↓ε That is, find a function F such that P1(x,y) iff P2(F(x),F(y)).
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε179␈↓-

␈↓ α,␈↓P2.␈α P1␈α
doesn't␈αhave␈αsuch␈α
a␈αrequirement.␈αThe␈α
above␈αrule␈αthen␈α
has␈αAM␈αseek␈α
out␈αa␈α
way␈αto
␈↓ α,␈↓guarantee␈α∩that␈α∩the␈α∩CARs␈α∩will␈α∩always␈α∩satisfy␈α∩Equality.␈α∩A␈α∩special␈α∩hand-crafted␈α∩piece␈α⊃of
␈↓ α,␈↓knowledge␈αtells␈αAM␈αthat␈αsince␈α"T=T"␈αis␈αan␈αexample␈αof␈αEquality,␈αone␈αsolution␈αis␈αfor␈αall␈αthe
␈↓ α,␈↓CARs␈α∞to␈α∞be␈α∞the␈α∞atom␈α∞T.␈α∞Then␈α∂the␈α∞operation␈α∞F␈α∞must␈α∞contain␈α∞a␈α∞procedure␈α∂for␈α∞converting
␈↓ α,␈↓each␈αember␈αof␈αa␈αstructure␈αto␈αthe␈αatom␈α"T".␈α Thus␈α(A␈αC␈α{Z␈αA␈αB}␈αQ␈αQ)␈αwould␈αbe␈αconverted
␈↓ α,␈↓to␈α
(T␈αT␈α
T␈α
T␈αT).␈α
 This␈αrule␈α
is␈α
a␈αspecialized,␈α
"compiled"␈αversion␈α
of␈α
the␈αidea␈α
expressed␈αin␈α
rule
␈↓ α,␈↓number 13.

␈↓ α,␈↓¬207. If the task is to Canonize P1 and P2 over AxA, trying to synthesize F, where A=Structure,
␈↓ α,␈↓¬␈↓ αlThen␈α
perhaps␈α
there␈α
is␈α
a␈α
distinguished␈α
type␈α∞of␈α
structure␈α
B␈α
which␈α
the␈α
argument␈α
to␈α∞F␈α
should
␈↓ α,␈↓¬␈↓ β,always be converted into.  In that case, consider P1 and P2 as two predicates over BxB.

␈↓ α,␈↓This␈αspecial-purpose␈αrule␈αis␈αused␈αto␈αguide␈αa␈αseries␈αof␈αexperiments,␈αto␈αdetermine␈αwhether␈αP1
␈↓ α,␈↓is␈α
a≥ected␈αby␈α
adding␈αmultiple␈α
copies␈αof␈α
existing␈α
elements␈αinto␈α
its␈αarguments,␈α
and␈αwhether␈α
its
␈↓ α,␈↓value␈α
is␈α
a≥ected␈α
by␈α∞rearranging␈α
some␈α
of␈α
its␈α
arguments'␈α∞elements.␈α
In␈α
the␈α
case␈α∞of␈α
P1=Same-
␈↓ α,␈↓size,␈αthe␈α
answers␈αare:␈αmultiple␈α
elements␈αdo␈α
make␈αa␈αdi≥erence,␈α
but␈αrearrangement␈αdoesn't.␈α
 So
␈↓ α,␈↓the␈αcanonical␈αtype␈αof␈αstructure␈αfor␈α
F=Size␈αmust␈αbe␈αone␈αwhich␈αis␈αMult-eles-allowed␈α
and␈αalso
␈↓ α,␈↓one␈α∞which␈α∂is␈α∞Unordered.␈α∞ Namely,␈α∂a␈α∞Bag.␈α∂Thus␈α∞F␈α∞is␈α∂modi≡ed␈α∞so␈α∞that␈α∂it␈α∞≡rst␈α∂converts␈α∞its
␈↓ α,␈↓argument␈αto␈α
a␈αBag.␈α
 Then␈αEquality␈α
and␈αSame-size␈αare␈α
viewed␈αas␈α
taking␈αa␈α
pair␈αof␈αBags,␈α
and
␈↓ α,␈↓Size is viewed as taking one Bag and giving back a canonical bag.

␈↓ α,␈↓¬208. After F is created from P1 and P2, as Canonize(P1,P2),
␈↓ α,␈↓¬␈↓ αlan acceptable value for the worth of F is the maximum of the worths of P1 and P2.

␈↓ α,␈↓In␈αthe␈α
actual␈αLisp␈αcode,␈α
an␈αextra␈α
small␈αterm␈αis␈α
added␈αwhich␈αtakes␈α
into␈αaccount␈α
the␈αoverall
␈↓ α,␈↓value of all the Canonizations which AM has recently produced.

␈↓ α,␈↓¬209.␈αIF␈α
the␈αcurrent␈α
task␈αhas␈αjust␈α
created␈αa␈α
canonical␈αspecialization␈αB␈α
of␈αconcept␈α
A,␈αwith␈αrespect␈α
to
␈↓ α,␈↓¬␈↓ β,predicates P1 and P2, [i.e., two members of B satisfy P1 iff they satisfy P2],
␈↓ α,␈↓¬␈↓ αlTHEN add the following entry to the Analogies facet of B:
␈↓ α,␈↓¬␈↓ αl␈↓ ¬≤<A␈↓ ¬lP1␈↓ ε<Operations-on-and-into(A)>
␈↓ α,␈↓¬␈↓ αl␈↓ ¬≤<B␈↓ ¬lP2␈↓ ε<Those operations restricted to B's>

␈↓ α,␈↓This␈αrather␈αincoherent␈αrule␈αsays␈αthat␈αit's␈αworth␈αtaking␈αthe␈αtrouble␈αto␈αstudy␈αthe␈αbehavior␈αof
␈↓ α,␈↓each␈αoperation␈αwhen␈αit␈αis␈αrestricted␈αto␈αworking␈αon␈αstandard␈αor␈α"canonical"␈αitems.␈αMoreover,
␈↓ α,␈↓some␈α∩of␈α⊃the␈α∩old␈α⊃relationships␈α∩may␈α⊃carry␈α∩over␈α∩¬␈α⊃or␈α∩at␈α⊃least␈α∩have␈α⊃analogues␈α∩¬␈α∩in␈α⊃this
␈↓ α,␈↓restricted␈α∞world.␈α∞ When␈α∞numbers␈α∞are␈α∞discovered␈α∞as␈α∞canonical␈α∞bags,␈α∞all␈α∞the␈α∞bag␈α
operations
␈↓ α,␈↓are␈α
restricted␈α
to␈αwork␈α
on␈α
only␈α
canonical␈αbags,␈α
and␈α
they␈α
magically␈αturn␈α
into␈α
what␈α
we␈αknow
␈↓ α,␈↓and␈αlove␈αas␈αnumeric␈αoperations.␈α Many␈αof␈αthe␈αold␈αbag-theoretic␈αrelationships␈αhave␈αnumeric
␈↓ α,␈↓analogues.␈α Thus␈α
we␈αknew␈α
that␈αthe␈α
bag-di≥erence␈αof␈α
x␈αand␈α
x␈αwas␈α
the␈αempty␈α
bag;␈αthis␈αis␈α
still
␈↓ α,␈↓true␈α
for␈α
x␈α
a␈α
canonical␈α
bag,␈α
but␈α
we␈α
would␈α
word␈α
it␈α
as␈α
"x␈α
minus␈α
x␈α
is␈α
zero".␈α
 This␈α
is␈α
because␈α
the
␈↓ α,␈↓restriction␈αof␈α
Bag-di≥erence␈αto␈αcanonical␈α
bags␈α(bags␈αof␈α
T's)␈αis␈αprecisely␈α
the␈αoperation␈αwe␈α
call
␈↓ α,␈↓subtraction.

␈↓ α,␈↓¬210. When Canonize works on P1, P2 (two predicates), and produces an operation, F,
␈↓ α,␈↓¬␈↓ αlBoth␈αpredicates␈αmust␈αshare␈αa␈αcommon␈αDomain,␈αof␈αthe␈αform␈αAxA␈αfor␈αsome␈αconcept␈αA,␈αand␈αthe
␈↓ α,␈↓¬␈↓ β,new operation F can have <A → A> as one of its Domain/range entries.
␈↓ α,␈↓¬␈↓ αlIf␈αa␈αcanonical␈αspecialization␈α(say␈α
`B')␈αof␈αA␈αis␈αdefined,␈α
then␈αthe␈αdomain/range␈αof␈αF␈αcan␈α
actually
␈↓ α,␈↓¬␈↓ β,be␈αtightened␈αto␈α<A␈α→␈αB>,␈αand␈αit␈αis␈αalso␈αworth␈αexplicitly␈αstoring␈αthe␈αredundant␈αentry␈α<B
␈↓ α,␈↓¬␈↓ β,→ B>.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε180␈↓-

␈↓ α,␈↓¬211. In the same situation as the last rule,
␈↓ α,␈↓¬␈↓ αlOne␈αconjecture␈αis␈αthat␈αP1␈αand␈αP2␈αare␈αequal,␈αwhen␈αrestricted␈αto␈αworking␈αon␈αpairs␈αof␈αB's␈α[i.e.,
␈↓ α,␈↓¬␈↓ β,pairs␈αof␈αCanonical␈αA's,␈αA's␈αwhich␈αare␈αin␈αF(A),␈αrange␈αitems␈αfor␈αF,␈αitems␈αx␈αwhich␈αare␈αthe
␈↓ α,␈↓¬␈↓ β,image F(a) of some aεA].

␈↓ α,␈↓After␈αcanonizing␈αEqual␈αand␈αSame-size␈αinto␈αthe␈αnew␈αoperation␈αLength,␈αAM␈αconjectures␈αthat
␈↓ α,␈↓two canonical bags are equal i≥ they have the same size.


␈↓ α,␈↓∧␈↓ α<Canonize . Suggest

␈↓ α,␈↓¬212.␈α⊂When␈α∂Canonize␈α⊂works␈α∂on␈α⊂P1,␈α⊂P2,␈α∂both␈α⊂predicates␈α∂over␈α⊂AxA,␈α∂and␈α⊂produces␈α⊂an␈α∂operation
␈↓ α,␈↓¬␈↓ β,F:A→A,
␈↓ α,␈↓¬␈↓ αlIt␈αis␈αworth␈αdefining␈αand␈αstudying␈αthe␈αimage␈αF(A);␈αi.e.,␈αthe␈αtotality␈αof␈αA's␈αwhich␈αare␈αcanonical,
␈↓ α,␈↓¬␈↓ β,already␈α
in␈α
standard␈α
form.␈α
 When␈α
this␈α
new␈α
concept␈α
Canonical-A␈α
is␈α
defined,␈α
suggest␈αthe
␈↓ α,␈↓¬␈↓ β,task "Fillin Dom/range entries for Canonical-A".

␈↓ α,␈↓Thus␈α∩AM␈α∪studied␈α∩Canonical-Bags,␈α∩which␈α∪turned␈α∩out␈α∩to␈α∪be␈α∩isomorphic␈α∩to␈α∪the␈α∩natural
␈↓ α,␈↓numbers.␈α∞What␈α∞we've␈α∞called␈α∂`Canonical-A'␈α∞in␈α∞this␈α∞rule,␈α∞we've␈α∂referred␈α∞to␈α∞simply␈α∞as␈α∂`B'␈α∞in
␈↓ α,␈↓earlier Canonizing rules.

␈↓ α,␈↓¬213. If P1 is a very interesting predicate over AxA, for some interesting concept A,
␈↓ α,␈↓¬␈↓ αlThen␈αlook␈αover␈αP1.Spec␈αfor␈αsome␈αother␈αpredicate␈αP2␈αwhich␈αis␈αalso␈αover␈αAxA,␈αand␈αwhich␈αhas
␈↓ α,␈↓¬␈↓ β,some␈α∞interesting␈α∞properties␈α∞P1␈α∞lacks.␈α
 For␈α∞each␈α∞such␈α∞predicate␈α∞P2,␈α∞consider␈α
applying
␈↓ α,␈↓¬␈↓ β,Canonize(P1,P2).


␈↓ α,␈↓¬214.␈α∪After␈α∪producing␈α∩F␈α∪as␈α∪Canonize(P1,P2)␈α∩[both␈α∪predicates␈α∪over␈α∩AxA],␈α∪and␈α∪after␈α∩defining
␈↓ α,␈↓¬␈↓ β,Canonical-A,
␈↓ α,␈↓¬␈↓ αlIt␈αis␈αworth␈αrestricting␈αoperations␈αin␈αIn-dom-of(A)␈αto␈αCanonical-A.␈α Some␈αnew␈αproperties␈αmay
␈↓ α,␈↓¬␈↓ β,emerge.

␈↓ α,␈↓Thus␈α∂after␈α∂de≡ning␈α⊂Canonical-Bags,␈α∂AM␈α∂looked␈α∂at␈α⊂Bags.In-dom-of.␈α∂ In␈α∂that␈α∂list␈α⊂was␈α∂the
␈↓ α,␈↓operation␈α
"Bag-union".␈α So␈α
AM␈α
considered␈αthe␈α
restriction␈αof␈α
Bag-union␈α
to␈αCanonical-bags.
␈↓ α,␈↓Instead␈α⊃of␈α⊃Bag-union␈α⊃mapping␈α⊃two␈α⊃bags␈α⊃into␈α⊃a␈α⊃new␈α⊃bag,␈α⊃this␈α⊃new␈α⊃operation␈α∩took␈α⊃two
␈↓ α,␈↓canonical-bags␈αand␈αmapped␈αthem␈α
into␈αa␈αnew␈αbag.␈αAM␈α
later␈αnoticed␈αthat␈αthis␈αnew␈α
bag␈αwas
␈↓ α,␈↓itself always canonical. Thus was born the operation we call "Addition".

␈↓ α,␈↓¬215. After Canonical-A is produced,
␈↓ α,␈↓¬␈↓ αlIt is marginally worth trying to restrict operations in In-range-of(A) to map into Canonical-A.

␈↓ α,␈↓This␈α
gives␈α
an␈α
added␈α
boost␈α
to␈α
picking␈α
Union␈α
to␈α
restrict,␈α
since␈α
it␈α
is␈α
in␈α∞both␈α
Bags.In-dom-of
␈↓ α,␈↓and␈α
Bags.In-ran-of.␈α
 This␈α
rule␈α
is␈α
much␈α
harder␈α
to␈α
implement,␈α
since␈α
it␈α
demands␈α
that␈α
the␈α
range
␈↓ α,␈↓be␈αrestricted.␈α There␈αare␈αjust␈αa␈αfew␈αspecial-purpose␈αtricks␈αAM␈αknows␈αto␈αdo␈αthis.␈α Restricting
␈↓ α,␈↓the␈α∞domain␈α∞is,␈α∞by␈α∞comparison,␈α∞much␈α∞cleaner.␈α∞ AM␈α∞simply␈α∞creates␈α∞a␈α∞new␈α∞concept␈α∂with␈α∞the
␈↓ α,␈↓same␈α∞de≡nition,␈α∞but␈α∞with␈α∞a␈α∞more␈α∞restricted␈α∞domain/range␈α∞facet.␈α∞ For␈α∞restricting␈α∞the␈α∞range,
␈↓ α,␈↓AM␈α
must␈α
insert␈α
into␈α
the␈α
de≡nition␈α
a␈α
check␈α
to␈α
ensure␈α
that␈α
the␈α
≡nal␈α
result␈α
is␈α
inside␈α
Canonical-
␈↓ α,␈↓A, not just in A.  This leads to a very ine≠cent de≡nition.

␈↓ α,␈↓¬216. After Canonical-A is produced,
␈↓ α,␈↓¬␈↓ αlIt is worthwhile to consider filling in examples (especially boundary) of that new concept.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε181␈↓-

␈↓ α,␈↓This is above and beyond the slight push which rule 12 gives that task.

␈↓ α,␈↓␈↓ αD␈↓↓␈↓&Appendix 2.10. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with the operation Substitute␈↓)αβ␈↓↓

␈↓ α,␈↓Note␈α
that␈α
substitution␈α
operations␈α
are␈α
produced␈α
via␈α
the␈α
initial␈α
concepts␈αcalled␈α
Parallel-replace
␈↓ α,␈↓and Parallel-replace2. The following rules are tacked on there.


␈↓ α,␈↓∧␈↓ α<Parallel-replace . Suggest

␈↓ α,␈↓¬217.␈α
If␈α
two␈α
different␈α
variables␈αare␈α
used␈α
to␈α
represent␈α
the␈αsame␈α
entity,␈↓	20␈↓¬␈α
then␈α
substitute␈α
one␈αfor
␈↓ α,␈↓¬␈↓ β,the other.
␈↓ α,␈↓¬␈↓ αlThis␈αis␈αvery␈αimportant␈αif␈αthe␈αtwo␈αoccurrences␈αare␈αwithin␈αthe␈αsame␈αentry␈αon␈αsome␈αfacet␈αof␈αa
␈↓ α,␈↓¬␈↓ β,single concept, and less so otherwise.
␈↓ α,␈↓¬␈↓ αlThe␈αdominant␈αvariable␈αshould␈αbe␈αthe␈αone␈αtyped␈αout␈αpreviously␈αto␈αthe␈αuser;␈αbarring␈αthat,␈αthe
␈↓ α,␈↓¬␈↓ β,older usage; barring that, the one closest to the letter "a" in the alphabet.

␈↓ α,␈↓This␈α⊂heuristic␈α⊂was␈α⊂used␈α⊃less␈α⊂often␈α⊂¬␈α⊂and␈α⊂proved␈α⊃less␈α⊂impressive␈α⊂¬␈α⊂than␈α⊃was␈α⊂originally
␈↓ α,␈↓anticipated␈α
by␈α
the␈α
author.␈α
Since␈α
most␈α
concepts␈α
were␈α
begotten␈α
from␈α
older␈α
ones,␈α
they␈αalways
␈↓ α,␈↓assumed␈αthe␈αsame␈αvariable␈αnamings,␈αhence␈α
there␈αwere␈αvery␈αfew␈αmismatches.␈α A␈α
special␈αtest
␈↓ α,␈↓was␈αneeded␈αto␈α
explicitly␈αcheck␈αfor␈α
"x=y"␈αoccurring␈αas␈αa␈α
conjunct␈αsomewhere,␈αin␈α
which␈αcase
␈↓ α,␈↓we removed it and y substituted for x throughout the conjunction.

␈↓ α,␈↓¬218.␈αIf␈αtwo␈αexpressions␈α(especially:␈αtwo␈αconjectures)␈αare␈αstructurally␈αsimilar,␈αand␈αappear␈αto␈αdiffer
␈↓ α,␈↓¬␈↓ β,by a certain substitution,
␈↓ α,␈↓¬␈↓ αlThen␈α⊃if␈α⊃the␈α⊂substitution␈α⊃is␈α⊃permissable␈α⊃we␈α⊂have␈α⊃just␈α⊃arrived␈α⊂at␈α⊃the␈α⊃same␈α⊃expression␈α⊂in
␈↓ α,␈↓¬␈↓ β,various ways, and tag it as such,
␈↓ α,␈↓¬␈↓ αlBut␈α
if␈αthe␈α
substitution␈αis␈α
not␈αseen␈α
to␈αbe␈α
tautologous,␈αthen␈α
a␈αnew␈α
analogy␈αis␈α
born.␈αAssociate
␈↓ α,␈↓¬␈↓ β,the␈αconstituent␈αparts␈αof␈αboth␈αexpressions.␈α This␈αis␈αmade␈αinteresting␈αif␈αthere␈αare␈αseveral
␈↓ α,␈↓¬␈↓ β,concepts involved which are assigned new analogues.

␈↓ α,␈↓The␈α
similar␈αstatements␈α
of␈α
the␈αassociativity␈α
of␈α
Add␈αand␈α
Times␈α
led␈αto␈α
this␈α
rule's␈αidentifying
␈↓ α,␈↓them␈α⊗as␈α↔analogous.␈α⊗If␈α↔AM␈α⊗had␈α⊗been␈α↔more␈α⊗sophisticated,␈α↔it␈α⊗might␈α↔have␈α⊗eventually
␈↓ α,␈↓formulated some abstract algebra concepts like "semigroup" from such analogies.

␈↓ α,␈↓␈↓ αZ␈↓↓␈↓&Appendix 2.11. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with the operation Restrict␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Restrict . Fillin

␈↓ α,␈↓¬219. When filling in definitions (algorithms) for a restriction R of the active concept F,
␈↓ α,␈↓¬␈↓ αlOne entry can simply be a call on F.Defn (F.Algs).

␈↓ α,␈↓Thus␈αone␈αde≡nition␈αof␈αAddition␈αwill␈αalways␈αbe␈αas␈αa␈αcall␈αon␈αthe␈αold,␈αgeneral␈αoperation␈α`Bag-
␈↓ α,␈↓union.'␈αOf␈αcourse␈αone␈αmajor␈αreason␈αfor␈αrestricting␈αthe␈αdomain/range␈αof␈αan␈αactivity␈αis␈αthat␈αit
␈↓ α,␈↓can␈α
be␈α∞performed␈α
using␈α
a␈α∞faster␈α
algorithm!␈α
 So␈α∞the␈α
above␈α
rule␈α∞was␈α
used␈α
frequently␈α∞if␈α
not
␈↓ α,␈↓dramatically.
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	20␈↓ε␈α	When␈αλwe␈α	say␈αλthat␈α	x␈αλand␈α	y␈αλrepresent␈α	the␈αλsame␈α	entity,␈αλwhat␈α	we␈αλreally␈α	mean␈αλis␈α	that␈αλthey␈α	have␈αλthe␈α	same␈αλdomain␈α	of␈αλidentity
␈↓ α,␈↓ε␈↓ βL(e.g.,␈α
∀xεBags)␈α	and␈α
they␈α	are␈α
equally␈α
free␈α	(there␈α
is␈α	a␈α
precise␈α	logical␈α
definition␈α
of␈α	all␈α
this,␈α	but␈α
there␈α
is␈α	little
␈↓ α,␈↓ε␈↓ βLpoint to presenting it here.)
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε182␈↓-

␈↓ α,␈↓¬220. When creating a restriction R of the active concept F,
␈↓ α,␈↓¬␈↓ αlNote that R.Genl should contain F, and that F.Spec should contain R.


␈↓ α,␈↓¬221.␈αWhen␈αcreating␈αin␈αa␈αrestriction␈αR␈αof␈αthe␈αactive␈αconcept␈αF,␈αby␈αrestricting␈αthe␈αdomain␈αor␈αrange
␈↓ α,␈↓¬␈↓ β,to some specialization S of its previous value C,
␈↓ α,␈↓¬␈↓ αlA␈αviable␈αinitial␈αguess␈αfor␈αR.Worth␈αis␈αF.Worth,␈αaugmented␈αby␈αthe␈αdifference␈αin␈αworth␈αbetween
␈↓ α,␈↓¬␈↓ β,S and C.  Hopefully, S.Worth is bigger than C.Worth!


␈↓ α,␈↓␈↓ αf␈↓↓␈↓&Appendix 2.12. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with the operation Invert␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Invert . Fillin

␈↓ α,␈↓¬222. When filling in definitions for an Inverse F␈↓	-1␈↓¬ of the active concept F,
␈↓ α,␈↓¬␈↓ αlOne "Sufficent Defn" entry can simply be a blind search through the examples of F.

␈↓ α,␈↓If␈αwe␈αalready␈αknew␈αthat␈α4→16␈αis␈αan␈αexample␈αof␈αSquare,␈αthen␈αAM␈αcan␈αuse␈αthe␈αabove␈αrule␈αto
␈↓ α,␈↓quickly␈α∀notice␈α∀that␈α∀Square-Inverse.Defn(16,4)␈α∀is␈α∀true.␈α∀ This␈α∀is␈α∀almost␈α∀the␈α∀"essence"␈α∀of
␈↓ α,␈↓inverting an operation, of course.


␈↓ α,␈↓∧␈↓ α<Invert . Suggest

␈↓ α,␈↓¬223. After creating an inverted form F␈↓	-1␈↓¬ of some operation F,
␈↓ α,␈↓¬␈↓ αlIf the only definition and algorithm entries are the "obvious" inefficient ones,
␈↓ α,␈↓¬␈↓ αlThen␈αconsider␈αthe␈αtask:␈α"Fill␈αin␈αalgorithms␈α
for␈αF␈↓	-1␈↓¬",␈αbecause␈αthe␈αold␈αblind␈αsearch␈αis␈α
just␈αtoo
␈↓ α,␈↓¬␈↓ β,awful to tolerate.


␈↓ α,␈↓␈↓ αe␈↓↓␈↓&Appendix 2.13. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Logical combinations␈↓)αβ␈↓↓

␈↓ α,␈↓Eventually,␈α⊂there␈α⊂may␈α⊂be␈α⊂separate␈α⊂concepts␈α⊂for␈α⊂each␈α⊂kind␈α⊂of␈α⊂logical␈α⊂connective.␈α⊃For␈α⊂the
␈↓ α,␈↓moment,␈αall␈αBoolean␈αoperators␈αare␈αlumped␈αtogether␈αhere.␈α Their␈αde≡nition␈αis␈αtoo␈αtrivial␈αfor
␈↓ α,␈↓a `Fillin' heuristic to be useful, and even `Check' heuristics are almost pointless.


␈↓ α,␈↓∧␈↓ α<Logical-combine . Check

␈↓ α,␈↓¬224. The user may sometimes indicate `Conjunction' when he really means `Repeating'.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε183␈↓-

␈↓ α,␈↓∧␈↓ α<Logical-combine . Suggest

␈↓ α,␈↓¬225.␈αIf␈αthere␈αis␈αsomething␈αinteresting␈αto␈αsay␈αabout␈αentities␈αsatisfying␈αthe␈αdisjunction␈α(conjunction)
␈↓ α,␈↓¬␈↓ β,of two concepts' definitions,
␈↓ α,␈↓¬␈↓ αlThen␈α
consider␈α
creating␈α
a␈α
new␈α
concept␈αdefined␈α
as␈α
that␈α
logical␈α
combination␈α
of␈α
the␈αtwo␈α
concepts'
␈↓ α,␈↓¬␈↓ β,definitions.


␈↓ α,␈↓¬226. Given an implication,
␈↓ α,␈↓¬␈↓ αlTry␈αto␈αweaken␈αthe␈αleft␈αside␈αas␈αmuch␈αas␈αpossible␈αwithout␈αdestroying␈αthe␈αvalidity␈αof␈αthe␈αwhole
␈↓ α,␈↓¬␈↓ β,implication.  Similarly, try to strengthen the right side of the implication.



␈↓ α,␈↓∧␈↓ α<Logical-combine . Interest

␈↓ α,␈↓¬227.␈α
A␈α
disjunction␈α
(conjunction)␈α
is␈α
interesting␈α
if␈α
there␈α
is␈α
a␈α
conjecture␈α
which␈α
is␈α
very␈αinteresting␈α
yet
␈↓ α,␈↓¬␈↓ β,which cannot be made about any one disjunct (conjunct).

␈↓ α,␈↓In other words, their logical combination implies more than any consituent.

␈↓ α,␈↓¬228. An implication is interesting if the right side is more interesting than the left side.


␈↓ α,␈↓¬229.␈α∞An␈α∞implication␈α∞is␈α∞interesting␈α∞if␈α∞the␈α∞right␈α∞side␈α∞is␈α∞interesting␈α∞yet␈α∞unexpected␈α∞based␈α∂only␈α∞on
␈↓ α,␈↓¬␈↓ β,assuming the left side.


␈↓ α,␈↓␈↓ β?␈↓↓␈↓&Appendix 2.14. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Structures␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Structure . Fillin

␈↓ α,␈↓¬230. To fill in examples of a kind of structure S,
␈↓ α,␈↓¬␈↓ αlStart␈α∪with␈α∪an␈α∪empty␈α∪S,␈α∪pluck␈α∪any␈α∪other␈α∪member␈α∪of␈α∪Examples(Structure),␈α∀and␈α∪transfer
␈↓ α,␈↓¬␈↓ β,members␈αone␈αat␈αa␈αtime␈αfrom␈αthe␈αrandom␈αstructure␈αinto␈αthe␈αembryonic␈αS.␈α Finally,␈αcheck
␈↓ α,␈↓¬␈↓ β,that the resultant S really satisfies S.Defn.

␈↓ α,␈↓This is useful, e.g., to convert examples of lists into examples of sets.

␈↓ α,␈↓¬231. To fill in specializations of a kind of structure,
␈↓ α,␈↓¬␈↓ αladd␈αa␈αnew␈αconstraint␈αthat␈αeach␈αmember␈αmust␈α
satisfy,␈αor␈αa␈αconstraint␈αon␈αall␈αpairs␈αof␈α
members,
␈↓ α,␈↓¬␈↓ β,or␈αa␈αconstraint␈αon␈αall␈α
pairs␈αof␈αdistinct␈αmembers,␈αor␈α
a␈αconstraint␈αwhich␈αthe␈αstructure␈αas␈α
a
␈↓ α,␈↓¬␈↓ β,whole␈αmust␈αsatisfy.␈α Such␈αa␈αconstraint␈αis␈αoften␈αmerely␈αa␈αstipulation␈αof␈αbeing␈αan␈αexample
␈↓ α,␈↓¬␈↓ β,of an X, for some interesting concept X.

␈↓ α,␈↓Thus␈αAM␈α
might␈αspecialize␈α
Bags␈αinto␈α
Bags-of-primes,␈αor␈α
into␈αBags-of-distinct-primes,␈αor␈α
into
␈↓ α,␈↓Bags-containing-a-prime.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε184␈↓-

␈↓ α,␈↓∧␈↓ α<Structure . Interest

␈↓ α,␈↓¬232.␈α
Structure␈α
S␈αis␈α
mildly␈α
interesting␈αif␈α
all␈α
members␈αof␈α
S␈α
satisfy␈αthe␈α
interesting␈α
predicate␈α
P,␈αor
␈↓ α,␈↓¬␈↓ β,(equivalently)␈α⊂if␈α⊂they␈α⊂are␈α⊂all␈α⊂accidentally␈α⊂examples␈α⊂of␈α⊂the␈α⊂interesting␈α⊂concept␈α⊃C,␈α⊂or
␈↓ α,␈↓¬␈↓ β,(similarly) if all pairs of members of S satisfy the interesting binary predicate P, etc.
␈↓ α,␈↓¬␈↓ αlAlso:␈α
a␈α
KIND␈α
of␈α
structure␈α
is␈α
interesting␈α
if␈αit␈α
appears␈α
that␈α
each␈α
instance␈α
of␈α
such␈α
a␈αstructure
␈↓ α,␈↓¬␈↓ β,satisfies the above condition (for a fixed P or C).

␈↓ α,␈↓Thus␈α∂a␈α∂singleton␈α∂is␈α∂interesting␈α∂because␈α∞all␈α∂pairs␈α∂of␈α∂members␈α∂satisfy␈α∂Equal.␈α∂ The␈α∞concept
␈↓ α,␈↓"Singletons"␈αis␈αinteresting␈αbecause␈αeach␈αsingleton␈αis␈αmildly␈αinteresting␈αin␈αjust␈αthat␈αsame␈αway.
␈↓ α,␈↓Similarly,␈αAM␈αde≡nes␈αthe␈αconcept␈αof␈α
a␈αbag␈αcontaining␈αonly␈αprimes,␈αbecause␈αthe␈α
above␈αrule
␈↓ α,␈↓says it might be interesting.

␈↓ α,␈↓¬233.␈α
A␈αstructure␈α
is␈α
mildly␈αinteresting␈α
if␈α
one␈αmember␈α
is␈α
very␈αinteresting.␈α
 Even␈α
better:␈αexactly␈α
one
␈↓ α,␈↓¬␈↓ β,member.
␈↓ α,␈↓¬␈↓ αlAlso:␈α
a␈α
KIND␈αof␈α
structure␈α
is␈αinteresting␈α
if␈α
each␈αinstance␈α
satisfies␈α
the␈αabove␈α
condition␈α
in␈αthe
␈↓ α,␈↓¬␈↓ β,same way.

␈↓ α,␈↓Thus␈α∞the␈α∞values␈α∂of␈α∞ADD␈↓	-1␈↓␈α∞are␈α∂interesting␈α∞because␈α∞they␈α∂always␈α∞contain␈α∞precisely␈α∂one␈α∞bag
␈↓ α,␈↓which is a Singleton.

␈↓ α,␈↓␈↓ αv␈↓↓␈↓&Appendix 2.15. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Ordered-structures␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Ordered-struc . Fillin

␈↓ α,␈↓¬234. To fill in some new examples of the ordered structure S, when some already exist,
␈↓ α,␈↓¬␈↓ αlPick an existing example and randomly permute its members.


␈↓ α,␈↓¬235. To fill in specializations of a kind of ordered structure,
␈↓ α,␈↓¬␈↓ αladd␈αa␈αnew␈αconstraint␈αthat␈αeach␈αpair␈αof␈αadjacent␈αmembers␈αmust␈αsatisfy,␈αor␈αa␈αconstraint␈αon␈αall
␈↓ α,␈↓¬␈↓ β,ordered␈αpairs␈αof␈αmembers␈αin␈αthe␈αorder␈αthey␈αappear␈αin␈αthe␈αstructure.␈α Such␈αa␈αconstraint
␈↓ α,␈↓¬␈↓ β,is␈αoften␈αmerely␈αa␈αstipulation␈αof␈αbeing␈αan␈αexample␈αof␈αan␈αX,␈αfor␈αsome␈αinteresting␈αconcept
␈↓ α,␈↓¬␈↓ β,X.

␈↓ α,␈↓Thus␈αLists-of-numbers␈αmight␈αbe␈αspecialized␈αinto␈αSorted-lists-of-numbers,␈αassuming␈αAM␈αhas
␈↓ α,␈↓discovered␈α∞"≤"␈α∞and␈α∞assuming␈α∞it␈α∞is␈α
chosen␈α∞as␈α∞the␈α∞`constraint'␈α∞relationship␈α∞between␈α
adjacent
␈↓ α,␈↓members of the list.


␈↓ α,␈↓∧␈↓ α<Ordered-struc . Check

␈↓ α,␈↓¬236.␈αIf␈αthe␈αstructure␈αis␈αto␈αbe␈αaccessed␈αsequentially␈αuntil␈αsome␈αcondition␈αis␈αmet,␈αand␈αif␈αthe␈αprecise
␈↓ α,␈↓¬␈↓ β,ordering is superfluous,
␈↓ α,␈↓¬␈↓ αlThen keep the structure ordered by frequency of use, the most useful element first.

␈↓ α,␈↓This␈α∂is␈α⊂a␈α∂simple␈α⊂data-structure␈α∂management␈α⊂trick.␈α∂If␈α⊂you␈α∂have␈α⊂several␈α∂rules␈α⊂to␈α∂use␈α⊂in␈α∂a
␈↓ α,␈↓certain␈α∞situation,␈α
and␈α∞rule␈α∞R␈α
is␈α∞one␈α∞which␈α
usually␈α∞succeeds,␈α∞then␈α
put␈α∞R␈α∞≡rst␈α
in␈α∞the␈α∞list␈α
of
␈↓ α,␈↓rules␈α⊃to␈α⊃try.␈α⊃Similarly,␈α⊃in␈α⊃a␈α⊃pattern-matcher,␈α⊃try␈α⊃≡rst␈α⊃the␈α⊃test␈α⊃most␈α⊃likely␈α⊃to␈α⊃detect␈α⊃non-
␈↓ α,␈↓matching arguments.
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε185␈↓-

␈↓ α,␈↓¬237. If structure S is always to be maintained in alphanumeric order,
␈↓ α,␈↓¬␈↓ αlThen AM can␈↓	21␈↓¬ actually maintain it as an unordered structure, if desired.

␈↓ α,␈↓Luckily this heavily implementation-dependent rule was never needed by AM.


␈↓ α,␈↓∧␈↓ α<Ordered-struc . Interest

␈↓ α,␈↓¬238.␈α⊃An␈α∩ordered␈α⊃structure␈α∩S␈α⊃is␈α⊃interesting␈α∩if␈α⊃each␈α∩adjacent␈α⊃pair␈α⊃of␈α∩members␈α⊃of␈α∩S␈α⊃satisfies
␈↓ α,␈↓¬␈↓ β,predicate P (for some rare, interesting P).

␈↓ α,␈↓When␈α
AM␈α
discovers␈α
the␈α
relation␈α
"≤",␈α
it␈α
immediately␈α
thinks␈α
that␈α
any␈α
␈↓βsorted␈↓␈α
list␈α
of␈αnumbers␈α
is
␈↓ α,␈↓more interesting, due to the above rule.

␈↓ α,␈↓␈↓ αa␈↓↓␈↓&Appendix 2.16. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Unordered-structures␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Unord-struc . Check

␈↓ α,␈↓¬239. To check an example of an unordered structure,
␈↓ α,␈↓¬␈↓ αlEnsure that it is in alphanumerically-sorted order. If not, then SORT it.

␈↓ α,␈↓All␈α∞unordered␈α∞objects␈α∞are␈α∞maintained␈α∞in␈α∞lexicographic␈α∞order,␈α∞so␈α∞that␈α∞two␈α∞of␈α∞them␈α∞can␈α
be
␈↓ α,␈↓tested␈αfor␈α
equality␈αusing␈αthe␈α
LISP␈αfunction␈α
EQUAL.␈α Because␈αof␈α
this␈αconvention,␈α
any␈αtwo
␈↓ α,␈↓structures can therefore be tested for equality using this fast list-structure comparator.

␈↓ α,␈↓␈↓ αJ␈↓↓␈↓&Appendix 2.17. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Multiple-eles-structures␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Mult-ele-struc . Fillin

␈↓ α,␈↓¬240.␈α
To␈α
fill␈α
in␈α
some␈αnew␈α
examples␈α
of␈α
the␈α
structure␈α
S,␈αwhere␈α
S␈α
is␈α
a␈α
structure␈α
admitting␈αmultiple
␈↓ α,␈↓¬␈↓ β,occurrences of the same element, when some examples already exist,
␈↓ α,␈↓¬␈↓ αlPick␈α
an␈α
existing␈α∞example␈α
and␈α
randomly␈α
change␈α∞the␈α
multiplicity␈α
with␈α
which␈α∞various␈α
members
␈↓ α,␈↓¬␈↓ β,occur within the structure.


␈↓ α,␈↓␈↓ βx␈↓↓␈↓&Appendix 2.18. ␈↓)αβ␈↓∧␈↓& Heuristics for dealing with Sets␈↓)αβ␈↓↓


␈↓ α,␈↓∧␈↓ α<Sets . Suggest

␈↓ α,␈↓¬241. If P is a very interesting predicate over X,
␈↓ α,␈↓¬␈↓ αlThen␈α
study␈αthese␈α
two␈αsets:␈α
the␈αmembers␈α
of␈αX␈α
for␈αwhich␈α
P␈αholds,␈α
and␈αthe␈α
ones␈αfor␈α
which␈αP
␈↓ α,␈↓¬␈↓ β,fails.


␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	21␈↓ε due to the current LISP implementation of data-structures
␈↓ α,␈↓␈↓εAppendix 2␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε186␈↓-

␈↓ α,␈↓While␈αwe␈αhumans␈α
know␈αthat␈αthis␈αpartitions␈α
X␈αinto␈αtwo␈αpieces,␈α
AM␈αis␈αnever␈α
explicitly␈αtold
␈↓ α,␈↓this.␈αIt␈αwould␈αoccasionally␈αbe␈αsurprised␈αto␈αdiscover␈αthat␈αthe␈αunion␈αof␈αtwo␈αsuch␈αcomplements
␈↓ α,␈↓"accidentally"␈α⊂coincided␈α⊂with␈α∂X.␈α⊂ Incidentally,␈α⊂this␈α∂rule␈α⊂is␈α⊂really␈α∂the␈α⊂key␈α⊂linkage␈α∂between
␈↓ α,␈↓predicates␈α
and␈α
sets;␈α
it␈αis␈α
close␈α
to␈α
the␈αentry␈α
on␈α
Set.View␈α
which␈αtells␈α
how␈α
to␈α
view␈αany␈α
predicate
␈↓ α,␈↓as a set.


␈↓ α,␈↓∧␈↓ α<Sets . Interest

␈↓ α,␈↓¬242. A set S is interesting if, for some interesting predicate P, whose domain is X,
␈↓ α,␈↓¬␈↓ αlS␈α∞accidentally␈α∞appears␈α∞to␈α∞be␈α∂related␈α∞strongly␈α∞to␈α∞{xεX␈α∞|␈α∂P(x)},␈α∞i.e.,␈α∞to␈α∞those␈α∞members␈α∂of␈α∞X
␈↓ α,␈↓¬␈↓ β,satisfying P.

␈↓ α,␈↓To the surprise of the author, this rule never found application in any of AM's runs.
␈↓ α,␈↓␈↓ ≥-␈↓ε187␈↓-

␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ∧6␈↓∧Appendix 3.  Traces of AM in Action␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓


␈↓ α,␈↓Here␈α∂is␈α∂the␈α∂way␈α⊂that␈α∂the␈α∂AM␈α∂program␈α∂begins.␈α⊂The␈α∂human␈α∂user's␈α∂typing␈α∂will␈α⊂appear␈α∂in
␈↓ α,␈↓italics.

␈↓ α,␈↓The␈αcareful␈αreader␈αwill␈αnotice␈αseveral␈αsmall␈αanomalies␈αin␈αthis␈αtranscript:␈αFor␈αone␈αthing,␈αthe
␈↓ α,␈↓task␈α∞numbering␈α∂here␈α∞is␈α∞not␈α∂precisely␈α∞the␈α∞same␈α∂as␈α∞in␈α∞the␈α∂rest␈α∞of␈α∞this␈α∂document.␈α∞A␈α∂task␈α∞is
␈↓ α,␈↓called␈αa␈α"Cand",␈αand␈αthe␈αagenda␈αis␈αcalled␈α"CANDS".␈α Only␈αsome␈αof␈αthe␈αreasons␈αare␈αprinted
␈↓ α,␈↓out,␈α
and␈αthey␈α
are␈αnot␈α
as␈α
"chatty"␈αas␈α
the␈αreasons␈α
in.␈α
e.g.,␈αChapter␈α
2's␈αexample␈α
trace.␈α The␈α
user
␈↓ α,␈↓has␈α∞asked␈α∞AM␈α∞to␈α∂type␈α∞out␈α∞the␈α∞top␈α∂three␈α∞tasks␈α∞on␈α∞the␈α∞agenda␈α∂at␈α∞each␈α∞"cycle".␈α∞In␈α∂a␈α∞better
␈↓ α,␈↓hardware␈α
environment,␈α∞the␈α
user␈α
could␈α∞dynamically␈α
watch␈α
the␈α∞top␈α
hundred␈α∞tasks␈α
bubbling
␈↓ α,␈↓around␈αon␈αone␈αside␈αof␈αa␈αCRT␈αscreen.␈α To␈αinterrupt␈αAM,␈αthe␈αuser␈αtypes␈αCONTROL-I.␈α At
␈↓ α,␈↓that moment he has a very limited syntax of questions he may ask. See (α) below (page 188).

␈↓ α,␈↓An␈α∞approximate␈α∂level␈α∞of␈α∞familiarity␈α∂of␈α∞the␈α∞user␈α∂with␈α∞the␈α∞AM␈α∂program␈α∞is␈α∂maintained␈α∞by
␈↓ α,␈↓AM,␈α∞as␈α∞a␈α∞numeric␈α∞variable.␈α∞Initially,␈α∞its␈α∂value␈α∞is␈α∞determined␈α∞by␈α∞the␈α∞number␈α∞of␈α∂times␈α∞the
␈↓ α,␈↓human␈α∞user␈α∞has␈α∞used␈α∞AM␈α∞in␈α∞the␈α∞past.␈↓	1␈↓␈α∞It␈α∞gradually␈α∞changes␈α∞in␈α∞value␈α∞as␈α∞a␈α∂single␈α∞session
␈↓ α,␈↓proceeds.␈αMany␈αprint␈αstatements␈αuse␈αthis␈αvariable␈αto␈αdetermine␈αthe␈αnecessary␈αlevel␈αof␈αdetail
␈↓ α,␈↓to␈α∂type.␈α∂ The␈α∂level␈α∂of␈α∂detail␈α∂needed␈α∂for␈α∞clarity␈α∂should␈α∂not␈α∂be␈α∂confused␈α∂with␈α∂the␈α∂level␈α∞of
␈↓ α,␈↓verbosity␈αof␈αoutput␈αthat␈αis␈α
desired.␈α Should␈αthe␈αuser␈αsee␈α
every␈αfunction␈αcall,␈αor␈αthe␈αresults␈α
of
␈↓ α,␈↓each␈αtask,␈αor␈αjust␈αmonitor␈αthe␈αoverall␈αcharacter␈αof␈αwhat␈αAM␈αis␈αdoing?␈α A␈αseparate␈αvariable
␈↓ α,␈↓is␈α→maintained␈α~for␈α→this␈α→verbosity␈α~indication.␈α→ Finally,␈α→another␈α~partially␈α→independent
␈↓ α,␈↓dimension␈αis␈αthe␈αamount␈αof␈αcontrol␈αthe␈αuser␈αwishes␈αto␈αhave␈αon␈αAM.␈αMust␈αhe␈αapprove␈αeach
␈↓ α,␈↓move␈α⊂AM␈α⊂makes,␈α⊂occasionally␈α⊂redirect␈α⊂AM,␈α⊂be␈α⊂able␈α⊂to␈α⊂ask␈α⊂questions?␈α⊃Another␈α⊂variable
␈↓ α,␈↓indicates the level of user control in e≥ect at the moment.



␈↓ α,␈↓αEntering AM's main loop now.


␈↓ α,␈↓α No Cand on CANDS is good enuf.
␈↓ α,␈↓α Do-thresh reduced from 500 to 333
␈↓ α,␈↓α Must find new candidates and merge them into CANDS.


␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Set-struc-intersect
␈↓ α,␈↓α    2: Fill in some examples of Set-struc-join
␈↓ α,␈↓α    3: Fill in some examples of Coalesce

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason for considering this Cand is: (We have no examples
␈↓ α,␈↓αfor SET-STRUC-INTERSECT yet)
␈↓ α,␈↓________________________________________________________________________________
␈↓ α,␈↓ε␈↓	1␈↓ε I shall resist the temptation to call this a simple "user model", even in a footnote.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε188␈↓-

␈↓ α,␈↓α      Beginning 1st cycle.

␈↓ α,␈↓αFailed.  Tried to fill in new examples of SET-STRUC-INTERSECT.

␈↓ α,␈↓α␈↓¬<At this moment, the user hit control-I and interrupted AM.>␈↓α
␈↓ α,␈↓α?:  (W, I, E, M, N, ?, Q) ␈↓β?␈↓α ␈↓ ¬␈↓π←αα␈↓α␈↓↓(α)␈↓α

␈↓ α,␈↓α Here are more detailed explanations of your options:
␈↓ α,␈↓α W       Why: AM gives Ed the explanation behind its last printed
␈↓ α,␈↓α             message.
␈↓ α,␈↓α I       Interest: Ed can modify the interest ratings of concepts and
␈↓ α,␈↓α             Candidates.
␈↓ α,␈↓α E       Evaluate: Ed types in an expression and AM runs EVAL on it.
␈↓ α,␈↓α M       Message: What was the last message that AM did NOT type out
␈↓ α,␈↓α             because the verbosity was too low?
␈↓ α,␈↓α N       Name: Rename some concept to whatever you want to call it.
␈↓ α,␈↓α Q       Quit: resume execution.

␈↓ α,␈↓α In general,  AM will automatically resume execution after answering one
␈↓ α,␈↓α query. You must hit ↑I again to interrupt.


␈↓ α,␈↓α?: ␈↓βW␈↓α
␈↓ α,␈↓α       Why: (No examples of SET-STRUC-INTERSECT were found; there
␈↓ α,␈↓αis no reason to even consider specializing it further)


␈↓ α,␈↓α This Cand used 11.159 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Set-struc
␈↓ α,␈↓α    2: Fill in some examples of Coalesce
␈↓ α,␈↓α    3: Fill in some examples of Nonempty-struc

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The 2 reasons for considering this Cand are:
␈↓ α,␈↓α      (Active-exs specifically asked for some examples of SET-STRUC
␈↓ α,␈↓α,  while trying to Fill in some Set-struc-intersect examples)
␈↓ α,␈↓α      (We have no examples for SET-STRUC yet)


␈↓ α,␈↓α      Beginning 2nd cycle.


␈↓ α,␈↓α Creating new Being,  similar to SET-STRUC,  named INT-SET-STRUC,  but
␈↓ α,␈↓αrestricted so as to make it more interesting.
␈↓ α,␈↓α       An INT-SET-STRUC is any SET-STRUC for which (Each pair of
␈↓ α,␈↓αelements satisfies the same interesting predicate P (for some P)).

␈↓ α,␈↓α Filled in examples of SET-STRUC.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε189␈↓-

␈↓ α,␈↓α       0 examples existed originally on SET-STRUC.
␈↓ α,␈↓α       11 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      (CLASS)
␈↓ α,␈↓α      (CLASS DOUG CORDELL BRUCE)
␈↓ α,␈↓α      (CLASS R0-7 R1-7 R2-7 R3-7 R4-7 R5-7 R6-7 R7-7)
␈↓ α,␈↓α      (CLASS A)
␈↓ α,␈↓α      (CLASS B)
␈↓ α,␈↓α      (CLASS A B)
␈↓ α,␈↓α      (CLASS 0 D F I M)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 7 new,  distinct examples of SET-STRUC had to be added.


␈↓ α,␈↓α Do-thresh raised from 332 to 346 because this last Cand succeeded,  so
␈↓ α,␈↓αwe raise our hopes-- and our standards-- temporarily.␈↓ π␈↓π←αα␈↓α␈↓↓(β)␈↓α

␈↓ α,␈↓α This Cand used 23.743 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Int-set-struc
␈↓ α,␈↓α    2: Fill in some examples of Coalesce
␈↓ α,␈↓α    3: Check all examples of Set-struc

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason for considering this Cand is: (Any example of
␈↓ α,␈↓αINT-SET-STRUC is automatically an interesting example of SET-STRUC)


␈↓ α,␈↓α      Beginning 3rd cycle.

␈↓ α,␈↓α Won't try to create a restricted interesting version of INT-SET-STRUC.

␈↓ α,␈↓α Filled in examples of INT-SET-STRUC.
␈↓ α,␈↓α       0 examples existed originally on INT-SET-STRUC.
␈↓ α,␈↓α       13 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      (CLASS)
␈↓ α,␈↓α      (CLASS A)
␈↓ α,␈↓α      (CLASS B)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 3 new,  distinct examples of INT-SET-STRUC had to be added.


␈↓ α,␈↓α Do-thresh raised from 346 to 358.␈↓ ∂␈↓π←αα␈↓α␈↓↓(ε)␈↓α

␈↓ α,␈↓α This Cand used 11.881 cpu seconds.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε190␈↓-

␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Obj-equal
␈↓ α,␈↓α    2: Check all examples of Int-set-struc
␈↓ α,␈↓α    3: Check all examples of Set-struc

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason for considering this Cand is: (We have no examples
␈↓ α,␈↓αfor OBJ-EQUAL yet)


␈↓ α,␈↓α      Beginning 4th cycle.

␈↓ α,␈↓α Record of attempts to find examples:-----------------------------------
␈↓ α,␈↓α An ex ( sought) is: ((CLASS A),(CLASS A) → T) +------------------
␈↓ α,␈↓α----+---------------------------------------+---------------------------
␈↓ α,␈↓α--------+----------+------+---

␈↓ α,␈↓α Found 6 examples ( and 151 non-exs),  in 11.644 secs.
␈↓ α,␈↓α Ratio of exs to non-exs is too low ( 6 / 151); Exs are too sparse.
␈↓ α,␈↓α       AM will sometime try to generalize OBJ-EQUAL.
␈↓ α,␈↓α Won't try to create a restricted interesting version of OBJ-EQUAL.

␈↓ α,␈↓α Filled in examples of OBJ-EQUAL.
␈↓ α,␈↓α       0 examples existed originally on OBJ-EQUAL.
␈↓ α,␈↓α       6 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      ((CLASS A)  (CLASS A)   →  T)
␈↓ α,␈↓α      ((CLASS O D F I M)  (CLASS O D F I M)   →  T)
␈↓ α,␈↓α      (FALSE    FALSE   →  T)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 3 new,  distinct examples of OBJ-EQUAL had to be added.


␈↓ α,␈↓α Do-thresh raised from 358 to 359.

␈↓ α,␈↓α This Cand used 17.886 cpu seconds.


␈↓ α,␈↓α No Cand on CANDS is good enuf.
␈↓ α,␈↓α Do-thresh reduced from 359 to 239
␈↓ α,␈↓α Must find new candidates and merge them into CANDS.


␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Set-struc-intersect
␈↓ α,␈↓α    2: Check all examples of Int-set-struc
␈↓ α,␈↓α    3: Fill in some generalizations of Obj-equal


␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason for considering this Cand is: (We have no examples
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε191␈↓-

␈↓ α,␈↓αfor SET-STRUC-INTERSECT yet)

␈↓ α,␈↓α AM recently tried this same Cand,  so let's skip it now.



␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Check all examples of Int-set-struc
␈↓ α,␈↓α    2: Fill in some generalizations of Obj-equal
␈↓ α,␈↓α    3: Check all examples of Set-struc

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason for considering this Cand is: (Some new ,  unchecked
␈↓ α,␈↓αexamples of INT-SET-STRUC have recently been added)


␈↓ α,␈↓α      Beginning 5th cycle.

␈↓ α,␈↓α AM is forgetting the entire SUGG facet of the INT-SET-STRUC concept.
␈↓ α,␈↓α       Because: (No sense using this suggestion more than once).

␈↓ α,␈↓α Checked examples of INT-SET-STRUC and all entries were confirmed

␈↓ α,␈↓α This Cand used 11.362 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Check all examples of Set-struc
␈↓ α,␈↓α    2: Fill in some generalizations of Obj-equal
␈↓ α,␈↓α    3: Fill in some examples of Coalesce

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason for considering this Cand is: (Some new ,  unchecked
␈↓ α,␈↓αexamples of SET-STRUC have recently been added)


␈↓ α,␈↓α      Beginning 6th cycle.


␈↓ α,␈↓α Based on empirical experiments,  AM believes that SET-STRUC may really
␈↓ α,␈↓αbe no more specialized than UNORD-OBJ.

␈↓ α,␈↓α Closer inspection reveals that the evidence for this was quite flimsy.
␈↓ α,␈↓α AM will wait until some examples of any of these have been found: (
␈↓ α,␈↓αBAG-STRUC),  and then see if they truly also are SET-STRUC's.


␈↓ α,␈↓α Based on empirical experiments,  AM believes that SET-STRUC may really
␈↓ α,␈↓αbe no more specialized than NONMULT-STRUC.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε192␈↓-

␈↓ α,␈↓α Closer inspection reveals that the evidence for this was quite flimsy.
␈↓ α,␈↓α AM will wait until some examples of any of these have been found: (
␈↓ α,␈↓αOSET-STRUC),  and then see if they truly also are SET-STRUC's.

␈↓ α,␈↓α Checked examples of SET-STRUC.
␈↓ α,␈↓α       5 entries were there initially.
␈↓ α,␈↓α       1 small modifications had to be made.
␈↓ α,␈↓α       5 entries are present now.


␈↓ α,␈↓α This Cand used 8.008 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Bag-struc
␈↓ α,␈↓α    2: Fill in some examples of Oset-struc
␈↓ α,␈↓α    3: Fill in some generalizations of Obj-equal

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason for considering this Cand is: (We have no examples
␈↓ α,␈↓αfor BAG-STRUC yet)


␈↓ α,␈↓α      Beginning 7th cycle.

␈↓ α,␈↓α Filled in examples of BAG-STRUC.
␈↓ α,␈↓α       0 examples existed originally on BAG-STRUC.
␈↓ α,␈↓α       19 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      (BAG)
␈↓ α,␈↓α      (BAG A)
␈↓ α,␈↓α      (BAG B)
␈↓ α,␈↓α      (BAG A B)
␈↓ α,␈↓α      (BAG A A)
␈↓ α,␈↓α      (BAG A A B)
␈↓ α,␈↓α      (BAG 0 D F I M)
␈↓ α,␈↓α      (BAG A B (BAG B) (CLASS))
␈↓ α,␈↓α      (BAG BRUCE CORDELL DOUG)
␈↓ α,␈↓α      (BAG R0-7 R1-7 R2-7 R3-7 R4-7 R5-7 R6-7 R7-7)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 10 new,  distinct examples of BAG-STRUC had to be added.


␈↓ α,␈↓αXEQ-CAND

␈↓ α,␈↓α Do-thresh raised from 239 to 264.

␈↓ α,␈↓α This Cand used 17.692 cpu seconds.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε193␈↓-

␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some generalizations of Obj-equal
␈↓ α,␈↓α    2: Fill in some examples of Oset-struc
␈↓ α,␈↓α    3: Fill in some examples of Coalesce

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason is: (The ratio of examples to non-examples of
␈↓ α,␈↓αOBJ-EQUAL is too low ; OBJ-EQUAL is too specialized ,  too narrow)


␈↓ α,␈↓α      Beginning 8th cycle.

␈↓ α,␈↓α Considering genlizing a recursive defn of OBJ-EQUAL
␈↓ α,␈↓α      Will try to remove a conjunct.
␈↓ α,␈↓α      2 possible conjuncts to choose from.
␈↓ α,␈↓α       AM generalizes OBJ-EQUAL into the new concept GENL-OBJ-EQUAL,  by
␈↓ α,␈↓α not recursing on the CAR of each arg.
␈↓ α,␈↓α i.e.,  GENL-OBJ-EQUAL will not have a recursive check
␈↓ α,␈↓α like this one,  which is present in OBJ-EQUAL:

␈↓ α,␈↓α      APPLYB
␈↓ α,␈↓α      (QUOTE OBJ-EQUAL)
␈↓ α,␈↓α      (QUOTE DEFN)
␈↓ α,␈↓α      (CAR BA1)
␈↓ α,␈↓α      (CAR BA2)
␈↓ α,␈↓α       AM generalizes OBJ-EQUAL into the new concept GENL-OBJ-EQUAL-1,
␈↓ α,␈↓αby not recursing on the CDR of each arg.
␈↓ α,␈↓α i.e.,  GENL-OBJ-EQUAL-1 will not have a recursive check
␈↓ α,␈↓α like this one,  which is present in OBJ-EQUAL:

␈↓ α,␈↓α      APPLYB
␈↓ α,␈↓α      (QUOTE OBJ-EQUAL)
␈↓ α,␈↓α      (QUOTE DEFN)
␈↓ α,␈↓α      (CDR BA1)
␈↓ α,␈↓α      (CDR BA2)

␈↓ α,␈↓α If any of (GENL-OBJ-EQUAL GENL-OBJ-EQUAL-1) ever seems to be too
␈↓ α,␈↓αspecialized,  AM will consider disjoining it with other members of that
␈↓ α,␈↓αset.

␈↓ α,␈↓α Filled in generalizations of OBJ-EQUAL.
␈↓ α,␈↓α       0 generalizations existed originally on OBJ-EQUAL.
␈↓ α,␈↓α       2 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed generalizations are:
␈↓ α,␈↓α      GENL-OBJ-EQUAL
␈↓ α,␈↓α      GENL-OBJ-EQUAL-1
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α all 2 new,  distinct generalizations of OBJ-EQUAL had to be added.



␈↓ α,␈↓α Do-thresh raised from 264 to 335.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε194␈↓-

␈↓ α,␈↓α This Cand used 6.667 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Genl-obj-equal-1
␈↓ α,␈↓α    2: Fill in some examples of Genl-obj-equal
␈↓ α,␈↓α    3: Fill in some examplls of Oset-struc

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason is: (The generalization GENL-OBJ-EQUAL-1 of OBJ-EQUAL
␈↓ α,␈↓αis relatively new and has no exs of its own yet ,  excepting those
␈↓ α,␈↓αof OBJ-EQUAL)


␈↓ α,␈↓α      Beginning 9th cycle.


␈↓ α,␈↓α?: ␈↓βN␈↓α

␈↓ α,␈↓α Rename which existing concept? ␈↓βGENL-OBJ-EQUAL␈↓α

␈↓ α,␈↓α What is its new name? ␈↓βSAME-SIZE␈↓α

␈↓ α,␈↓α Done.



␈↓ α,␈↓α Record of attempts to find examples:
␈↓ α,␈↓α-
␈↓ α,␈↓α An ex ( sought) is: ((VECTOR BAG) (VECTOR B (BAG B) (CLASS) A))+-----++
␈↓ α,␈↓α-+---+-------++-+--++-+------------++-------+--+--------+-----------+-+-
␈↓ α,␈↓α---------++--+--------------------++------+-+----+

␈↓ α,␈↓α Found 26 examples ( and 105 non-exs),  in 8.037 secs.
␈↓ α,␈↓α A nice ratio of exs/non-exs was encountered for GENL-OBJ-EQUAL-1
␈↓ α,␈↓α Won't try to create a restricted interesting version of
␈↓ α,␈↓αGENL-OBJ-EQUAL-1.

␈↓ α,␈↓α Filled in examples of GENL-OBJ-EQUAL-1.
␈↓ α,␈↓α       0 examples existed originally on GENL-OBJ-EQUAL-1.
␈↓ α,␈↓α       26 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      ((VECTOR BAG) (VECTOR B (BAG B) (CLASS) A) → T)
␈↓ α,␈↓α      ((OSET 0 D F I M) (OSET 0 D F I M)  → T)
␈↓ α,␈↓α      ((BAG) (BAG DON ED) → T)
␈↓ α,␈↓α      ((OSET D M I F 0) (OSET D M I F 0) → T)
␈↓ α,␈↓α      ((PAIR DOUG BRUCE) (PAIR DOUG BRUCE) →  T)
␈↓ α,␈↓α      ((VECTOR BAG) (VECTOR D M I F 0) → T)
␈↓ α,␈↓α      ((VECTOR B) (VECTOR D M I F 0) → T)
␈↓ α,␈↓α      ((BAG B) (BAG B) → T)
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε195␈↓-

␈↓ α,␈↓α      ((VECTOR D M I F 0) (VECTOR A A B) → T)
␈↓ α,␈↓α      ((BAG A) (BAG A B) → T)
␈↓ α,␈↓α      ((VECTOR) (VECTOR B (BAG B) (CLASS) A) → T)
␈↓ α,␈↓α      ((OSET BRUCE DON) (OSET B A) → T)
␈↓ α,␈↓α      ((PAIR COMPOSE-EXS COMPOSE-EXS) (PAIR LIST-STRUC-INTERSECT
␈↓ α,␈↓αANYB-SPEC) → T)
␈↓ α,␈↓α      ((OSET R2-1 R2-2 R2-3 R2-4 R2-5 R2-6 R3-1 R3-2 R3-3 R3-4 R3-5
␈↓ α,␈↓αR3-6 R4-1 R4-2 R4-3 R4-4 R4-5 R4-6 R5-1 R5-2 R5-3 R5-4 R5-5 R5-6 R6-1
␈↓ α,␈↓αR6-2 R6-3 R6-4 R6-5 R6-6) (OSET 0 D F I M) → T)
␈↓ α,␈↓α      ((OSET A B (BAG B) (CLASS)) (OSET B (BAG B) (CLASS) A) → T)
␈↓ α,␈↓α      ((OSET 0 D F I M) (OSET B) → T)
␈↓ α,␈↓α      ((VECTOR A A) (VECTOR A B) → T)
␈↓ α,␈↓α      ((OSET DON ED) (OSET BAG) → T)
␈↓ α,␈↓α      ((BAG A A B) (BAG) → T)
␈↓ α,␈↓α      ((OSET B) (OSET BRUCE DON) → T)
␈↓ α,␈↓α      ((CLASS DON ED) (CLASS A) → T)
␈↓ α,␈↓α      ((PAIR LIST-STRUC-INSERT CANONIZE) (PAIR LIST-STRUC-INTERSECT
␈↓ α,␈↓αANYB-SPEC) → T)
␈↓ α,␈↓α      ((VECTOR) (VECTOR BAG) → T)
␈↓ α,␈↓α      ((OSET A) (OSET D M I F 0) → T)
␈↓ α,␈↓α      ((VECTOR BAG) (VECTOR BAG) → T)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 25 new,  distinct examples of GENL-OBJ-EQUAL-1 had to be added.



␈↓ α,␈↓α Do-thresh raised from 335 to 367.

␈↓ α,␈↓α This Cand used 29.095 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Same-size
␈↓ α,␈↓α    2: Check all examples of Genl-obj-equal-1
␈↓ α,␈↓α    3: Fill in some examples of Coalesce

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The 2 reasons are:
␈↓ α,␈↓α      (Interestingness of SAME-SIZE has changed recently)
␈↓ α,␈↓α      (The generalization SAME-SIZE of OBJ-EQUAL is relatively new
␈↓ α,␈↓αand has no exs of its own yet ,  excepting those of OBJ-EQUAL)


␈↓ α,␈↓α      Beginning 10th cycle.

␈↓ α,␈↓α Record of attempts to find examples:
␈↓ α,␈↓α-
␈↓ α,␈↓α An ex ( sought) is: ((VECTOR A) (OSET B))+---+--+--------+----++++-----
␈↓ α,␈↓α----+-+---+-+-----+-------+--+-+---+-+----------+-+----------+---+-----+
␈↓ α,␈↓α----+---+---------------+
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε196␈↓-

␈↓ α,␈↓α Found 26 examples ( and 102 non-exs),  in 8.032 secs.
␈↓ α,␈↓α A nice ratio of exs/non-exs was encountered for SAME-SIZE
␈↓ α,␈↓α Won't try to create a restricted interesting version of SAME-SIZE.

␈↓ α,␈↓α Filled in examples of SAME-SIZE.
␈↓ α,␈↓α       0 examples existed originally on SAME-SIZE.
␈↓ α,␈↓α       36 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      ((OSET 0 D F I M) (OSET 0 D F I M) → T)
␈↓ α,␈↓α      ((OSET D M I F 0) (OSET D M I F 0) → T)
␈↓ α,␈↓α      ((PAIR DOUG BRUCE) (PAIR DOUG BRUCE) → T)
␈↓ α,␈↓α      ((BAG B) (BAG B) → T)
␈↓ α,␈↓α      ((OSET BRUCE DON) (OSET B A) → T)
␈↓ α,␈↓α      ((PAIR COMPOSE-EXS COMPOSE-EXS) (PAIR LIST-STRUC-INTERSECT
␈↓ α,␈↓αANYB-SPEC) → T)
␈↓ α,␈↓α      ((OSET A B (BAG B) (CLASS)) (OSET B (BAG B) (CLASS) A) → T)
␈↓ α,␈↓α      ((VECTOR A A) (VECTOR A B) → T)
␈↓ α,␈↓α      ((PAIR LIST-STRUC-INSERT CANONIZE) (PAIR LIST-STRUC-INTERSECT
␈↓ α,␈↓αANYB-SPEC) → T)
␈↓ α,␈↓α      ((VECTOR BAG) (VECTOR BAG) → T)
␈↓ α,␈↓α      ((VECTOR A) (OSET B) → T)
␈↓ α,␈↓α      ((BAG A B) (OSET B A) → T)
␈↓ α,␈↓α      ((CLASS 0 D F I M) (BAG 0 D F I M) → T)
␈↓ α,␈↓α      ((VECTOR B) (BAG A) → T)
␈↓ α,␈↓α      ((PAIR LIST-STRUC-INTERSECT ANYB-SPEC) (PAIR DOUG BRUCE) → T)
␈↓ α,␈↓α      ((OSET DON ED) (PAIR LIST-STRUC-INTERSECT ANYB-SPEC) → T)
␈↓ α,␈↓α      ((BAG 0 D F I M) (VECTOR D M I F 0) → T)
␈↓ α,␈↓α      ((VECTOR B) (BAG B) → T)
␈↓ α,␈↓α      ((OSET BAG) (OSET A) → T)
␈↓ α,␈↓α      ((VECTOR A A) (BAG A A) → T)
␈↓ α,␈↓α      ((CLASS A) (VECTOR BAG) → T)
␈↓ α,␈↓α      ((CLASS A B) (OSET A B) → T)
␈↓ α,␈↓α      ((PAIR COMPOSE-EXS COMPOSE-EXS) (OSET DON ED) → T)
␈↓ α,␈↓α      ((VECTOR A) (OSET A) → T)
␈↓ α,␈↓α      ((OSET BAG) (CLASS A) → T)
␈↓ α,␈↓α      ((OSET A) (CLASS A) → T)
␈↓ α,␈↓α      ((OSET B) (OSET A) → T)
␈↓ α,␈↓α      ((BAG 0 D F I M) (OSET 0 D F I M) → T)
␈↓ α,␈↓α      ((OSET DON ED) (OSET ED CORDELL) → T)
␈↓ α,␈↓α      ((OSET ED CORDELL) (OSET B A) → T)
␈↓ α,␈↓α      ((OSET A) (BAG B) → T)
␈↓ α,␈↓α      ((OSET B A) (OSET A B) → T)
␈↓ α,␈↓α      ((VECTOR B A) (OSET ED CORDELL) → T)
␈↓ α,␈↓α      ((OSET A) (VECTOR BAG) → T)
␈↓ α,␈↓α      ((OSET B A) (OSET DON ED) → T)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 35 new,  distinct examples of SAME-SIZE had to be added.



␈↓ α,␈↓α Do-thresh raised from 367 to 406.

␈↓ α,␈↓α This Cand used 21.725 cpu seconds.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε197␈↓-

␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Check all examples of Same-size
␈↓ α,␈↓α    2: Check all examples of Genl-obj-equal-1
␈↓ α,␈↓α    3: Check all things which just barely miss being examples of
␈↓ α,␈↓αSame-size

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason is: (Some new ,  unchecked examples of SAME-SIZE
␈↓ α,␈↓αhave recently been added)


␈↓ α,␈↓α      Beginning 11st cycle.

␈↓ α,␈↓α Checked examples of SAME-SIZE.
␈↓ α,␈↓α       35 entries were there initially.
␈↓ α,␈↓α       1 had to be completely discarded.
␈↓ α,␈↓α       4 had to be transferred elsewhere.
␈↓ α,␈↓α       30 entries are present now.


␈↓ α,␈↓α Do-thresh raised from 406 to 421.

␈↓ α,␈↓α This Cand used 6.917 cpu seconds.



␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Check all examples of Genl-obj-equal-1
␈↓ α,␈↓α    2: Check all things which just barely miss being examples of
␈↓ α,␈↓αSame-size
␈↓ α,␈↓α    3: Fill in some examples of Coalesce

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason is: (Some new ,  unchecked examples of
␈↓ α,␈↓αGENL-OBJ-EQUAL-1 have recently been added)


␈↓ α,␈↓α      Beginning 12nd cycle.

␈↓ α,␈↓α Checked examples of GENL-OBJ-EQUAL-1.
␈↓ α,␈↓α       25 entries were there initially.
␈↓ α,␈↓α       1 had to be completely discarded.
␈↓ α,␈↓α       4 had to be transferred elsewhere.
␈↓ α,␈↓α       20 entries are present now.


␈↓ α,␈↓α This Cand used 4.711 cpu seconds.

␈↓ α,␈↓α No Cand on CANDS is good enuf.
␈↓ α,␈↓α Do-thresh reduced from 421 to 333
␈↓ α,␈↓α Must find new candidates and merge them into CANDS.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε198␈↓-

␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Canonize these 2 arguments:  Genl-obj-equal-1 and Obj-equal
␈↓ α,␈↓α    2: Canonize these 2 arguments:  Same-size and Obj-equal
␈↓ α,␈↓α    3: Fill in some examples of Coalesce

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason is: (It would be nice to find a canonical  ( with
␈↓ α,␈↓αrespect to Genl-obj-equal-1 and Obj-equal ) representation C for any
␈↓ α,␈↓αObject X ; that is ,
␈↓ α,␈↓α  ( GENL-OBJ-EQUAL-1 x y ) iff
␈↓ α,␈↓α  ( OBJ-EQUAL  ( C x )    ( C y ) ) .
␈↓ α,␈↓α)


␈↓ α,␈↓α      Beginning 13rd cycle.

␈↓ α,␈↓α Experiments indicate that GENL-OBJ-EQUAL-1 is affected by the varying
␈↓ α,␈↓αthe type of structure of its arguments.

␈↓ α,␈↓α GENL-OBJ-EQUAL-1 doesn't look at any elements of OBJECT except possibly
␈↓ α,␈↓α the car of the structure which denotes its type,  so AM replaces the
␈↓ α,␈↓αtail of OBJECT by a canonical distinguished tail,  say NIL.

␈↓ α,␈↓αSucceeded!

␈↓ α,␈↓α Some conjectures that AM considers believable:

␈↓ α,␈↓α OBJ-EQUAL,  restricted to canonical OBJECT's,  is indistinguishable
␈↓ α,␈↓αfrom GENL-OBJ-EQUAL-1.

␈↓ α,␈↓α There is a powerful analogy between

␈↓ α,␈↓αGENL-OBJ-EQUAL-1.................OBJ-EQUAL
␈↓ α,␈↓αOBJECT...........................CANONICAL-OBJECT
␈↓ α,␈↓αoperators on and into            those operators restricted to
␈↓ α,␈↓α      OBJECT...........................CANONICAL-OBJECT
␈↓ α,␈↓αstatements involving these.......statements involving these


␈↓ α,␈↓α Do-thresh raised from 333 to 341.

␈↓ α,␈↓α This Cand used 9.02 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Canonical-object
␈↓ α,␈↓α    2: Restrict the following:  Genl-obj-equal-1 Canonical-object Domain
␈↓ α,␈↓α    3: Canonize these 2 arguments:  Same-size and Obj-equal

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε199␈↓-

␈↓ α,␈↓α       The reason is: (Any example of CANONICAL-OBJECT is a canonical
␈↓ α,␈↓αexample of OBJECT)


␈↓ α,␈↓α      Beginning 14th cycle.


␈↓ α,␈↓α AM will now try to produce examples of CANONICAL-OBJECT by running the
␈↓ α,␈↓αfollowing operations:
␈↓ α,␈↓α       (CANONIZE-GENL-OBJ-EQUAL-1&OBJ-EQUAL).

␈↓ α,␈↓α Won't try to create a restricted interesting version of
␈↓ α,␈↓αCANONICAL-OBJECT.

␈↓ α,␈↓α Filled in examples of CANONICAL-OBJECT.
␈↓ α,␈↓α       0 examples existed originally on CANONICAL-OBJECT.
␈↓ α,␈↓α       165 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      (VECTOR)
␈↓ α,␈↓α      (BAG)
␈↓ α,␈↓α      (CLASS)
␈↓ α,␈↓α      (OSET)
␈↓ α,␈↓α      FALSE
␈↓ α,␈↓α      T
␈↓ α,␈↓α      TRUE
␈↓ α,␈↓α      (PAIR)
␈↓ α,␈↓α      (T)
␈↓ α,␈↓α      (NIL)
␈↓ α,␈↓α      (TRUE)
␈↓ α,␈↓α      (FALSE)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 12 new,  distinct examples of CANONICAL-OBJECT had to be added.



␈↓ α,␈↓α Do-thresh raised from 341 to 391.

␈↓ α,␈↓α This Cand used 23.827 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Restrict the following:  Genl-obj-equal-1 Canonical-object Domain
␈↓ α,␈↓α    2: Canonize these 2 arguments:  Same-size and Obj-equal
␈↓ α,␈↓α    3: Fill in examples of Coalesce

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason is: (GENL-OBJ-EQUAL-1 was one of the predicates
␈↓ α,␈↓αwhich defined the new concept CANONICAL-OBJECT ,  so it is worth
␈↓ α,␈↓αconsidering the restriction of GENL-OBJ-EQUAL-1 to that subset of
␈↓ α,␈↓αOBJECT 's)
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε200␈↓-

␈↓ α,␈↓α      Beginning 15th cycle.


␈↓ α,␈↓αSucceeded!

␈↓ α,␈↓α Do-thresh raised from 391 to 431.

␈↓ α,␈↓α This Cand used 3.562 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Canonize these 2 arguments:  Same-size and Obj-equal
␈↓ α,␈↓α    2: Fill in some examples of Coalesce
␈↓ α,␈↓α    3: Restrict the following:  Obj-equal Canonical-object Domain

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes.␈↓α

␈↓ α,␈↓α       The reason is: (It would be nice to find a canonical  ( with
␈↓ α,␈↓αrespect to Same-size and Obj-equal ) representation C for any Object
␈↓ α,␈↓αX ; that is ,
␈↓ α,␈↓α  ( SAME-SIZE x y ) iff
␈↓ α,␈↓α  ( OBJ-EQUAL  ( C x )    ( C y ) ) .
␈↓ α,␈↓α)


␈↓ α,␈↓α      Beginning 16th cycle.


␈↓ α,␈↓α Experiments indicate that SAME-SIZE is not affected by varying the type
␈↓ α,␈↓α of structure of its arguments.

␈↓ α,␈↓α Experiments indicate that SAME-SIZE is not affected by reordering
␈↓ α,␈↓αelements of its structural arguments.
␈↓ α,␈↓α       So any canonical arguments can be Bags and Sets.

␈↓ α,␈↓α Experiments indicate that SAME-SIZE is affected by the presence of
␈↓ α,␈↓αmultiple elements in its structural arguments.
␈↓ α,␈↓α       So any canonical arguments can be Bags and Lists.

␈↓ α,␈↓α SAME-SIZE doesn't look at the specific elements in OBJECT,  like
␈↓ α,␈↓αOBJ-EQUAL does,  so AM can replace them all by a single distinguished
␈↓ α,␈↓αelement,  say T.

␈↓ α,␈↓αSucceeded!

␈↓ α,␈↓α Some conjectures that AM considers believable:

␈↓ α,␈↓α OBJ-EQUAL,  restricted to canonical BAG-STRUC's,  is indistinguishable
␈↓ α,␈↓αfrom SAME-SIZE.

␈↓ α,␈↓α There is a powerful analogy between
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε201␈↓-

␈↓ α,␈↓αSAME-SIZE........................OBJ-EQUAL
␈↓ α,␈↓αBAG-STRUC........................CANONICAL-BAG-STRUC
␈↓ α,␈↓αoperators on and into............those operators restricted to
␈↓ α,␈↓α      BAG-STRUC..................      CANONICAL-BAG-STRUC
␈↓ α,␈↓αstatements involving these.......statements involving these


␈↓ α,␈↓α Do-thresh raised from 431 to 457.

␈↓ α,␈↓α This Cand used 17.297 cpu seconds.




␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Canonical-bag-struc
␈↓ α,␈↓α    2: Restrict the following:  Same-size Canonical-bag-struc Domain
␈↓ α,␈↓α    3: Restrict the following:  Bag-struc-join Canonical-bag-struc Domain

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes␈↓α.

␈↓ α,␈↓α       The reason is: (Any example of CANONICAL-BAG-STRUC is a canonical
␈↓ α,␈↓αexample of BAG-STRUC)


␈↓ α,␈↓α      Beginning 17th cycle.


␈↓ α,␈↓α AM will now try to produce examples of CANONICAL-BAG-STRUC by running
␈↓ α,␈↓αthe following operations:
␈↓ α,␈↓α       (CANONIZE-SAME-SIZE&OBJ-EQUAL).

␈↓ α,␈↓α Filled in examples of CANONICAL-BAG-STRUC.
␈↓ α,␈↓α       0 examples existed originally on CANONICAL-BAG-STRUC.
␈↓ α,␈↓α       211 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      (BAG)
␈↓ α,␈↓α      (BAG T T)
␈↓ α,␈↓α      (BAG T T T)
␈↓ α,␈↓α      (BAG T)
␈↓ α,␈↓α      (BAG T T T T T)
␈↓ α,␈↓α      (BAG T T T T)
␈↓ α,␈↓α      (BAG T T T T T T T T T T T T T T T T T T T T T T T T T T T T
␈↓ α,␈↓αT T)
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 7 new,  distinct examples of CANONICAL-BAG-STRUC had to be added.



␈↓ α,␈↓α Do-thresh raised from 457 to 478.

␈↓ α,␈↓α This Cand used 35.918 cpu seconds.
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε202␈↓-

␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Restrict the following:  Same-size Canonical-bag-struc Domain
␈↓ α,␈↓α    2: Restrict the following:  Bag-struc-join Canonical-bag-struc Domain
␈↓ α,␈↓α    3: Restrict the following:  Obj-equal Canonical-object Domain

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes␈↓α.

␈↓ α,␈↓α       The reason is: (SAME-SIZE was one of the predicates which defined
␈↓ α,␈↓αthe new concept CANONICAL-BAG-STRUC ,  so it is worth considering
␈↓ α,␈↓αthe restriction of SAME-SIZE to that subset of BAG-STRUC 's)


␈↓ α,␈↓α      Beginning 18th cycle.


␈↓ α,␈↓αSucceeded!

␈↓ α,␈↓α Do-thresh raised from 478 to 495.

␈↓ α,␈↓α This Cand used 3.311 cpu seconds.


␈↓ α,␈↓α?: ␈↓βN␈↓α

␈↓ α,␈↓α Rename which existing concept? ␈↓βCANONICAL-BAG-STRUC␈↓α

␈↓ α,␈↓α What is its new name? ␈↓βNUMBER␈↓α

␈↓ α,␈↓α Done.


␈↓ α,␈↓α?:  (W, I, E, M, N, ?, Q) ␈↓βN␈↓α

␈↓ α,␈↓α Rename which existing concept? ␈↓βCANONIZE-SAME-SIZE&OBJ-EQUAL␈↓α

␈↓ α,␈↓α What is its new name? ␈↓βSIZE␈↓α

␈↓ α,␈↓α Done.



␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Check all examples of Number
␈↓ α,␈↓α    2: Restrict the following:  Obj-equal Canonical-object Domain
␈↓ α,␈↓α    3: Check all examples of Canonical-object

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes␈↓α.

␈↓ α,␈↓α       The 2 reasons are:
␈↓ α,␈↓α      (Interestingness of NUMBER has changed recently)
␈↓ α,␈↓α      (Some new ,  unchecked examples of NUMBER have recently been
␈↓ α,␈↓αadded)
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε203␈↓-

␈↓ α,␈↓α      Beginning 19th cycle.

␈↓ α,␈↓α Checked examples of NUMBER and all entries were confirmed

␈↓ α,␈↓α This Cand used 1.909 cpu seconds.






␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Check all examples of Canonical-object
␈↓ α,␈↓α    2: Check all things which just barely miss being examples of Number
␈↓ α,␈↓α    3: Restrict the following: Bag-struc-join Number Domain

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes␈↓α.

␈↓ α,␈↓α       The reason is: (Some new ,  unchecked examples of
␈↓ α,␈↓αCANONICAL-OBJECT have recently been added)


␈↓ α,␈↓α      Beginning 20th cycle.


␈↓ α,␈↓α CANONICAL-OBJECT has 7 examples which occupy 11 list cells,  but is not
␈↓ α,␈↓α interesting enough to warrant taking up that much space; so about 2
␈↓ α,␈↓αwill be selected at random and forgotten.
␈↓ α,␈↓α Checked examples of CANONICAL-OBJECT.
␈↓ α,␈↓α       12 entries were there initially.
␈↓ α,␈↓α       10 were never confirmed or rejected.
␈↓ α,␈↓α       2 had to be completely discarded.
␈↓ α,␈↓α       5 entries are present now.

␈↓ α,␈↓α This Cand used 16. 626 cpu seconds.

␈↓ α,␈↓α No Cand on CANDS is good enuf.
␈↓ α,␈↓α Do-thresh reduced from 495 to 340
␈↓ α,␈↓α Must find new candidates and merge them into CANDS.





␈↓ α,␈↓α The top 3 Cands are:
␈↓ α,␈↓α    1: Fill in some examples of Size
␈↓ α,␈↓α    2: Fill in some examples of Coalesce
␈↓ α,␈↓α    3: Restrict the following: Bag-struc-join Number Domain

␈↓ α,␈↓α I choose first Cand.       OK?   ␈↓βyes␈↓α.

␈↓ α,␈↓α       The reason is: (We have no examples for SIZE yet)
␈↓ α,␈↓␈↓εAppendix 3␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε204␈↓-

␈↓ α,␈↓α      Beginning 21st cycle.


␈↓ α,␈↓α Record of attempts to find examples:
␈↓ α,␈↓α An ex ( sought) is: (BAG T T)++++++++++++++++++++++++++

␈↓ α,␈↓α Found 26 examples ( and 0 non-exs),  in .996 secs.
␈↓ α,␈↓α A nice ratio of exs/non-exs was encountered for SIZE
␈↓ α,␈↓α Won't try to create a restricted interesting version of SIZE.

␈↓ α,␈↓α Filled in examples of SIZE.
␈↓ α,␈↓α       13 examples existed originally on SIZE.
␈↓ α,␈↓α       26 potential new entries were just proposed.

␈↓ α,␈↓α Eliminating duplicates,  the newly constructed examples are:
␈↓ α,␈↓α      ((BAG T T) → (BAG T T))
␈↓ α,␈↓α      ((BAG T T T T T) → (BAG T T T T T))
␈↓ α,␈↓α      ((BAG B) → (BAG T))
␈↓ α,␈↓α      ((BAG A A) → (BAG T T))
␈↓ α,␈↓α      ((BAG T T T) → (BAG T T T))
␈↓ α,␈↓α      ((BAG T T T T) → (BAG T T T T))
␈↓ α,␈↓α      ((BAG A B) → (BAG T T))
␈↓ α,␈↓α      ((BAG R2-1 R2-2 R2-3 R2-4 R2-5 R2-6 R3-1 R3-2 R3-3 R3-4 R3-5
␈↓ α,␈↓αR3-6 R4-1 R4-2 R4-3 R4-4 R4-5 R4-6 R5-1 R5-2 R5-3 R5-4 R5-5 R5-6 R6-1
␈↓ α,␈↓αR6-2 R6-3 R6-4 R6-5 R6-6) → (BAG T T T T T T T T T T T T T T T T T T
␈↓ α,␈↓αT T T T T T T T T T T T))
␈↓ α,␈↓α      ((BAG A A B) → (BAG T T T))
␈↓ α,␈↓α      ((BAG 0 D F I M) → (BAG T T T T T))
␈↓ α,␈↓α      ((BAG A) → (BAG T))
␈↓ α,␈↓α      ((BAG T T T T T T T T T T T T T T T T T T T T T T T T T T T
␈↓ α,␈↓αT T T) → (BAG T T T T T T T T T T T T T T T T T T T T T T T T T T T
␈↓ α,␈↓αT T T))
␈↓ α,␈↓α      ((BAG DON ED) → (BAG T T))
␈↓ α,␈↓α      ((BAG A B (BAG B) → (CLASS)) (BAG T T T T))
␈↓ α,␈↓α      ((BAG A B) → (BAG T T))
␈↓ α,␈↓α After eliminating duplicate and already-known entries,  AM finds that.
␈↓ α,␈↓α only 14 new,  distinct examples of SIZE had to be added.



␈↓ α,␈↓α Do-thresh raised from 340 to 414.

␈↓ α,␈↓α This Cand used 9.2 cpu seconds.
␈↓ α,␈↓␈↓ ≥-␈↓ε205␈↓-

␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓
␈↓ α,␈↓␈↓ ¬∨␈↓∧Appendix 4.  Bibliography␈↓
␈↓ α,␈↓␈↓ β∧␈↓π≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡␈↓


␈↓ α,␈↓Of␈α⊂all␈α⊂the␈α⊂articles,␈α⊂books,␈α∂and␈α⊂memos␈α⊂which␈α⊂were␈α⊂read␈α∂as␈α⊂background␈α⊂for␈α⊂AM,␈α⊂I␈α∂have
␈↓ α,␈↓selected␈αthose␈α
which␈αhad␈αsome␈α
impact␈αon␈αthat␈α
work␈α(or␈αat␈α
least,␈αon␈αthis␈α
document).␈α While
␈↓ α,␈↓numerous,␈αthey␈αform␈αa␈αfar␈αfrom␈αcomprehensive␈αlist␈αof␈αpublications␈αdealing␈αwith␈αautomated
␈↓ α,␈↓theory formation, with AI in general, and with how mathematicians do research.


␈↓ α,␈↓Adams, James L., ␈↓βConceptual Blockbusting␈↓, W.H. Freeman and Co., San Francisco, 1974.

␈↓ α,␈↓Amarel,␈α∀Saul,␈α∀␈↓βOn␈α∀Representations␈α∀and␈α∀Modelling␈α∀in␈α∀Problem␈α∀Solving␈α∀and␈α∀On␈α∀Future
␈↓ α,␈↓β␈↓ αlDirections for Intelligent Systems␈↓, RCA Labs Scienti≡c Report No. 2, Princeton, 1967.

␈↓ α,␈↓Atkin,␈α
A.␈α
O.␈α
L.,␈α
and␈α
B.␈α
J.␈αBirch,␈α
eds.,␈α
␈↓βComputers␈α
in␈α
Number␈α
Theory␈↓,␈α
Proceedings␈α
of␈αthe␈α
1969
␈↓ α,␈↓␈↓ αlSRCA Oxford Symposium, Academic Press, New York, 1971.

␈↓ α,␈↓Badre,␈α∂Nagib␈α∂A.,␈α∂␈↓βComputer␈α∂Learning␈α∂From␈α∂English␈α∂Text␈↓,␈α∂Memorandum␈α∂No.␈α∞ERL-M372,
␈↓ α,␈↓␈↓ αlElectronics␈α∀Research␈α∀Laboratory,␈α∀UCB,␈α∀December␈α∀20,␈α∀1972.␈α∀ Also␈α∀summarized␈α∀in
␈↓ α,␈↓␈↓ αl␈↓βCLET␈α∂¬␈α∂A␈α∞Computer␈α∂Program␈α∂that␈α∂Learns␈α∞Arithmetic␈α∂from␈α∂an␈α∂Elementary␈α∞Textbook␈↓,
␈↓ α,␈↓␈↓ αlIBM Research Report RC 4235, February 21, 1973.

␈↓ α,␈↓Berliner,␈α∞H.,␈α∂␈↓βChess␈α∞as␈α∞Problem␈α∂Solving:␈α∞The␈α∞Development␈α∂of␈α∞a␈α∞Tactics␈α∂Analyzer␈↓,␈α∞Carnegie-
␈↓ α,␈↓␈↓ αlMellon University Computer Science Department Thesis, March, 1974.

␈↓ α,␈↓Beth,␈αE.␈αW.,␈αand␈αJ.␈αPiaget,␈α␈↓βMathematical␈αEpistemology␈αand␈αPsychology␈↓,␈αGordon␈αand␈α
Breach,
␈↓ α,␈↓␈↓ αlNew York, 1966.

␈↓ α,␈↓Beveridge, W. I., ␈↓βThe Art of Scienti≡c Investigation␈↓, Vintage Books, N. Y. 1950.

␈↓ α,␈↓Biermann,␈αA.␈αW.,␈α␈↓βApproaches␈αto␈αAutomatic␈αProgramming␈↓,␈αin␈α␈↓&Advances␈αin␈αComputers␈↓)αβ,␈αv.␈α15,
␈↓ α,␈↓␈↓ αlAcademic Press, 1976.

␈↓ α,␈↓Black, M., ␈↓βMargins of Precision␈↓, Cornell University Press, Ithaca, New York, 1970.

␈↓ α,␈↓Blalock, H. M., ␈↓βTheory Construction␈↓, Prentice-Hall, Englewood Cli≥s, N.J., 1969.

␈↓ α,␈↓Bledsoe,␈α
W.␈αW.,␈α
␈↓βSplitting␈α
and␈αReduction␈α
Heuristics␈α
in␈αAutomatic␈α
Theorem␈αProving␈↓,␈α
Arti≡cial
␈↓ α,␈↓␈↓ αlIntelligence 2, 1971, pp. 55-77.

␈↓ α,␈↓Bledsoe,␈α∩W.␈α∪W.,␈α∩and␈α∪Bruell,␈α∩Peter,␈α∩␈↓βA␈α∪Man-Machine␈α∩Theorem-Proving␈α∪System␈↓,␈α∩Arti≡cial
␈↓ α,␈↓␈↓ αlIntelligence 5, 1974, 51-72.

␈↓ α,␈↓Bobrow,␈α⊂D.,␈α∂and␈α⊂A.␈α∂Collins,␈α⊂editors,␈α∂␈↓βRepresentation␈α⊂and␈α∂Understanding␈↓,␈α⊂Academic␈α∂Press,
␈↓ α,␈↓␈↓ αlS.F., 1975.

␈↓ α,␈↓Bobrow,␈αD.,␈αand␈αD.␈αNorman,␈α␈↓βSome␈αPrinciples␈αof␈αMemory␈αSchemata␈↓,␈αXEROX␈αPARC␈αMemo
␈↓ α,␈↓␈↓ αlCSL 75-4, Palo Alto, July, 1975.
␈↓ α,␈↓␈↓εAppendix 4␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε206␈↓-

␈↓ α,␈↓Bobrow,␈α∩D.␈α∩G.,␈α∩and␈α∩T.␈α∩Winograd,␈α∩␈↓βAn␈α∩Overview␈α∩of␈α∩KRL,␈α∩A␈α∩Knowledge␈α∩Representation
␈↓ α,␈↓β␈↓ αlLanguage␈↓, Journal of Cognitive Science, Vol 1, No 1, January 1977.

␈↓ α,␈↓Bourbaki,␈αN.,␈α␈↓βThe␈αArchitechture␈αof␈αMathematics␈↓,␈αAmerican␈αMathematics␈αMonthly,␈αv.␈α57,␈αpp.
␈↓ α,␈↓␈↓ αl221-232, Published by the MAA, Albany, N. Y., 1950.

␈↓ α,␈↓Boyer,␈αR.␈αS.,␈αand␈α
J␈αS.␈αMoore,␈α␈↓βProving␈α
Theorems␈αabout␈αLISP␈αFunctions␈↓,␈α
JACM,␈αv.␈α22,␈αNo.␈α
1,
␈↓ α,␈↓␈↓ αlJanuary, 1975, pp. 129-144.

␈↓ α,␈↓Brotz,␈α∩D.␈α∪K.,␈α∩␈↓βEmbedding␈α∪Heuristic␈α∩Problem␈α∪Solving␈α∩Methods␈α∪in␈α∩a␈α∪Mechanical␈α∩Theorem
␈↓ α,␈↓β␈↓ αlProver␈↓,␈αPh.D.␈αdissertation␈αpublished␈αas␈αStanford␈αComputer␈αScience␈αReport␈αSTAN-CS-
␈↓ α,␈↓␈↓ αl74-443, August, 1974.

␈↓ α,␈↓Bruijn,␈αN.␈αG.␈αde,␈α␈↓βAUTOMATH,␈αa␈α
language␈αfor␈αmathematics␈↓,␈αLes␈αPresses␈αde␈αL'Universite␈α
de
␈↓ α,␈↓␈↓ αlMontreal, Montreal, 1973.

␈↓ α,␈↓Buchanan,␈α∂B.␈α∂G.,␈α∂G.␈α∂Sutherland,␈α⊂and␈α∂E.␈α∂Feigenbaum,␈α∂␈↓βHeuristic␈α∂Dendral:␈α∂A␈α⊂Program␈α∂for
␈↓ α,␈↓β␈↓ αlGenerating␈αExplanatory␈α
Hypotheses␈αin␈αOrganic␈α
Chemistry␈↓,␈αin␈α
(Meltzer␈αand␈αMichie,␈α
eds.)
␈↓ α,␈↓␈↓ αlMachine Intelligence 4, American Elsevier Pub., N. Y., 1969, pp. 209-254.

␈↓ α,␈↓Buchanan,␈α⊃B.␈α⊂G.,␈α⊃E.␈α⊃Feigenbaum,␈α⊂and␈α⊃Sridharan,␈α⊂␈↓βHeuristic␈α⊃Theory␈α⊃Formation␈↓,␈α⊂Machine
␈↓ α,␈↓␈↓ αlIntelligence 7, 1972, pp. 267-290.

␈↓ α,␈↓Buchanan,␈α∃B.␈α∃G.,␈α∃␈↓βScienti≡c␈α∃Theory␈α∃Formation␈α∃by␈α∃Computer␈↓,␈α∃NATO␈α⊗Advanced␈α∃Study
␈↓ α,␈↓␈↓ αlInstitute on Computer Oriented Learning Processes, Bonas, France, 1974.

␈↓ α,␈↓Buchanan,␈α∞Bruce␈α∞G.,␈α∞␈↓βApplications␈α∞of␈α∞Arti≡cial␈α∞Intelligence␈α∞to␈α∞Scienti≡c␈α∂Reasoning␈↓,␈α∞Second
␈↓ α,␈↓␈↓ αlUSA-Japan␈α∂Computer␈α∂Conference,␈α∂Tokyo,␈α∂August␈α∂26-28.␈α∂ Published␈α∂by␈α⊂AFIPS␈α∂and
␈↓ α,␈↓␈↓ αlIPSJ, Tokyo, 1975, pp. 189-194.

␈↓ α,␈↓Bundy,␈α
A.,␈α␈↓βDoing␈α
Arithmetic␈α
with␈αDiagrams␈↓,␈α
3rd␈α
International␈αJoint␈α
Conference␈αon␈α
Arti≡cial
␈↓ α,␈↓␈↓ αlIntelligence (3rd IJCAI), Stanford, 1973, pp. 130-138.

␈↓ α,␈↓Burstall,␈α_R.,␈α_and␈α↔J.␈α_Darlington,␈α_␈↓βA␈α↔Transformation␈α_System␈α_for␈α_Developing␈α↔Recursive
␈↓ α,␈↓β␈↓ αlPrograms␈↓, University of Edinburgh AI Research Report, March, 1976.

␈↓ α,␈↓Church, A., ␈↓βThe calculi of Lambda-conversion␈↓, Princeton University Press, Princeton, 1941.

␈↓ α,␈↓Cohen,␈α∂P.␈α∂J.,␈α∂␈↓βSet␈α∂Theory␈α∂and␈α∞the␈α∂Continuum␈α∂Hypothesis␈↓,␈α∂W.A.Benjamin,␈α∂Inc.,␈α∂New␈α∞York,
␈↓ α,␈↓␈↓ αl1966.

␈↓ α,␈↓Colby, K. M., ␈↓β"Simulations of belief systems"␈↓, in [Schank and Colby 73].

␈↓ α,␈↓Copeland,␈α⊂R.␈α∂W.,␈α⊂␈↓βHow␈α∂Children␈α⊂Learn␈α∂Mathematics␈↓,␈α⊂The␈α∂MacMillan␈α⊂Company,␈α∂London,
␈↓ α,␈↓␈↓ αl1970.

␈↓ α,␈↓Courant,␈α∂R.,␈α∂and␈α⊂H.␈α∂Robins,␈α∂␈↓βWhat␈α∂is␈α⊂Mathematics␈↓,␈α∂Oxford␈α∂University␈α∂Press,␈α⊂New␈α∂York,
␈↓ α,␈↓␈↓ αl1941.
␈↓ α,␈↓␈↓εAppendix 4␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε207␈↓-

␈↓ α,␈↓Dahl,␈α⊂O.,␈α∂et.␈α⊂al.,␈α∂␈↓βSIMULA-67:␈α⊂A␈α∂Common␈α⊂Base␈α∂Language␈↓,␈α⊂Norwegian␈α⊂Computing␈α∂Center
␈↓ α,␈↓␈↓ αlPublication No. S-2, Oslo, 1968.

␈↓ α,␈↓Darlington,␈α∩J.,␈α∩and␈α∩R.␈α∩Burstall,␈α⊃␈↓βA␈α∩System␈α∩Which␈α∩Automatically␈α∩Improves␈α∩Programs␈↓,␈α⊃3rd
␈↓ α,␈↓␈↓ αlIJCAI, 1973, pp. 479-485.

␈↓ α,␈↓Davis,␈α∞R.,␈α∞and␈α∞J.␈α∞King,␈α∞␈↓βAn␈α∞Overview␈α
of␈α∞Production␈α∞Systems␈↓,␈α∞Stanford␈α∞AI␈α∞Lab␈α∞Memo␈α
271,
␈↓ α,␈↓␈↓ αlOctober, 1975.

␈↓ α,␈↓Davis,␈αR.,␈α␈↓βApplications␈α
of␈αMeta␈αLevel␈αKnowledge␈α
to␈αthe␈αConstruction,␈αMaintenance␈α
and␈αUse
␈↓ α,␈↓β␈↓ αlof Large Knowledge Bases,␈↓ Stanford AI Lab Memo 283, July, 1976.

␈↓ α,␈↓Dijkstra,␈α∂E.␈α∂W.,␈α∂␈↓βA␈α∂Discipline␈α∂of␈α∂Programming␈↓,␈α∂Prentice-Hall,␈α∂Inc.,␈α∂Englewood␈α∂Cli≥s,␈α∂N.J.,
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␈↓ α,␈↓Eddington, Sir A. S., ␈↓βNew Pathways in Science␈↓, Macmillan Co., N. Y., 1935.

␈↓ α,␈↓Engelman,␈α
C.,␈α
␈↓βMATHLAB:␈αA␈α
Program␈α
for␈α
On-Line␈αAssistance␈α
in␈α
Symbolic␈α
Computation␈↓,␈αin
␈↓ α,␈↓␈↓ αlProceedings of the FJCC, v. 2, Spartan Books, 1965.

␈↓ α,␈↓Engelman, C., ␈↓βMATHLAB '68␈↓, in IFIP, Edinburgh, 1968.

␈↓ α,␈↓Evans,␈α
T.␈α
G.,␈α
␈↓βA␈α
Program␈α
for␈αthe␈α
Solution␈α
of␈α
Geometric-Analogy␈α
Intelligence␈α
Test␈αQuestions␈↓,␈α
in
␈↓ α,␈↓␈↓ αl[Minsky 68], pp. 271-353.

␈↓ α,␈↓Eynden,␈αC.␈αV.,␈α␈↓βNumber␈αTheory:␈αAn␈αIntroduction␈αto␈αProof␈↓,␈αInternational␈αTextbook␈αCompany,
␈↓ α,␈↓␈↓ αlScranton, Pennsylvania, 1970.

␈↓ α,␈↓Feigenbaum,␈αE.␈αA.,␈α␈↓βEPAM:␈αThe␈αSimulation␈αof␈αVerbal␈αLearning␈αBehavior␈↓,␈αin␈α[Feigenbaum␈α&
␈↓ α,␈↓␈↓ αlFeldman 63], Part 2, Section 2, pp. 297-309.

␈↓ α,␈↓Feigenbaum,␈α∂E.␈α∞A.,␈α∂and␈α∞J.␈α∂Feldman,␈α∞editors,␈α∂␈↓βComputers␈α∞and␈α∂Thought␈↓,␈α∂McGraw-Hill␈α∞Book
␈↓ α,␈↓␈↓ αlCompany, New York, New York, 1963.

␈↓ α,␈↓Feigenbaum,␈α
E.,␈α
B.␈α
Buchanan,␈αand␈α
J.␈α
Lederberg,␈α
␈↓βOn␈αGenerality␈α
and␈α
Problem␈α
Solving:␈αA␈α
Case
␈↓ α,␈↓β␈↓ αlStudy␈α⊗Using␈α⊗The␈α⊗DENDRAL␈α⊗Program␈↓,␈α∃in␈α⊗(Meltzer␈α⊗and␈α⊗Michie,␈α⊗eds.)␈α∃Machine
␈↓ α,␈↓␈↓ αlIntelligence 6, 1971, pp 165-190.

␈↓ α,␈↓Fogel,␈α⊂L.,␈α⊂A.␈α⊂Owens,␈α⊂and␈α⊂M.␈α⊂Walsh,␈α⊂␈↓βArti≡cial␈α⊂Intelligence␈α⊂Through␈α⊂Simulated␈α∂Evolution␈↓,
␈↓ α,␈↓␈↓ αlJohn␈αWiley␈α&␈αSons,␈αInc.,␈αN.␈αY.,␈α1966.␈α Fuller,␈αR.␈αB.,␈α␈↓βSynergetics␈↓,␈αMacmillan␈αCo.,␈αN.␈αY.,
␈↓ α,␈↓␈↓ αl1975.

␈↓ α,␈↓Gardner,␈α⊗M.,␈α⊗␈↓βMathematical␈α⊗Games␈↓,␈α⊗Scienti≡c␈α⊗American␈α⊗[numerous␈α⊗columns,␈α⊗including
␈↓ α,␈↓␈↓ αlespecially: February, 1975.]

␈↓ α,␈↓Gelernter,␈α∪H.,␈α∪␈↓βRealization␈α∪of␈α∀a␈α∪Geometry-Theorem␈α∪Proving␈α∪Machine␈↓,␈α∪in␈α∀[Feigenbaum␈α∪&
␈↓ α,␈↓␈↓ αlFeldman 63], Part 1, Section 3, pages 134-152.

␈↓ α,␈↓Goldstein, I., ␈↓βElementary Geometry Theorem Proving␈↓, MIT AI Memo 280, April, 1973.
␈↓ α,␈↓␈↓εAppendix 4␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε208␈↓-

␈↓ α,␈↓Goodstein, R. L., ␈↓βFundamental Concepts of Mathematics␈↓, Pergamon Press, New York, 1962.

␈↓ α,␈↓Green,␈αC.C.,␈αR.␈αWaldinger,␈αD.␈αBarstow,␈αR.␈αElschlager,␈αD.␈αLenat,␈αB.␈αMcCune,␈αD.␈αShaw,␈αand
␈↓ α,␈↓␈↓ αlL.␈αSteinberg,␈α␈↓βProgress␈α
Report␈αon␈αProgram-Understanding␈α
Systems␈↓,␈αMemo␈αAIM-240,␈α
CS
␈↓ α,␈↓␈↓ αlReport␈α≡STAN-CS-74-444,Arti≡cial␈α≡Intelligence␈α≡Laboratory,␈α∨Stanford␈α≡University,
␈↓ α,␈↓␈↓ αlAugust, 1974.

␈↓ α,␈↓Guard,␈α
J.␈α
R.,␈α
Eastman␈αet␈α
al.,␈α
␈↓βSemi-Automated␈α
Mathematics␈↓,␈αJACM␈α
16,␈α
January,␈α
1969,␈αpp.␈α
49-
␈↓ α,␈↓␈↓ αl62.

␈↓ α,␈↓Hadamard,␈α
J.,␈α
␈↓βThe␈α
Psychology␈α
of␈α∞Invention␈α
in␈α
the␈α
Mathematical␈α
Field␈↓,␈α∞Dover␈α
Publications,
␈↓ α,␈↓␈↓ αlNew York, 1945.

␈↓ α,␈↓Halmos,␈αP.␈αR.,␈α␈↓βInnovation␈αin␈αMathematics␈↓,␈αin␈α[Kline␈α68].␈α Originally␈αin␈αScienti≡c␈αAmerican,
␈↓ α,␈↓␈↓ αlSeptember, 1958.

␈↓ α,␈↓Hardy,␈α∂G.␈α∂H.,␈α∂and␈α∂E.␈α∂M.␈α∂Wright,␈α∂␈↓βAn␈α∂Introduction␈α∂to␈α∂the␈α∂Theory␈α∂of␈α∂Numbers␈↓,␈α⊂Oxford␈α∂U.
␈↓ α,␈↓␈↓ αlPress, London, 1938. (Fourth edition, 1960)

␈↓ α,␈↓Hayes-Roth,␈αF.,␈αand␈αV.␈αR.␈αLesser,␈α␈↓βFocus␈αof␈αAttention␈αin␈αa␈αDistributed␈αSpeech␈αUnderstanding
␈↓ α,␈↓β␈↓ αlSystem␈↓,␈α∂Computer␈α∂Science␈α∂Dept.␈α∂Memo,␈α∂Carnegie-Mellon␈α∂University,␈α∂Pittsburgh,␈α∂Pa.,
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␈↓ α,␈↓Hempel,␈α⊃C.␈α⊂G.,␈α⊃␈↓βFundamentals␈α⊃of␈α⊂Concept␈α⊃Formation␈α⊃in␈α⊂Empirical␈α⊃Science␈↓,␈α⊃University␈α⊂of
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␈↓ α,␈↓Hewitt,␈α∀C.,␈α∃␈↓βA␈α∀Universal␈α∃Modular␈α∀ACTOR␈α∀Formalism␈α∃for␈α∀Arti≡cial␈α∃Intelligence␈↓,␈α∀Third
␈↓ α,␈↓␈↓ αlInternational Joint Conference on Arti≡cial Intelligence, 1973, pp. 235-245.

␈↓ α,␈↓Hewitt,␈α∩C.,␈α⊃␈↓βViewing␈α∩Control␈α⊃Structures␈α∩as␈α⊃Patterns␈α∩of␈α⊃Passing␈α∩Messages␈↓,␈α⊃MIT␈α∩AI␈α⊃Lab
␈↓ α,␈↓␈↓ αlWorking Paper 92, April, 1976.

␈↓ α,␈↓Hilpinen,␈αR.,␈α␈↓βRules␈αof␈αAcceptance␈αand␈αInductive␈αLogic␈↓,␈αActa␈αPhilosophica␈αFennica,␈αFasc.␈α22,
␈↓ α,␈↓␈↓ αlNorth-Holland Publishing Company, Amsterdam, 1968.

␈↓ α,␈↓Hintikka, J., ␈↓βKnowledge and Belief␈↓, Cornell U. Press, Ithaca, N. Y., 1962.

␈↓ α,␈↓Hintikka,␈α
J.,␈α
and␈α
P.␈α
Suppes,␈α
editors,␈α
␈↓βAspects␈α
of␈α
Inductive␈α
Logic␈↓,␈α
North-Holland␈αPublishing
␈↓ α,␈↓␈↓ αlCompany, Amsterdam, 1966.

␈↓ α,␈↓Iberall,␈α∞A.␈α∂S.,␈α∞␈↓βToward␈α∞a␈α∂General␈α∞Science␈α∞of␈α∂Viable␈α∞Systems␈↓,␈α∞McGraw-Hill␈α∂Book␈α∞Co.,␈α∂N.␈α∞Y.
␈↓ α,␈↓␈↓ αl1972.

␈↓ α,␈↓Kershner,␈αR.B.,␈α
and␈αL.R.Wilcox,␈α␈↓βThe␈α
Anatomy␈αof␈αMathematics␈↓,␈α
The␈αRonald␈αPress␈α
Company,
␈↓ α,␈↓␈↓ αlNew York, 1950.

␈↓ α,␈↓Kline,␈α∪M.␈α∪(ed),␈α∪␈↓βMathematics␈α∪in␈α∪the␈α∪Modern␈α∪World:␈α∪Readings␈α∪from␈α∪Scienti≡c␈α∩American␈↓,
␈↓ α,␈↓␈↓ αlW.H.Freeman and Co., San Francisco, 1968.
␈↓ α,␈↓␈↓εAppendix 4␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε209␈↓-

␈↓ α,␈↓Kling,␈α
R.␈αE.,␈α
␈↓βReasoning␈α
by␈αAnalogy␈α
with␈αApplications␈α
to␈α
Heuristic␈αProblem␈α
Solving:␈α
A␈αCase
␈↓ α,␈↓β␈↓ αlStudy␈↓,␈α
Stanford␈α
Arti≡cial␈α
Intelligence␈α∞Project␈α
Memo␈α
AIM-147,␈α
CS␈α∞Department␈α
report
␈↓ α,␈↓␈↓ αlCS-216, August, 1971.

␈↓ α,␈↓Knuth,␈α∪D.␈α∪E.,␈α∪␈↓βFundamental␈α∪Algorithms␈↓,␈α∀v.␈α∪1␈α∪of␈α∪␈↓&The␈α∪Art␈α∪of␈α∀Computer␈α∪Programming␈↓)αβ,
␈↓ α,␈↓␈↓ αlAddison-Wesley Publishing Company, Menlo Park, 1968.

␈↓ α,␈↓Knuth,␈α⊃D.␈α⊃ E.,␈α⊂␈↓βSurreal␈α⊃Numbers␈↓,␈α⊃Addison-Wesley␈α⊂Publishing␈α⊃Company,␈α⊃Reading,␈α⊂Mass.,
␈↓ α,␈↓␈↓ αll974.

␈↓ α,␈↓Knuth, D. E., ␈↓βAncient Babylonian Algorithms␈↓, CACM 15, July, 1972, pp. 671-677.

␈↓ α,␈↓Koestler, A., ␈↓βThe Act of Creation␈↓, New York, Dell Pub., 1967.

␈↓ α,␈↓Koppelman,␈α⊂E.,␈α⊂"Progress␈α⊂in␈α⊂Mathematics",␈α⊃in␈α⊂the␈α⊂proceedings␈α⊂of␈α⊂the␈α⊂Workshop␈α⊃on␈α⊂the
␈↓ α,␈↓␈↓ αlHistorical Development of Modern Mathematics, July, 1975.

␈↓ α,␈↓Lamon,␈αW.␈αE.,␈α␈↓βLearning␈αand␈αthe␈αNature␈αof␈αMathematiccs␈↓,␈αScience␈αResearch␈αAssociates,␈αPalo
␈↓ α,␈↓␈↓ αlAlto, 1972.

␈↓ α,␈↓Lederberg,␈α∀J.,␈α∃␈↓βDENDRAL-64:␈α∀A␈α∀System␈α∃for␈α∀Computer␈α∀Construction,␈α∃Enumeration,␈α∀and
␈↓ α,␈↓β␈↓ αlNotation␈α
of␈αOrganic␈α
Molecules␈α
as␈αTree␈α
Structures␈αand␈α
Cyclic␈α
Graphs␈↓,␈αParts␈α
I-V␈α
of␈αthe
␈↓ α,␈↓␈↓ αlInterim Report to NASA, 1964.

␈↓ α,␈↓Lederberg, J.; New York Times review of [Weizenbaum 76]; 1976.

␈↓ α,␈↓Lefrancois,␈α∩G.␈α⊃R.,␈α∩␈↓βPsychological␈α⊃Theories␈α∩and␈α⊃Human␈α∩Learning␈↓,␈α∩Wadsworth␈α⊃Publishing,
␈↓ α,␈↓␈↓ αlBelmont, Ca., 1972.

␈↓ α,␈↓Lenat,␈α⊗D.,␈α⊗␈↓βSynthesis␈α⊗of␈α⊗Large␈α∃Programs␈α⊗from␈α⊗Speci≡c␈α⊗Dialogues␈↓,␈α⊗Proceedings␈α⊗of␈α∃the
␈↓ α,␈↓␈↓ αlInternational␈α∞Symposium␈α∞on␈α∞Proving␈α∞and␈α∞Improving␈α∞Programs,␈α∞Le␈α∞Chesnay,␈α∞France,
␈↓ α,␈↓␈↓ αlJuly, 1975a.

␈↓ α,␈↓Lenat,␈α
D.,␈α∞␈↓βBEINGs:␈α
Knowledge␈α∞as␈α
Interacting␈α∞Experts␈↓,␈α
4th␈α∞IJCAI,␈α
Tbilisi,␈α∞Georgian␈α
SSR,
␈↓ α,␈↓␈↓ αlUSSR, 1975b.

␈↓ α,␈↓Lesser,␈α∂V.,␈α∂R.D.␈α∂Fennell,␈α∂L.␈α∂D.␈α∂Erman,␈α∂and␈α∂D.␈α∂R.␈α∂Reddy,␈α∂␈↓βOrganization␈α∂of␈α⊂the␈α∂Hearsay-II
␈↓ α,␈↓β␈↓ αlSpeech␈αUnderstanding␈αSystem␈↓,␈αin␈αIEEE␈αTransactions␈αon␈αAcoustics,␈αSpeech,␈αand␈αSignal
␈↓ α,␈↓␈↓ αlProcessing, v. ASSP-23, 1975, pp. 11-23.

␈↓ α,␈↓Linderholm, C. E., ␈↓βMathematics Made Di≠cult␈↓, World Publishing Co., N. Y. 1972.

␈↓ α,␈↓Lombardi,␈αL.A.,␈αand␈αB.␈αRaphael,␈α␈↓βLISP␈αas␈αthe␈αlanguage␈αfor␈αan␈αincremental␈αcomputer␈↓,␈αin␈α(E.
␈↓ α,␈↓␈↓ αlC.␈αBerkeley␈αand␈α
D.␈αG.␈αBobrow,␈α
eds.)␈α␈↓&The␈αProgramming␈α
Language␈αLISP:␈αIts␈α
Operation␈↓'≠
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␈↓ α,␈↓McDermott,␈αD.,␈α␈↓βArti≡cial␈αIntelligence␈αMeets␈αNatural␈αStupidity␈↓,␈αin␈αSigart␈αNewsletter,␈αNo.␈α57,
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␈↓ α,␈↓␈↓εAppendix 4␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε210␈↓-

␈↓ α,␈↓Martin,␈α
W.,␈α
and␈α
R.␈α
Fateman,␈α
␈↓βThe␈α
MACSYMA␈α
System␈↓,␈α
in␈α
(S.␈α
Petrick,␈α
ed.)␈α
Second␈α
Symposium
␈↓ α,␈↓␈↓ αlon␈αSymbolic␈αand␈αAlgebraic␈αManipulation,␈αACM␈αSIGSAM,␈αN.␈αY.␈α(conference␈αwas␈αheld
␈↓ α,␈↓␈↓ αlin Los Angeles), 1971, pp. 59-75.

␈↓ α,␈↓Minsky,␈α~M.,␈α~editor,␈α→␈↓βSemantic␈α~Information␈α~Processing␈↓,␈α→The␈α~MIT␈α~Press,␈α→Cambridge,
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␈↓ α,␈↓Minsky, M., ␈↓βFrames␈↓, in [Winston 75].

␈↓ α,␈↓Mirsky, L., ␈↓βStudies in Pure Mathematics␈↓, Academic Press, New York, 1971.

␈↓ α,␈↓Moore,␈α⊂J␈α∂S.,␈α⊂␈↓βIntroducing␈α∂Iteration␈α⊂into␈α∂the␈α⊂Pure␈α∂LISP␈α⊂Theorem␈α∂Prover␈↓,␈α⊂XEROX␈α∂PARC
␈↓ α,␈↓␈↓ αlreport CSL-74-3, Palo Alto, 1975.

␈↓ α,␈↓Moore,␈αJ.␈αand␈αA.␈αNewell,␈α"How␈αcan␈αMERLIN␈αunderstand?,"␈αin␈α(Gregg,␈αed.)␈α␈↓&Knowledge␈αand␈↓'≠
␈↓ α,␈↓␈↓ αl␈↓&Cognition␈↓)αβ, Lawrence Erlbaum Associates, 1973.

␈↓ α,␈↓Moore,␈α⊂R.␈α∂C.,␈α⊂␈↓βD-SCRIPT:␈α∂A␈α⊂Computational␈α⊂Theory␈α∂of␈α⊂Descriptions␈↓,␈α∂MIT␈α⊂AI␈α⊂Memo␈α∂278,
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␈↓ α,␈↓Neumann,␈α
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in␈α
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(ed),␈α
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the␈α
Mind␈↓)αβ,␈αU.
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␈↓ α,␈↓Nevins,␈α
Arthur␈αJ.,␈α
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Human␈αOriented␈α
Logic␈αfor␈α
Automatic␈α
Theorem␈αProving␈↓,␈α
MIT␈αAI␈α
Memo
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␈↓ α,␈↓Nevins,␈α⊃Arthur␈α⊂J.,␈α⊃␈↓βPlane␈α⊃Geometry␈α⊂Theorem␈α⊃Proving␈α⊂Using␈α⊃Forward␈α⊃Chaining␈↓,␈α⊂Arti≡cial
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␈↓ α,␈↓Newell,␈αA.␈α␈↓βHeuristic␈α
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in␈α(A.␈αAronofsky,␈α
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␈↓ α,␈↓Newell,␈αA.,␈α␈↓βProduction␈αSystems:␈αModels␈αof␈αControl␈αStructures␈↓,␈αMay,␈α1973␈αCMU␈αReport,␈αalso
␈↓ α,␈↓␈↓ αlpublished␈αin␈α(W.G.␈αChase,␈αed.)␈α␈↓&Visual␈αInformation␈αProcessing␈↓)αβ,␈αN.␈αY.:␈α
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␈↓ α,␈↓Newell,␈αA.,␈αJ.␈αShaw,␈αand␈αH.␈αSimon,␈α␈↓βEmpirical␈αExplorations␈αof␈αthe␈αLogic␈αTheory␈αMachine:␈αA
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Search␈↓,␈αthe
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ACM␈αTuring␈α
Award␈αLecture,␈α
printed␈αin␈α
CACM␈α
19,␈αNo.␈α
3,␈αMarch,␈α
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␈↓ α,␈↓Nilsson,␈α↔N.␈α↔J.,␈α↔␈↓βProblem-solving␈α↔Methods␈α↔in␈α↔Arti≡cial␈α↔Intelligence␈↓,␈α↔McGraw-Hill␈α⊗Book
␈↓ α,␈↓␈↓ αlCompany, New York, New York, 1971
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␈↓ α,␈↓Norman,␈α∂D.,␈α∂and␈α∂D.␈α∂Bobrow,␈α⊂␈↓βOn␈α∂Data-limited␈α∂and␈α∂Resource-limited␈α∂Processes␈↓,␈α⊂Journal␈α∂of
␈↓ α,␈↓␈↓ αlCognitive Psychology, v. 7, 1975, pp. 44-64.

␈↓ α,␈↓Norman, D., and D. Rumelhart, ␈↓βExplorations in Cognition␈↓, W. H. Freeman & Co., S.F., 1975.

␈↓ α,␈↓Ore, O., ␈↓βNumber Theory and its History␈↓, McGraw-Hill, New York, 1948.

␈↓ α,␈↓Papert,␈α
S.,␈α
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Children␈α
to␈α
be␈α
Mathematicians␈α
Versus␈α
Teaching␈α
About␈α
Mathematics␈↓,␈α
in
␈↓ α,␈↓␈↓ αlthe␈αInternational␈αJournal␈αof␈αMathematical␈αEducation␈αin␈αScience␈αand␈αTechnology,␈αv.␈α3,
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␈↓ α,␈↓Piaget, J., ␈↓βThe Language and Thought of the Child␈↓, The World Publishing Co., N. Y., 1955.

␈↓ α,␈↓Pietarinen,␈α⊗J.,␈α∃␈↓βLawlikeness,␈α⊗Analogy,␈α∃and␈α⊗Inductive␈α∃Logic␈↓,␈α⊗North-Holland,␈α∃Amsterdam,
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␈↓ α,␈↓Pitrat,␈α J.,␈α ␈↓βHeuristic␈α Interest␈α of␈α using␈α Metatheorems␈↓,␈α Symposium␈α on␈α Automatic
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␈↓ α,␈↓Poincare',␈α⊂H.,␈α∂␈↓βThe␈α⊂Foundations␈α∂of␈α⊂Science:␈α⊂Science␈α∂and␈α⊂Hypothesis,␈α∂The␈α⊂Value␈α⊂of␈α∂Science,
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␈↓ α,␈↓Polya,␈αG.,␈α
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Plausible␈αReasoning␈↓,␈α
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␈↓ α,␈↓Polya,␈α
G.,␈α
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To␈α
Solve␈α
It␈↓,␈α
Second␈α
Edition,␈α
Doubleday␈α
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␈↓ α,␈↓Polya, G., ␈↓βMathematical Discovery␈↓, John Wiley & Sons, New York, Vol. 1, 1962; Vol. 2, 1965.

␈↓ α,␈↓Ramanujan,␈α
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and␈α∞Wilson,␈α
eds.),␈α∞Chelsea␈α
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␈↓ α,␈↓Rulifson,␈α∪J.,␈α∪J.␈α∀Derksen,␈α∪and␈α∪R.␈α∪Waldinger,␈α∀␈↓βQA4:␈α∪A␈α∪Procedural␈α∪Calculus␈α∀for␈α∪Intuitive
␈↓ α,␈↓β␈↓ αlReasoning␈↓,␈α⊂SRI␈α⊂Project␈α⊂8721,␈α⊂Technical␈α⊂Note␈α⊂73,␈α⊂Arti≡cial␈α⊂Intelligence␈α⊂Center,␈α∂SRI,
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␈↓ α,␈↓Saaty,␈αT.␈αL.,␈α
and␈αF.␈αJ.␈αWeyl,␈α
editors,␈α␈↓βThe␈αSpirit␈αand␈α
the␈αUses␈αof␈αthe␈α
Mathematical␈αSciences␈↓,
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␈↓ α,␈↓Samuel,␈α⊂A.,␈α⊂␈↓βSome␈α⊃Studies␈α⊂in␈α⊂Machine␈α⊃Learning␈α⊂Using␈α⊂the␈α⊃Game␈α⊂of␈α⊂Checkers␈α⊃II.␈α⊂ Recent
␈↓ α,␈↓β␈↓ αlProgress␈↓,␈α
in␈α
the␈α
IBM␈α
Journal␈α
of␈α
Research␈α
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Development,␈α
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6,␈αpp.␈α
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␈↓ α,␈↓Schank,␈α∞R.␈α∞and␈α
K.␈α∞Colby,␈α∞␈↓βComputer␈α∞Models␈α
of␈α∞Thought␈α∞and␈α
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␈↓ α,␈↓Schminke,␈α∂C.␈α∂W.,␈α∂and␈α∂W.␈α∂R.␈α⊂Arnold,␈α∂editors,␈α∂␈↓βMathematics␈α∂is␈α∂a␈α∂Verb␈↓,␈α∂The␈α⊂Dryden␈α∂Press,
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␈↓ α,␈↓Simon,␈αH.␈α
A.,␈α␈↓βThe␈α
Heuristic␈αCompiler␈↓,␈α
in␈α[Simon␈α&␈α
Siklossy␈α72],␈α
Part␈α1,␈α
Chapter␈α1,␈αpp.␈α
9-43,
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␈↓ α,␈↓Simon,␈α
H.␈α
A.,␈α␈↓βDoes␈α
Scienti≡c␈α
Discovery␈α
Have␈αa␈α
Logic?␈↓,␈α
Philosophy␈α
of␈αScience,␈α
v.␈α
40,␈α
No.␈α4,
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␈↓ α,␈↓Simon,␈α∩H.␈α∩A.,␈α⊃and␈α∩L.␈α∩Siklossy,␈α∩editors,␈α⊃␈↓βRepresentation␈α∩and␈α∩Meaning:␈α∩Experiments␈α⊃with
␈↓ α,␈↓β␈↓ αlInformation Processing Systems␈↓, Prentice-Hall Inc., Englewood Cli≥s, N.J., 1972.

␈↓ α,␈↓Skemp,␈α∂R.␈α∂R.,␈α∞␈↓βThe␈α∂Psychology␈α∂of␈α∂Learning␈α∞Mathematics␈↓,␈α∂Penguin␈α∂Books,␈α∂Ltd.,␈α∞Middlesex,
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␈↓ α,␈↓Sloman,␈αA.,␈α␈↓βInteractions␈αBetween␈αPhilosophy␈α
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Intuition
␈↓ α,␈↓β␈↓ αland Non-Logical Reasoning in Intelligence␈↓, Arti≡cial Intelligence 2, 1971, pp. 209-225.

␈↓ α,␈↓Smith,␈αN.␈αW.,␈α␈↓βA␈αQuestion-Answering␈αSystem␈αfor␈αElementary␈αMathematics␈↓,␈α
Stanford␈αInstitute
␈↓ α,␈↓␈↓ αlfor␈αMathematical␈αStudies␈αin␈αthe␈αSocial␈αSciences␈α(IMSSS),␈αTechnical␈αReport␈α227,␈αApril
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␈↓ α,␈↓Wang,␈αH.,␈α␈↓βToward␈αMechanical␈αMathematics␈↓,␈αIBM␈αJournal␈αof␈αResearch␈αand␈αDevelopment,␈αv.
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␈↓ α,␈↓Weizenbaum, J., ␈↓βComputer Power and Human Reason␈↓, W. H. Freeman, S.F., 1976.

␈↓ α,␈↓Wickelgren,␈α
W.␈α
A.,␈α
␈↓βHow␈α
to␈α
Solve␈α∞Problems:␈α
Elements␈α
of␈α
a␈α
Theory␈α
of␈α
Problems␈α∞and␈α
Problem
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␈↓ α,␈↓Wilder, R. L., ␈↓βEvolution of Mathematical Concepts␈↓, John Wiley & Sons, Inc., N. Y., 1968.

␈↓ α,␈↓Winograd,␈α⊂T.,␈α⊂␈↓βUnderstanding␈α⊂Natural␈α⊂Language␈↓,␈α⊂Academic␈α⊂Press,␈α⊂Inc.,␈α⊂New␈α⊂York,␈α⊂New
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␈↓ α,␈↓Winston,␈α⊗P.,␈α∃␈↓βLearning␈α⊗Structural␈α∃Descriptions␈α⊗from␈α∃Examples␈↓,␈α⊗Ph.D.␈α∃thesis,␈α⊗Dept.␈α∃of
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␈↓ α,␈↓␈↓εAppendix 4␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ≥␈↓-␈↓ε213␈↓-

␈↓ α,␈↓Winston,␈αP.,␈αeditor,␈α␈↓βNew␈αProgress␈αin␈αArti≡cial␈αIntelligence␈↓,␈αMIT␈αAI␈αLab␈αMemo␈αAI-TR-310,
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␈↓ α,␈↓Wittner,␈αG.␈αE.,␈α
␈↓βThe␈αStructure␈αof␈αMathematics␈↓,␈α
Xerox␈αCollege␈αPublishing,␈α
Lexington,␈αMass.,
␈↓ α,␈↓␈↓ αl1972.
␈↓ α,␈↓␈↓εAppendix ␈↓ ¬¬␈↓↓AM  ␈↓ε Discovery in Mathematics as Heuristic Search␈↓ ;␈↓-␈↓ε1␈↓-

␈↓ α,␈↓␈↓ ∧S␈↓↓␈↓&Appendix .. ␈↓)αβ␈↓∧␈↓&Index to Initial Concepts␈↓)αβ␈↓↓



␈↓ α,␈↓∧␈↓&CONCEPT␈↓)αβ␈↓ ¬|␈↓&PAGE␈↓)αβ     ␈↓&CONCEPT␈↓)αβ␈↓ 
q␈↓&PAGE␈↓)αβ


␈↓ α,␈↓␈↓ ¬g␈↓∧␈↓&Things still to do␈↓)αβ␈↓



␈↓ α,␈↓␈↓↓␈↓&CHAPTER 1␈↓)αβ ¬ ␈↓ ␈↓εOverview␈↓

␈↓ α,␈↓␈↓↓␈↓&CHAPTER 2␈↓)αβ ¬ ␈↓ ␈↓εAn Example: Discovering Prime Numbers␈↓

␈↓ α,␈↓␈↓↓␈↓&CHAPTER 3␈↓)αβ ¬ ␈↓ ␈↓εControl Structure␈↓

␈↓ α,␈↓␈↓↓␈↓&CHAPTER 4␈↓)αβ ¬ ␈↓ ␈↓εHeuristic Rules␈↓

␈↓ α,␈↓␈↓↓␈↓&CHAPTER 5␈↓)αβ ¬ ␈↓ ␈↓εAM's Concepts␈↓

␈↓ α,␈↓␈↓↓␈↓&CHAPTER 6␈↓)αβ ¬ ␈↓ ␈↓εResults␈↓

␈↓ α,␈↓␈↓↓␈↓&CHAPTER 7␈↓)αβ ¬ ␈↓ ␈↓εEvaluating AM␈↓
␈↓ α,␈↓   Page 106:  ␈↓εin mid last para: new, p2, SOMEWHERE␈↓
␈↓ α,␈↓   Page 111:  ␈↓εMAYBE:   Separate   into   3   lists:   definite/probable/potential applications.␈↓

␈↓ α,␈↓␈↓↓␈↓&APPENDIX 1␈↓)αβ ¬ ␈↓ ␈↓εAM's Concepts␈↓
␈↓ α,␈↓   Page 134:  ␈↓εShould there be more explanation of the bits of this code? ␈↓

␈↓ α,␈↓␈↓↓␈↓&APPENDIX 2␈↓)αβ ¬ ␈↓ ␈↓εAM's Heuristics␈↓

␈↓ α,␈↓␈↓↓␈↓&APPENDIX 3␈↓)αβ ¬ ␈↓ ␈↓εTraces of AM in Action␈↓

␈↓ α,␈↓␈↓↓␈↓&APPENDIX 4␈↓)αβ ¬ ␈↓ ␈↓εBibliography␈↓
␈↓ α,␈↓   Page 211:  ␈↓εSchlick ref?␈↓