/LMAR=0/XLINE=3/FONT#0=BASL30/FONT#1=BASB30/FONT#3=BASI30/FONT#4=BDR40/FONT#5=NGR25/FONT#6=NGR20/FONT#7=GRFX35/FONT#9=SUP/FONT#10=SUB






␈↓ ↓*␈↓∧␈↓ α
␈↓ β{AUTOMATED THEORY FORMATION

␈↓ ↓*␈↓∧␈↓ α
␈↓ ¬*IN MATHEMATICS







␈↓ ↓*␈↓↓␈↓ α
␈↓ ¬oDouglas B. Lenat



␈↓ ↓*␈↓↓␈↓ α
␈↓ ¬βArti≡cial Intelligence Laboratory

␈↓ ↓*␈↓↓␈↓ α
␈↓ ¬[Stanford University








␈↓ ↓*␈↓↓␈↓ α
␈↓ ¬␈↓βDescription as of:  January 7, 1976␈↓↓





␈↓ ↓*␈↓↓␈↓ α
␈↓ ¬2␈↓εPh.D. Dissertation Research Proposal␈↓↓


␈↓ ↓*␈↓↓␈↓ α
␈↓ ¬A␈↓∧Table of Contents␈↓↓



␈↓"β␈↓ ↓*␈↓␈↓ αz1. Objective ..................................................................................................................................␈↓ 
1 1
␈↓"β␈↓ ↓*␈↓␈↓ αz2. Why Choose Math Research? .......................................................................................␈↓ 
1 4
␈↓"β␈↓ ↓*␈↓␈↓ αz3. Potential Applications .......................................................................................................␈↓ 
1 5
␈↓"β␈↓ ↓*␈↓␈↓ αz4. Measuring Success ...............................................................................................................␈↓ 
1 6
␈↓"β␈↓ ↓*␈↓␈↓ αz5. Experimenting with AM ..................................................................................................␈↓ 
1 7
␈↓"β␈↓ ↓*␈↓␈↓ αz6. Model of Math Research .................................................................................................␈↓ 
1 8
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Details␈↓ ..............................................................................................................................␈↓ 
1 9
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Textbook-Order vs Research-Order␈↓ ..............................................................................␈↓ 
" 10
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Metaphors to other processes␈↓ ......................................................................................␈↓ 
" 19
␈↓"β␈↓ ↓*␈↓␈↓ αz7. Knowledge AM Starts With ........................................................................................␈↓ 
" 22
␈↓"β␈↓ ↓*␈↓␈↓ αz8. AM's Representation of Knowledge ........................................................................␈↓ 
" 24
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Facets that each concept might have␈↓ ............................................................................␈↓ 
" 24
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Overall View of Representation in AM␈↓ ..........................................................................␈↓ 
" 25
␈↓"β␈↓ ↓*␈↓␈↓ αz9. Flow of Control in AM ..................................................................................................␈↓ 
" 30
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Belief and Justification␈↓ ..................................................................................................␈↓ 
" 30
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               The Environment␈↓ ............................................................................................................␈↓ 
" 31
␈↓"β␈↓ ↓*␈↓␈↓ αz10. Details: Getting AM going .........................................................................................␈↓ 
" 38
␈↓"β␈↓ ↓*␈↓␈↓ αz11. Examples of Individual Modules ...........................................................................␈↓ 
" 42
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               The "EXAMPLE" module␈↓ .................................................................................................␈↓ 
" 42
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               The "COMPOSE" module␈↓ ................................................................................................␈↓ 
" 43
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Intuition for a Set␈↓ ..........................................................................................................␈↓ 
" 44
␈↓"β␈↓ ↓*␈↓␈↓ αz12. Example: The Modules Interacting .......................................................................␈↓ 
" 48
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Some Early Numerical Concepts␈↓ ....................................................................................␈↓ 
" 48
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               A Hypothetical Session␈↓ ..................................................................................................␈↓ 
" 49
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Comments on the Session␈↓ .............................................................................................␈↓ 
" 50
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               The Session Revisited: An In-Depth Example␈↓ ................................................................␈↓ 
" 51
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               A Few Other Examples␈↓ ..................................................................................................␈↓ 
" 58
␈↓"β␈↓ ↓*␈↓␈↓ αz13. Parameters: Size / Timetable / Results so far ..................................................␈↓ 
" 69
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Parameters Characterizing the Magnitude of AM␈↓ ..........................................................␈↓ 
" 69
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Timetable for the AM Project␈↓ ........................................................................................␈↓ 
" 69
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Results as of 5/14/75␈↓ .................................................................................................␈↓ 
" 70
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Results as of 11/7/75␈↓ .................................................................................................␈↓ 
" 70
␈↓"β␈↓ ↓*␈↓␈↓ αz Bibliography
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Comparison to Other Systems␈↓
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Books and Memos␈↓
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Articles␈↓
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Acknowledgements␈↓
␈↓"β␈↓ ↓*␈↓␈↓ αz Appendix 1: Background for readers unfamiliar with Beings
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               BEINGs and Experts␈↓
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Internal Design of BEINGs␈↓
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               BEINGs Interacting␈↓
␈↓"β␈↓ ↓*␈↓␈↓ αz␈↓ε               Aspects of BEINGs Systems␈↓





␈↓ ↓*␈↓␈↓ ε␈↓∧␈↓&1. Objective␈↓)αβ␈↓


␈↓ ↓*␈↓Throughout␈αall␈αof␈α
science,␈αone␈αof␈αthe␈α
most␈αimportant␈αissues␈αis␈α
that␈αof␈α␈↓β␈↓&theory␈αformation:␈↓)αβ␈↓␈α
 concept␈αacquisition,

␈↓ ↓*␈↓how␈α∂to␈α⊂recognize␈α∂potentially␈α⊂related␈α∂concepts,␈α⊂how␈α∂to␈α⊂tie␈α∂such␈α⊂concepts␈α∂together␈α⊂productively,␈α∂how␈α⊂to␈α∂use

␈↓ ↓*␈↓intuition,␈αhow␈αto␈αchoose,␈αwhen␈αto␈αgive␈αup␈αand␈αtry␈αanother␈αapproach,␈αhow␈αto␈αextend,␈αwhen␈αto␈αde≡ne,␈αwhat␈αto

␈↓ ↓*␈↓examine, what to ignore,...  These questions are di≠cult to ␈↓βask␈↓ precisely, even in a single domain.


␈↓ ↓*␈↓For␈α⊂my␈α⊂dissertation,␈α⊃I␈α⊂am␈α⊂investigating␈α⊃creative␈α⊂theory␈α⊂formation␈α⊃in␈α⊂mathematics:␈α⊂how␈α⊃math␈α⊂researchers

␈↓ ↓*␈↓propose interesting yet plausible hypotheses, test them, and develop them into interesting theories.␈↓	1␈↓


␈↓ ↓*␈↓The␈α∩experimental␈α∩␈↓βvehicle␈↓␈α∩of␈α∩my␈α∩research␈α∩is␈α∩a␈α∩system,␈α∩a␈α∩computer␈α∩program␈α∩called␈α∩AM␈α∩(for␈α∩␈↓↓␈↓&A␈↓)αβ␈↓utomated

␈↓ ↓*␈↓␈↓↓␈↓&M␈↓)αβ␈↓athematician),␈α↔which␈α↔can␈α↔actually␈α↔do␈α_simple␈α↔mathematical␈α↔research:␈α↔propose␈α↔new␈α_de≡nitions␈↓	2␈↓␈α↔and

␈↓ ↓*␈↓axiomatizations,␈↓	3␈↓␈α
 propose␈α
and␈α∞(less␈α
importantly)␈α
prove␈α∞conjectures,␈↓	4␈↓␈α
use␈α
intuition␈↓	5␈↓␈α∞to␈α
justify␈α
beliefs␈α∞and␈α
to

␈↓ ↓*␈↓understand␈α∪new␈α∩observations,␈α∪and␈α∪evaluate␈α∩concepts␈α∪and␈α∪theories␈α∩for␈α∪aesthetic␈↓	6␈↓␈α∪interestingness␈↓	7␈↓.␈α∩ AM's

␈↓ ↓*␈↓activities␈αall␈αserve␈α
to␈αexpand␈αAM␈αitself,␈α
to␈αenlarge␈αupon␈α
a␈αgiven␈αbody␈αof␈α
mathematical␈αknowledge;␈αdue␈αto␈α
the

␈↓ ↓*␈↓enormity␈αof␈αthe␈α"search␈αspace"␈αinvolved,␈αAM␈αmust␈αuse␈αjudgmental␈αcriteria␈αto␈αguide␈αdevelopment␈αin␈αthe␈αmost
␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓promising direction.␈↓ ↓*␈↓ε␈↓	1␈↓ε␈α
A␈α
mathematical␈α
theory␈αencompasses␈α
(i)␈α
a␈α
basis␈α
of␈αprimitive␈α
objects␈α
and␈α
activities,␈α(ii)␈α
a␈α
foundation␈α
of␈α
axiomatic␈αconstraints␈α
on␈α
the␈α
basis,␈α(iii)␈α
a
␈↓ ↓*␈↓ε␈↓ αJbody␈α	of␈α	definitions␈αλtying␈α	together␈α	basic␈α	and␈αλalready-defined␈α	concepts,␈α	(iv)␈α	a␈αλbody␈α	of␈α	theorems␈α	which␈αλare␈α	implied␈α	by␈α	the␈αλfoundation
␈↓ ↓*␈↓ε␈↓ αJand the definitions, (v) some interpretations in terms of either reality or  other theories.
␈↓ ↓*␈↓ε␈↓	2␈↓ε  By combining or modifying existing concepts
␈↓ ↓*␈↓ε␈↓	3␈↓ε  Based on AM's earlier successes and/or on simulated real-world situations
␈↓ ↓*␈↓ε␈↓	4␈↓ε␈α
 Using␈α
the␈α
heuristics␈α
discussed␈α
by␈α
Polya␈α
(e.g.,␈α
analogy,␈α∞inductive␈α
inference␈α
from␈α
empirical␈α
evidence,␈α
planning␈α
in␈α
an␈α
abstract␈α∞space,␈α
double
␈↓ ↓*␈↓ε␈↓ αJinduction)
␈↓ ↓*␈↓ε␈↓	5␈↓ε␈α	 Intuition␈α	will␈α	mean␈α	 simulated␈α	real-world␈α	scenarios.␈α	These␈α	are␈α	predominantly␈α	visual␈α	analogies␈α	of␈α	current␈α	system␈α	problems.␈α	From␈α	them,␈α	the␈αλsystem
␈↓ ↓*␈↓ε␈↓ αJcan extract "answers" which, while not justifiable, are quite often correct.
␈↓ ↓*␈↓ε␈↓	6␈↓ε Aesthetics has components of harmony, unity, elegance, simplicity, closure, etc.
␈↓ ↓*␈↓ε␈↓	7␈↓ε␈α∞ Interestingness␈α∞will␈α
be␈α∞evaluated␈α∞locally;␈α
e.g.,␈α∞the␈α∞concept␈α
named␈α∞"COMPOSITION"␈α∞knows␈α
several␈α∞features␈α∞which␈α
indicate␈α∞when␈α∞any␈α
given
␈↓ ↓*␈↓ε␈↓ αJcomposition␈α	of␈α
relations␈α	 is␈α
interesting;␈α	one␈α
of␈α	these␈α
is␈α	"if␈α
the␈α	domain␈α
and␈α	range␈α
of␈α	the␈α
resultant␈α	composition␈α
are␈α	equal␈α
sets,␈α	and
␈↓ ↓*␈↓ε␈↓ αJsuch domain/range equality does ␈↓¬␈↓)αβ hold true for either of the two constituent relations of the composition".
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  2␈↓

␈↓ ↓*␈↓More␈α⊂speci≡cally,␈α∂AM␈α⊂will␈α∂be␈α⊂given␈α⊂150␈α∂particular␈α⊂␈↓βpremathematical␈↓␈↓	8␈↓␈α∂concepts,␈↓	9␈↓␈α⊂and␈α∂will␈α⊂then␈α⊂study␈α∂their

␈↓ ↓*␈↓facets, occasionally deciding that some facet is worth naming and storing as a new concept.


␈↓ ↓*␈↓There␈α
is␈αno␈α
speci≡c␈α
target␈αor␈α
goal␈αthat␈α
AM␈α
should␈αacheive␈↓	10␈↓␈α
The␈αbasic␈α
criteria␈α
for␈αevaluating␈α
AM␈α
will␈αbe

␈↓ ↓*␈↓the␈αtraditional␈αharsh␈αstandards␈αused␈αto␈α
judge␈αany␈αmathematical␈αwork:␈αare␈αthe␈αnew␈α
mini-theories␈αinteresting,

␈↓ ↓*␈↓aesthetic, useful,...?


␈↓ ↓*␈↓To␈α∞indicate␈α∞the␈α∂level␈α∞of␈α∞sophistication␈α∞--␈α∂and␈α∞naivete␈α∞--␈α∞expected␈α∂of␈α∞AM,␈α∞a␈α∞sketch␈α∂of␈α∞one␈α∞possible␈α∂line␈α∞of

␈↓ ↓*␈↓development␈αwill␈αbe␈αpresented.␈α Some␈α
guidance␈αby␈αa␈αhuman␈αuser␈αmight␈α
be␈αrequired␈αto␈αpush␈αAM␈αin␈α
such␈αa

␈↓ ↓*␈↓"traditional"␈α⊃direction.␈α⊃ AM␈α⊃soon␈α∩develops␈α⊃the␈α⊃concepts␈α⊃of␈α∩singleton,␈α⊃function,␈α⊃inverse␈α⊃relation,␈α∩and␈α⊃set-

␈↓ ↓*␈↓equivalence␈α
(Cardinality).␈α
 These␈α
pave␈α
 the␈α
way␈α
to␈α
the␈α
domain␈α
of␈α
(elementary)␈α
number␈α
theory.␈α Eventually,

␈↓ ↓*␈↓AM␈α∀ develops␈α∪enough␈α∀mathematical␈α∪systems␈α∀to␈α∪notice␈α∀a␈α∪common␈α∀structure,␈α∪which␈α∀are␈α∀abstracted␈α∪into

␈↓ ↓*␈↓something like the group axioms. AM then begins exploring elementary abstract algebra.


␈↓ ↓*␈↓AM␈α∂is␈α∂almost␈α∂completely␈α⊂speci≡ed␈α∂on␈α∂paper␈↓	11␈↓.␈α∂ The␈α∂control␈α⊂structure␈α∂for␈α∂the␈α∂system␈α∂is␈α⊂programmed,␈α∂and

␈↓ ↓*␈↓many␈αconcepts␈α
have␈αalready␈α
been␈αintroduced␈↓	12␈↓.␈α Some␈α
nontrivial␈αbehavior␈α
already␈αevinced␈α(November,␈α
1975)

␈↓ ↓*␈↓includes␈α(1)␈αthe␈α
concept␈αof␈α"a␈αset,␈α
all␈αpairs␈αof␈αwhose␈α
elements␈αsatisfy␈αa␈αgiven␈α
predicate";␈α(2)␈αthe␈αconcept␈α
of␈αa

␈↓ ↓*␈↓composition␈α
of␈αrelations␈α
whose␈α
initial␈αdomain␈α
and␈α
≡nal␈αrange␈α
intersect␈αeach␈α
other;␈α
(3)␈αthe␈α
predicate␈α
"sets␈αof

␈↓ ↓*␈↓equal␈αlength"␈α--␈αi.e.,␈αCardinality.␈αOf␈αcourse,␈α
many␈αuninteresting␈αnew␈αconcepts␈αhave␈αdeveloped,␈αand␈α
these␈αare

␈↓ ↓*␈↓all still too primitive to form any conclusions.  Hopefully, some de≡nite results will be obtained this Winter.

␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	8␈↓ε␈αUniversal,␈αfundamental,␈αprenumerical␈αknowledge.␈αThis␈αincludes:␈α(i)␈αmathematics:␈αelementary␈αnotions␈αof␈αsets,␈αrelations,␈αproperties,␈α (ii)␈αmethods:
␈↓ ↓*␈↓ε␈↓ αJproblems,␈α
problem-solving,␈α
proof␈αtechniques,␈α
analogy,␈α
programming,␈αstrategies,␈α
communication,␈α
(iii)␈αopaque␈α
abilities:␈α
 criteria␈αto
␈↓ ↓*␈↓ε␈↓ αJevaluate interestingness, aesthetics, utility, etc; locate relevant knowledge, manipulate abstract simulated "intuition" scenarios.
␈↓ ↓*␈↓ε␈↓	9␈↓ε␈αWhat␈αdo␈αwe␈αmean␈αby␈αa␈α"mathematical␈αconcept"?␈α A␈αfew␈αmoments␈αof␈αpondering␈αthis␈αshould␈αconvince␈αthe␈αreader␈αthat␈αthis␈αcannot␈αbe␈αanswered
␈↓ ↓*␈↓ε␈↓ αJcleanly.␈α
A␈α
circular␈α	definition␈α
of␈α
a␈α
mathematical␈α	concept␈α
might␈α
be␈α
"all␈α	the␈α
things␈α
discussed␈α
in␈α	math␈α
books";␈α
an␈α
operational␈α	definition
␈↓ ↓*␈↓ε␈↓ αJmight␈αbe␈α"whatever␈α
AM␈αknows␈αabout␈α
initially,␈αplus␈αwhatever␈α
new␈αknowledge␈αit␈α
acquires."␈α Later,␈αwe␈α
shall␈αindicate␈αthe␈αclasses␈α
of
␈↓ ↓*␈↓ε␈↓ αJconcepts␈αinvolved;␈αfor␈αnow,␈αlet␈α
us␈αjust␈αmention␈αa␈αfew␈αspecific␈α
ones␈αto␈αindicate␈αthe␈αbreadth␈α
involved.␈αEach␈αof␈αthe␈αfollowing␈αis␈α
a
␈↓ ↓*␈↓ε␈↓ αJmathematical␈αconcept:␈αSet,␈αRelation,␈α{1,␈αfrob,␈α{}},␈αZorn's␈αLemma,␈αTheory,␈αUnion,␈αProve,␈αProof,␈αTheorem,␈αThe␈αUnique␈αFactorization
␈↓ ↓*␈↓ε␈↓ αJTheorem (UFT), The proof of UFT, The methods used to prove UFT, Constructively proving existence, Associativity.
␈↓ ↓*␈↓ε␈↓	10␈↓ε␈α
 Prior␈α
research␈α
[Lenat,␈α
IJCAI75]␈α	has␈α
convinced␈α
me␈α
of␈α
the␈α	danger␈α
in␈α
choosing␈α
a␈α
too-well-defined␈α	goal.␈α
Even␈α
subconsciously,␈α
one␈α
"builds␈α
in"␈α	too
␈↓ ↓*␈↓ε␈↓ αJmuch  task-specific knowledge, instead of more general domain-specific abilities.
␈↓ ↓*␈↓ε␈↓	11␈↓ε Each of the 150 concepts to be initially supplied has about one page of information; all of this is found in the file GIVEN[TLK,DBL] at SAIL.
␈↓ ↓*␈↓ε␈↓	12␈↓ε␈α	 The␈α	latest␈α	Interlisp␈α	program␈α	is␈α	stored␈α	on␈α	the␈α	<LENAT>␈α	directory␈α	at␈α	SUMEX;␈α	it␈α	it␈α	is␈α	a␈α	group␈α	of␈α	2␈α	files:␈α	TOP5␈α	(the␈α	control␈α	functions)␈α	and␈αλCON5
␈↓ ↓*␈↓ε␈↓ αJ(the actual concepts). To run  it, simply enter Interlisp and load  those two files.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  3␈↓

␈↓ ↓*␈↓Such␈α
a␈α
project␈α∞immediately␈α
raises␈α
many␈α
issues;␈α∞they␈α
are␈α
listed␈α
below,␈α∞and␈α
each␈α
one␈α
is␈α∞then␈α
dealt␈α
with␈α∞in␈α
a

␈↓ ↓*␈↓separate section.



␈↓ ↓*␈↓↓␈↓ ↓ZWhy choose mathematics research as the domain for investigating theory formation?
␈↓ ↓*␈↓↓␈↓ ↓ZWhat are some potential concrete applications of this project?
␈↓ ↓*␈↓↓␈↓ ↓ZHow will the success of the system be measured?
␈↓ ↓*␈↓↓␈↓ ↓ZWhat experiments can be done on AM?
␈↓ ↓*␈↓↓␈↓ ↓ZWhat is our model of math research? ␈↓εTo automate something, you must have a good model for it.␈↓↓
␈↓ ↓*␈↓↓␈↓ ↓ZWhat given knowledge will AM initially start with?
␈↓ ↓*␈↓↓␈↓ ↓ZHow will this knowledge be represented?
␈↓ ↓*␈↓↓␈↓ ↓ZWhat is the control mechanism; what does AM "do"?
␈↓ ↓*␈↓↓␈↓ ↓ZExamples of individual knowledge modules. ␈↓εOne module for each concept.␈↓↓
␈↓ ↓*␈↓↓␈↓ ↓ZExamples of communities of modules interacting, developing new modules.
␈↓ ↓*␈↓↓␈↓ ↓ZWhat is the timetable for this project?   What is the current state?
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  4␈↓

␈↓ ↓*␈↓␈↓ ∧↑␈↓∧␈↓&2. Why Choose Math Research?␈↓)αβ␈↓


␈↓ ↓*␈↓As␈α
the␈α
DENDRAL␈αproject␈α
illustrated␈α
so␈α
clearly␈↓	13␈↓,␈αchoice␈α
of␈α
subject␈α
domain␈αis␈α
quite␈α
important␈αwhen␈α
studying

␈↓ ↓*␈↓how␈α∞researchers␈α∞discover␈α
and␈α∞develop␈α∞their␈α∞theories.␈α
 Mathematics␈α∞has␈α∞been␈α∞chosen␈α
as␈α∞the␈α∞domain␈α∞of␈α
this

␈↓ ↓*␈↓investigation, because


␈↓ ↓*␈↓(i)␈αIn␈αdoing␈αmath␈αresearch,␈αone␈αneedn't␈αcope␈αwith␈αthe␈αuncertainties␈αand␈αfallability␈αof␈αtesting␈αequipment;␈αthat

␈↓ ↓*␈↓is, there are no uncertainties in the data (compared to, e.g., chemical identi≡cation from mass spectrograms).


␈↓ ↓*␈↓(ii)␈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
codifying␈α
the␈αjudgmental

␈↓ ↓*␈↓criteria␈α
necessary␈αto␈α
do␈α
e≥ective␈αwork␈α
in␈α
a␈α≡eld;␈α
I␈αam␈α
enough␈α
 of␈αan␈α
expert␈α
in␈αelementary␈α
mathematics␈αso␈α
that

␈↓ ↓*␈↓I won't have to rely on external sources for guidance in formulating such heuristic rules.


␈↓ ↓*␈↓(iii)␈αA␈αhard␈αscience␈αis␈αof␈αcourse␈αeasier␈αto␈αwork␈αin␈αthan␈αa␈αsoft␈αone;␈αto␈αautomate␈αresearch␈αin␈αpsychology␈αwould

␈↓ ↓*␈↓require␈αmore␈αknowledge␈αabout␈αhuman␈αinformation␈α
processing␈αthan␈αnow␈αis␈αknown,␈αbecause␈α
psychology␈αdeals

␈↓ ↓*␈↓with␈αentities␈α
as␈αcomplex␈α
as␈αyou␈α
and␈αI.␈αAlso,␈α
in␈αa␈α
hard␈αscience,␈α
the␈α␈↓βlanguages␈↓␈α
to␈αcommunicate␈αinformation␈α
can

␈↓ ↓*␈↓be simple even though the ␈↓βmessages␈↓ themselves be sophisticated.


␈↓ ↓*␈↓(iv)␈α∪Since␈α∪mathematics␈α∀can␈α∪deal␈α∪with␈α∪any␈α∀conceivable␈α∪constructs,␈α∪a␈α∪researcher␈α∀there␈α∪is␈α∪not␈α∀limited␈α∪to

␈↓ ↓*␈↓explaining␈αobserved␈α
data.␈α This␈αreason␈α
is␈αmuch␈α
less␈αin∨uential␈αin␈α
practice␈αthan␈αwould␈α
be␈αexpected:␈α
It␈αturns

␈↓ ↓*␈↓out␈αthat␈αmost␈α
unmotivated␈αforays␈αare␈α
fruitless.␈α Most␈αof␈αthe␈α
successful,␈αinteresting␈αmathematical␈α
theories␈αare

␈↓ ↓*␈↓inspired by reality␈↓	14␈↓.








␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	13␈↓ε  see Buchanan, Feigenbaum, et. al.  Choice of subject was enabled by J. Lederberg in 1965.
␈↓ ↓*␈↓ε␈↓	14␈↓ε␈α	Or␈α
by␈α	other␈α	math␈α
theories␈α	which␈α
are␈α	well-established␈α	already.␈α
It␈α	is␈α
too␈α	philosophical␈α	an␈α
issue␈α	to␈α
argue␈α	whether␈α	or␈α
not␈α	such␈α
theories␈α	constitute
␈↓ ↓*␈↓ε␈↓ αJpart of reality, so I merely note them as a possible exeception.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  5␈↓

␈↓ ↓*␈↓␈↓ ¬~␈↓∧␈↓&3. Potential Applications␈↓)αβ␈↓


␈↓ ↓*␈↓(i)␈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⊃e≥ective␈α⊃in␈α∩other␈α⊃large

␈↓ ↓*␈↓knowledge-based systems.


␈↓ ↓*␈↓(ii)␈α∂Most␈α⊂of␈α∂the␈α⊂particular␈α∂heuristic␈α∂judgmental␈α⊂criteria␈α∂for␈α⊂interestingness,␈α∂utility,␈α∂etc.,␈α⊂might␈α∂be␈α⊂valid␈α∂in

␈↓ ↓*␈↓developing theories in other sciences.


␈↓ ↓*␈↓(iii)␈αIf␈αthe␈αrepertoire␈α
of␈αintuition␈α(simulated␈αreal-world␈αscenarios)␈α
is␈αsu≠cient␈αfor␈αAM␈αto␈α
develop␈αelementary

␈↓ ↓*␈↓concepts␈αof␈αmath,␈αthen␈αeducators␈αshould␈αensure␈αthat␈αchildren␈α(4-6␈αyears␈αold)␈αare␈αthoroughly␈αexposed␈αto␈αthose

␈↓ ↓*␈↓scenarios.␈↓	15␈↓


␈↓ ↓*␈↓(iv)␈α∩We␈α∩might␈α∩learn␈α∩something␈α∪about␈α∩how␈α∩the␈α∩theory␈α∩formation␈α∪task␈α∩changes␈α∩as␈α∩the␈α∩theory␈α∪grows␈α∩in

␈↓ ↓*␈↓sophistication.␈α∪For␈α∩example,␈α∪can␈α∩the␈α∪same␈α∩methods␈α∪which␈α∩lead␈α∪AM␈α∩from␈α∪premathematical␈α∪concepts␈α∩to

␈↓ ↓*␈↓arithmetic also lead AM from there to abstract algebra? Or are a new set of intuitions or criteria required?


␈↓ ↓*␈↓(v)␈αAn␈αunanticipated␈αbut␈αpossible␈αresult␈αwould␈αbe␈αthe␈αproofs␈αof␈α--␈αor␈αat␈αleast␈αthe␈αstatements␈αof␈α
--␈αinteresting

␈↓ ↓*␈↓new␈αtheorems␈αor␈αeven␈αwhole␈αtheories.␈α This␈αmight␈αalso␈αtake␈αthe␈αform␈αof␈αa␈αredivision␈αof␈αexisting␈αconcepts,␈αan

␈↓ ↓*␈↓alternate formulation of some established theory, etc.















␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	15␈↓ε␈αλ These␈αλtentatively␈αλinclude␈αλsituations␈αλlike␈αλseesaws,␈αλslides,␈αλpiling␈αλmarbles␈αλinto␈αλpans␈αλof␈αλa␈αλbalance␈αλscale,␈αλcomparing␈αλthe␈αλheights␈αλof␈αλtowers␈αλbuilt␈αλout␈αλof
␈↓ ↓*␈↓ε␈↓ αJcubical blocks, solving a jigsaw puzzle, etc.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  6␈↓

␈↓ ↓*␈↓␈↓ ¬;␈↓∧␈↓&4. Measuring Success␈↓)αβ␈↓


␈↓ ↓*␈↓(i)␈αBy␈αAM's␈αachievements:␈αCompare␈αthe␈αlist␈αof␈αconcepts␈αand␈αmethods␈αit␈αdevelops␈αagainst␈αthe␈αlist␈αof␈αconcepts

␈↓ ↓*␈↓and methods it began with.  Did AM ever discover anything interesting yet unknown to the user?␈↓	16␈↓


␈↓ ↓*␈↓(ii)␈α∂By␈α⊂the␈α∂route␈α∂AM␈α⊂took␈α∂to␈α∂accomplish␈α⊂these␈α∂advances:␈α∂ How␈α⊂clever,␈α∂how␈α∂circuitous,␈α⊂how␈α∂many␈α⊂of␈α∂the

␈↓ ↓*␈↓detours were quickly identi≡ed as such and abandoned?


␈↓ ↓*␈↓(iii)␈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∞guidance?␈α∂How␈α∞closely

␈↓ ↓*␈↓must␈α
he␈αguide␈α
AM?␈αWhat␈α
happens␈α
if␈αhe␈α
doesn't␈αsay␈α
anything␈αever?␈α
 When␈α
he␈αdoes␈α
want␈αto␈α
say␈αsomething,␈α
is

␈↓ ↓*␈↓there␈αan␈αeasy␈αway␈αto␈αexpress␈αthat␈αto␈αAM,␈αand␈αdoes␈αAM␈αrespond␈αwell␈αto␈αit?␈α Given␈αa␈αreasonable␈αkick␈αin␈αthe

␈↓ ↓*␈↓right␈α∂direction,␈α∞can␈α∂AM␈α∞develop␈α∂the␈α∞mini-theories␈α∂which␈α∞the␈α∂user␈α∞intended,␈α∂or␈α∞at␈α∂least␈α∂something␈α∞equally

␈↓ ↓*␈↓interesting?


␈↓ ↓*␈↓(iv)␈α∂By␈α∂its␈α∂intuitive␈α∂powers:␈α⊂when␈α∂forced␈α∂to␈α∂make␈α∂a␈α∂snap␈α⊂judgment␈α∂on␈α∂some␈α∂issue,␈α∂which␈α∂side␈α⊂does␈α∂AM

␈↓ ↓*␈↓choose,␈αand␈αwhy?␈α
 Are␈αthe␈αconjectures␈α
it␈αtries␈αto␈αprove␈α
usually␈αtrue?␈↓	17␈↓␈α How␈α
accurately␈αdoes␈αAM␈αestimate␈α
the

␈↓ ↓*␈↓di≠culty␈α∞of␈α∞tasks␈α∞it␈α∞is␈α∞considering?␈α∞ How␈α∞big␈α∞a␈α∞help␈α∞is␈α∞the␈α∞intuitive␈α∞belief␈α∞in␈α∞a␈α∞conjecture␈α∞when␈α∞trying␈α
to

␈↓ ↓*␈↓formulate␈αa␈αconstructive␈αproof␈αof␈αthat␈αconjecture?␈α Does␈αAM␈αtie␈αtogether␈α(e.g.,␈αas␈αanalogous)␈α
concepts␈αwhich

␈↓ ↓*␈↓are formally unrelated yet which bene≡t from such a tie?


␈↓ ↓*␈↓(v)␈α∞By␈α
the␈α∞ability␈α
to␈α∞perform␈α
the␈α∞experiments␈α
outlined␈α∞in␈α
the␈α∞next␈α
section.␈α∞ Regardless␈α
of␈α∞the␈α
experiments'

␈↓ ↓*␈↓outcomes, the features of AM which allow them to be carried out at all would be interesting in themselves.






␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	16␈↓ε␈α	  The␈α	"user"␈α	is␈α
a␈α	human␈α	works␈α	with␈α
AM␈α	interactively,␈α	giving␈α	it␈α
hints,␈α	commands,␈α	questions,␈α	etc.␈α
Notice␈α	that␈α	by␈α	"new"␈α
we␈α	mean␈α	new␈α	to␈α
the␈α	user,
␈↓ ↓*␈↓ε␈↓ αJnot␈α	new␈α	to␈αλMankind.␈α	 This␈α	might␈αλoccur␈α	if␈α	the␈αλuser␈α	were␈α	a␈α	child,␈αλand␈α	AM␈α	discovered␈αλsome␈α	elementary␈α	facts␈αλof␈α	arithmetic.␈α	This␈α	is␈αλnot
␈↓ ↓*␈↓ε␈↓ αJreally␈α	so␈α	provincial:␈α	 mathematicians␈α	take␈α	"new"␈α	to␈α	mean␈α	new␈α	to␈αλMankind,␈α	not␈α	new␈α	in␈α	the␈α	Universe.␈α	 I␈α	feel␈α	philosophy␈α	slipping␈α	in,␈αλso
␈↓ ↓*␈↓ε␈↓ αJthis footnote is terminated.
␈↓ ↓*␈↓ε␈↓	17␈↓ε  In fact, I hope AM tries to prove an intuitively clear yet false proposition.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  7␈↓

␈↓ ↓*␈↓␈↓ ¬α␈↓∧␈↓&5. Experimenting with AM␈↓)αβ␈↓


␈↓ ↓*␈↓Assume␈αthat␈αAM␈αhas␈αbeen␈αwritten,␈αand␈αhas␈αin␈αfact␈αdeveloped␈αsome␈αnew␈↓	18␈↓␈αconcepts␈αon␈αits␈αown.␈αHere␈αis␈αa␈αlist

␈↓ ↓*␈↓of some of the more interesting experiments that would be possible to perform, using AM:


␈↓ ↓*␈↓(i)␈αRemove␈αindividual␈αconcept␈αmodules.␈αIs␈α
performance␈αnoticably␈αdegraded?␈↓	19␈↓␈αWhich␈αconcepts␈αdoes␈αAM␈α
now

␈↓ ↓*␈↓"miss"␈α
discovering?␈α∞Is␈α
the␈α∞removed␈α
concept␈α∞later␈α
discovered␈α
anyway␈α∞by␈α
those␈α∞which␈α
are␈α∞left␈α
in␈α∞AM?␈α
 This

␈↓ ↓*␈↓should indicate the importance of each kind of concept (and method) which AM starts with.


␈↓ ↓*␈↓(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,␈αworth,

␈↓ ↓*␈↓utility, etc.  See how important each factor is in directing AM along successful routes.


␈↓ ↓*␈↓(iii)␈α∩Add␈α∩several␈α∩new␈α∩concept␈α∩modules␈α∩(and␈α∩a␈α∩few␈α∩new␈α∩intuitions)␈α∩and␈α∩see␈α∩if␈α∩AM␈α∩can␈α∩work␈α∪in␈α∩some

␈↓ ↓*␈↓unanticipated ≡eld of mathematics (like graph theory or calculus).


␈↓ ↓*␈↓(iv)␈α
Add␈α
several␈α∞new␈α
intuitions,␈α
and␈α∞see␈α
if␈α
AM␈α
can␈α∞develop␈α
nonmathematical␈α
theories␈α∞(elementary␈α
physics,

␈↓ ↓*␈↓program␈α
veri≡cation).␈α
This␈α
would␈αalso␈α
require␈α
limiting␈α
AM's␈αfreedom␈α
to␈α
"ignore␈α
a␈αgiven␈α
body␈α
of␈α
data␈αand

␈↓ ↓*␈↓move on to something more interesting".









␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	18␈↓ε New to AM, not to the world. It would be quite accidental if AM proved  theorems unknown to Mankind.
␈↓ ↓*␈↓ε␈↓	19␈↓ε␈α
AM␈α
should␈αoperate␈α
even␈α
if␈αmost␈α
of␈α
its␈αcriteria␈α
have␈α
been␈α"turned␈α
off"␈α
and␈αmost␈α
of␈α
its␈αmodules␈α
excised,␈α
so␈αlong␈α
as␈α
␈↓&any␈↓)αβ␈αparts␈α
of␈α
any␈αof␈α
the
␈↓ ↓*␈↓ε␈↓ αJmodules␈α	remain␈α	enabled.␈α	 If␈α
the␈α	remaining␈α	fragment␈α	of␈α
AM␈α	is␈α	too␈α	small,␈α	however,␈α
it␈α	may␈α	not␈α	be␈α
able␈α	to␈α	find␈α	anything␈α
interesting␈α	to
␈↓ ↓*␈↓ε␈↓ αJdo.␈α	In␈αλfact,␈α	this␈α	situation␈αλhas␈α	been␈α	encountered␈αλexperimentally,␈α	when␈αλthe␈α	first␈α	few␈αλpartially␈α	complete␈α	modules␈αλwere␈α	inserted.␈α	If␈αλonly
␈↓ ↓*␈↓ε␈↓ αJsome␈α
abilities,␈α
criteria␈α
are␈α	turned␈α
off,␈α
AM␈α
may␈α	in␈α
fact␈α
keep␈α
operating␈α	without␈α
this␈α
"uninteresting␈α
collapse".␈α	For␈α
example,␈α
if␈α
all␈α	but
␈↓ ↓*␈↓ε␈↓ αJthe␈α	formal␈α	manipulation␈α	knowledge␈α	is␈α	removed,␈α	the␈α	system␈α	should␈α	still␈α	grind␈α	out␈α	(simple)␈α	proofs.␈α	If␈α	all␈α	but␈α	the␈α	analogy␈α	and␈α	intuition
␈↓ ↓*␈↓ε␈↓ αJcriteria␈α	are␈α	excised,␈α
some␈α	plausible␈α	(but␈α	uncertain)␈α
conjectures␈α	should␈α	still␈α	be␈α
produced␈α	and␈α	built␈α	upon.␈α
 If␈α	these␈α	forces␈α
are␈α	buried
␈↓ ↓*␈↓ε␈↓ αJdeep␈αin␈αan␈αEnvironment,␈αthey␈αshould␈αbe␈αtunable␈α(by␈αthe␈αcreators)␈αto␈αalmost␈αnegligibility,␈αso␈αthe␈αsame␈αexperiments␈αcan␈αstill␈αbe
␈↓ ↓*␈↓ε␈↓ αJcarried␈α	out.␈αλThe␈α	converse␈α	situation␈αλshould␈α	also␈αλhold:␈α	although␈α	still␈αλfunctional␈α	with␈αλany␈α	module␈α	unplugged,␈αλthe␈α	performance␈α	␈↓&should␈↓)αβ␈αλbe
␈↓ ↓*␈↓ε␈↓ αJnoticably␈αλdegraded.␈αλ That␈αλis,␈αλwhile␈αλnot␈α	indispensible,␈αλeach␈αλmodule␈αλshould␈αλnontrivially␈αλhelp␈α	the␈αλothers.␈αλFor␈αλexample,␈αλthe␈αλjob␈α	of␈αλproving
␈↓ ↓*␈↓ε␈↓ αJan␈αassertion␈αshould␈αbe␈α
made␈αmuch␈αeasier␈αby␈αthe␈α
presence␈αof␈αintuitive␈αunderstanding.␈αIf␈α
a␈αconstructive␈αproof␈αis␈α
available,␈αthe
␈↓ ↓*␈↓ε␈↓ αJnecessary materials will already be sketched out for the formal methods to build upon.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  8␈↓

␈↓ ↓*␈↓␈↓ ¬∧␈↓∧␈↓&6. Model of Math Research␈↓)αβ␈↓


␈↓ ↓*␈↓In␈α
order␈α
to␈α
intelligently␈α
discuss␈α∞how␈α
to␈α
automate␈α
an␈α
activity,␈α
we␈α∞must␈α
be␈α
very␈α
clear␈α
about␈α
how␈α∞humans␈α
do

␈↓ ↓*␈↓that␈α∂activity.␈α∂Thus,␈α∂for␈α⊂AM,␈α∂we␈α∂must␈α∂begin␈α∂by␈α⊂hypothesizing␈α∂a␈α∂particular␈α∂model␈α∂of␈α⊂how␈α∂mathematicians

␈↓ ↓*␈↓actually␈αgo␈α
about␈αdoing␈α
their␈αresearch.␈α After␈α
presenting␈αour␈α
model,␈αwe␈αcan␈α
then␈αdiscuss␈α
how␈αto␈αautomate␈α
the

␈↓ ↓*␈↓processes␈α∞involved.␈α∞ Thanks␈α∞to␈α∞Polya,␈α∞Kersher,␈α∞Hadamard,␈α∞Skemp,␈α∞many␈α∞others,␈α∞and␈α∞introspection,␈α∞I␈α
have

␈↓ ↓*␈↓pieced together a tentative such information processing model for math theory formation:


␈↓ ↓*␈↓π⊂ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ *⊃ 
␈↓ ↓*␈↓␈↓ ↓J1.␈α∞The␈α∞order␈α∞of␈α∞events␈α∞in␈α∞a␈α
typical␈α∞mathematical␈α∞investigation␈α∞is:␈α∞(a)␈α∞OBSERVE:␈α∞The␈α∞observation␈α
is
␈↓ ↓*␈↓␈↓ ↓zeither␈α⊗of␈α⊗reality␈α⊗or␈α⊗of␈α∃an␈α⊗analogous,␈α⊗already-established␈α⊗mathematical␈α⊗theory.␈α⊗ (b)␈α∃NOTICE
␈↓ ↓*␈↓␈↓ ↓zREGULARITY:␈α↔Perceive␈α↔some␈α↔patterns,␈α↔some␈α↔interesting␈α↔relationships␈α↔that␈α↔appear␈α↔to␈α↔hold
␈↓ ↓*␈↓␈↓ ↓zsometimes.␈α
 Thus␈α
math␈αresearch␈α
is␈α
an␈α
␈↓βempirical␈↓␈αprocess.␈α
 (c)␈α
FORMALIZE:␈αDecide␈α
on␈α
some␈α
of␈αthe
␈↓ ↓*␈↓␈↓ ↓zobjects,␈α∞operators,␈α∂de≡nitions,␈α∞and␈α∂statements␈α∞that␈α∞the␈α∂theory␈α∞will␈α∂contain.␈α∞ (d)␈α∂FINALIZE:␈α∞Decide
␈↓ ↓*␈↓␈↓ ↓zwhich␈α⊂concepts␈α∂are␈α⊂to␈α∂be␈α⊂primitve␈α∂and␈α⊂which␈α∂aren't;␈α⊂decide␈α∂which␈α⊂statements␈α∂will␈α⊂be␈α∂considered
␈↓ ↓*␈↓␈↓ ↓zaxioms,␈α∩and␈α⊃ensure␈α∩that␈α⊃the␈α∩others␈α⊃can␈α∩in␈α⊃fact␈α∩be␈α⊃derived␈α∩from␈α⊃them.␈α∩ (e)␈α∩DEVELOP:␈α⊃What
␈↓ ↓*␈↓␈↓ ↓zadditional␈α∩theorems␈α∩can␈α⊃be␈α∩proven␈α∩from␈α⊃this␈α∩formal␈α∩system;␈α⊃do␈α∩they␈α∩correspond␈α∩to␈α⊃observable
␈↓ ↓*␈↓␈↓ ↓zphenomena␈αin␈αthe␈αdomain␈αwhich␈αmotivated␈αthis␈αnew␈αtheory?␈α When␈αnew␈αobservations␈αare␈αmade␈αin
␈↓ ↓*␈↓␈↓ ↓zthat␈αmotivating␈α
domain,␈αcan␈αthey␈α
be␈αnaturally␈αphrased␈α
as␈αformal␈α
statements␈αin␈αthis␈α
theory;␈αand␈αif␈α
so,
␈↓ ↓*␈↓␈↓ ↓zare they provable from the existing axioms and theorems?
␈↓ ↓*␈↓␈↓ ↓J2.␈α∞Notice␈α∞that␈α∞each␈α∞step␈α∞in␈α∞(1)␈α∞involves␈α∞choosing␈α
from␈α∞a␈α∞large␈α∞set␈α∞of␈α∞alternatives␈α∞--␈α∞that␈α∞is,␈α
searching.
␈↓ ↓*␈↓␈↓ ↓zThe␈α∞key␈α∞to␈α∞keeping␈α∞this␈α∞from␈α∞becoming␈α∞a␈α
blind,␈α∞explosive␈α∞search␈α∞is␈α∞the␈α∞proper␈α∞use␈α∞of␈α
evaluation
␈↓ ↓*␈↓␈↓ ↓zcriteria.␈α⊃That␈α⊃is,␈α⊃one␈α⊃must␈α⊃constantly␈α⊂choose␈α⊃the␈α⊃most␈α⊃interesting,␈α⊃aesthetically␈α⊃pleasing,␈α⊂useful,...
␈↓ ↓*␈↓␈↓ ↓zalternative␈α∂available.␈α∂ This␈α∂is␈α∞analogous␈α∂to␈α∂Dendral's␈α∂reliance␈α∞on␈α∂good␈α∂heuristics␈α∂to␈α∂constrain␈α∞the
␈↓ ↓*␈↓␈↓ ↓zstructure generator.
␈↓ ↓*␈↓␈↓ ↓J3.␈α
But␈α
many␈αof␈α
those␈α
criteria␈α
are␈αusually␈α
opposed␈α
to␈αeach␈α
other␈α
(e.g.,␈α
one␈αoften␈α
must␈α
sacri≡ce␈αelegance␈α
for
␈↓ ↓*␈↓␈↓ ↓zutility,␈αinterestingness␈αfor␈αsafety,␈αetc.).␈αHow␈αshould␈αone␈αweight␈αthese␈αfeatures␈αwhen␈αdeciding␈αwhat␈αto
␈↓ ↓*␈↓␈↓ ↓zdo␈αnext␈α
 during␈αan␈α
investigation?␈α We␈αbelieve␈α
(and␈αone␈α
goal␈αof␈αAM␈α
is␈αto␈α
test)␈αthat␈α
the␈αnon-formal
␈↓ ↓*␈↓␈↓ ↓zcriteria␈α∞(aesthetics,␈α∞interestingness,␈α∞inductive␈α∞inference␈α∞(from␈α∞empirical␈α∞evidence),␈α∂analogy,␈α∞intuitive
␈↓ ↓*␈↓␈↓ ↓zclarity,␈α≠utility)␈α≠are␈α≠much␈α≠more␈α≤important␈α≠than␈α≠formal␈α≠deductive␈α≠methods␈α≤in␈α≠developing
␈↓ ↓*␈↓␈↓ ↓zmathematically␈α∩worthwhile␈α∩theories,␈α∪and␈α∩in␈α∩avoiding␈α∪barren␈α∩diversions.␈α∩ Among␈α∪the␈α∩subjective
␈↓ ↓*␈↓␈↓ ↓zcriteria,␈α
the␈α order␈α
listed␈α
above␈αis␈α
roughly␈αtheir␈α
order␈α
of␈αimportance.␈α
However,␈αAM␈α
should␈α
have␈αa
␈↓ ↓*␈↓␈↓ ↓zdynamically␈αvariable␈α"orientation",␈αwhich␈αunder␈αcertain␈αcircumstances␈αmight␈αinduce␈αit␈αto␈αseek␈αsafety
␈↓ ↓*␈↓␈↓ ↓z(e.g., utility) rather than uncertainty (e.g., proposing an analogous proposition).
␈↓ ↓*␈↓␈↓ ↓J4.␈αThe␈α
above␈αcriteria␈α
are␈αvirtually␈α
the␈αsame␈αin␈α
all␈αdomains␈α
of␈αmathematics,␈α
and␈αat␈α
all␈αlevels.␈αEach␈α
factor
␈↓ ↓*␈↓␈↓ ↓zencourages␈αsome␈αpursuits␈αand␈αdiscourages␈α
others.␈α It␈αis␈αhoped␈αthat␈α
no␈αmodi≡cations␈αneed␈αbe␈αmade␈α
to
␈↓ ↓*␈↓␈↓ ↓zAM's judgmental scheme, as AM acquires more and more new concepts.
␈↓ ↓*␈↓␈↓ ↓J5.␈α∂For␈α∂true␈α∂understanding,␈α∂AM␈α∂should␈α∂grasp␈↓	20␈↓␈α∂each␈α∂concept␈α∂in␈α∂several␈α∂ways:␈α⊂declarative␈α∂(de≡nition),
␈↓ ↓*␈↓␈↓ ↓zoperational␈α
(how␈α
to␈α
use␈α
it),␈α
demonic␈α
(recognizing␈α
when␈α
it␈α
is␈α
relevant),␈α
exemplary␈α(especially␈α
boundary
␈↓ ↓*␈↓␈↓ ↓zexamples), and intuitive (simulated image of a real-world interpretation).
␈↓ ↓*␈↓␈↓ ↓J6.␈αProgress␈αin␈α␈↓βany␈↓␈α≡eld␈αof␈αmathematics␈α
demands␈αmuch␈αintuition␈α(and␈αsome␈αformal␈αknowledge)␈α
of␈α␈↓βmany␈↓

␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓ ↓J␈↓	20␈↓ε   Have access to, relate to, store, be able to manipulate, be able to answer questions about
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
'January 7, 1976     page  9␈↓

␈↓ ↓*␈↓␈↓ ↓zdi≥erent␈α∞"nearby"␈α∞mathematical␈α∞≡elds.␈α∞So␈α∞a␈α∂broad,␈α∞universal␈α∞core␈α∞of␈α∞intuition␈α∞must␈α∂be␈α∞established
␈↓ ↓*␈↓␈↓ ↓zbefore␈α
any␈α∞single␈α
theory␈α
can␈α∞meaningfully␈α
be␈α∞developed.␈α
 Intuition␈α
is␈α∞contrasted␈α
with␈α∞more␈α
formal
␈↓ ↓*␈↓␈↓ ↓zrepresentations␈α
by␈α
the␈α∞fact␈α
that␈α
it␈α
is␈α∞opaque␈α
(AM␈α
cannot␈α
introspect␈α∞to␈α
determine␈α
how␈α
the␈α∞result␈α
is
␈↓ ↓*␈↓␈↓ ↓zproduced) and fallable.
␈↓ ↓*␈↓π%ααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ *$ 


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&6.1. Details␈↓)αβ␈↓


␈↓ ↓*␈↓This␈α∂last␈α∞point␈α∂(6)␈α∞merits␈α∂elaboration.␈α∂I␈α∞believe␈α∂that␈α∞to␈α∂develop␈α∂any␈α∞given␈α∂≡eld␈α∞of␈α∂mathematics,␈α∂one␈α∞must

␈↓ ↓*␈↓possess␈α∂much␈α∂intuition␈α∂about␈α∂each␈α∂␈↓βpsychologically␈↓␈α⊂prerequisite␈α∂≡eld,␈↓	21␈↓␈α∂and␈α∂some␈α∂de≡nite␈α∂facts␈α⊂ about␈α∂each

␈↓ ↓*␈↓␈↓βformally␈↓␈α∞preceding␈α∞≡eld␈↓	22␈↓␈α∞ of␈α∞mathematics.␈α∞ The␈α
diagram␈α∞here␈α∞indicates␈α∞the␈α∞prerequisites␈α∞for␈α∞each␈α
domain

␈↓ ↓*␈↓which␈αmight␈α
conceivably␈αbe␈αworked␈α
in␈αby␈αAM.␈α
  To␈αstart␈αin␈α
a␈αgiven␈αdomain,␈α
some␈αknowledge␈α
should␈αexist

␈↓ ↓*␈↓about all domains which point to the given one, about all domains which point to ␈↓βthem␈↓, etc.





␈↓ ↓*␈↓␈↓βNOTE: This section, and especially the one after it, are supplementary, and may be skipped on ≡rst reading.␈↓



















␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	21␈↓ε␈α
 One␈α
which␈α
makes␈αthe␈α
new␈α
field␈α
easier␈α
to␈αlearn,␈α
which␈α
contains␈α
more␈α
concrete␈αanalogues␈α
of␈α
the␈α
ideas␈α
of␈αthe␈α
new␈α
field.␈α
For␈αexample,␈α
knowing
␈↓ ↓*␈↓ε␈↓ αJabout␈α	geometry␈α	makes␈α	it␈α	easier␈α	to␈α	learn␈α	about␈α	topology,␈α	even␈α	though␈α	topology␈α	never␈α	formally␈α	uses␈α	any␈α	results␈α	or␈α	definitions␈αλfrom
␈↓ ↓*␈↓ε␈↓ αJgeometry.
␈↓ ↓*␈↓ε␈↓	22␈↓ε  For example, arithmetic is usually formally based upon set theory, although most of us learn about them in the other order
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  10␈↓



␈↓"↓␈↓ ↓*␈↓πElementary Logic  ααα→  Theorem-Proving  ααααααααααααααα⊃
␈↓"↓␈↓ ↓*␈↓π    ↑                                                   ~
␈↓"↓␈↓ ↓*␈↓π    ~                                                   ~
␈↓"↓␈↓ ↓*␈↓π    ~                                                   ~
␈↓"↓␈↓ ↓*␈↓π    εαααααααα→  Geometry  ααα→  Topology                ~
␈↓"↓␈↓ ↓*␈↓π    ~               ~           ↑      ~                ~
␈↓"↓␈↓ ↓*␈↓π    ~               ~           ~      ~                ~
␈↓"↓␈↓ ↓*␈↓π    ~               ↓           ~      ↓                ~
␈↓"↓␈↓ ↓*␈↓π    ~      Analytic Geometry    ~   Algebraic Topology  ~
␈↓"↓␈↓ ↓*␈↓π    ~            ↑              ↓      ↑                ~
␈↓"↓␈↓ ↓*␈↓π    ~            ~ Measure Theory      ~                ~
␈↓"↓␈↓ ↓*␈↓π    ↓            ~ ↑                   ~                ~
␈↓"↓␈↓ ↓*␈↓πBoolean Algebra αβαβαα→  Abstract Algebra               ~
␈↓"↓␈↓ ↓*␈↓π    ↑            ~ ~      ~                             ↓
␈↓"↓␈↓ ↓*␈↓π    ~            ~ ↓      ~               Program Verification
␈↓"↓␈↓ ↓*␈↓π    ~       Analysis      ↓                             ↑
␈↓"↓␈↓ ↓*␈↓π    ~              ↑     Concrete Algebra               ~
␈↓"↓␈↓ ↓*␈↓π    ~              ~      ↑                             ~
␈↓"↓␈↓ ↓*␈↓π    ~              ~      ~                             ~
␈↓"↓␈↓ ↓*␈↓π    ↓              ~      ~                             ~
␈↓"↓␈↓ ↓*␈↓πSet Theory  ααα→  Arithmetic  ααα→  Number Theory       ~
␈↓"↓␈↓ ↓*␈↓π                      ~                                 ~
␈↓"↓␈↓ ↓*␈↓π                      ~                                 ~
␈↓"↓␈↓ ↓*␈↓π                      ↓                                 ~
␈↓"↓␈↓ ↓*␈↓π                Combinatorics  ←ααα→  Graph Theory  αααα$


␈↓ ↓*␈↓Each␈α
arrow␈α
represents␈α
either␈α
a␈α
psychological␈α
or␈α
a␈α
formal␈α
prerequisite:␈α
To␈α
work␈α
in␈α
the␈α
≡eld␈α
pointed␈α
␈↓&to␈↓)αβ,␈αone

␈↓ ↓*␈↓should␈α∞know␈α∞something␈α∞about␈α∞the␈α∞≡eld␈α∞pointed␈α∞␈↓&from␈↓)αβ.␈α∞ Notice␈α∞that␈α∞almost␈α∞all␈α∞the␈α∞"elementary"␈α∞branches␈α∞of

␈↓ ↓*␈↓mathematics␈α∞are␈α∂either␈α∞formally␈α∂or␈α∞psychologically␈α∂prerequisite␈α∞to␈α∞each␈α∂other.␈↓	23␈↓␈α∞So␈α∂a␈α∞broad␈α∂foundation␈α∞of

␈↓ ↓*␈↓intuition,␈αspanning␈α␈↓β␈↓&several␈↓)αβ␈↓␈αmathematical␈αand␈αreal-world␈αconcepts,␈αis␈αprerequisite␈αto␈αsophisticated␈αbehavior␈αin

␈↓ ↓*␈↓␈↓βany␈↓␈α∃branch␈α∃of␈α∃mathematics.␈α∃ This␈α⊗smacks␈α∃of␈α∃ the␈α∃idea␈α∃of␈α⊗a␈α∃"critical␈α∃mass"␈α∃of␈α∃knowledge,␈α⊗so␈α∃often

␈↓ ↓*␈↓sensationalized␈α∩in␈α⊃science␈α∩≡ction.␈α⊃In␈α∩addition␈α∩to␈α⊃expecting␈α∩that␈α⊃the␈α∩corpus␈α⊃must␈α∩exceed␈α∩some␈α⊃minimum

␈↓ ↓*␈↓absolute size, I am claiming that it must also exceed some minimum degree of richness, of breadth.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&6.2. Textbook-Order vs Research-Order␈↓)αβ␈↓


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	23␈↓ε␈α	 For␈α
example,␈α	one␈α
should␈α	have␈α
some␈α	intuition␈α
about␈α	 sets␈α
before␈α	doing␈α	Number␈α
theory,␈α	and␈α
one␈α	should␈α
have␈α	some␈α
intuition␈α	about␈α
numbers␈α	 and
␈↓ ↓*␈↓ε␈↓ αJcounting before doing Set theory.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  11␈↓

␈↓ ↓*␈↓Let␈α∂us␈α⊂elaborate␈α∂on␈α∂point␈α⊂(1)␈α∂of␈α∂the␈α⊂model.␈α∂The␈α∂idea␈α⊂there␈α∂is␈α∂that␈α⊂the␈α∂order␈α∂in␈α⊂which␈α∂a␈α⊂math␈α∂textbook

␈↓ ↓*␈↓presents a theory is almost the exact opposite of the order in which it was actually discovered and developed.

␈↓"↓␈↓ ↓*␈↓π␈↓ αJPRIMITIVES, AXIOMS ααααα→ DEFINITIONS    ←αααααααααααααα⊃
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                        STATEMENT OF LEMMA              ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                          PROOF OF LEMMA                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                     STATEMENT OF NEXT LEMMA            ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ⊂αααααααα⊃                      ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ~textbook~                      |                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ~ order  ~                      |                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ%αααααααα$                      |                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                       PROOF OF LAST LEMMA              ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                      STATEMENT OF THEOREM              ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                        PROOF OF THEOREM ααααααααααααααα$



␈↓ ↓*␈↓In␈α∞a␈α
textbook,␈α∞one␈α
introduces␈α∞some␈α
primitive␈α∞elements␈α
of␈α∞the␈α
theory,␈α∞postulates␈α
some␈α∞axioms␈α∞relating␈α
these

␈↓ ↓*␈↓objects␈α
and␈α
operations,␈α
then␈α
enters␈α
a␈α
␈↓β"De≡ne␈α
→␈αState␈α
→␈α
Prove␈↓"␈α
loop.␈α
Lemmas␈α
are␈α
stated␈α
and␈α
proved␈αbefore

␈↓ ↓*␈↓the␈α∃theorems␈α∀which␈α∃require␈α∃them.␈α∀Motivations␈α∃of␈α∀any␈α∃kind␈α∃are␈α∀a␈α∃rareity,␈α∀and␈α∃often␈α∃are␈α∀mentioned

␈↓ ↓*␈↓parenthetically␈αas␈αan␈αapplication␈αof␈αa␈αtheorem␈αwhich␈αhas␈αjust␈αbeen␈αproved.␈α As␈αan␈αexample,␈αhere␈αis␈αhow␈αmy

␈↓ ↓*␈↓sophomore textbook was organized:
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  12␈↓


␈↓"↓␈↓ ↓*␈↓π␈↓ αJSet N, Op S:N→N,
␈↓"↓␈↓ ↓*␈↓π␈↓ αJPeano␈↓'␈↓πs Axioms   αααααα→   Define +
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                    State some property of +
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                      Prove this propterty
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                           Define ␈↓¬≤␈↓π
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                           Define g.l.b
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                   State that glb always exists
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                        Prove this claim
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                          Define  TIMES
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                          Define ␈↓∧Z␈↓π
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                |
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                          Define  DIVIDES
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                          Define  PRIME
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                     State Euclid␈↓'␈↓πs Algorithm
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                                ↓
␈↓"↓␈↓ ↓*␈↓π␈↓ αJ                     Prove Euclid␈↓'␈↓πs Algorithm


␈↓ ↓*␈↓It␈αbegan␈αby␈αpresenting␈αPeano's␈αaxioms,␈αcompletely␈αunmotivated,␈αthen␈αentered␈αthe␈αthe␈αDe≡ne␈α→␈αState␈α→␈αProve

␈↓ ↓*␈↓loop.␈α
 For␈α
example,␈α
the␈α
two␈α
concepts␈α
of␈α
␈↓βprime␈↓␈α
and␈α
␈↓βnon-factorizable␈α
number␈↓␈α
are␈α
de≡ned␈α
separately,␈α
even␈α
though
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  13␈↓

␈↓ ↓*␈↓a␈α∪hundred␈α∪pages␈α∪elapse␈α∪before␈α∩any␈α∪mathematical␈α∪system␈α∪is␈α∪presented␈α∩in␈α∪which␈α∪the␈α∪two␈α∪notions␈α∩don't

␈↓ ↓*␈↓completely coincide.


␈↓ ↓*␈↓This␈αpsychic␈αordering␈αis␈αrepeated,␈αin␈αmicrocosm,␈αin␈αeach␈αtextbook␈αproof.␈αFor␈αexample,␈αa␈αtypical␈αepsilon/delta

␈↓ ↓*␈↓argument␈α
will␈α
start␈αby␈α
magically␈α
proposing␈αsome␈α
function␈α
delta(epsilon),␈αwhich␈α
is␈α
just␈αthe␈α
right␈α
one␈α
for␈αthe

␈↓ ↓*␈↓approximations taken later.


␈↓ ↓*␈↓In contrast, a mathematician doing research works in almost the opposite order.

␈↓"↓␈↓ ↓*␈↓πOBSERVE x  ααααα→  NOTICE SOME INTERESTING REGULARITIES
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ↓
␈↓"↓␈↓ ↓*␈↓π             GATHER UP ALL RELATED INTERESTING RELATIONSHIPS
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ~                                      ⊂αααααααα⊃
␈↓"↓␈↓ ↓*␈↓π                                 ↓                                      ~research~
␈↓"↓␈↓ ↓*␈↓π                        STATE THESE FORMALLY                            ~ order  ~
␈↓"↓␈↓ ↓*␈↓π                                 ~                                      %αααααααα$
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ↓
␈↓"↓␈↓ ↓*␈↓π                CHOOSE THE MOST INTUITIVELY OBVIOUS OF THESE
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ↓
␈↓"↓␈↓ ↓*␈↓π                 TRY TO DERIVE ALL THE REST FROM THIS CORE
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ↓
␈↓"↓␈↓ ↓*␈↓π       IF THE CORE IS INCONSISTENT, ELIMINATE THE LEAST BELIEVED MEMBERS
␈↓"↓␈↓ ↓*␈↓π       IF ANY CAN␈↓¬'␈↓πT BE DERIVED: ENLARGE THE CORE SO THAT THEY CAN BE
␈↓"↓␈↓ ↓*␈↓π       IF ANY MEMBER OF THE CORE CAN BE DERIVED FROM THE OTHERS, ELIMINATE IT
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ↓
␈↓"↓␈↓ ↓*␈↓π           CALL THE CORE "AXIOMS", AND THE REST "THEOREMS"
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ~
␈↓"↓␈↓ ↓*␈↓π                                 ↓
␈↓"↓␈↓ ↓*␈↓π       TRY TO DERIVE PREVIOUSLY UNSUSPECTED THEOREMS, FORMALLY
␈↓"↓␈↓ ↓*␈↓π                IF SO: TRY TO REINTERPRET IN TERMS OF x
␈↓"↓␈↓ ↓*␈↓π       TRY TO FIND NEW RELATIONSHIPS OCCURRING IN x
␈↓"↓␈↓ ↓*␈↓π                IF SO: TRY TO PROVE THEM USING THE EXISTING AXIOMS AND THEOREMS
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  14␈↓

␈↓ ↓*␈↓The␈α∂researcher␈α∂begins␈α∂by␈α∂observing␈α∂the␈α∂world:␈α∂he␈α∂looks␈α∂at␈α∂scenarios␈α∂from␈α∂reality␈α∂and␈α∂perhaps␈α∂also␈α∂some

␈↓ ↓*␈↓earlier,␈αalready-developed,␈αinteresting␈αmathematical␈αtheories.␈α He␈αnotices␈αsome␈αinteresting␈αfacts,␈αeither␈αby␈αraw

␈↓ ↓*␈↓perception␈αof␈αregularities␈αor␈αby␈αanalogy␈αto␈αsome␈αinteresting␈αpattern␈αin␈αanother␈αtheory.␈αOf␈αthese␈αobservations,

␈↓ ↓*␈↓he␈αchooses␈α
a␈αsmall␈αset␈α
of␈αthe␈α
most␈αintuitively␈αclear␈α
ones,␈αand␈α
tries␈αto␈αderive␈α
the␈αother␈α
observations␈αformally

␈↓ ↓*␈↓from␈αthis␈α
chosen␈αcore.␈αAfter␈α
several␈αpasses,␈αhe␈α
will␈αsucceed;␈α
the␈α≡nal␈αchosen␈α
core␈αhe␈αcalls␈α
axioms,␈αand␈αthe␈α
rest

␈↓ ↓*␈↓he calls theorems.  ␈↓	24␈↓


␈↓ ↓*␈↓Let's␈αmake␈αthis␈αmore␈αconcrete,␈αby␈αconsidering␈αhow␈αa␈αmathematician␈αmight␈αdiscover␈αand␈αdevelop␈αthe␈αsimplest

␈↓ ↓*␈↓numerical concepts, the ones my textbook introduced and developed in such a polished way.


␈↓ ↓*␈↓Assume␈αthat␈αhe␈αpossesses␈αour␈αpreviously-mentioned␈αprenumerical␈αbackground,␈αbut␈αnothing␈αmore.␈α He␈αbegins

␈↓ ↓*␈↓by␈α⊃considering␈α⊃the␈α⊂two␈α⊃activities:␈α⊃Comparing␈α⊃piles␈α⊂of␈α⊃marbles␈α⊃using␈α⊃a␈α⊂pan␈α⊃balance,␈α⊃and␈α⊃comparing␈α⊂the

␈↓ ↓*␈↓heights of towers constructed from toy blocks.
























␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	24␈↓ε␈αλ Once␈α	axiomatized,␈αλnew␈α	theorems␈αλmay␈α	be␈αλproposed␈αλsyntactically:␈α	the␈αλresearcher␈α	then␈αλmay␈α	check␈αλeach␈αλconjecture␈α	against␈αλthe␈α	roots␈αλof␈α	the␈αλtheory
␈↓ ↓*␈↓ε␈↓ αJ(both␈α	real-world␈αλand␈α	analogous).␈α	Additional␈αλobservations␈α	can␈αλlater␈α	be␈α	checked␈αλagainst␈α	the␈αλtheory,␈α	to␈α	see␈αλif␈α	they␈αλalso␈α	can␈α	be␈αλderived
␈↓ ↓*␈↓ε␈↓ αJfrom the axioms.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  15␈↓


␈↓"↓␈↓ ↓*␈↓π␈↓ α~               /~\                                     /~\
␈↓"↓␈↓ ↓*␈↓π␈↓ α~              / ~ \                                   / ~ \
␈↓"↓␈↓ ↓*␈↓π␈↓ α~             /  ~  \                                 /  ~  \
␈↓"↓␈↓ ↓*␈↓π␈↓ α~            /   ~   \                               /   ~   \
␈↓"↓␈↓ ↓*␈↓π␈↓ α~           /    ~    \                             /    ~    \
␈↓"↓␈↓ ↓*␈↓π␈↓ α~          /     ~     \                           /     ~     \
␈↓"↓␈↓ ↓*␈↓π␈↓ α~         /      ~      \                     oo  /      ~      \  oo
␈↓"↓␈↓ ↓*␈↓π␈↓ α~    αααα/       ~       \αααα               αααα/       ~       \αααα
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                ~                                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                ~                                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                ~                                       ~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~    αααααααααααα∀αααααααααααα               αααααααααααα∀αααααααααααα

␈↓"↓␈↓ ↓*␈↓π␈↓ α~                               /~\
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                              / ~ \
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                             /  ~  \
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                            /   ~   \   o
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                           /    ~    \αααα
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                          /     ~                       ⊂αααααααα⊃
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                         /      ~                       ~weighing~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                        /       ~                       ~ marbles~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                       /        ~                       %αααααααα$
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                 ooo  /         ~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                 αααα/          ~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                                ~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                    ααααααααααααααααααααααααα

␈↓"↓␈↓ ↓*␈↓π␈↓ α~*********************************************************************************

␈↓"↓␈↓ ↓*␈↓π␈↓ α~          αααααα⊃
␈↓"↓␈↓ ↓*␈↓π␈↓ α~                |
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        ⊂ααααα⊃ |
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        ~     ~ |
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        ~     ~ |                                       ⊂αααααα⊃
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        εαααααλ |                                       ~piling~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        ~     ~ |                 αααααα⊃               ~blocks~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        ~     ~ |                       |               %αααααα$
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        εαααααλ |               ⊂ααααα⊃ |
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        ~     ~ |               ~     ~ |
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        ~     ~                 ~     ~
␈↓"↓␈↓ ↓*␈↓π␈↓ α~        %ααααα$                 %ααααα$
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  16␈↓

␈↓ ↓*␈↓When he tires of playing, the mathematician makes a list of the most basic features of these situations.


␈↓ ↓*␈↓␈↓ ∧c␈↓∧FUNDAMENTAL CONCEPTS␈↓

␈↓ ↓*␈↓ABOUT WEIGHING MARBLES:␈↓ εZABOUT PILING BLOCKS: ␈↓ 
-COMMON NAME:

␈↓ ↓*␈↓There are some marbles to play with.␈↓ εZThere are some blocks to play with.␈↓ _ele

␈↓ ↓*␈↓The marbles are indistinguishable.␈↓ εZThe blocks are indistinguishable.␈↓ →set

␈↓ ↓*␈↓Besides individual marbles, we can␈↓ εZBesides individual blocks, we can␈↓ svar n
␈↓ ↓*␈↓talk about a pile of marbles as a unit.␈↓ εZtalk about a tower of blocks as unit.

␈↓ ↓*␈↓We can copy a given pile.␈↓ εZWe can copy a given tower.␈↓ `copy C

␈↓ ↓*␈↓A pile of marbes and a copy of␈↓ εZA tower of blocks and a copy of
␈↓ ↓*␈↓that pile are indistinguishable.␈↓ εZthat tower are indistinguishable.␈↓ Iignore C

␈↓ ↓*␈↓To make pile heavier, add a marble.␈↓ εZWe can stack another block onto a tower.␈↓ ksucc S

␈↓ ↓*␈↓Lighten pile by removing a marble.␈↓ εZCan shorten tower by removing top block.␈↓ apred P

␈↓ ↓*␈↓We can tell if piles x,y balance.␈↓ εZCan see if towers x,y are same size.␈↓ Zequal =

␈↓ ↓*␈↓Can tell if pile x is heavier than y.␈↓ εZWe can see if tower x is higher than y.␈↓ ,␈↓¬>␈↓

␈↓ ↓*␈↓We can add one pile to another.␈↓ εZWe can stack one tower on top of another.␈↓ radd +

␈↓ ↓*␈↓Can remove a pile from a big one.␈↓ εZCan lift a tower o≥ top of a big one.␈↓ zsub -

␈↓ ↓*␈↓Can remove any marble from pile.␈↓ εZWe can only remove topmost blocks.␈↓ 	pop

␈↓ ↓*␈↓Pile might have no marbles.␈↓ εZThere might be no blocks in a tower.␈↓ lzero 0

␈↓ ↓*␈↓Can replace each marble in a pile␈↓ εZWe can replace each block in a tower
␈↓ ↓*␈↓by a copy of some other pile.␈↓ εZby a copy of some other tower.␈↓ emult x

␈↓ ↓*␈↓A lone marble is also a pile.␈↓ εZA lone block is also a tower.␈↓ tone 1

␈↓ ↓*␈↓etc.␈↓ εZetc.␈↓ ∩etc.

␈↓ ↓*␈↓***********************************************************************************


␈↓ ↓*␈↓These␈αcorrespond␈αto␈α
the␈αobjects␈αand␈α
operators␈αof␈αthe␈α
theory␈αhe␈αis␈α
about␈αto␈αdevelop.␈α
 He␈αgives␈αthem␈α
each␈αa

␈↓ ↓*␈↓name;␈αI␈αhave␈αpermitted␈αhim␈αto␈αchoose␈αthe␈αcommon␈αname␈αfor␈αeach␈αconcept,␈αbut␈αthat␈αdoesn't␈αmatter,␈αof␈αcourse.

␈↓ ↓*␈↓Notice that most of these are not usually considered primitive at all.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  17␈↓

␈↓ ↓*␈↓The␈α∃next␈α∃list␈α∀he␈α∃draws␈α∃up␈α∀contains␈α∃several␈α∃speci≡c␈α∀observations,␈α∃translated␈α∃from␈α∃the␈α∀blocks/marbles

␈↓ ↓*␈↓manipulation␈α
language␈αinto␈α
these␈αnew␈α
symbols.␈α
 Each␈αrelationship␈α
is␈αtranslated␈α
from␈αthese␈α
symbols␈α
into␈αthe

␈↓ ↓*␈↓␈↓βother␈↓␈α∩scenario's␈α∩language,␈α∩and␈α∩checked.␈α∩Thus␈α∩all␈α∩the␈α∩observations␈α∩here␈α∩are␈α∩meaningful␈α∩in␈α∩both␈α∪of␈α∩the

␈↓ ↓*␈↓manipulative domains:

␈↓ ↓*␈↓¬************************************************************************************************

␈↓ ↓*␈↓¬0=0␈↓ β
1=1␈↓ ∧z¬(0=1)␈↓ εj¬(1=0)␈↓ λZ¬(S(1)=1)␈↓ 
J¬(0=S(0))

␈↓ ↓*␈↓¬S(0)=1␈↓ β
1=S(0)␈↓ ∧zS(0)=S(0)␈↓ εjS(1)=S(1)␈↓ λZ¬(S(0)=S(1))␈↓ 
J¬(S(1)=S(0))

␈↓ ↓*␈↓¬0+0=0␈↓ β
1+1=S(1)␈↓ ∧z0+1=1␈↓ εj1+0=1␈↓ λZS(1)+0=S(1)␈↓ 
JS(0)+S(0)=S(1)

␈↓ ↓*␈↓¬0x0=0␈↓ β
1x1=1␈↓ ∧z0x1=0␈↓ εj1x0=0␈↓ λZS(S(1))x0=0␈↓ 
JS(S(1)x1=S(S(1))

␈↓ ↓*␈↓¬1>0␈↓ β
S(1)>1␈↓ ∧zS(1)>0␈↓ εj¬(0>1)␈↓ λZ¬(1>1)␈↓ 
J¬(0>0)

␈↓ ↓*␈↓¬************************************************************************************************



␈↓ ↓*␈↓Now␈α⊂comes␈α⊂some␈α⊂simple␈α⊂generalization,␈α⊂either␈α⊂directly␈α⊂from␈α⊂the␈α⊂scenarios,␈α⊂perceptually␈α⊂from␈α⊂the␈α⊃list␈α⊂just

␈↓ ↓*␈↓compiled,␈α⊂or␈α⊃syntactically␈α⊂from␈α⊃that␈α⊂list␈α⊂(e.g.,␈α⊃by␈α⊂replacing␈α⊃constants␈α⊂by␈α⊂variables).␈α⊃Again,␈α⊂ each␈α⊃of␈α⊂these

␈↓ ↓*␈↓generalizations is intuitively checked in both real-world domains.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  18␈↓


␈↓ ↓*␈↓¬*************************************************************************************************

␈↓ ↓*␈↓¬If n>0, then  D(n) can be performed; else D(n) cannot be performed.

␈↓ ↓*␈↓¬If n>0, then n>D(n)␈↓ β
S(n)>n␈↓ ∧z␈↓ εjS(n)=n+1␈↓ λZS(S(n))>n

␈↓ ↓*␈↓¬If n=m then S(n)=S(m)␈↓ ∧z␈↓ εj␈↓ λZIf S(n)=S(m) then n=m

␈↓ ↓*␈↓¬D(S(n))=n␈↓ β
␈↓ ∧zIf n>0 then S(D(n))=n␈↓ λZIf n>0 then n+m>m

␈↓ ↓*␈↓¬m+S(n)>m␈↓ β
␈↓ ∧zIf n=m then ¬(n>m)␈↓ εjIf n>m then ¬(n=m)

␈↓ ↓*␈↓¬If n=m ∧ n>r then m>r␈↓ ∧zn=n␈↓ εj¬(n>n)␈↓ λZ¬(n=S(n))

␈↓ ↓*␈↓¬If n>m ∧ m>r then n>r␈↓ ∧zIf n=m ∧ m=r then n=r␈↓ λZIf n=m ∧ r=m then r=n

␈↓ ↓*␈↓¬If n=m then m=n␈↓ β
If n>m then ¬(m>n)␈↓ ∧zn>m ∨ m>n ∨ n=m

␈↓ ↓*␈↓¬nxm = mxn␈↓ β
␈↓ ∧znxS(m)=nxm + n␈↓ εj␈↓ λZn+m=m+n

␈↓ ↓*␈↓¬n+S(m)=S(n)+m␈↓ β
␈↓ ∧z(n+m)+(r+s) = n+((s+m)+r)

␈↓ ↓*␈↓¬nx(mxr)=(nxr)xm␈↓ β
S(n)+D(m)=m+n

␈↓ ↓*␈↓¬etc.

␈↓ ↓*␈↓¬*************************************************************************************************



␈↓ ↓*␈↓Some␈α
of␈αthese␈α
are␈αnow␈α
considered␈αas␈α
axiomatic,␈α
as␈αde≡ning␈α
the␈αoperations␈α
and␈αstructures␈α
involved;␈α
the␈αrest

␈↓ ↓*␈↓are␈α
considered␈α
secondary,␈α
as␈α
theorems␈αfollowing␈α
from␈α
the␈α
axioms.␈α
This␈αprocess␈α
is␈α
a␈α
slow␈α
search.␈α
 The␈αmost

␈↓ ↓*␈↓important␈α∞game␈α∂is␈α∞not␈α∂"≡nding␈α∞minimal␈α∂axiom␈α∞sets",␈α∂but␈α∞rather␈α∞"what␈α∂else␈α∞can␈α∂we␈α∞prove␈α∂from␈α∞this␈α∂set␈α∞of

␈↓ ↓*␈↓axioms and theorems".  The AM research goal is to learn how to play this game of theory development.


␈↓ ↓*␈↓To␈α
summarize␈αidea␈α
(1):␈αThe␈α
␈↓βmotivation␈↓␈αfor␈α
a␈α
new␈αmathematical␈α
development␈αis␈α
either:␈α(a)␈α
looking␈α
at␈αreality

␈↓ ↓*␈↓and␈αat␈αthe␈αmathematics␈αyou␈αhave␈αalready␈↓	25␈↓,␈αor␈α(b)␈αgiven␈αan␈αexisting␈αtheory,␈αpropose␈αsome␈αnew␈αconjecture␈αor




␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	25␈↓ε␈α	 Then␈α	abstract␈α	out␈α	some␈α	interesting␈α	features;␈α	 formalize␈α	those␈α	into␈α	a␈α	specific␈α	set␈α	of␈α	primitive␈α	structures␈α	and␈α	relations␈α	 and␈α	axioms␈α
about␈α	those
␈↓ ↓*␈↓ε␈↓ αJbasic entities
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  19␈↓

␈↓ ↓*␈↓de≡nition,␈α⊃based␈α⊃on␈α⊃analogy␈α⊂to␈α⊃the␈α⊃real-world␈α⊃roots␈α⊂of␈α⊃this␈α⊃theory,␈↓	26␈↓␈α⊃ or␈α⊂based␈α⊃on␈α⊃analogy␈α⊃to␈α⊃a␈α⊂known

␈↓ ↓*␈↓interesting␈α⊂theorem␈α⊂in␈α⊂another␈α⊂theory,␈α⊂or␈α⊂occasionally␈α⊂based␈α⊂only␈α⊂on␈α⊂applying␈α⊂some␈α⊂fruitful␈α⊂method␈α∂and

␈↓ ↓*␈↓observing the result.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&6.3. Metaphors to other processes␈↓)αβ␈↓


␈↓ ↓*␈↓Let's␈α
consider␈α
some␈α
alternate␈α
ways␈α
of␈α
looking␈α
at␈α
such␈α
theory␈α
development,␈α
some␈α
analogies␈α
to␈α
processes␈αwith

␈↓ ↓*␈↓which you're probably more familiar.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&6.3.1. Growing a tree -- using heuristic constraints␈↓)αβ␈↓


␈↓ ↓*␈↓Once␈αmotivated,␈αthe␈αidea␈αis␈αdeveloped␈αand␈αevaluated.␈αAt␈αeach␈αmoment,␈αthe␈αresearcher␈αshould␈αbe␈αworking␈αon

␈↓ ↓*␈↓the␈αmost␈αpromising␈α
of␈αall␈αthe␈α
proposed␈αideas.␈α This␈α
process␈αis␈αthus␈α
a␈αsophisticated␈αexpansion␈α
of␈αa␈αtree,␈α
where

␈↓ ↓*␈↓new␈α
conceptual␈α
nodes␈α
are␈α
"grown"␈α
in␈α
the␈α
most␈α
promising␈α
area,␈α
and␈α
where␈α
barren␈α
dead-ends␈α∞are␈α
eventually

␈↓ ↓*␈↓"pruned".␈α∂To␈α∂do␈α∂mathematical␈α∂research␈α∂well,␈α∂it␈α⊂is␈α∂thus␈α∂necessary␈α∂and␈α∂su≠cent␈α∂to␈α∂have␈α∂good␈α⊂methods␈α∂for

␈↓ ↓*␈↓proposing␈α∂new␈α⊂concepts␈α∂from␈α⊂existing␈α∂ones,␈α⊂and␈α∂for␈α⊂deciding␈α∂how␈α⊂interesting␈α∂each␈α⊂candidate␈α∂is.␈α⊂That␈α∂is:

␈↓ ↓*␈↓e≥ective growing and pruning strategies.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&6.3.2. Exploring a space using an evaluation function␈↓)αβ␈↓


␈↓ ↓*␈↓Consider␈αour␈αcore␈αof␈αpremathematical␈αknowledge.␈α By␈αcompounding␈αthe␈αknown␈αconcepts␈αand␈α
methods,␈↓	27␈↓␈αwe

␈↓ ↓*␈↓can␈αextend␈α
this␈αfoundation␈α
a␈αlittle␈α
wherever␈αwe␈αwish.␈α
 ␈↓β<Visualize:␈αSEVERAL␈α
SHORT␈αLINES␈α
IN␈αBLACK,

␈↓ ↓*␈↓βEMANATING␈αFROM␈αA␈αCENTRAL␈αCORE>␈↓.␈α The␈αincredible␈αvariety␈αof␈αalternatives␈αto␈αinvestigate␈αincludes

␈↓ ↓*␈↓all␈α
known␈α
mathematics,␈α
much␈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

␈↓ ↓*␈↓are␈α∞the␈α∞narrow␈α
ribbons␈α∞of␈α∞known␈α∞mathematics␈α
(perhaps␈α∞with␈α∞a␈α
few␈α∞undiscovered␈α∞other␈α∞slivers).␈α
␈↓β<Visualize

␈↓ ↓*␈↓βSNAKE-LIKE LINES IN RED, twisting away from the core, intersecting>␈↓.


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	26␈↓ε  Based on patterns perceived in empirical evidence
␈↓ ↓*␈↓ε␈↓	27␈↓ε Using  abstraction from reality, analogy with existing theories, the postulational method, and problem-solving techniques
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  20␈↓

␈↓ ↓*␈↓How␈α
should␈αwe␈α
walk␈αthrough␈α
this␈αimmense␈α
space,␈αwith␈α
any␈αhope␈α
of␈αfollowing␈α
the␈αfew,␈α
slender␈α
branches␈αof

␈↓ ↓*␈↓already-established␈α∂mathematics␈α∞(or␈α∂some␈α∞equally␈α∂successful␈α∞new␈α∂≡elds)?␈α∞We␈α∂must␈α∞do␈α∂hill-climbing;␈α∂as␈α∞new

␈↓ ↓*␈↓concepts␈α∩are␈α∩formed,␈α∩decide␈α∩how␈α∩promising␈α∩they␈α∩are,␈α∩always␈α∩explore␈α∩the␈α∩currently␈α∪most-promising␈α∩new

␈↓ ↓*␈↓concept.␈αThe␈αevaluation␈αfunction␈αis␈αquite␈αnontrivial,␈αand␈αthis␈αresearch␈αmay␈αbe␈αviewed␈αas␈αan␈αattempt␈αto␈αstudy

␈↓ ↓*␈↓and␈α↔explain␈α↔and␈α⊗duplicate␈α↔the␈α↔judgmental␈α⊗criteria␈α↔people␈α↔employ.␈α⊗ My␈α↔attempts␈α↔at␈α↔codifying␈α⊗such

␈↓ ↓*␈↓"mysterious"␈αemotive␈αforces␈αas␈αintuition,␈αaesthetics,␈αutility,␈αrichness,␈αinterestingness,␈αrelevance...␈αindicate␈αthat␈αa

␈↓ ↓*␈↓large but not unmanagable collection of heuristic rules should su≠ce.


␈↓ ↓*␈↓The␈αimportant␈αvisualization␈αto␈αmake␈αis␈αthat␈αwith␈αproper␈αevaluation␈αcriteria,␈αwe␈αconvert␈αthe␈α∨at␈αpicture␈α to␈αa

␈↓ ↓*␈↓breath-taking␈α
relief␈α
map:␈α
the␈α
known␈α
lines␈α
of␈αdevelopment␈α
become␈α
mountain␈α
ranges,␈α
soaring␈α
above␈α
the␈αvast

␈↓ ↓*␈↓∨at␈α⊂plains␈α⊂of␈α∂trivia␈α⊂and␈α⊂inconsistency␈α∂below.␈α⊂ Occasionally␈α⊂an␈α∂isolated␈α⊂hill␈α⊂is␈α∂discovered␈α⊂near␈α⊂the␈α∂core;␈↓	28␈↓

␈↓ ↓*␈↓certainly␈αwhole␈α
ranges␈αlie␈α
undiscovered␈αfor␈α
long␈αperiods␈α
of␈αtime:␈↓	29␈↓␈α
the␈αterrrain␈α
far␈αfrom␈α
the␈αinitial␈α
core␈αis␈α
not

␈↓ ↓*␈↓at all explored.


␈↓ ↓*␈↓Intuition␈α∂is␈α∞like␈α∂vision,␈α∞letting␈α∂the␈α∞explorer␈α∂observe␈α∞a␈α∂distant␈α∞mountain,␈α∂long␈α∞before␈α∂he␈α∞has␈α∂conquered␈α∞its

␈↓ ↓*␈↓intricate, speci≡c challenges.


␈↓ ↓*␈↓If␈αthe␈αcriteria␈αfor␈αevaluating␈αinterestingness␈αand␈αpromise␈αare␈αgood␈αenough,␈αthen␈αit␈αshould␈αbe␈αstraightforward

␈↓ ↓*␈↓to␈α∪simply␈α∪push␈α∀o≥␈α∪in␈α∪any␈α∪direction,␈α∀locate␈α∪some␈α∪nearby␈α∪peak,␈α∀and␈α∪follow␈α∪the␈α∪mountain␈α∀range␈α∪along

␈↓ ↓*␈↓(duplicating␈α∩the␈α∩development␈α∩in␈α∩some␈α∩≡eld).␈α∩In␈α∩fact,␈α∩by␈α∩intentionally␈α∩pushing␈α∩o≥␈α∩in␈α∩apparently␈α∩barren

␈↓ ↓*␈↓directions,␈α
new␈α
ranges␈α
might␈α
be␈αenountered.␈α
If␈α
the␈α
criteria␈α
are␈α"correct",␈α
then␈α
␈↓βany␈↓␈α
new␈α
discovery␈α
the␈αsystem

␈↓ ↓*␈↓makes and likes will necessarily be interesting to humans.


␈↓ ↓*␈↓If,␈αas␈αis␈α
more␈αlikely,␈αthe␈αcriteria␈α
are␈αde≡cient,␈αthis␈αtoo␈α
will␈αtell␈αus␈αmuch.␈α
Before␈αbeginning,␈αwe␈αshall␈α
strive␈αto



␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	28␈↓ε  E.g., Knuth's ␈↓&Surreal Numbers␈↓)αβ
␈↓ ↓*␈↓ε␈↓	29␈↓ε  E.g., non-Euclidean geometries
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  21␈↓

␈↓ ↓*␈↓include␈α∞all␈α
the␈α∞obvious␈α∞factors␈α
which␈α∞enter␈α
into␈α∞judgmental␈α∞decisions,␈α
with␈α∞appropriate␈α
weights,␈α∞etc.␈α∞If␈α
the

␈↓ ↓*␈↓criteria␈αfail,␈αthen␈αwe␈αcan␈αanalyze␈αthat␈αfailure␈αand␈αlearn␈αabout␈αone␈αnonobvious␈αfactor␈αin␈αevaluating␈αsuccess␈αin

␈↓ ↓*␈↓mathematics␈α(and␈αin␈α
any␈αcreative␈αendeavor).␈α
After␈αmodifying␈αthe␈α
criteria␈αto␈αinclude␈α
this␈αnew␈αfactor,␈α
we␈αcan

␈↓ ↓*␈↓proceed again.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&6.3.3. Syntax and Semantics in Natural Language Processing␈↓)αβ␈↓


␈↓ ↓*␈↓One␈α⊂≡nal␈α⊃analogy:␈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

␈↓ ↓*␈↓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

␈↓ ↓*␈↓purely␈αsyntactic␈αprocess.␈α Then␈α
these␈αjudgmental␈αcriteria␈αI␈αam␈α
alluding␈αto␈αwould␈αcorrespond␈αto␈α
the␈αsemantic

␈↓ ↓*␈↓knowledge␈α↔associated␈α⊗with␈α↔these␈α⊗more␈α↔formal␈α⊗methods.␈α↔ Just␈α⊗as␈α↔one␈α⊗can␈α↔upgrade␈α⊗natural-language-

␈↓ ↓*␈↓understanding␈α∀by␈α∀encorporating␈α∀semantic␈α∀knowledge,␈α∀so␈α∪AM␈α∀ensures␈α∀that␈α∀it␈α∀knows␈α∀the␈α∀intuitive␈α∪and

␈↓ ↓*␈↓empirical roots of each concept in its repertoire.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  22␈↓

␈↓ ↓*␈↓␈↓ ∧e␈↓∧␈↓&7. Knowledge AM Starts With␈↓)αβ␈↓


␈↓ ↓*␈↓At␈α∞each␈α∞stage,␈α∂AM␈α∞has␈α∞some␈α∂set␈α∞of␈α∞concepts␈α∂(represented␈α∞as␈α∞modules,␈α∂clumps␈α∞of␈α∞information).␈α∂AM's␈α∞basic

␈↓ ↓*␈↓activity␈α
is␈αto␈α
≡ll␈αout␈α
the␈αexisting␈α
clumps␈αmore,␈α
and␈αdevelop␈α
new␈αclumps.␈α
Each␈αclump␈α
is␈αcalled␈α
a␈αBEING.␈α
Here

␈↓ ↓*␈↓are the names of the BEINGs (concepts, clumps, ... ) which AM will start with:



␈↓ ↓*␈↓¬(i) ␈↓↓Objects␈↓¬
␈↓ ↓*␈↓¬␈↓ ↓ZOrdered-pair,␈α~Variable,␈α~Propositional-constant,␈α~Structure,␈α~List-structure,␈α~Bag-structure,␈α~Set-structure,
␈↓ ↓*␈↓¬␈↓ α~Assertion, Axiom


␈↓ ↓*␈↓¬(ii) ␈↓↓Actives␈↓¬
␈↓ ↓*␈↓¬␈↓ ↓ZOperations:␈α≡ Compose,␈α≡Insert,␈α≡Delete,␈α≡Convert-structure,␈α≡Substitute,␈α≡Assign,␈α≡Map-structure,␈α≡Undo,
␈↓ ↓*␈↓¬␈↓ α~Reverse-ordered-pair,␈α%Rule-of-inference,␈α$Disjoin,␈α%Conjoin,␈α$Negate,␈α%Imply,␈α%Unite,␈α$Set-union,
␈↓ ↓*␈↓¬␈↓ α~Cross-product,␈α∂Common-parts,␈α∞Set-intersection,␈α∂Set-difference,␈α∂Put-in-order,␈α∞Abstract-out-similarities,
␈↓ ↓*␈↓¬␈↓ α~Contrast
␈↓ ↓*␈↓¬␈↓ ↓ZRelations:␈α→Equality,␈α→Membership,␈α→Containment,␈α→Equivalence,␈α→Equipollence,␈α→Scope,␈α~Quantification,␈α→␈↓εFor-all,
␈↓ ↓*␈↓ε␈↓ α~There-exists, Not-for-all, Never, Only-sometimes␈↓¬
␈↓ ↓*␈↓¬␈↓ ↓ZProperties: Ordered, Extreme, Properties-of-Activities


␈↓ ↓*␈↓¬(iii) ␈↓↓Higher-order Actives␈↓¬
␈↓ ↓*␈↓¬␈↓ ↓ZFind, Select, Guess, Ellipsis, Analogize, Conserve, Approximate
␈↓ ↓*␈↓¬␈↓ ↓ZExamine,␈α⊂Test,␈α⊂Assume,␈α⊂Judge,␈α⊂Define,␈α⊂Prove,␈α⊂␈↓εLogically-deduce,␈α⊂Prove-directly,␈α⊂Cases,␈α⊂Working-backwards,␈α∂Prove-indirectly,
␈↓ ↓*␈↓ε␈↓ α~Prove-universal-claims,␈α⊃Mathematical-induction,␈α⊂Prove-existential-claims,␈α⊃Prove-existence-constructively,␈α⊂Prove-existence-deductively,␈↓¬
␈↓ ↓*␈↓¬␈↓ α~Disprove, ␈↓εDisprove-constructively, Disprove-indirectly,␈↓¬
␈↓ ↓*␈↓¬␈↓ ↓ZSolve-problem, Debug, Trial-and-error, Hill-climb, Subgoal, Work-indirectly, Relations-between-problems
␈↓ ↓*␈↓¬␈↓ ↓ZCommunicate,␈α∞Communicate-with-user,␈α∞Translate-into-English-for-User,␈α
Translate-from-English-for-concepts,
␈↓ ↓*␈↓¬␈↓ α~User-model, Communicate-among-concepts, Infer-from-examples, Select-representation
␈↓ ↓*␈↓¬␈↓ ↓ZIsomorphism, Categoricity, Canonical, Interpretation-of-theory


␈↓ ↓*␈↓¬(iv) ␈↓↓Higher-order Objects␈↓¬
␈↓ ↓*␈↓¬␈↓ ↓ZStatement,␈αConjecture,␈α␈↓εStatement-of-generality,␈αStatement-of-existence,␈↓¬␈αTheorem,␈αProof,␈αCounterexample,␈αContradiction,
␈↓ ↓*␈↓¬␈↓ α~Analogy, Assumption
␈↓ ↓*␈↓¬␈↓ ↓ZProblem,␈α*Problem-to-find,␈α*Problem-to-prove,␈α*Problem-to-creatively-define,␈α*Action-problem,␈α*Bug,
␈↓ ↓*␈↓¬␈↓ α~Inference-problem
␈↓ ↓*␈↓¬␈↓ ↓ZRepresentation, Symbolic-representation, Diagram, Scenario
␈↓ ↓*␈↓¬␈↓ ↓ZMathematical-theory, Axiomatic-Foundation, Primitive-Basis, Formal-system



␈↓ ↓*␈↓The␈αmore␈αbasic␈αthe␈αinitial␈αcore␈αconcepts,␈αthe␈αmore␈α
chance␈αthere␈αis␈αthat␈αthe␈αresearch␈αwill␈αgo␈αo≥␈α
in␈αdirections

␈↓ ↓*␈↓di≥erent␈αfrom␈αhumans,␈αthe␈αmore␈αchance␈αit␈αwill␈αbe␈αa␈αwaste␈αof␈αtime,␈αand␈αthe␈αmore␈αvalid␈αthe␈αtest␈αof␈αthe␈αsearch-

␈↓ ↓*␈↓pruning␈α⊗forces.␈α∃ Thus␈α⊗we␈α∃won't␈α⊗even␈α⊗provide␈α∃AM␈α⊗with␈α∃the␈α⊗"simple"␈α∃concepts␈α⊗of␈α⊗function,␈α∃inverse,
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  23␈↓

␈↓ ↓*␈↓commutativity,␈α∞counting,␈α∞etc.␈α∞We␈α∞do␈α∞hope␈α∂that␈α∞AM␈α∞discovers␈α∞them,␈α∞or␈α∞some␈α∞equally␈α∂interesting␈α∞"low-level"

␈↓ ↓*␈↓building blocks.


␈↓ ↓*␈↓Of␈αcourse,␈αwe␈αmust␈αask␈α␈↓βprecisely␈↓␈αwhat␈αAM␈αknows␈α
about␈αeach␈αof␈αthese␈αconcepts.␈α But␈αto␈αfully␈αanswer␈αthat,␈α
we

␈↓ ↓*␈↓must␈αdigress␈αand␈α
discuss␈αhow␈αthe␈αknowledge␈α
is␈αrepresented.␈αWe␈αmust␈α
explain␈αthe␈αmodular␈α
BEINGs␈αscheme

␈↓ ↓*␈↓for␈α
representing␈α
information␈α∞about␈α
a␈α
concept.␈α∞This␈α
is␈α
treated␈α∞curtly␈α
in␈α
the␈α∞next␈α
section;␈α
a␈α∞longer,␈α
smoother

␈↓ ↓*␈↓introduction to BEINGs is found in Appendix 1.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  24␈↓

␈↓ ↓*␈↓␈↓ ∧%␈↓∧␈↓&8. AM's Representation of Knowledge␈↓)αβ␈↓


␈↓ ↓*␈↓AM␈α∞will␈α∞represent␈α∞each␈α∞concept␈α∞in␈α∞its␈α
repertoire␈α∞as␈α∞a␈α∞bundle␈α∞of␈α∞facets␈α∞␈↓β(De≡nitions,␈α∞Intuitions,␈α
Recognition,

␈↓ ↓*␈↓βUsefulness,␈α
Interestingness,␈α
Properties,...)␈↓,␈α
and␈α
each␈α
facet␈α
will␈α
be␈α
stored␈α
internally␈α
as␈α
a␈α
little␈α
program.␈α Each

␈↓ ↓*␈↓concept␈α∞will␈α∞have␈α∞precisely␈α∞the␈α∞same␈α∞set␈α∞of␈α∞25␈α∞facets.␈α∞ This␈α∞enables␈α∞us␈α∞to␈α∞assemble,␈α∞in␈α∞advance,␈α∞a␈α∞body␈α
of

␈↓ ↓*␈↓knowledge␈α
(called␈α␈↓βstrategic␈↓␈α
knowledge)␈αabout␈α
each␈αfacet.␈α
Each␈αconcept␈α
is␈αcalled␈α
a␈αBEING␈↓	30␈↓.␈α
 Depending␈αon

␈↓ ↓*␈↓how␈αyou␈αwant␈αto␈αvisualize␈αa␈αBEING,␈αits␈α  subdivisions␈αcan␈αbe␈αcalled␈αparts,␈αquestions,␈αfacets,␈αor␈αslots.␈α Part␈αP

␈↓ ↓*␈↓of BEING B will be abbreviated as B.P.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&8.1. Facets that each concept might have␈↓)αβ␈↓


␈↓ ↓*␈↓Each␈αfacet␈αprogram␈αcan␈αbe␈αviewed␈αas␈αanswering␈αa␈αcertain␈αfamily␈αof␈αquestions␈αabout␈αthe␈αBEING␈α(concept)␈αin

␈↓ ↓*␈↓which␈α∂it␈α∂is␈α∂embedded␈↓	31␈↓.␈α∂ Below␈α∂is␈α∂the␈α∂tentative␈α∂set␈α∂of␈α∂facets␈α∂that␈α∂concepts␈α∂can␈α∂have,␈α∂followed␈α∂by␈α∂a␈α∞brief

␈↓ ↓*␈↓description of what question(s) each facet answers:






















␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	30␈↓ε␈α
For␈α
historical␈α
reasons.␈α
BEINGs␈α
is␈α
a␈α
modular␈α
scheme␈α
for␈α
representing␈α
knowledge,␈α
akin␈α
to␈α
ACTORs,␈α
 PANDEMONIUM,␈α
SMALLTALK,␈αCLASSes,␈α
and
␈↓ ↓*␈↓ε␈↓ αJFRAMEs. Details about the origin of the BEINGs ideas, can be found in [Green et al], in [Lenat], or in Appendix 1 of this paper.
␈↓ ↓*␈↓ε␈↓	31␈↓ε  For example, the DEFINITION facet of COMPOSE should be able to tell if any given entity is a composition.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  25␈↓

␈↓ ↓*␈↓ε␈↓βRECOGNITION GROUPING␈↓ε  Notice when this BEING, call it B, is relevant.
␈↓ ↓*␈↓ε␈↓ β~ CHANGES                Is this rele. to producing the desired change in the world?
␈↓ ↓*␈↓ε␈↓ β~ FINAL                  What situations is this BEING rele. to bringing about?
␈↓ ↓*␈↓ε␈↓ β~ PAST                   Where is this used frequently, to advantage?
␈↓ ↓*␈↓ε␈↓ β~ IDEN {not}{quick}      {fast} tests to see if this BEING is {not} currently referred to
␈↓ ↓*␈↓ε␈↓βALTER-ONESELF GROUPING␈↓ε  Know about variations of yourself, how good you are, etc.
␈↓ ↓*␈↓ε␈↓ β~ GENERALIZATIONS        What is this a special case of? How to make this more general.
␈↓ ↓*␈↓ε␈↓ β~ SPECIALIZATIONS        Special cases of this? What new properties exist only there?
␈↓ ↓*␈↓ε␈↓ β~ BOUNDARY               What marks the limits of this concept? Why exactly there?
␈↓ ↓*␈↓ε␈↓ β~ DOMAIN/RANGE {not} Set of (what one can{'t} apply it to, what kind of thing one {never} gets)
␈↓ ↓*␈↓ε␈↓ β~ ORDERING(Complete)     What order should the parts be concentrated on (default)
␈↓ ↓*␈↓ε␈↓ β~ WORTH  Aesthetic, efficency, complexity, ubiquity, certainty, analogic utility, survival basis
␈↓ ↓*␈↓ε␈↓ β~ INTEREST               What special factors make this type of BEING interesting?
␈↓ ↓*␈↓ε␈↓ β~ JUSTIFICATION   Why believe this? Formal/intu. For thms and conjecs. What has been tried?
␈↓ ↓*␈↓ε␈↓ β~ OPERATIONS  Properties associated with BEING. What can one do to it, what happens then?
␈↓ ↓*␈↓ε␈↓βACT-ON-ANOTHER GROUPING␈↓ε  Look at part of another BEING, and perhaps do something to it.
␈↓ ↓*␈↓ε␈↓ β~␈↓CHANGE␈↓ε subgrouping of parts:
␈↓ ↓*␈↓ε␈↓ β~ BOUNDARY-OPERATIONS {not}  Ops rele. to patching {messing}up not-bdy-entities {bdy-entities}
␈↓ ↓*␈↓ε␈↓ β~ FILLIN  How to initially fill it in, when and how to augment what is there already.
␈↓ ↓*␈↓ε␈↓ β~ STRUCTURE              Whether, When, How to retructure (or split) this part.
␈↓ ↓*␈↓ε␈↓ β~ ALGORITHMS             How to compute this, do this activity. Related to Representation.
␈↓ ↓*␈↓ε␈↓ β~␈↓INTERPRET␈↓ε subgrouping of parts:
␈↓ ↓*␈↓ε␈↓ β~ CHECK                  How to examine and test out what is already there.
␈↓ ↓*␈↓ε␈↓ β~ REPRESENTATION  How should entities of type B be structured internally? Contents' format.
␈↓ ↓*␈↓ε␈↓ β~ VIEWS  (e.g., How to view any Active as an operator, function, relation, property, corres., set of tuples)
␈↓ ↓*␈↓ε␈↓βINFO GROUPING␈↓ε General information about this BEING, B, and how it fits in.
␈↓ ↓*␈↓ε␈↓ β~ DEFINITION             Several alternative formal definitions of this concept. Can be axiomatic, recursive.
␈↓ ↓*␈↓ε␈↓ β~ INTU           Analogic interp., ties to simpler objects, to reality. Opaque.
␈↓ ↓*␈↓ε␈↓ β~ TIES           Alterns. Parents/offspring. Analogies. Associated thms, conjecs, axioms, specific BEING's.
␈↓ ↓*␈↓ε␈↓ β~ EXAMPLES {not} {bdy}   Includes trivial, typical, and advanced cases of each type.
␈↓ ↓*␈↓ε␈↓ β~ CONTENTS       What is the value stored here, the actual contents of this entity.


␈↓ ↓*␈↓Let␈α⊂us␈α∂take␈α⊂the␈α∂organization␈α⊂sketched␈α⊂above␈α∂as␈α⊂␈↓βliteral␈↓;␈α∂thus␈α⊂all␈α⊂knowledge␈α∂in␈α⊂the␈α∂system␈α⊂is␈α⊂organized␈α∂in

␈↓ ↓*␈↓packets␈α∂called␈α∂"BEINGs",␈α⊂with␈α∂each␈α∂BEING␈α∂containing␈α⊂all␈α∂the␈α∂relevant␈α∂facts␈α⊂and␈α∂ideas␈α∂about␈α⊂one␈α∂single

␈↓ ↓*␈↓concept.␈α
 Each␈α
BEING␈α
has␈α
about␈α
25␈α
di≥erent␈α
slots␈α∞or␈α
"parts",␈α
with␈α
each␈α
part␈α
having␈α
a␈α
name␈α
and␈α∞a␈α
value.

␈↓ ↓*␈↓One␈α
interpretation␈α∞of␈α
this␈α∞is␈α
that␈α
the␈α∞name␈α
represents␈α∞a␈α
question,␈α
and␈α∞the␈α
value␈α∞represents␈α
how␈α∞to␈α
answer

␈↓ ↓*␈↓that␈α∩question.␈α⊃ The␈α∩BEINGs␈α∩are␈α⊃organized␈α∩into␈α∩≡ve␈α⊃categories␈α∩or␈α∩"families",␈α⊃each␈α∩containing␈α∩about␈α⊃30

␈↓ ↓*␈↓BEINGs to start with. The slots are organized into four categories, each with about 6 di≥erently-named slots.


␈↓ ↓*␈↓There␈αis␈αone␈αdistinguished␈αstructure,␈αcalled␈αCS␈α(for␈αCurrent␈αSituation),␈αwhich␈αholds␈αpointers␈αto␈αall␈αthe␈αrecent

␈↓ ↓*␈↓system␈α⊂activities,␈α∂notably␈α⊂those␈α∂which␈α⊂have␈α⊂been␈α∂suspended,␈α⊂relevant␈α∂new␈α⊂information,␈α⊂goals␈α∂outstanding,

␈↓ ↓*␈↓BEINGs␈α∂which␈α∞were␈α∂recently␈α∞judged␈α∂interesting␈α∞but␈α∂haven't␈α∞been␈α∂worked␈α∞on␈α∂since␈α∞then,␈α∂recent␈α∞comments

␈↓ ↓*␈↓from the user, etc.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&8.2. Overall View of Representation in AM␈↓)αβ␈↓


␈↓ ↓*␈↓The␈α∞two␈α∞broad␈α∞categories␈α∞of␈α∂knowledge␈α∞are␈α∞de≡nite␈α∞and␈α∞intuitive.␈α∂To␈α∞represent␈α∞the␈α∞former,␈α∞we␈α∂employ␈α∞(i)
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  26␈↓

␈↓ ↓*␈↓rules␈α∀and␈α∪assertions,␈α∀(ii)␈α∀BEINGs␈α∪grouped␈α∀into␈α∪families,␈α∀and␈α∀(iii)␈α∪opaque␈α∀Environment␈α∀functions.␈α∪To

␈↓ ↓*␈↓represent␈α∞the␈α∞latter,␈α∞we␈α∞employ␈α∞(i)␈α∞abstract␈α∞rules,␈α∞(ii)␈α∞pictures␈α∞and␈α∞examples,␈α∞and␈α∞(iii)␈α∞opaque␈α
Environment

␈↓ ↓*␈↓functions.


␈↓ ↓*␈↓Each␈αcurrently␈αpopular␈αArti≡cial␈αIntelligence␈αformalism␈αfor␈αrepresenting␈αknowledge␈αlies␈αsomewhere␈αalong␈α(or

␈↓ ↓*␈↓very␈αnear␈αto)␈αthe␈αideal␈α"intelligence␈αvs.␈αsimplicity/speed"␈αtradeo≥␈αcurve.␈α Another␈αway␈αto␈αsee␈αthis␈αis␈αto␈αpicture

␈↓ ↓*␈↓each␈α
representation␈α
as␈α∞some␈α
tradeo≥␈α
between␈α
structure␈α∞and␈α
uniformity,␈α
between␈α
declarative␈α∞and␈α
procedural

␈↓ ↓*␈↓formulations.␈α
 Each␈α
idea␈α
has␈α
its␈α
uses,␈α
and␈α
it␈α∞would␈α
be␈α
unwise␈α
to␈α
demand␈α
that␈α
any␈α
single␈α∞representation␈α
be

␈↓ ↓*␈↓used␈α⊂everywhere␈α⊂in␈α⊃a␈α⊂given␈α⊂system.␈α⊂ One␈α⊃problem␈α⊂with␈α⊂the␈α⊂alternative␈α⊃is␈α⊂how␈α⊂to␈α⊂interface␈α⊃the␈α⊂multiple

␈↓ ↓*␈↓representations.␈α
 Usually␈α
this␈αis␈α
enough␈α
to␈α
persuade␈αsystem-builders␈α
to␈α
make␈α
do␈αwith␈α
a␈α
single␈αformalism.␈α
 The

␈↓ ↓*␈↓proposed␈α
system␈α
will␈α
be␈α
pushing␈α
the␈α
limits␈α
of␈αthe␈α
available␈α
machinery␈α
in␈α
both␈α
the␈α
time␈α
and␈αspace␈α
dimensions,

␈↓ ↓*␈↓and␈α∪therefore␈α∪cannot␈α∪indulge␈α∪in␈α∪such␈α∪luxuries!␈α∩ Knowledge␈α∪used␈α∪for␈α∪di≥erent␈α∪types␈α∪of␈α∪tasks␈α∪must␈α∩be

␈↓ ↓*␈↓represented by the most suitable formalism for each task.


␈↓ ↓*␈↓BEINGs␈α∂are␈α∂higly␈α∂structured,␈α∂intelligent,␈α∂but␈α∂slow.␈α∂Rules␈α∂and␈α∂assertions␈α∂are␈α∂more␈α∂uniform␈α∂and␈α∂swift,␈α∂but

␈↓ ↓*␈↓frequently␈α∩awkward.␈α⊃Compiled␈α∩functions␈α⊃win␈α∩on␈α⊃speed␈α∩but␈α⊃lose␈α∩on␈α⊃accessability␈α∩of␈α⊃the␈α∩knowledge␈α⊃they

␈↓ ↓*␈↓contain.␈α∂ Pictures␈α⊂and␈α∂examples␈α∂are␈α⊂universal␈α∂but␈α∂ine≠cient␈α⊂in␈α∂communicating␈α∂a␈α⊂small,␈α∂speci≡c␈α⊂chunk␈α∂of

␈↓ ↓*␈↓information.


␈↓ ↓*␈↓Let␈α∞us␈α∞now␈α∞partition␈α∞the␈α
types␈α∞of␈α∞tasks␈α∞in␈α∞our␈α
system␈α∞among␈α∞these␈α∞various␈α∞representations.␈α∞ The␈α
frequent,

␈↓ ↓*␈↓opaque␈α
system␈αtasks␈α
(like␈αevaluating␈α
interestingness,␈αaesthetic␈α
appeal,␈α
deciding␈αwho␈α
should␈αbe␈α
in␈αcontrol␈α
next)

␈↓ ↓*␈↓can␈αbe␈αprogrammed␈αas␈αcompiled␈αfunctions␈α(the␈αonly␈αloss␈α
--␈αthat␈αof␈αaccessability␈αof␈αthe␈αinformation␈αinside␈α--␈α
is

␈↓ ↓*␈↓not relevant since they should be opaque anyway).


␈↓ ↓*␈↓The␈α
speci≡c␈α
math␈α
knowledge␈αhas␈α
many␈α
sophisticated␈α
duties,␈αincluding␈α
recognizing␈α
its␈α
own␈αrelevance,␈α
knowing

␈↓ ↓*␈↓how␈α∂to␈α∂apply␈α∂itself,␈α∂how␈α∂to␈α∂modify␈α∂itself,␈α∂how␈α∞to␈α∂relate␈α∂itself␈α∂to␈α∂other␈α∂chunks␈α∂of␈α∂knowledge,␈α∂etc.␈α∂It␈α∞seems

␈↓ ↓*␈↓appropriate␈αthat␈αBEINGs␈αbe␈αused␈αto␈αhold␈αand␈αorganize␈αthis␈αinformation.␈αThe␈αmain␈αcost,␈αthat␈αof␈αslowness,␈αis
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  27␈↓

␈↓ ↓*␈↓not␈α∞critical␈α∞here,␈α∞since␈α
each␈α∞individual␈α∞chunk␈α∞is␈α
used␈α∞infrequently,␈α∞and␈α∞a␈α
wrong␈α∞usage␈α∞is␈α∞far␈α∞more␈α
serious

␈↓ ↓*␈↓than␈α⊃a␈α⊂slow␈α⊃usage.␈α⊃ One␈α⊂≡nal␈α⊃factor␈α⊃in␈α⊂favor␈α⊃of␈α⊂using␈α⊃BEINGs␈α⊃here␈α⊂is␈α⊃that␈α⊃all␈α⊂the␈α⊃knowledge␈α⊃that␈α⊂is

␈↓ ↓*␈↓available␈αat␈αthe␈αtime␈αof␈αcreation␈αof␈αa␈αnew␈αBEING␈α
will␈α≡nd␈αits␈αway␈αto␈αthe␈αright␈αplace;␈αany␈αmissing␈α
knowledge

␈↓ ↓*␈↓will be conspicuous as a blank or incomplete BEING part.


␈↓ ↓*␈↓The␈αcontents␈αof␈αeach␈αpart␈αof␈αeach␈αBEING␈αis␈αcomposed␈αof␈αspecialized␈αrules,␈αassertions,␈αsimple␈αprograms,␈αand

␈↓ ↓*␈↓pointers␈αto␈αother␈αparts␈αand␈αother␈αBEINGs.␈αThe␈αknowledge␈αmay␈αhave␈αto␈αbe␈αaltered␈αfrom␈αtime␈αto␈αtime,␈αhence

␈↓ ↓*␈↓must␈α
be␈α
inspectable␈α
and␈α
interpretable␈α
meaningfully␈α
and␈αeasily,␈α
so␈α
compiled␈α
code␈α
is␈α
ruled␈α
out.␈α
To␈αdemand␈α
that

␈↓ ↓*␈↓each␈α∞part␈α∞of␈α∞each␈α∞BEING␈α∂be␈α∞itself␈α∞a␈α∞BEING␈α∞would␈α∞trivially␈α∂cause␈α∞an␈α∞in≡nite␈α∞regress.␈α∞Hence␈α∂the␈α∞reliance

␈↓ ↓*␈↓upon "intermediate" representations.


␈↓ ↓*␈↓Communication␈αbetween␈αvery␈αdi≥erent␈αentities,␈αfor␈αexample␈αbetween␈αthe␈αUser␈αand␈αa␈αBEING␈αnot␈αdesigned␈αto

␈↓ ↓*␈↓talk␈α
with␈αhim,␈α
are␈α
best␈αe≥ected␈α
via␈α
a␈αpicture␈α
language␈α
and/or␈αan␈α
examples␈α
language␈α(from␈α
which␈αthe␈α
receiver

␈↓ ↓*␈↓must␈α
infer␈α
the␈α
message).␈α
Such␈α
universal␈α
media␈α
are␈α
proposed␈α
for␈α
faltering␈α
communications,␈α
for␈α
holding␈αand

␈↓ ↓*␈↓relating intuitions of the essences of the knowledge chunks stored in the BEINGs.


␈↓ ↓*␈↓The␈α⊂representation␈α∂of␈α⊂intuitive␈α⊂knowledge␈α∂as␈α⊂pictures␈α⊂and␈α∂examples␈α⊂is␈α⊂certainly␈α∂not␈α⊂original.␈α⊂ Set␈α∂theory

␈↓ ↓*␈↓books␈α
usually␈α∞have␈α
pictures␈α∞of␈α
blobs,␈α∞or␈α
dots␈α∞with␈α
a␈α∞closed␈α
curve␈α∞around␈α
them,␈α∞representing␈α
sets.␈α∞ For␈α
our

␈↓ ↓*␈↓purposes,␈α⊃a␈α⊃set␈α⊃will␈α⊃be␈α⊂represented␈α⊃in␈α⊃many␈α⊃ways.␈α⊃ These␈α⊃include␈α⊂pointer␈α⊃structures␈α⊃for␈α⊃␈↓¬ε␈↓,␈α⊃⊂,␈α⊃and␈α⊂their

␈↓ ↓*␈↓inverses;␈α∩analytic␈α∪geometric␈α∩functions␈α∪dealing␈α∩with␈α∪sets␈α∩as␈α∪equations␈α∩representing␈α∪regions␈α∩in␈α∪the␈α∩plane;

␈↓ ↓*␈↓prototypical␈αexamples␈αof␈αsets;␈αa␈αcollection␈α
of␈αabstract␈αrules␈αfor␈αsimulating␈αthe␈αconstruction␈α
and␈αmanipulation

␈↓ ↓*␈↓of␈αsets;␈αand,␈α≡nally,␈αa␈αset␈αmight␈αbe␈αintuitively␈αrepresented␈α
as␈αa␈αsquare␈αin␈αthe␈αcartesian␈αplane.␈α All␈αthese␈αare␈α
in

␈↓ ↓*␈↓addition␈αto␈αthe␈αde≡nite␈αknowlege␈αabout␈αsets␈α(de≡nition,␈αaxioms␈αand␈αtheorems␈αabout␈αsets,␈αspeci≡c␈αanalogies␈αto

␈↓ ↓*␈↓other concepts).


␈↓ ↓*␈↓The␈α∩notion␈α∩of␈α⊃a␈α∩fuzzy␈α∩rule␈α∩will␈α⊃remain␈α∩fuzzy␈α∩throughout␈α⊃this␈α∩document.␈α∩The␈α∩basic␈α⊃idea␈α∩is␈α∩that␈α∩of␈α⊃a

␈↓ ↓*␈↓production␈αsystem,␈αwith␈αleft␈αsides␈αthat␈αcan␈αlatch␈αonto␈αalmost␈αanything,␈αwhich␈αeventually␈αgenerate␈αlots␈αof␈αlow-
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  28␈↓

␈↓ ↓*␈↓certainty␈α
results.␈α
These␈α
would␈α
augment␈α
some␈α
BEINGs'␈α
intuition␈α
parts,␈α
and␈α
when␈α
trying␈α
to␈α
relate␈α
two␈α
given

␈↓ ↓*␈↓BEINGs␈αwhich␈αboth␈αhad␈αsuch␈αfuzzy␈αabstract␈αrules,␈αone␈αmight␈αtry␈αto␈α"run"␈αthe␈αcombined␈α
production␈αsystem,

␈↓ ↓*␈↓or␈α⊂merely␈α∂to␈α⊂"compare"␈α∂the␈α⊂two␈α∂systems.␈α⊂As␈α⊂with␈α∂pictures␈α⊂and␈α∂examples,␈α⊂the␈α∂bene≡ts␈α⊂of␈α⊂universality␈α∂and

␈↓ ↓*␈↓adaptability outweigh the ine≠cencies.


␈↓ ↓*␈↓Opaque␈α∀simulations␈α∀of␈α∃(about␈α∀a␈α∀dozen)␈α∃real-world␈α∀situations␈α∀is␈α∃another␈α∀important␈α∀component␈α∃of␈α∀the

␈↓ ↓*␈↓representation␈αof␈αintuitive␈α
knowledge.␈αFor␈αexample,␈α
there␈αmight␈αbe␈α
a␈αsimulated␈αjigsaw␈α
puzzle,␈αwith␈αmodels␈α
of

␈↓ ↓*␈↓pieces,␈α
goals,␈α
rules,␈α
hints,␈α
progress,␈α
extending,␈α
etc.␈α There␈α
will␈α
be␈α
a␈α
simulated␈α
playground,␈α
with␈α
a␈αseesaw␈α
model

␈↓ ↓*␈↓that␈αwill␈αrespond␈αwith␈αwhat␈αhappens␈αwhenever␈αanything␈α
is␈αdone␈αto␈αeither␈αside␈αof␈αthe␈αseesaw.␈αThere␈α
will␈αbe

␈↓ ↓*␈↓∨amboyant␈α∪models,␈α∪like␈α∪mountain-climbing;␈α∪mundane␈α∀ones␈α∪like␈α∪playing␈α∪with␈α∪blocks;␈α∪etc.␈α∀ The␈α∪obvious

␈↓ ↓*␈↓representation for these simulations is compiled functions, which are automatically opaque.


␈↓ ↓*␈↓The␈α∃next␈α∃items␈α∃of␈α∃interest␈α∃are␈α∃which␈α∃parts␈α∃each␈α∃BEING␈α∃must␈α∃have.␈α∃In␈α∃PUP6,␈α∃each␈α∃BEING␈α∃had

␈↓ ↓*␈↓(theoretically)␈α
exactly␈α∞the␈α
same␈α
set␈α∞of␈α
parts.␈α∞In␈α
AM,␈α
each␈α∞␈↓βfamily␈↓␈α
will␈α∞have␈α
the␈α
same␈α∞set.␈α
 For␈α∞each␈α
possible

␈↓ ↓*␈↓part, we list below those families having that part:
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  29␈↓

␈↓ ↓*␈↓ε␈↓↓   ␈↓&Part Name␈↓)αβ␈↓ ∧zStatic␈↓ ε~Active␈↓ π:Static Meta␈↓ λzActive Meta␈↓ 
:Archetypical

␈↓ ↓*␈↓↓␈↓ α
␈↓¬RECOGNITION GROUPING␈↓↓
␈↓ ↓*␈↓↓␈↓ α
 CHANGES␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 FINAL␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 PAST␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 IDEN {not}{quick}␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~X

␈↓ ↓*␈↓↓␈↓ α
␈↓¬ALTER GROUPING␈↓↓
␈↓ ↓*␈↓↓␈↓ α
 GENERALIZATIONS␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX
␈↓ ↓*␈↓↓␈↓ α
 SPECIALIZATIONS␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX
␈↓ ↓*␈↓↓␈↓ α
 BOUNDARY␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	j
␈↓ ↓*␈↓↓␈↓ α
 DOMAIN/RANGE {not}␈↓ ¬:␈↓ εZX␈↓ λ*␈↓ 	jX
␈↓ ↓*␈↓↓␈↓ α
 ORDERING(Complete)␈↓ ¬:␈↓ εZX␈↓ λ*X␈↓ 	jX
␈↓ ↓*␈↓↓␈↓ α
 WORTH␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX
␈↓ ↓*␈↓↓␈↓ α
 INTEREST␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX
␈↓ ↓*␈↓↓␈↓ α
 JUSTIFICATION␈↓ ¬:␈↓ εZ␈↓ λ*X␈↓ 	j
␈↓ ↓*␈↓↓␈↓ α
 OPERATIONS␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX

␈↓ ↓*␈↓↓␈↓ α
␈↓¬ACT GROUPING␈↓↓
␈↓ ↓*␈↓↓␈↓ α
                CHANGE subgrouping of parts
␈↓ ↓*␈↓↓␈↓ α
 BOUNDARY OPERATIONS {not}␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~
␈↓ ↓*␈↓↓␈↓ α
 FILLIN␈↓ ¬:␈↓ εZ␈↓ λ*␈↓ 	j␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 STRUCTURE␈↓ ¬:␈↓ εZ␈↓ λ*␈↓ 	j␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 CHECK␈↓ ¬:␈↓ εZ␈↓ λ*␈↓ 	j␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 ALGORITHMS␈↓ ¬:␈↓ εZX␈↓ λ*␈↓ 	jX␈↓ ~
␈↓ ↓*␈↓↓␈↓ α
                INTERPRET subgrouping of parts
␈↓ ↓*␈↓↓␈↓ α
 REPRESENTATION␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~
␈↓ ↓*␈↓↓␈↓ α
 VIEWS␈↓ ¬:␈↓ εZX␈↓ λ*X␈↓ 	j␈↓ ~

␈↓ ↓*␈↓↓␈↓ α
␈↓¬INFO GROUPING␈↓↓
␈↓ ↓*␈↓↓␈↓ α
 DEFINITION␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 INTU␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 TIES␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~X
␈↓ ↓*␈↓↓␈↓ α
 EXAMPLES {not} {bdy}␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~
␈↓ ↓*␈↓↓␈↓ α
 CONTENTS␈↓ ¬:X␈↓ εZX␈↓ λ*X␈↓ 	jX␈↓ ~

␈↓ ↓*␈↓↓␈↓ α
␈↓**********************************************************************␈↓↓
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  30␈↓

␈↓ ↓*␈↓␈↓ ¬
␈↓∧␈↓&9. Flow of Control in AM␈↓)αβ␈↓


␈↓ ↓*␈↓Control␈αin␈αthe␈αsystem:␈αAs␈αAM␈αruns,␈αat␈αeach␈αmoment,␈αeach␈αconcept␈αwill␈αhave␈αonly␈αa␈αfew␈αof␈αits␈αfacet␈α
programs

␈↓ ↓*␈↓≡lled␈α⊂in;␈α⊂most␈α⊂of␈α⊃the␈α⊂facets␈α⊂of␈α⊂most␈α⊂of␈α⊃the␈α⊂concepts␈α⊂will␈α⊂be␈α⊂unknown.␈α⊃AM's␈α⊂only␈α⊂control␈α⊂structure␈α⊃is␈α⊂to

␈↓ ↓*␈↓repeatedly␈α∂choose␈α∞some␈α∂facet␈α∂of␈α∞some␈α∂concept␈α∂(some␈α∞part␈α∂of␈α∞some␈α∂BEING),␈α∂and␈α∞then␈α∂use␈α∂the␈α∞appropriate

␈↓ ↓*␈↓strategic␈α∞knowledge␈α∂to␈α∞≡ll␈α∞in␈α∂that␈α∞missing␈α∞program.␈α∂ The␈α∞strategic␈α∞knowledge␈α∂will␈α∞typically␈α∞access␈α∂and␈α∞run

␈↓ ↓*␈↓many␈α
other␈α
facet␈α
programs␈αfrom␈α
many␈α
other␈α
concepts.␈α In␈α
the␈α
course␈α
of␈αthis,␈α
new␈α
concepts␈α
may␈αbe␈α
constructed

␈↓ ↓*␈↓and␈α∂worth␈α⊂giving␈α∂names␈α∂to␈↓	32␈↓.␈α⊂ Whenever␈α∂this␈α⊂happens,␈α∂the␈α∂new␈α⊂concept␈α∂has␈α∂almost␈α⊂all␈α∂his␈α⊂facets␈α∂blank,

␈↓ ↓*␈↓which␈α⊂means␈α⊂AM␈α⊂now␈α⊂has␈α⊂about␈α⊂20␈α⊃speci≡c␈α⊂tasks␈α⊂to␈α⊂eventually␈α⊂attend␈α⊂to␈α⊂(if␈α⊂they're␈α⊃deemed␈α⊂interesting

␈↓ ↓*␈↓enough).␈αSo␈α
the␈αsystem␈αnever␈α
runs␈αout␈αof␈α
things␈αto␈α
do;␈αrather,␈αthat␈α
number␈αkeeps␈αgrowing␈α
rapidly.␈αIt␈α
is␈αthe

␈↓ ↓*␈↓judgmental criteria which must limit this otherwise-explosive growth.


␈↓ ↓*␈↓AM␈αis␈αinteractive:␈α
AM␈αinforms␈αa␈α
human␈αuser␈αof␈α
the␈αthings␈αit␈α
≡nds␈αwhich␈αit␈α
thinks␈αare␈αinteresting.␈α The␈α
user

␈↓ ↓*␈↓can␈αinterrupt␈αAM␈αand␈αinterrogate␈αit,␈αredirect␈αits␈αenergies,␈αand␈αso␈αon.␈αSince␈αthe␈αuser␈αhas␈αseveral␈αroles␈↓	33␈↓,␈αAM

␈↓ ↓*␈↓should␈αhave␈αseveral␈αmodes␈αof␈αcommunicating␈αwith␈αhim.␈↓	34␈↓␈αProtocols␈αindicate␈αthis␈αis␈αfeasable;␈αone␈αshould␈αonly

␈↓ ↓*␈↓require a few minutes' instruction before comfortably using the system.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&9.1. Belief and Justi≡cation␈↓)αβ␈↓


␈↓ ↓*␈↓The␈α∪system␈α∩will␈α∪justify␈α∪what␈α∩it␈α∪believes␈α∪with␈α∩␈↓βintuition␈↓␈α∪and␈α∩␈↓βempirical␈α∪evidence␈↓␈α∪as␈α∩well␈α∪as␈α∪with␈α∩formal

␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	32␈↓ε␈α	If␈α	one␈α
part␈α	of␈α	a␈α	BEING␈α
becomes␈α	large␈α	and␈α	interesting,␈α
it␈α	may␈α	be␈α
split␈α	into␈α	several␈α	brand␈α
new␈α	BEINGs.␈α	This␈α	is␈α
how␈α	new␈α	concepts␈α
develop.␈α	  For
␈↓ ↓*␈↓ε␈↓ αJexample,␈αλin␈αλfilling␈α	out␈αλthe␈αλExamples␈α	part␈αλof␈αλthe␈αλBEING␈α	 dealing␈αλwith␈αλComposition,␈α	the␈αλsystem␈αλwill␈αλhave␈α	to␈αλthink␈αλup␈α	new␈αλcompositions
␈↓ ↓*␈↓ε␈↓ αJand␈α
relations.␈α
If␈α
one␈α
of␈α	them␈α
turns␈α
out␈α
to␈α
be␈α	interesting,␈α
it␈α
will␈α
be␈α
made␈α
into␈α	a␈α
new␈α
BEING,␈α
and␈α
almost␈α	all␈α
of␈α
its␈α
25␈α
parts␈α
will␈α	be
␈↓ ↓*␈↓ε␈↓ αJblank.␈αSo␈αthe␈αsystem␈αwill␈αthen␈αhave␈α25␈α well-defined␈αquestions␈αto␈αtry␈αto␈αanswer␈αabout␈αa␈αspecific,␈αpromising␈αnew␈αconcept.␈αThe
␈↓ ↓*␈↓ε␈↓ αJExamples part of the Composition BEING would now contain a pointer to this new BEING.
␈↓ ↓*␈↓ε␈↓	33␈↓ε␈αCreator␈α(decide␈α
what␈αknowledge␈αAM␈α
starts␈αwith),␈αGuide␈α
(by␈αencouragement,␈αdiscouragement,␈α
suggestion,␈αhint),␈αAbsolute␈α
Authority␈α(provide
␈↓ ↓*␈↓ε␈↓ αJneeded facts which AM couldn't derive itself), Co-researcher (if AM is operating in a domain unknown to the user).
␈↓ ↓*␈↓ε␈↓	34␈↓ε␈αTraditional␈α
math␈αnotation,␈αtextbook␈α
English,␈αformal␈α(e.g.␈α
AUTOMATH␈αor␈αpredicate␈α
calculus),␈αpictorial␈α(simulated␈α
visual␈αintuitions),␈αexamples␈α
(I/O
␈↓ ↓*␈↓ε␈↓ αJpairs, traces).
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  31␈↓

␈↓ ↓*␈↓reasoning.␈α⊃This␈α⊃justi≡cation␈α⊃would␈α∩be␈α⊃recomputed␈α⊃only␈α⊃when␈α∩needed␈α⊃(e.g.,␈α⊃if␈α⊃an␈α∩apparent␈α⊃contradiction

␈↓ ↓*␈↓arose).␈↓	35␈↓␈αIntuition␈αwill␈αbe␈αsimulated␈αby␈αfunctions␈αwhich␈αare␈αopaque␈↓	36␈↓␈αand␈αfallible.␈α The␈αfootnote␈αbelow␈αgives

␈↓ ↓*␈↓a␈α
brief␈α
example␈α∞of␈α
how␈α
they␈α
might␈α∞work.␈↓	37␈↓␈α
Here␈α
is␈α∞the␈α
most␈α
serious␈α
attack␈α∞on␈α
the␈α
reliance␈α∞upon␈α
divinely-

␈↓ ↓*␈↓provided␈α
intuitive␈α
abilities:␈α
we␈α
creators␈α
might␈α
stack␈α
the␈α
deck;␈α
we␈α
might␈α
contrive␈α
just␈α
the␈α
right␈α
intuitions␈αto

␈↓ ↓*␈↓drive␈α⊃the␈α⊃worker␈α⊃toward␈α⊃making␈α⊃the␈α∩"proper"␈α⊃discoveries.␈α⊃ The␈α⊃rebuttal␈α⊃is␈α⊃two-pronged:␈α⊃≡rst,␈α∩one␈α⊃must

␈↓ ↓*␈↓assume␈α
the␈αintegrity␈α
of␈αthe␈α
creators;␈αthey␈α
must␈αstrive␈α
not␈αto␈α
anticipate␈αthe␈α
precise␈αuses␈α
that␈αwill␈α
be␈α
made␈αof

␈↓ ↓*␈↓the␈α
intuition␈α
functions.␈α∞Second,␈α
regardless␈α
of␈α∞how␈α
contrived␈α
it␈α
was,␈α∞if␈α
a␈α
small␈α∞set␈α
of␈α
intutition␈α∞models␈α
were

␈↓ ↓*␈↓found␈α∂which␈α∂are␈α∂su≠cient␈α∂to␈α∂drive␈α∂a␈α∂researcher␈α∂to␈α∂discover␈α∂a␈α∂signi≡cant␈α∂part␈α∂of␈α∂mathematics,␈α⊂that␈α∂alone

␈↓ ↓*␈↓would␈α⊂be␈α⊂an␈α⊃interesting␈α⊂discovery␈α⊂(educators␈α⊂would␈α⊃like␈α⊂to␈α⊂ensure␈α⊂that␈α⊃children␈α⊂understood␈α⊂this␈α⊃core␈α⊂of

␈↓ ↓*␈↓images, for example).


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&9.2. The Environment␈↓)αβ␈↓


␈↓ ↓*␈↓A␈αrather␈αspecialized␈α␈↓βenvironment␈↓␈αexists␈αto␈αsupport␈αthese␈αBEINGs.␈αEncoded␈αas␈αe≠cient␈αopaque␈αfunctions,␈αthe

␈↓ ↓*␈↓environment␈α∞must␈α∞oversee␈α∂the␈α∞∨ow␈α∞of␈α∞control␈α∂in␈α∞the␈α∞system␈α∞(although␈α∂the␈α∞BEINGs␈α∞themselves␈α∂make␈α∞each



␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	35␈↓ε␈αThe␈αaim␈αis␈αnot␈αto␈αbuild␈αa␈αtheorem-prover,␈αyet␈αthe␈αleap␈αfrom␈α"very␈αprobable"␈αto␈α "certain"␈αis␈αnot␈αignorable.␈α Many␈αstatements␈αare␈αinfinitely
␈↓ ↓*␈↓ε␈↓ αJprobable␈α	yet␈α	silly.␈α	Some␈αλsophisticated␈α	studies␈α	into␈α	this␈α	problem␈αλhave␈α	been␈α	done␈α	[Pietarinen]␈αλand␈α	may␈α	prove␈α	usable.␈α	 The␈αλmechanism
␈↓ ↓*␈↓ε␈↓ αJfor␈α∞belief␈α
in␈α∞each␈α∞fact,␈α
its␈α∞certainty,␈α∞should␈α
be␈α∞descriptive␈α
(a␈α∞collection␈α∞of␈α
supporting␈α∞reasons)␈α∞with␈α
a␈α∞vector␈α∞of␈α
numerical
␈↓ ↓*␈↓ε␈↓ αJprobabiities␈α	(estimated␈α	for␈α	each␈α	factor)␈α	attached.␈α	These␈α	numbers␈α	would␈α	be␈α	computed␈α	at␈α	creation␈α	of␈α	this␈α	entity,␈α	recomputed␈α	only␈α	as
␈↓ ↓*␈↓ε␈↓ αJrequired.␈α  The␈αmost␈αfundamental␈αentities␈αmay␈αhave␈α␈↓&only␈↓)αβ␈αnumerical␈αweights.␈α If␈αthe␈αweight␈αof␈αany␈αentity␈αchanges,␈α
no␈α"chasing
␈↓ ↓*␈↓ε␈↓ αJaround"␈α	need␈α	be␈α	done.␈αλContradictions␈α	are␈α	not␈α	catastrophic:␈α	they␈αλsimply␈α	indicate␈α	that␈α	the␈αλreasons␈α	supporting␈α	each␈α	of␈α	the␈αλconflicting
␈↓ ↓*␈↓ε␈↓ αJideas␈α
should␈α
be␈αreexamined,␈α
their␈α
intuitive␈αand␈α
formal␈α
justifications␈αscrutinized,␈α
until␈α
the␈α"sum"␈α
of␈α
the␈αultimate␈α
beliefs␈α
in␈αthe
␈↓ ↓*␈↓ε␈↓ αJcontradictory␈α	statements␈α	falls␈α	below␈α	unity,␈α	and␈α	until␈α	some␈α	intuitive␈α	visualization␈α	of␈α	the␈α	situation␈α	is␈α	accepted.␈α	If␈α	this␈α
never␈α	happens,
␈↓ ↓*␈↓ε␈↓ αJthen␈αa␈α
problem␈αreally␈αexists␈α
here,␈αand␈α
might␈αhave␈αto␈α
be␈αassimilated␈αas␈α
an␈αexception␈α
to␈αsome␈αrule,␈α
might␈αdecrease␈αthe␈α
reliability
␈↓ ↓*␈↓ε␈↓ αJplaced␈αλon␈αλcertain␈αλfacts␈αλand␈αλmethods,␈αλmight␈αλcause␈αλthe␈α	rejection␈αλof␈αλsome␈αλmathematical␈αλtheory␈αλas␈αλinconsistent␈αλafter␈αλall,␈αλetc.␈α	This␈αλbelief
␈↓ ↓*␈↓ε␈↓ αJalgorithm,␈α	whatever␈α	its␈α	details,␈α	should␈α	be␈α
embedded␈α	implicitly␈α	in␈α	the␈α	control␈α	environment;␈α
AM␈α	should␈α	not␈α	have␈α	the␈α	power␈α
to␈α	inspect
␈↓ ↓*␈↓ε␈↓ αJor modify it.
␈↓ ↓*␈↓ε␈↓	36␈↓ε  Their knowledge is not inspectable by the rest of the system; e.g., compiled code
␈↓ ↓*␈↓ε␈↓	37␈↓ε␈α∞ Consider␈α∞the␈α∞intuition␈α∞about␈α∞a␈α∂seesaw.␈α∞  This␈α∞is␈α∞useful␈α∞for␈α∞visualizing␈α∂anti-symmetry␈α∞and␈α∞symmetry,␈α∞associativity,␈α∞and␈α∞the␈α∂concept␈α∞of
␈↓ ↓*␈↓ε␈↓ αJmultiplication␈α
(distance␈α
x␈α	weight).␈α
Our␈α
seesaw␈α	function␈α
will␈α
accept␈α	some␈α
features␈α
of␈α	a␈α
seesaw␈α
scenario:␈α	details␈α
about␈α
each␈α	person
␈↓ ↓*␈↓ε␈↓ αJinvolved␈α
in␈α
the␈α
scene␈α
(exactly␈αwhere␈α
they␈α
were,␈α
what␈α
their␈αnames␈α
were,␈α
how␈α
much␈α
they␈αweighed)␈α
 and␈α
about␈α
the␈α
initial␈αand␈α
final
␈↓ ↓*␈↓ε␈↓ αJposition␈α	of␈α	the␈α	seesaw␈α	board.␈α	The␈α	intuition␈α	function␈α	SEESAW␈α	 would␈α	accept␈α	a␈α	␈↓&partial␈↓)αβ␈α	list␈α	of␈α	such␈α	features,␈α	and␈α	then␈α	return␈α	a␈α	fully-
␈↓ ↓*␈↓ε␈↓ αJfilled␈αλout␈αλlist.␈α	If␈αλthe␈αλfinal␈αλposition␈α	of␈αλthe␈αλseesaw␈αλ is␈α	one␈αλof␈αλthe␈αλomitted␈α	parameters,␈αλthe␈αλfunction␈αλwill␈α	compute␈αλthe␈αλsums␈αλof␈α	(weight␈αλx
␈↓ ↓*␈↓ε␈↓ αJdistance)␈α	on␈α	each␈α	side,␈α	and␈α	decide␈α	what␈α	happened␈αλto␈α	the␈α	board.␈α	If␈α	some␈α	feature␈α	like␈α	a␈αλperson's␈α	name␈α	was␈α	omitted,␈α	it␈α	will␈α	fill␈α	that␈αλin
␈↓ ↓*␈↓ε␈↓ αJby␈α	randomly␈αλchoosing␈α	some␈αλname.␈α	Of␈αλcourse,␈α	the␈αλcaller␈α	of␈αλthe␈α	function␈αλdoesn't␈α	know␈αλwhich␈α	features␈αλare␈α	irrelevant;␈αλhe␈α	doesn't␈αλknow
␈↓ ↓*␈↓ε␈↓ αJwhich details of the gestalt are computed and which are random.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  32␈↓

␈↓ ↓*␈↓speci≡c␈αdecision␈α
as␈αto␈αwho␈α
goes␈αnext).␈α
It␈αmust␈αalso␈α
include␈αevaluations␈αof␈α
belief,␈αinterest,␈α
supe≡ciality,␈αsafety,

␈↓ ↓*␈↓utility;␈α∩it␈α∩must␈α⊃keep␈α∩brief␈α∩statistics␈α∩on␈α⊃when␈α∩and␈α∩how␈α⊃each␈α∩part␈α∩of␈α∩each␈α⊃BEING␈α∩is␈α∩accessed;␈α∩and␈α⊃the

␈↓ ↓*␈↓environment␈α∞must␈α∞maintain␈α∂a␈α∞rigidly-formatted␈α∞description␈α∞of␈α∂the␈α∞Current␈α∞Situation␈α∞(abbreviated␈α∂CS;␈α∞this

␈↓ ↓*␈↓structure␈α
also␈α
includes␈αsummaries␈α
of␈α
recent␈α
system␈αhistory).␈α
 When␈α
a␈α
part␈αis␈α
big␈α
and␈α
heavily␈αaccessed,␈α
detailed

␈↓ ↓*␈↓records␈α
must␈α
be␈αkept␈α
of␈α
each␈α
usage␈α(how,␈α
why,␈α
when,␈α
≡nal␈αresult)␈α
of␈α
each␈α
␈↓βsubpart␈↓.␈α Based␈α
on␈α
this,␈α
the␈αpart

␈↓ ↓*␈↓may␈αbe␈αsplit␈αinto␈αa␈αgroup␈αof␈αnew␈αBEINGs,␈αand␈αthe␈αvalue␈αof␈αthe␈αpart␈αreplaced␈αby␈αa␈αpointer␈αto␈αa␈αlist␈αof␈αthese

␈↓ ↓*␈↓new BEINGs.


␈↓ ↓*␈↓The␈α
environment␈αwould␈α
have␈αto␈α
accept␈α
the␈αreturning␈α
messages␈αof␈α
the␈αattempt␈α
to␈α
deal␈αwith␈α
a␈αcertain␈α
part␈αof␈α
a

␈↓ ↓*␈↓certain␈αBEING.␈αA␈αsuccess␈αor␈αa␈αfailure␈αwould␈αmean␈αbacking␈αup␈αto␈αthe␈αlast␈αdecision␈αand␈αre-making␈αit␈α(usually

␈↓ ↓*␈↓the␈αtop-level␈α"select␈α(P,B)␈αto␈αwork␈αon␈αnext"␈αdecision).␈α An␈α"interrupt"␈αfrom␈αa␈αtrial␈αwould␈αmean␈α"here␈αis␈αsome

␈↓ ↓*␈↓possibly␈αmore␈αinteresting␈αinfo".␈αThe␈αenvironment␈αmust␈αdecide␈αif␈αit␈αis␈αin␈αfact␈αjudged␈αto␈αbe␈αmore␈αinteresting;␈α
if

␈↓ ↓*␈↓not,␈α
control␈α
is␈α
returned␈α
to␈α
the␈α
interrupted␈α
process.␈α
 If␈α
so,␈α
control␈α
automatically␈α
switches␈α
to␈α
that␈αnew,␈α
interesting

␈↓ ↓*␈↓part.␈α Later,␈αthere␈αwill␈αbe␈αno␈αautomatic␈αreturn␈αto␈αthe␈αinterrupted␈αprocess,␈αbut␈αwhatever␈αsequence␈αof␈αdecisions

␈↓ ↓*␈↓led␈αto␈αits␈αinitiation␈αmay␈αvery␈αprobably␈αlead␈αthere␈αagain.␈α Two␈αtricks␈αare␈αplanned␈αhere.␈αOne␈αis␈αa␈αcache:␈αeach

␈↓ ↓*␈↓BEING␈α⊂will␈α⊂let␈α⊂ its␈α⊂RECOG␈α⊂parts␈α⊂store␈α⊂the␈α⊃last␈α⊂value␈α⊂computed,␈α⊂and␈α⊂let␈α⊂each␈α⊂such␈α⊂part␈α⊂have␈α⊃a␈α⊂quick

␈↓ ↓*␈↓predicate␈αwhich␈αcan␈αtell␈αif␈αany␈αfeature␈αof␈αthe␈αworld␈αhas␈αchanged␈αwhich␈αmight␈αa≥ect␈αthis␈αvalue.␈α If␈αnot,␈αthen

␈↓ ↓*␈↓no␈α
work␈α
is␈α
done;␈α
the␈α∞old␈α
value␈α
is␈α
simply␈α
returned.␈α
If␈α∞x␈α
is␈α
interrupted,␈α
an␈α
auxilliary␈α
development␈α∞is␈α
begun,

␈↓ ↓*␈↓and␈αthen␈αwork␈αon␈αx␈αshould␈αcontinue,␈αmost␈αof␈αthe␈αdecisions␈αleading␈αback␈αto␈αx␈αwill␈αprobably␈αnot␈αinvolve␈αany

␈↓ ↓*␈↓real␈α∞work,␈α
since␈α∞most␈α∞of␈α
the␈α∞world␈α∞hasn't␈α
changed.␈α∞The␈α
second␈α∞trick␈α∞is␈α
that␈α∞to␈α∞evaluate␈α
a␈α∞part,␈α∞one␈α
moves

␈↓ ↓*␈↓down␈α
its␈α
code␈αwith␈α
a␈α
cursor,␈α
evalling.␈αWhen␈α
interrupted,␈α
that␈α
cursor␈αis␈α
left␈α
just␈α
at␈αthe␈α
point␈α
one␈α
wants␈αto␈α
start

␈↓ ↓*␈↓at when the work resumes.


␈↓ ↓*␈↓New␈αBEINGs␈αare␈αcreated␈αautomatically␈αif,␈αwhen␈αa␈αpart␈αis␈αevaluated␈αand␈αa␈αnew␈αentity␈αformed,␈αthe␈αentity␈αhas

␈↓ ↓*␈↓su≠cient␈α⊂interest␈α⊂to␈α⊂make␈α⊂it␈α⊂worth␈α∂keeping␈α⊂by␈α⊂name.␈α⊂ Also,␈α⊂an␈α⊂existing␈α∂part␈α⊂P␈α⊂of␈α⊂a␈α⊂BEING␈α⊂B␈α⊂may␈α∂be

␈↓ ↓*␈↓replaced␈αby␈αa␈αlist␈αof␈αnew␈αBEINGs.␈α The␈αdetails␈αof␈αwhen␈αand␈αhow␈αto␈αdo␈αthis␈αrestructuring␈αof␈αB.P␈αare␈αstored
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  33␈↓

␈↓ ↓*␈↓under␈αthe␈αStructure␈αpart␈αof␈αthe␈αarchetypical␈αBEING␈αwhose␈αname␈αis␈αP.␈αThe␈αtypical␈αprocess␈αis␈αas␈αfollows:␈αThe

␈↓ ↓*␈↓environment␈α
keeps␈αloose␈α
checks␈αon␈α
the␈αsize␈α
and␈αusage␈α
of␈α
each␈αpart;␈α
if␈αone␈α
ever␈αgrows␈α
and␈αeats␈α
up␈αmuch␈α
time,

␈↓ ↓*␈↓it␈α
is␈α
carefully␈α
monitored.␈α
Eventually,␈α
its␈α
subparts␈α
may␈α
partition␈α
into␈α
a␈α
set␈α
whose␈α
usage␈α
is␈α
nearly␈α
disjoint.␈αIf

␈↓ ↓*␈↓this␈αis␈αseen,␈αthen␈αthe␈αpart␈αis␈αactually␈αsplit␈αinto␈αa␈αnew␈αset␈αof␈αBEINGs.␈α If␈αa␈αnew␈αBEING␈αdoesn't␈αlive␈αup␈αto␈αits

␈↓ ↓*␈↓expectations,␈αit␈α
may␈αbe␈α
destroyed␈αor␈α
enslaved␈α(overwritten␈αand␈α
forgotten;␈αperhaps␈α
enough␈αis␈α
remembered␈αto

␈↓ ↓*␈↓not␈α
waste␈α
time␈αlater␈α
on␈α
the␈αsame␈α
concept;␈α
perhaps␈αit␈α
is␈α
denegrated␈α
into␈αjust␈α
a␈α
subpart␈αof␈α
some␈α
part␈αof␈α
another

␈↓ ↓*␈↓BEING.)


␈↓ ↓*␈↓Here is a more detailed (but still tentative) description of some of these top-level environment routines:


␈↓ ↓*␈↓COMPLETE(P,B) means ≡ll in material in part P of BEING B.


␈↓ ↓*␈↓␈↓↓1.␈α∞Locate␈α∞P␈α∞and␈α∂B.␈↓␈α∞If␈α∞P␈α∞is␈α∂unknown␈α∞but␈α∞B␈α∞is␈α∂known,␈α∞ask␈α∞B.ORDERING␈α∞and␈α∂up␈↓	*␈↓(B).ORDERING.␈α∞Also,

␈↓ ↓*␈↓there␈α
may␈α∞be␈α
special␈α∞information␈α
stored␈α
in␈α∞some␈α
part(s)␈α∞of␈α
B␈α
earlier,␈α∞by␈α
other␈α∞BEINGs,␈α
which␈α∞make␈α
them

␈↓ ↓*␈↓more or less promising to work on now␈↓	38␈↓.


␈↓ ↓*␈↓If␈αB␈α
is␈αunknown␈α
but␈αP␈α
is␈αknown,␈αask␈α
P␈αand␈α
ask␈αeach␈α
BEING␈αabout␈αinterest␈α
of␈α≡lling␈α
in␈αP.␈α
 Each␈αBEING

␈↓ ↓*␈↓runs␈α
a␈α
quick␈α
test␈α
to␈αsee␈α
if␈α
it␈α
is␈α
worth␈αdoing␈α
a␈α
detailed␈α
examination.␈α
 Sometimes␈αthe␈α
family␈α
of␈α
B␈α
will␈αbe␈α
known

␈↓ ↓*␈↓(or at least constrained).


␈↓ ↓*␈↓If␈α
neither␈αis␈α
known,␈α
each␈αBEING␈α
must␈α
see␈αhow␈α
rele.␈αit␈α
is;␈α
the␈αwinner␈α
decides␈α
on␈αP.␈α
 If␈α
there␈αis␈α
more␈αthan␈α
one

␈↓ ↓*␈↓BEING␈αtied␈αfor␈αtop␈αrecognition,␈αthen␈αplace␈αthe␈αresults␈αin␈αorder␈αusing␈αthe␈αenvironment␈αfunction␈αORD,␈α
which

␈↓ ↓*␈↓examines␈α
the␈αWorth␈α
components␈αof␈α
each,␈α
and␈αby␈α
using␈αthe␈α
value␈αof␈α
the␈α
most␈αpromising␈α
part␈αto␈α
work␈αon␈α
next

␈↓ ↓*␈↓for␈α∩each␈α∪BEING.␈α∩The␈α∪frequent␈α∩access␈α∩to␈α∪the␈α∩(best␈α∪part,␈α∩value)␈α∩pair␈α∪for␈α∩each␈α∪BEING␈α∩means␈α∪that␈α∩its

␈↓ ↓*␈↓calculation␈αshould␈αbe␈αquick;␈αin␈αgeneral,␈αeach␈αBEING␈αwill␈α
recompute␈αit␈αonly␈αwhen␈αnew␈αinfo.␈αis␈αadded␈αto␈α
some

␈↓ ↓*␈↓part,␈α
or␈α
at␈αrare␈α
intervals␈α
otherwise.␈α After␈α
ranking␈α
this␈α
list,␈αchop␈α
it␈α
o≥␈αat␈α
the␈α
≡rst␈αbig␈α
break␈α
in␈α
values,␈αand


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	38␈↓ε up␈↓#
n␈↓#(B).P means the set of BEINGs named P, B.P,  (B.Ties.Up).P, ((B.Ties.Up).Ties.Up).P, etc.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  34␈↓

␈↓ ↓*␈↓print␈α
it␈α
out␈α
to␈α
the␈α
user␈α
to␈α
inspect.␈α
Pause␈α
WAIT␈α
seconds,␈α
then␈α
commence␈α
work␈α
on␈α
the␈α
≡rst␈α
in␈α
the␈α
list.␈αWAIT␈α
is

␈↓ ↓*␈↓a␈αparameter␈αset␈αby␈αthe␈αuser␈αinitially␈↓	39␈↓.␈α When␈αyou␈α
≡nish,␈αdon't␈αthrow␈αthe␈αlist␈αaway␈αuntil␈αafter␈αthe␈αnext␈α
B␈αis

␈↓ ↓*␈↓chosen,␈α∞since␈α∂the␈α∞list␈α∞might␈α∂immediately␈α∞need␈α∞to␈α∂be␈α∞recomputed!␈α∞If␈α∂the␈α∞user␈α∞doesn't␈α∂like␈α∞the␈α∂choice␈α∞you've

␈↓ ↓*␈↓made,␈α
he␈α∞can␈α
interrupt␈α∞and␈α
switch␈α∞you␈α
over.␈α∞ A␈α
similar␈α∞process␈α
occurs␈α∞if␈α
P␈α∞is␈α
unknown,␈α∞(except␈α
the␈α∞list␈α
is

␈↓ ↓*␈↓never saved).


␈↓ ↓*␈↓␈↓↓2.␈α∞Collect␈α∞pointers␈α∞to␈α∞helpful␈α∞information:␈α∞␈↓␈α∞ Create␈α∞a␈α∞(partialy␈α∞ordered)␈α∞plan␈α∞for␈α∞dealing␈α∞with␈α∞part␈α∞P␈α
of

␈↓ ↓*␈↓BEING␈αB␈α(abbreviated␈αB.P).␈α  This␈αincludes␈αthe␈αP.FILLIN␈αpart,␈αand␈αin␈αfact␈αany␈αexisting␈αup␈↓	*␈↓(B).P.FILLIN,

␈↓ ↓*␈↓and␈α∀ also␈α∀some␈α∀use␈α∃of␈α∀the␈α∀representation,␈α∀defn,␈α∀views,␈α∃dom/range␈α∀parts␈α∀of␈α∀the␈α∀P␈α∃BEING.␈α∀  Consult

␈↓ ↓*␈↓ALGORITHMS and FILLIN parts of B and all upward-tied BEING's to B.


␈↓ ↓*␈↓␈↓↓3.␈α∞Decide␈α
what␈α∞must␈α
be␈α∞done␈α∞now␈↓;␈α
 which␈α∞of␈α
the␈α∞above␈α
pieces␈α∞of␈α∞information␈α
is␈α∞"best".␈α
Tag␈α∞it␈α∞as␈α
having

␈↓ ↓*␈↓been tried.   If it is precisely = one currently active goal, then forget it and go to 3.


␈↓ ↓*␈↓␈↓↓4.␈α∞Carry␈α
out␈α∞the␈α
step.␈↓␈α∞(Evaluate␈α∞the␈α
interest␈α∞of␈α
any␈α∞new␈α
BEING␈α∞when␈α∞it␈α
is␈α∞created)␈α
 Notice␈α∞that␈α∞the␈α
step

␈↓ ↓*␈↓might␈αin␈αturn␈αcall␈αfor␈αaccessing␈αand␈α(rarely)␈α≡lling␈α in␈αparts␈αof␈αother␈αBEINGs.␈αThis␈αactivity␈αwill␈αbe␈αstandard

␈↓ ↓*␈↓heirarchical calling.   As parts of other BEINGs are modi≡ed, update their (best part, value) estimate.


␈↓ ↓*␈↓␈↓↓5.␈α
When␈α
done,␈α
update.␈↓␈α
 Update␈α
statistics␈α
in␈α
B,␈α
P,␈αand␈α
current␈α
situation.␈α
(worth␈α
and␈α
recog␈α
parts)␈α
 If␈α
we␈αare

␈↓ ↓*␈↓through␈α∞dealing␈α∞with␈α∞B.P␈α∞(because␈α∞of␈α
higher␈α∞interest␈α∞entity␈α∞∃,␈α∞ or␈α∞because␈α
the␈α∞part␈α∞is␈α∞≡lled␈α∞in␈α∞enough␈α
for

␈↓ ↓*␈↓now) goto 1; else goto 3.   If you stop because of higher interest entity, save the plan for P.B inside P.B.


␈↓ ↓*␈↓CURRENT SITUATION is a vector of weights and features of the recent behavior of the system.


␈↓ ↓*␈↓The␈α∞Environment␈α∞also␈α∞maintains␈α∞a␈α
list␈α∞of␈α∞records␈α∞and␈α∞statistics␈α∞of␈α
the␈α∞recent␈α∞past␈α∞activities,␈α∞in␈α∞a␈α
structure

␈↓ ↓*␈↓called␈α∂CS,␈α⊂for␈α∂"Current␈α⊂Situation".␈α∂ Each␈α⊂Recognition␈α∂grouping␈α⊂part␈α∂is␈α⊂prefaced␈α∂by␈α⊂a␈α∂vector␈α⊂of␈α∂numbers



␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	39␈↓ε    0 would mean go on unless user interrupts you, infinity would mean always wait for user's reply, etc.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  35␈↓

␈↓ ↓*␈↓which␈α
are␈αdot-multiplied␈α
into␈αCS,␈α
to␈α
produce␈αa␈α
rapid␈αrough␈α
guess␈α
of␈αrelevance.␈α
 Only␈αthe␈α
best␈αperformers␈α
are

␈↓ ↓*␈↓examined␈α∩more␈α⊃closely␈α∩for␈α∩relevance.␈α⊃ The␈α∩representation␈α⊃of␈α∩each␈α∩CS␈α⊃component␈α∩is␈α∩(identi≡cation␈α⊃info,

␈↓ ↓*␈↓motivation, safety, interest, work done so far on it, ≡nal result or outlook). The actual components might be:

␈↓ ↓*␈↓Recent Accesses.   For each, save (B, P, contents of subpart used).
␈↓ ↓*␈↓Recent Fillins.    Save (B, P, old contents which were altered).
␈↓ ↓*␈↓Current Hierarchical History Stack.  Save  (B, P, why).
␈↓ ↓*␈↓Recent Top-level B,P pairs.
␈↓ ↓*␈↓A couple signi≡cant recent but not current hierarchical (B,P,why) records.
␈↓ ↓*␈↓A backward-sorted list of the most interesting but currently-deferred (B,P) ≡llins.
␈↓ ↓*␈↓A few recent or collossal ≡ascos (B, P, what, why this was a huge waste).


␈↓ ↓*␈↓ORD(B,C)  Which of the recognition-tied BEINGs B,C is potentially more worthwhile?



␈↓ ↓*␈↓This␈α⊂simple␈α⊂ordering␈α∂function␈α⊂will␈α⊂probably␈α⊂examine␈α∂the␈α⊂Worth␈α⊂vectors,␈α∂ perhaps␈α⊂involving␈α⊂the␈α⊂sum␈α∂of

␈↓ ↓*␈↓weighted factors, perhaps even cross-terms such as (probability of success)*(interest rating).


␈↓ ↓*␈↓¬PLAUSIBILITY(z)       How believable is z?    INTEREST(z)    How interesting is z?

␈↓ ↓*␈↓¬         each statement has a probability weight attached to it, the degree of belief
␈↓ ↓*␈↓¬         this number is a fn. of a list of justifications
␈↓ ↓*␈↓¬         Polya's plausibility axioms and rules of inference
␈↓ ↓*␈↓¬         if there are several alternate justifs., it is more plausible
␈↓ ↓*␈↓¬         if some consequences are verified, it is more plaus.
␈↓ ↓*␈↓¬         if an analogous prop. is verified, it is more plaus.
␈↓ ↓*␈↓¬         if the consequences of analogue are verif., it is slightly more plaus.
␈↓ ↓*␈↓¬         the converses of the above also hold
␈↓ ↓*␈↓¬         believe in those things with high enough prob. of belief (rely on them)
␈↓ ↓*␈↓¬         this level should fluctuate just above the point of belief in contradictions
␈↓ ↓*␈↓¬         the higher the prob., the higher the reliability
␈↓ ↓*␈↓¬         the amt. one bets should be prop. to the reliability
␈↓ ↓*␈↓¬         the interest increases as the odds do
␈↓ ↓*␈↓¬         Zadeh: p(∧) is min, p(∨) is max, p(¬) is 1-.
␈↓ ↓*␈↓¬         Hintikka's formulae (λ, α)
␈↓ ↓*␈↓¬         Carnap's formulas (λ)
␈↓ ↓*␈↓¬         p=1 iff both the start and the methods are certain ←← truth
␈↓ ↓*␈↓¬         p=0 iff both start is false and method is false-preserving ←← falsity
␈↓ ↓*␈↓¬         p is higher as the plausibility is higher, and as the interest is lower
␈↓ ↓*␈↓¬         if ∃ several alternative plaus. justifs., p is higher
␈↓ ↓*␈↓¬         don't update p value unless you have to
␈↓ ↓*␈↓¬         update p values of contradictory props.
␈↓ ↓*␈↓¬         update p values of new props
␈↓ ↓*␈↓¬         maybe update p value if it is a reason for a new prop
␈↓ ↓*␈↓¬      empiricism, experiment, random sampling, statistics
␈↓ ↓*␈↓¬         true ideas will be "verified" in (consistent with) any and all experiments
␈↓ ↓*␈↓¬         false ideas may only have a single exceptional case
␈↓ ↓*␈↓¬         a single exception makes a universal idea false
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  36␈↓

␈↓ ↓*␈↓¬         nature is fair, uniform, nice, regular; coincidences have meaning
␈↓ ↓*␈↓¬         more plaus. the more cases verified
␈↓ ↓*␈↓¬         more plaus. the more diff. types of cases verified
␈↓ ↓*␈↓¬         central tendency (mean, mode, median)
␈↓ ↓*␈↓¬         standard deviation, normal distribution
␈↓ ↓*␈↓¬         other distributions (binomial, Poisson, flat, bimodal)
␈↓ ↓*␈↓¬         statistical formulae for significance of hypothesis
␈↓ ↓*␈↓¬      regularity, order, form, arrangement
␈↓ ↓*␈↓¬         economy of description means regularity exists
␈↓ ↓*␈↓¬         aesthetic desc (ana. to known descs. elsewhere)
␈↓ ↓*␈↓¬         each part of desc. is organized regularly
␈↓ ↓*␈↓¬         the parts are related regularly

␈↓ ↓*␈↓¬  Below, α means ␈↓βincreases with increasing␈↓¬ (proportionality), and
␈↓ ↓*␈↓¬  α␈↓	-1␈↓¬ means ␈↓βdecreases with increasing␈↓¬ (inversely proportional).
␈↓ ↓*␈↓¬  Perhaps one should distribute these morsels among the various concerned BEING's:
␈↓ ↓*␈↓¬   Completeness of an analogy  α  safety of using it for prediction
␈↓ ↓*␈↓¬   Completeness of an analogy  α␈↓	-1␈↓¬ how interesting it is
␈↓ ↓*␈↓¬   How expected a relationship is  α␈↓	-1␈↓¬  how interesting it is
␈↓ ↓*␈↓¬   How intuitive a conjec/relationship is  α␈↓	-1␈↓¬  how interesting it is
␈↓ ↓*␈↓¬   How intuitive a conjec/relationship is  α  how certain/safe it is
␈↓ ↓*␈↓¬   How superficial something is  α  how intuitive it is
␈↓ ↓*␈↓¬   How superficial something is  α  how certain it is
␈↓ ↓*␈↓¬   How superficial something is  α␈↓	-1␈↓¬ how interesting it is

␈↓ ↓*␈↓¬  Perhaps included here should be universally applicable algorithms for applying these rules
␈↓ ↓*␈↓¬  to choosing the best strategies, as a function of the situation.

␈↓ ↓*␈↓¬   One crude estimate of interest level is the interest component of the current BEING's
␈↓ ↓*␈↓¬   Modify this estimate in close cases using the above relations
␈↓ ↓*␈↓¬   Generally, choose the most specific strategies possible
␈↓ ↓*␈↓¬   If the estimated value of applying one of these falls too low, try a more general one
␈↓ ↓*␈↓¬   Rework the current BEING slightly, if that enables a much more specific strategy to be used
␈↓ ↓*␈↓¬   Locate specific concepts which partially instantiate general strategies
␈↓ ↓*␈↓¬   The more specific new strategies are associated with the specific info. used
␈↓ ↓*␈↓¬   Once chosen, use the strategies on the most promising specific information
␈↓ ↓*␈↓¬   If a strat. falters: Collect the names of the specific, needed but blank (sub)parts
␈↓ ↓*␈↓¬      Each such absence lowers int. and raises cost, and may cause switch to new strategy
␈↓ ↓*␈↓¬      If too costly, low int, store pointer to partial results in blank parts
␈↓ ↓*␈↓¬         The partial results maintain set of still-blank needed parts

␈↓ ↓*␈↓¬   Competing goals: On the one hand, desire to maximize certainty,
␈↓ ↓*␈↓¬      safety, complete analogies, advance the level of intuition.
␈↓ ↓*␈↓¬      On the other hand, desire to maximize interestingness, find poss. and poten. interesting ana.
␈↓ ↓*␈↓¬       find unexpected, nonsuperficial, and unintuitive relationships.
␈↓ ↓*␈↓¬   If an entity is used frequently, it should be made efficient.
␈↓ ↓*␈↓¬      Conversely, try to use efficient entities over nearly
␈↓ ↓*␈↓¬      equivalent (w.r.t. given purpose) but inefficient ones.
␈↓ ↓*␈↓¬   If an entity is formally justified but resists intuitive comprehension, its use is
␈↓ ↓*␈↓¬      dangerous but probably very interesting; ibid for intuitive but unprovable.
␈↓ ↓*␈↓¬   Resolve choices in favor of aesthetic superiority

␈↓ ↓*␈↓¬   Maximize net behavior
␈↓ ↓*␈↓¬    Maximize desired effects
␈↓ ↓*␈↓¬      In this case, prefer hi interest over hi safety.
␈↓ ↓*␈↓¬      Generally preferred to the folowing case.
␈↓ ↓*␈↓¬    Minimize costs, conserve resources
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  37␈↓

␈↓ ↓*␈↓¬      In this case, prefer safety to interest.
␈↓ ↓*␈↓¬      Locate the most inefficient, highest-usage entity, and improve or replace it
␈↓ ↓*␈↓¬      Use: If time/space become a problem, worry about conservation until this relaxes.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  38␈↓

␈↓ ↓*␈↓␈↓ ∧i␈↓∧␈↓&10. Details: Getting AM going␈↓)αβ␈↓


␈↓ ↓*␈↓This␈α
diversion␈α
is␈α
only␈α
for␈α
the␈α
interested␈α
reader.␈α
It␈α
may␈α
be␈α
skipped␈α
without␈α
loss␈α
of␈α
continuity.␈α
It␈α
deals␈αwith

␈↓ ↓*␈↓short-cuts which can be taken by myself to get AM running faster and more easily.


␈↓ ↓*␈↓When␈αAM␈αproposes␈αa␈αconjecture,␈αthe␈αuser␈αmay␈αinterrupt␈αand␈αsay␈α"Assume␈αit␈αis␈αtrue,␈αand␈αgo␈αon".␈αIf␈αhe␈αdoes

␈↓ ↓*␈↓this␈α⊂consistently,␈α⊂we␈α⊂can␈α⊂debug␈α⊂AM's␈α⊂higher-level␈α⊂proposing␈α⊂abilities␈α⊂even␈α⊂before␈α⊂its␈α⊃lower-level␈α⊂proving

␈↓ ↓*␈↓abilities are perfected.


␈↓ ↓*␈↓We␈α∪may␈α∪be␈α∪able␈α∩to␈α∪beat␈α∪the␈α∪"critical␈α∩mass"␈α∪constraint␈α∪slightly.␈α∪ The␈α∩vast␈α∪amount␈α∪of␈α∪necessary␈α∩initial

␈↓ ↓*␈↓knowledge␈α∩can␈α∩be␈α∩generated␈α∩from␈α⊃a␈α∩much␈α∩smaller␈α∩core␈α∩of␈α⊃intuition␈α∩and␈α∩de≡nite␈α∩facts,␈α∩using␈α∩the␈α⊃same

␈↓ ↓*␈↓collection␈αof␈αstrategies␈αand␈α
wisdom␈αwhich␈αalso␈αdrive␈α
the␈αlater␈αdiscovery␈αand␈α
the␈αdevelopment.␈α Since␈αAM␈α
has

␈↓ ↓*␈↓been␈αgiven␈αgood␈αknowledge-acquisition␈α(strategic)␈αpowers,␈αAM␈αitself␈αshould␈αbe␈αable␈αto␈α≡ll␈αin␈αthe␈αblank␈αparts

␈↓ ↓*␈↓of␈α
the␈α
concepts␈αit␈α
starts␈α
with.␈↓	40␈↓␈α
The␈αonly␈α
constraint␈α
on␈αsuch␈α
a␈α
partially-equipped␈α
system␈αis␈α
that␈α
it␈α≡rst␈α
develop

␈↓ ↓*␈↓a␈α∞broad␈α∞base␈α∞of␈α∞intuition,␈α
examples,␈α∞and␈α∞interrelations,␈α∞spanning␈α∞several␈α
areas␈α∞of␈α∞interest,␈α∞before␈α∞trying␈α
to

␈↓ ↓*␈↓develop any one particular area in depth.


␈↓ ↓*␈↓A␈α⊂balanced␈α⊂distribution␈α⊂of␈α⊂intelligence␈α⊂ought␈α⊂to␈α∂exist:␈α⊂related␈α⊂facets␈α⊂which␈α⊂are␈α⊂rarely␈α⊂accessed␈α∂separately

␈↓ ↓*␈↓should␈α
be␈α
coalesced,␈α
and␈αany␈α
single␈α
type␈α
of␈α
facet␈αused␈α
very␈α
heavily␈α
should␈α
be␈αsplit.␈α
Notice␈α
that␈α
this␈αtheme␈α
has

␈↓ ↓*␈↓two␈α∂quite␈α∂di≥erent␈α⊂realizations.␈α∂During␈α∂run␈α∂time,␈α⊂this␈α∂policy␈α∂would␈α∂refer␈α⊂to␈α∂the␈α∂contents␈α∂held␈α⊂in␈α∂existing

␈↓ ↓*␈↓BEINGs'␈α∂parts␈↓	41␈↓.␈α⊂Before␈α∂the␈α∂system␈α⊂is␈α∂actually␈α∂created,␈α⊂however,␈α∂this␈α∂policy␈α⊂means␈α∂altering␈α⊂the␈α∂proposed





␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	40␈↓ε␈α
When␈α
you␈α
are␈α
preparing␈α
the␈α
knowledge␈α
to␈α
initially␈α
get␈α
AM␈α
going,␈α
the␈α
way␈α
to␈α
save␈α
some␈α
effort␈α
is␈α
by␈α
 omitting␈α
whole␈α
categories␈α
of␈α
knowledge
␈↓ ↓*␈↓ε␈↓ αJwhich␈αλyou␈αλcan␈α	expect␈αλthe␈αλsystem␈α	to␈αλdiscover␈αλquickly␈αλfor␈α	itself.␈αλAM␈αλhas␈α	to␈αλhave␈αλgood␈αλmethods␈α	for␈αλgenerating␈αλexamples␈α	of␈αλwhatever
␈↓ ↓*␈↓ε␈↓ αJconcepts␈αit␈αlater␈αcomes␈αup␈αwith;␈αthose␈αsame␈αmethods␈αcan␈αbe␈αused␈αto␈αproduce␈αexamples␈αof␈αthe␈αinitially-supplied␈αconcepts.␈αThis
␈↓ ↓*␈↓ε␈↓ αJmeans␈α
that␈α
the␈α	creators␈α
can␈α
omit␈α
all␈α	examples␈α
of␈α
the␈α
concepts␈α	originally␈α
supplied;␈α
the␈α
system␈α	will␈α
be␈α
able␈α
to␈α	fill␈α
in␈α
all␈α
such␈α	gaps.
␈↓ ↓*␈↓ε␈↓ αJNotice␈αλthat␈αλAM␈α	has␈αλto␈αλbe␈α	able␈αλto␈αλfill␈αλin␈α	␈↓&all␈↓)αβ␈αλthe␈αλparts␈α	(slots)␈αλof␈αλeach␈αλnewly␈α	discovered␈αλconcept,␈αλso␈α	perhaps␈αλonly␈αλthe␈α	barest␈αλdefinition
␈↓ ↓*␈↓ε␈↓ αJor intuition need be initially taught about each concept.
␈↓ ↓*␈↓ε␈↓	41␈↓ε  E.g., the Structure part exists only to indicate what to do about a BEING's  part getting too big.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  39␈↓

␈↓ ↓*␈↓anatomy␈α⊃of␈α⊃BEINGs␈α⊃␈↓	42␈↓.␈α⊃ The␈α⊃runtime␈α⊃restructurings␈α⊂occur␈α⊃based␈α⊃on␈α⊃knowledge␈α⊃recently␈α⊃created␈α⊃by␈α⊂the

␈↓ ↓*␈↓system;␈α
the␈αplanning-stage␈α
restructurings␈α
are␈αnow␈α
being␈α
based␈αon␈α
results␈α
from␈αhand␈α
simulations␈α
of␈αthe␈α
system.

␈↓ ↓*␈↓Remember:␈α
during␈α∞runtime,␈α
the␈α
set␈α∞of␈α
parts␈α
that␈α∞any␈α
speci≡c␈α
BEING␈α∞can␈α
ever␈α
have␈α∞is␈α
≡xed␈α
by␈α∞the␈α
family

␈↓ ↓*␈↓(profession) to which he belongs.


␈↓ ↓*␈↓One␈αdi≥erence␈αfrom␈αPUP6␈αis␈αthat␈αin␈αAM␈αthe␈αBEINGs␈αare␈αgrouped␈αinto␈αfamilies.␈α Each␈αfamily␈αhas␈α
its␈αown

␈↓ ↓*␈↓set␈αof␈αparts␈α(although␈αthere␈αwill␈αbe␈αmany␈αparts␈αpresent␈αin␈αmany␈αfamilies,␈αe.g.␈αIden).␈αFor␈αeach␈αfamily␈αF␈αthere

␈↓ ↓*␈↓will␈α
be␈α
a␈α∞fairly␈α
general␈α
BEING␈α
named␈α∞F.␈α
Under␈α
each␈α∞part␈α
of␈α
F␈α
is␈α∞general␈α
information␈α
which,␈α∞though␈α
not

␈↓ ↓*␈↓applicable␈αto␈αall␈αBEINGs,␈αis␈αapplicable␈αto␈αall␈αBEINGs␈αbelonging␈αto␈αfamily␈αF.␈α Similarly,␈αif␈αP␈αis␈αa␈αpart␈αname,

␈↓ ↓*␈↓then␈α∞the␈α∂BEING␈α∞named␈α∂P␈α∞contains␈α∂information␈α∞which␈α∂is␈α∞useful␈α∂for␈α∞dealing␈α∂with␈α∞part␈α∂P␈α∞of␈α∂any␈α∞BEING.

␈↓ ↓*␈↓There␈αmight␈αalso␈αexist␈αan␈αarchetypical␈αBEING␈α
named␈αF.P,␈αwho␈αwould␈αhave␈αspecial␈αinformation␈αfor␈α
working

␈↓ ↓*␈↓with␈αpart␈α
P␈αof␈α
any␈αBEING␈αin␈α
family␈αF.␈α
 There␈αmight␈α
even␈αbe␈αa␈α
BEING␈αcalled␈α
B.P,␈αwhere␈α
B␈αis␈αsome␈α
speci≡c

␈↓ ↓*␈↓BEING,␈α∩with␈α∩information␈α⊃that␈α∩just␈α∩deals␈α⊃with␈α∩part␈α∩P␈α∩of␈α⊃B␈α∩and␈α∩any␈α⊃future␈α∩specializations␈α∩of␈α∩B.␈α⊃The

␈↓ ↓*␈↓information␈αstored␈αinside␈αa␈αpart␈αof␈αa␈αBEING,␈αfor␈α
example␈αthe␈αactual␈αcontents␈αof␈αB.P,␈αwould␈αbe␈αcode␈α
capable

␈↓ ↓*␈↓of␈αcomputing␈αB's␈αanswer␈αto␈αthe␈αquestion␈α
P;␈αthe␈αpreviously␈αmentioned␈αarchetypical␈αBEING␈αnamed␈αB.P␈α
would

␈↓ ↓*␈↓contain␈αstrategies␈αfor␈αdealing␈αwith␈αsuch␈αanswering␈αcode␈α(how␈αto␈α≡ll␈αit␈αin,␈αhow␈αto␈αcheck␈αit,␈αetc.).␈α To␈αreiterate:

␈↓ ↓*␈↓the␈αcontents␈αof␈αa␈αpart␈αare␈αspeci≡c␈αknowledge,␈αa␈αlittle␈αprogram␈αwhich␈αcan␈αanswer␈αa␈αspeci≡c␈αquery,␈αwhereas␈αthe

␈↓ ↓*␈↓contents␈αof␈αthe␈αparts␈αof␈αan␈αarchetypical␈αBEING␈αare␈αpartially␈αordered␈αsets␈αof␈αstrategies␈αfor␈αdealing␈α
with␈αthat

␈↓ ↓*␈↓part␈αof␈αthat␈α
type␈αof␈αBEING␈α(how␈α
to␈αextend␈αit,␈αwhat␈α
its␈αstructure␈αis,␈α
and␈αso␈αon).␈α Notice␈α
we␈αare␈αsaying␈αthat␈α
all

␈↓ ↓*␈↓the parts with the same part name, of BEINGs in the same family, must all have the same structure␈↓	43␈↓.


␈↓ ↓*␈↓When␈α⊂part␈α⊂p␈α⊂of␈α⊂BEING␈α⊂B␈α⊂is␈α⊂≡lled␈α⊂out,␈α⊃at␈α⊂some␈α⊂point␈α⊂in␈α⊂the␈α⊂sequence␈α⊂S␈α⊂of␈α⊂strategies␈α⊂listed␈α⊃under␈α⊂the

␈↓ ↓*␈↓archetypical␈α∂BEING␈α∂named␈α∂B.p␈α∂or␈α∞p,␈α∂some␈α∂new␈α∂information␈α∂may␈α∞be␈α∂discovered.␈α∂If␈α∂S␈α∂cannot␈α∂handle␈α∞this

␈↓ ↓*␈↓knowledge,␈α
then␈α∞ it␈α
will␈α∞simply␈α
return␈α∞with␈α
the␈α∞message␈α
"I␈α
am␈α∞not␈α
through,␈α∞but␈α
here␈α∞is␈α
some␈α∞fact(s)␈α
which


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	42␈↓ε␈αλ Proclaiming␈αλthat␈αλsome␈αλfamily␈αλhas␈αλto␈αλhave␈αλthis␈αλextra␈αλpart␈αλpresent,␈αλthat␈αλthese␈αλthree␈αλparts␈αλcan␈αλbe␈αλreplaced␈αλ(in␈αλall␈αλBEINGs␈αλof␈αλa␈αλgiven␈αλfamily)␈αλby␈αλthe
␈↓ ↓*␈↓ε␈↓ αJfollowing more general part, etc.
␈↓ ↓*␈↓ε␈↓	43␈↓ε This is one additional level of structure from the BEINGs proposed in PUP6.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  40␈↓

␈↓ ↓*␈↓may␈α
mean␈α
that␈α≡lling␈α
out␈α
part␈αp␈α
of␈α
B␈α
is␈αno␈α
longer␈α
the␈αbest␈α
activity".␈α
 The␈α
selected␈αpart␈α
and␈α
BEING␈αmay␈α
turn

␈↓ ↓*␈↓out␈α
the␈αsame,␈α
or␈αmay␈α
change␈α
due␈αto␈α
the␈αnew␈α
info␈α
which␈αwas␈α
just␈αuncovered.␈α
 The␈α
∨avor␈αof␈α
the␈αreturn␈α
should

␈↓ ↓*␈↓thus␈α∞be␈α∞one␈α∞of:␈α∞Not␈α∞Done␈α
because␈α∞x␈α∞is␈α∞possibly␈α∞more␈α∞interesting;␈α
Not␈α∞Done␈α∞because␈α∞x␈α∞is␈α∞a␈α∞prerequisite␈α
to

␈↓ ↓*␈↓doing me; Done because I succeeded; Done because I failed utterly.


␈↓ ↓*␈↓The␈α→lower-level␈α_BEINGs␈α→will␈α_provide␈α→fast␈α_access␈α→to␈α_well-organized␈α→information.␈α→ The␈α_background

␈↓ ↓*␈↓environment␈α_provides␈α_the␈α_necessary␈α↔evaluation␈α_services␈α_at␈α_high␈α↔speeds␈α_(though␈α_the␈α_system␈α↔cannot

␈↓ ↓*␈↓meaningfully␈α
examine,␈α
modify,␈α
or␈α∞add␈α
to␈α
what␈α
environment␈α∞functions␈α
the␈α
creators␈α
provide).␈α∞ The␈α
BEINGs

␈↓ ↓*␈↓hold␈α⊃"what␈α⊂to␈α⊃think";␈α⊂the␈α⊃environment␈α⊂implicitly␈α⊃controls␈α⊂"how␈α⊃to␈α⊂think",␈α⊃and␈α⊂the␈α⊃archetypical␈α⊂BEINGs

␈↓ ↓*␈↓explicitly␈α∞contain␈α∞hints␈α∂for␈α∞"how␈α∞to␈α∂think".␈α∞ The␈α∞big␈α∂assumption␈α∞is␈α∞that␈α∂one␈α∞may␈α∞think␈α∂creatively␈α∞without

␈↓ ↓*␈↓completely knowing how his thought processes operate; intelligence does not demand absolute introspection.


␈↓ ↓*␈↓Each␈αclump␈αis␈α(at␈αleast␈αpartially)␈αordered,␈αhence␈αcan␈α
be␈αexecuted␈αsequentially.␈α The␈αresult␈αmay␈αbe␈αto␈αchoose␈α
a

␈↓ ↓*␈↓lower-level␈α
clump,␈α
and/or␈α
modify␈α
some␈α
strategies␈α
at␈α
some␈α
level␈α
(some␈α
part␈α
of␈α
some␈α
BEING),␈α
and/or␈α
create␈α
new

␈↓ ↓*␈↓strategies␈α
at␈αsome␈α
level␈α(perhaps␈α
even␈α
to␈αcreate␈α
a␈αnew␈α
BEING).␈αThese␈α
lattter␈α
creations␈αand␈α
calls␈αwill␈α
be␈αin␈α
the

␈↓ ↓*␈↓form of strong suggestions to the environment.


␈↓ ↓*␈↓Common␈αknowledge␈αshould␈αin␈αsome␈αcases␈αbe␈αfactored␈αout.␈αPossibilities:␈α(i)␈αalways␈αask␈αa␈αspeci≡c␈αBEING,␈αwho

␈↓ ↓*␈↓sometimes␈α⊂queries␈α⊂a␈α⊃more␈α⊂general␈α⊂one␈α⊃if␈α⊂some␈α⊂knowledge␈α⊃is␈α⊂missing;␈α⊂(ii)␈α⊃always␈α⊂query␈α⊂the␈α⊃most␈α⊂general

␈↓ ↓*␈↓BEING␈α∂relevant,␈α∂who␈α⊂then␈α∂asks␈α∂some␈α∂speci≡c␈α⊂ones␈α∂(This␈α∂sounds␈α∂bad);␈α⊂(iii)␈α∂ask␈α∂all␈α∂the␈α⊂BEINGS␈α∂pseudo-

␈↓ ↓*␈↓simultaneously,␈α⊃and␈α⊃examine␈α∩the␈α⊃responders␈α⊃(this␈α∩sounds␈α⊃too␈α⊃costly.)␈α∩The␈α⊃organization␈α⊃of␈α∩BEINGs␈α⊃into

␈↓ ↓*␈↓hierarchical␈αgroupings␈αre∨ects␈αthe␈αspirit␈αof␈α(ii).␈αA␈αBEING␈αonly␈αcontains␈αadditions␈αand␈αexceptions␈αto␈αwhat␈αits

␈↓ ↓*␈↓generalization contains, so (i) is actually the dominant scheme now envisioned.


␈↓ ↓*␈↓There␈αare␈αtwo␈αkinds␈αof␈αdiscovery,␈αevolution␈αof␈αabilities,␈αwhich␈αare␈αspeci≡cally␈αdisallowed:␈α(i)␈αthe␈αmodi≡cation
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  41␈↓

␈↓ ↓*␈↓of␈α⊃the␈α⊃strategies␈α⊃and␈α∩control␈α⊃mechanism␈α⊃themselves␈↓	44␈↓,␈α⊃and␈α⊃(ii)␈α∩the␈α⊃introduction␈α⊃of␈α⊃brand␈α⊃new␈α∩␈↓βkinds␈↓␈α⊃of

␈↓ ↓*␈↓questions, of new parts of a type not forseen at time of system creation.␈↓	45␈↓




































␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	44␈↓ε␈αSince␈α
the␈αprecise␈αstrategies␈α
are␈αso␈α
crucial,␈αit␈αmight␈α
be␈αadvantageous␈α
to␈αallow␈αthem␈α
to␈αevolve.␈α
This␈αincludes␈αchanging␈α
the␈αsystem's␈αnotion␈α
of
␈↓ ↓*␈↓ε␈↓ αJinterestingness␈αas␈αits␈αexperience␈αgrows.␈αThis␈α was␈αdecided␈αagainst␈αbecause␈αour␈αthesis␈αis␈αthat␈αthe␈αsame␈αstrategies␈αuseful␈αfor
␈↓ ↓*␈↓ε␈↓ αJdealing␈αwith␈α
premathematical␈αconcepts␈α
should␈αsuffice␈α
no␈αmatter␈α
how␈αfar␈α
the␈αsystem␈α
progresses.␈α  One␈α
mechanism␈αto␈αeffect␈α
this
␈↓ ↓*␈↓ε␈↓ αJkind␈αλof␈αλdevelopment␈α	is␈αλthe␈αλstandard␈α	sort␈αλof␈αλfeedback␈αλparadigm,␈α	which␈αλin␈αλAM's␈α	case␈αλseems␈αλone␈αλfull␈α	step␈αλtoo␈αλambitious.␈α	Intuitions␈αλand
␈↓ ↓*␈↓ε␈↓ αJstrategies␈αmay␈αbe␈αinspected␈αand␈αaltered,␈α
just␈αas␈αany␈αother␈αfacets␈αof␈αBEINGs,␈α
but␈αthe␈αnotions␈αof␈αhow␈αto␈αjudge␈α
interestingness,
␈↓ ↓*␈↓ε␈↓ αJbelief,␈αsafety,␈αdifficulty,␈αetc.␈α (plus␈αall␈α
the␈αalgorithms␈αfor␈αsplit-msecond␈αestimating␈αand␈α
updating␈αthese)␈αare␈αfixed␈αfor␈αAM␈αby␈α
its
␈↓ ↓*␈↓ε␈↓ αJcreator. If they are unsatisfactory, he must retune them.
␈↓ ↓*␈↓ε␈↓	45␈↓ε␈α
The␈α	postulating␈α
of␈α	new␈α
kinds␈α
of␈α	parts␈α
is␈α	a␈α
very␈α	tricky␈α
process,␈α
and␈α	like␈α
the␈α	first␈α
exclusion,␈α	I␈α
feel␈α
that␈α	the␈α
kinds␈α	of␈α
things␈α	one␈α
wants␈α
to␈α	know
␈↓ ↓*␈↓ε␈↓ αJabout␈α	a␈α	concept␈α	shouldn't␈α	vary␈α	too␈α	much.␈αλA␈α	mechanism␈α	for␈α	introducing␈α	new␈α	kinds␈α	of␈αλBEING␈α	parts␈α	is␈α	the␈α	following:␈α	whenever␈α	a␈αλvery
␈↓ ↓*␈↓ε␈↓ αJinteresting␈α
property␈α	P␈α
is␈α
discovered,␈α	with␈α
interesting␈α
variations,␈α	which␈α
applies␈α
to␈α	many␈α
BEINGs,␈α
then␈α	assert␈α
that␈α
P␈α	is␈α
a␈α
new␈α	part
␈↓ ↓*␈↓ε␈↓ αJ(slot,␈α	question),␈α	and␈α	that␈α	every␈α	BEING␈α	which␈α	has␈α	some␈α	variation␈α	of␈α	 this␈α	property␈α	should␈α	fill␈α	in␈α	a␈α	slot␈α	labelled␈α	P,␈α	with␈α	a␈αλdescription
␈↓ ↓*␈↓ε␈↓ αJof␈α	its␈α	variation.␈α	 The␈α	common␈αλlink␈α	between␈α	all␈α	the␈α	BEINGs␈αλhaving␈α	 property␈α	P␈α	should␈α	be␈αλfound,␈α	and␈α	any␈α	new␈α	BEING␈α	possessing␈αλthis
␈↓ ↓*␈↓ε␈↓ αJcharacteristic should eventually try to see what variation of P it  possesses. That is, it should try to fill in a slot labelled P.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  42␈↓

␈↓ ↓*␈↓␈↓ ∧8␈↓∧␈↓&11. Examples of Individual Modules␈↓)αβ␈↓


␈↓ ↓*␈↓Perhaps␈α⊃it␈α⊂would␈α⊃be␈α⊃bene≡cial␈α⊂to␈α⊃glance␈α⊃over␈α⊂the␈α⊃≡le␈α⊃GIVEN␈α⊂at␈α⊃this␈α⊃point.␈α⊂ Each␈α⊃page␈α⊃there␈α⊂contains

␈↓ ↓*␈↓information␈αrelevant␈αto␈αone␈αconcept,␈αbroken␈αdown␈αby␈αfacets.␈α Since␈αthat's␈αa␈αcouple␈αhundred␈αpages␈αlong,␈αwe'll

␈↓ ↓*␈↓just␈α
glance␈α
at␈α
two␈α
typical␈α
BEINGs:␈α
EXAMPLES␈α
and␈α
COMPOSITION.␈α
 All␈α
the␈α
150␈α
BEINGs␈α∞in␈α
GIVEN

␈↓ ↓*␈↓will␈α
eventually␈α
be␈α
coded␈α
by␈α
hand␈α
and␈α
fed␈α
to␈α
AM.␈α
 Since␈α
the␈α
BEINGs␈α
are␈α
all␈α
considered␈α
equal,␈α
there␈α
is␈αin␈α
fact

␈↓ ↓*␈↓no␈αatypically␈α
interesting␈αone␈α
which␈αcan␈αbe␈α
exhibited.␈αSo␈α
the␈αtwo␈αbelow␈α
are␈αjust␈α
mediocre;␈αthey␈α
aren't␈αworth

␈↓ ↓*␈↓poring over.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&11.1. The "EXAMPLE" module␈↓)αβ␈↓


␈↓ ↓*␈↓<Look␈α∞at␈α
Examples␈α∞BEING,␈α∞from␈α
the␈α∞Given␈α
≡le>␈α∞This␈α∞is␈α
the␈α∞BEING␈α∞named␈α
ANY-BEING.EXAMPLES.

␈↓ ↓*␈↓Notice␈α⊂the␈α⊂general␈α⊂hints␈α⊂listed␈α⊂under␈α⊂FILLIN,␈α⊂for␈α⊂≡lling␈α∂in␈α⊂the␈α⊂examples␈α⊂of␈α⊂any␈α⊂given␈α⊂concept.␈α⊂β␈α⊂is␈α∂an

␈↓ ↓*␈↓abbreviation␈α⊂for␈α∂BEING.␈α⊂CHECK␈α∂explains␈α⊂how␈α⊂to␈α∂check␈α⊂any␈α∂unclear␈α⊂example,␈α∂and␈α⊂  REPR␈α⊂      is␈α∂the

␈↓ ↓*␈↓format␈αto␈αkeep␈αeach␈αExamples␈αpart␈αin.␈αThe␈αStructure␈αpart␈αsays␈αthat␈αwe␈αcan␈αsplit␈αo≥␈αany␈αinteresting␈αexample

␈↓ ↓*␈↓into a new BEING with no bookkeeping required.



␈↓ ↓*␈↓ε␈↓↓␈↓&EXAMPLES␈↓)αβ␈↓ε {not} {bdy}    Includes trivial, typical, and advanced cases of each type.

␈↓ ↓*␈↓ε␈↓ACT GROUPING␈↓ε
␈↓ ↓*␈↓ε FILLIN    Specialize β (in various ways) until only one entity is specified.
␈↓ ↓*␈↓ε        Any kind: instantiate specializations, and/or defn, and/or intu.
␈↓ ↓*␈↓ε                Inefficient: scan related specific entities until an ex. is found
␈↓ ↓*␈↓ε                        also: look for op. F: α→β; then use f(α.Examples).
␈↓ ↓*␈↓ε                Consider ana. β's examples, map them back. Examine similar β's exs.
␈↓ ↓*␈↓ε        Special: To get an example that satisfies P, bear P in mind at each spec. choice point.
␈↓ ↓*␈↓ε                Try setting various parameters s.t. they are equal to each other.
␈↓ ↓*␈↓ε        Simplest: plug in simple distinguished concepts into defn/intu. until singleton.
␈↓ ↓*␈↓ε                Consider the trivial case, the empty or minimal extreme, the base step of a recursive defn.
␈↓ ↓*␈↓ε        Big: Plug sophisticated, big, varied  examples of each variable concept into defn/intu.
␈↓ ↓*␈↓ε                If recursive, try to invert it so as to be able to build up harder and harder exs.
␈↓ ↓*␈↓ε                        Done: only use it to find a few actual examples.
␈↓ ↓*␈↓ε                        Save the inverted procedure for later usage, though.
␈↓ ↓*␈↓ε                Pick some decent example, find a chain of simpler and simpler examples
␈↓ ↓*␈↓ε                leading backward, then turn around and extrapolate off the hard end.
␈↓ ↓*␈↓ε        Prototypical: so representative that they will be useful in the future for
␈↓ ↓*␈↓ε                testing a conjecture for empirical plausibility.
␈↓ ↓*␈↓ε                Potentially: formally prove that any tie to this ex. is a valid tie.
␈↓ ↓*␈↓ε        Boundary: keep an eye out for two similar examples, one in and the other out.
␈↓ ↓*␈↓ε                        (these can come from work on this part and/or from similar B's exs.)
␈↓ ↓*␈↓ε                When such a pair is found, begin transforming them into each other,
␈↓ ↓*␈↓ε                and zero in on the most similar pair of in/out entities. Examine carefully.
␈↓ ↓*␈↓ε        Afterwards: if ¬∃ exs. before, ∃ exs. now, ∃ op O on β with no O.Exs, then
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  43␈↓

␈↓ ↓*␈↓ε                place into O.Exs. this suggestion:  can plug eles. of β.Exs into O.
␈↓ ↓*␈↓ε STRUCTURE   Split off any indiv. or class of high-int. examples, keep the rest here.
␈↓ ↓*␈↓ε CHECK   Each ex. should satisfy the defn. and intu parts.
␈↓ ↓*␈↓ε        If any type of new procedure (e.g., inverted recursive defn) has been created,
␈↓ ↓*␈↓ε        which supposedly generates β's, try to conjecture about the uniqueness of the
␈↓ ↓*␈↓ε        members of the sequence produced, the totality of their covering of all possible β's, etc.
␈↓ ↓*␈↓ε REPRESENTATION  (name, type, (part present, value, change from β)*, origin, complexity,
␈↓ ↓*␈↓ε        distance from the boundary of β)*
␈↓ ↓*␈↓ε        Data about examples should be stored under part headings (easy to make them BEINGs).
␈↓ ↓*␈↓ε        The types of type are: +, + boundary, -, - boundary.
␈↓ ↓*␈↓ε        The types of origin are: by specializing defn, by intu, to satisfy property.

␈↓ ↓*␈↓ε␈↓INFO GROUPING␈↓ε
␈↓ ↓*␈↓ε DEFINITION  Specific entities which are (not)(barely) instances of β's.
␈↓ ↓*␈↓ε INTU  Zero in on specific representatives, individuals.
␈↓ ↓*␈↓ε EXAMPLES  Access the Examples part of any BEING, and it contains examples of that.
␈↓ ↓*␈↓ε TIES  Up: Info group.   Side: Specializations.




␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&11.2. The "COMPOSE" module␈↓)αβ␈↓


␈↓ ↓*␈↓<Look at Compose BEING, from the Given ≡le>

␈↓ ↓*␈↓RECOG:Here are productions which might match against the current situation.

␈↓ ↓*␈↓INTEREST:␈αHere␈αare␈α
some␈αspecial␈αfeatures␈α
that␈αmake␈αa␈α
composition␈αinteresting.␈α Since␈α
composition␈αis␈αa␈α
type

␈↓ ↓*␈↓of␈α
␈↓βoperation␈↓␈αwhich␈α
is␈αa␈α
type␈αof␈α
␈↓βactivity␈↓,␈αwhich␈α
is␈αa␈α
type␈αof␈α
␈↓βany-Being␈↓,␈α these␈α
hints␈αare␈α
all␈αin␈α
addition␈α
to␈α(or

␈↓ ↓*␈↓exceptions␈α→to)␈α→the␈α→hints␈α→listed␈α→under␈α→OPERATION.INTEREST␈α→and␈α→ACTIVITY.INTEREST␈α_and

␈↓ ↓*␈↓ANY-BEING.INTEREST.

␈↓ ↓*␈↓VIEWS: Here is how to view this as a set, an operator, a relation, a property.

␈↓ ↓*␈↓INTU:␈αThis␈αspeci≡es␈αthat␈α
we␈αmay␈αabstractly␈αmanipulate␈αexpressions␈α
about␈αcompositions␈αby␈αthinking␈αin␈α
terms

␈↓ ↓*␈↓of␈αarrows␈αlanding␈αand␈αbeing␈αre≡red␈αagain.␈α This␈αmeshes␈αwith␈αthe␈αintuition␈αof␈αan␈αOPERATION␈αas␈αa␈αset␈αof

␈↓ ↓*␈↓arrows being ≡red from one set into another.


␈↓ ↓*␈↓ε␈↓↓␈↓&COMPOSITION␈↓)αβ␈↓ε    Apply an operation to the result of a previous operation.

␈↓ ↓*␈↓ε␈↓RECOGNITION GROUPING␈↓ε
␈↓ ↓*␈↓ε CHANGES  {(eles. of dom. of 1st changed to some eles. of range of 2nd., .90, .80, )}
␈↓ ↓*␈↓ε FINAL {(2nd range ele., .20, .70, intu: final product of a 2-step manufac. process)}
␈↓ ↓*␈↓ε PAST
␈↓ ↓*␈↓ε IDEN {not}{quick}    This must be filled in for AM initially, but hasn't been yet.

␈↓ ↓*␈↓ε␈↓ALTER GROUPING␈↓ε
␈↓ ↓*␈↓ε GENERALIZATIONS    Sequence of actions, an ordered pair of operations to perform.
␈↓ ↓*␈↓ε        FILLIN: Increase 1st true domain, typically by modifying the definition.
␈↓ ↓*␈↓ε        Increase 2nd range.
␈↓ ↓*␈↓ε SPECIALIZATIONS  Given an Active F:AxB→C, and an Active G:CxD→E, consider the new Active
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  44␈↓

␈↓ ↓*␈↓ε        h:(AxB)xD→E, written g*(fxi), defined as h((a,b),c)= g(f((a,b)),c). Alternatively,
␈↓ ↓*␈↓ε        given f:BxD→C and g:AxC→E, construct h:(AxB)xD→E as h((a,b),c)= g(a,f((b,c))).
␈↓ ↓*␈↓ε        An even further specialization: let some of A,B,C,D,E coincide.
␈↓ ↓*␈↓ε        An even further specialization: let all of A,B,C,D,E coincide.
␈↓ ↓*␈↓ε BOUNDARY  {(second operation doesn't care what the result of the first was, sequenceing unnec))}
␈↓ ↓*␈↓ε DOMAIN/RANGE {not}   (AOP␈↓	2␈↓ε, OP, op)
␈↓ ↓*␈↓ε        FILLIN: domain is a pair of operations, range is a new operation whose domain
␈↓ ↓*␈↓ε                is the dom. of the 1st pair element, and whose range is the range of the 2nd.
␈↓ ↓*␈↓ε                int. depends on the 2 ops and on props which this particular op possesses;
␈↓ ↓*␈↓ε                if high enough and no ptr, BEINGize this new op.
␈↓ ↓*␈↓ε ORDERING(Complete)
␈↓ ↓*␈↓ε WORTH (.8, .95, .5, .5, .6, .9, .5, (1.0 formal, .8 intu), .8, (1.0 basis))
␈↓ ↓*␈↓ε        FILLIN: To forestall an infinite regress, decrease the activation energy of this investigation.
␈↓ ↓*␈↓ε INTEREST  Int. property of result which is not true of either argument relation.
␈↓ ↓*␈↓ε        Int. properties of both argument relations are preserved, undesirable ones lost.
␈↓ ↓*␈↓ε        Int. subsets (cases) of domain of 1st map into interesting subsets of range of 2nd.
␈↓ ↓*␈↓ε        Preimages of int subsets (cases) of range(2nd) are themselves interesting subsets of domain(1st).
␈↓ ↓*␈↓ε        The range of the first is equal to, not just a subset of, the domain of the second.
␈↓ ↓*␈↓ε OPERATIONS

␈↓ ↓*␈↓ε␈↓ACT GROUPING␈↓ε
␈↓ ↓*␈↓ε BOUNDARY-OPERATIONS {not}
␈↓ ↓*␈↓ε ALGORITHMS
␈↓ ↓*␈↓ε        FILLIN: Sequence: do 1st op (ALG), then take result and do 2nd op on it (ALG).
␈↓ ↓*␈↓ε REPRESENTATION  repr. as any operation, or perhaps just as 2nd o 1st.
␈↓ ↓*␈↓ε VIEWS  to view as a reln: view each arg. op. as a reln, R␈↓#v1␈↓# ⊂ AxB, R␈↓#v2␈↓# ⊂ BxC, composition
␈↓ ↓*␈↓ε                is a relation R␈↓#v2␈↓# o R␈↓#v1␈↓#  ⊂  AxC, where (a,c) is in composition iff
␈↓ ↓*␈↓ε                (a,b) ε R␈↓#v1␈↓#   and   (b,c) ε R␈↓#v2␈↓#.
␈↓ ↓*␈↓ε        to view as op: domain is dom of 1st, intermed. activity, then range is range of 2nd.
␈↓ ↓*␈↓ε        to view as a prop:  prop. master set is (same notation) AxC; R␈↓#v2␈↓#oR␈↓#v1␈↓#(a,c)) ↔
␈↓ ↓*␈↓ε                R␈↓#v2␈↓#(R␈↓#v1␈↓#(a), c).
␈↓ ↓*␈↓ε        to view as a set: set of ordred pairs which is a subset of dom␈↓#v1␈↓# x range␈↓#v2␈↓#.

␈↓ ↓*␈↓ε␈↓INFO GROUPING␈↓ε
␈↓ ↓*␈↓ε DEFINITION   The sequencing of two operations, f and g, where domain of f contains the
␈↓ ↓*␈↓ε        range of g as a subset. E.g., f:A→D, and g:W→B with B⊂A.
␈↓ ↓*␈↓ε        Then the new combined operation is the composition of f and g, written
␈↓ ↓*␈↓ε        f o g, f(g), fg. The computation is straightforward; apply g and then apply
␈↓ ↓*␈↓ε        f to the result.

␈↓ ↓*␈↓ε        If f:AxB→C,   g:D→A,  and h:E→B, then one can consider the composition f o gxh,
␈↓ ↓*␈↓ε        also written f(g,h), whose value on (x,y) ε DxE is f(g(x),h(y)) ε C.
␈↓ ↓*␈↓ε        This is combinatorially explosive. Suggestions for containment: possibility of
␈↓ ↓*␈↓ε        considering f o gxg;  f o ixj  where at least one of i,j is the identity,
␈↓ ↓*␈↓ε        especially if the other one of i,j is equal to f itself; f o ixj, where one
␈↓ ↓*␈↓ε        of i,j is f, occasionally both are f. In general, f o gxh is about as intersting
␈↓ ↓*␈↓ε        as the ties between the three functions are already (equality is quite high).

␈↓ ↓*␈↓ε        To expand this flexibility, view any Active as an operation for these purposes.
␈↓ ↓*␈↓ε        The composition FoG is more formally defined as the specific set of ordered pairs
␈↓ ↓*␈↓ε        (a,B) where aεdom(G) and B⊂range(F), and B={F(c) | cεG(a)}, together with an
␈↓ ↓*␈↓ε        explicit statement of the domain and range of F and of G.
␈↓ ↓*␈↓ε INTU   Arrows emanate from one set, go into another set, new arrows go from there to
␈↓ ↓*␈↓ε        a third set. The composition means follow along and transfer to new arrows.
␈↓ ↓*␈↓ε TIES   Up: Operation
␈↓ ↓*␈↓ε EXAMPLES {not} {bdy}
␈↓ ↓*␈↓ε CONTENTS


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&11.3. Intuition for a Set␈↓)αβ␈↓


␈↓ ↓*␈↓Here␈αare␈α
some␈αof␈α
the␈αintuitive␈α
images␈αwhich␈α
must␈αbe␈α
simulated␈αby␈α
the␈αprogram␈α
in␈αslot␈α
"INTUITIONS"␈αof

␈↓ ↓*␈↓the BEING named "SET".
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  45␈↓

␈↓ ↓*␈↓ Our␈α
≡rst␈α
characterization␈α
of␈α
a␈α
set␈α
will␈α
be␈α
as␈αa␈α
solid␈α
rectangle␈α
in␈α
the␈α
Cartesian␈α
plane;␈α
the␈α
opaque␈αintuition

␈↓ ↓*␈↓function␈α
knows␈α∞about␈α
numerical␈α∞equality␈α
and␈α∞inequality,␈α
hence␈α∞about␈α
borders␈α∞of␈α
such␈α∞sets.␈α
The␈α∞notions␈α
of

␈↓ ↓*␈↓intersection,␈α
union,␈α
complement,␈αsetdi≥erence,␈α
disjointness,␈α
projection␈αonto␈α
each␈α
axis,␈αetc.␈α
are␈α
also␈αintuitively

␈↓ ↓*␈↓available.␈α
 Notice␈α
that␈α
the␈α
sophisticated␈α
operations␈αrequired␈α
(e.g.,␈α
projection)␈α
will␈α
exist␈α
as␈α
opaque␈αfunctions,

␈↓ ↓*␈↓totally inaccessable to the rest of the system␈↓	46␈↓.


␈↓ ↓*␈↓This␈α"square"␈αrepresentation␈αis␈αnot␈αwell␈αsuited␈αto␈αall␈αconcepts␈αinvolving␈αsets.␈α For␈αthat␈αreason,␈αthe␈αsystem␈αwill

␈↓ ↓*␈↓simultaneously␈αmaintain␈αseveral␈αof␈αthe␈αother␈α
forms␈αof␈αintuitive␈αstorage␈αmentioned␈αpreviously.␈α
 Consider,␈αfor

␈↓ ↓*␈↓example,␈αthe␈α
possibility␈αof␈αfuzzy␈α
rules,␈αwhich␈αcan␈α
latch␈αonto␈αalmost␈α
anything␈αand␈αproduce␈α
some␈αtype␈αof␈α
result

␈↓ ↓*␈↓(but␈αwith␈αlow␈αcertainty).␈αThat␈αis,␈αthey␈αoperate␈αat␈αa␈αhigher␈αlevel␈αof␈αabstraction␈αthan␈αde≡nite␈αrules,␈αby␈αignoring

␈↓ ↓*␈↓many␈αdetails.␈αAnother␈αpossibility␈αis␈αthe␈α
use␈αof␈αexamples.␈αIf␈αa␈αsmall␈αset␈α
of␈αthem␈αcan␈αbe␈αfound␈αwhich␈α
is␈αtruly

␈↓ ↓*␈↓representative␈αof␈αa␈αconcept,␈αthen␈αfuture␈αreferences␈αto␈αthat␈αconcept␈αcan␈αbe␈αcompared␈αto␈αthese␈αexamples.␈α This

␈↓ ↓*␈↓may sound very crude, but I believe that people rely heavily (and successfully!) on it.


␈↓ ↓*␈↓Euler,␈α∂to␈α∂overcome␈α∂language␈α∂problems␈α∂when␈α∂lecturing␈α∂a␈α∞princess␈α∂of␈α∂Sweden,␈α∂devised␈α∂the␈α∂use␈α∂of␈α∂circles␈α∞to

␈↓ ↓*␈↓represent␈αsets.␈αVenn␈αand␈αothers␈αhave␈αfrequently␈αadopted␈αthis␈αimage.␈αFor␈αa␈αmachine,␈αit␈αseems␈αmore␈αa␈αpropos

␈↓ ↓*␈↓to␈α
use␈α
a␈αrectangle,␈α
not␈α
a␈α
circle.␈α Consider␈α
 the␈α
lattice␈α
of␈αintegral␈α
points␈α
in␈α
two␈αdimensions.␈α
Now␈α
a␈α
set␈αis␈α
viewed

␈↓ ↓*␈↓as␈αa␈αrectangle␈α
--␈αor␈αa␈αcombination␈α
of␈αa␈αfew␈αrectangles␈α
--␈αin␈αthis␈α
space.␈αThis␈αmakes␈αit␈α
hard␈αto␈αget␈αany␈α
intuition

␈↓ ↓*␈↓about␈αcontinuity␈αor␈αboundary␈αor␈αopenness,␈αbut␈αworks␈α≡ne␈αfor␈αthe␈αdiscrete␈αsets␈αwhich␈αare␈αdealt␈αwith␈αin␈αlogic,

␈↓ ↓*␈↓elementary␈α
set␈α
theory,␈α
arithmetic,␈α
number␈αtheory,␈α
and␈α
algebra.␈α
It␈α
is␈αprobable␈α
that␈α
the␈α
system␈α
will␈αtherefore␈α
not

␈↓ ↓*␈↓be␈αtried␈αin␈αthe␈αdomains␈αof␈αreal␈αanalysis,␈αgeometry,␈αtopology,␈αetc.␈αwith␈αonly␈αthis␈αprimitive␈αnotion␈αof␈αspace␈αand

␈↓ ↓*␈↓con≡nement.␈α Speci≡cly,␈αa␈αset␈αin␈αthis␈αworld␈αis␈αan␈αordered␈αpair␈αof␈αpairs␈αof␈αnatural␈αnumbers.␈αProjection␈αis␈αthus

␈↓ ↓*␈↓trivial␈α∂in␈α∂LISP␈α∂(CAR␈α∂or␈α∂CADR),␈α∂as␈α∂is␈α⊂test␈α∂for␈α∂intersection,␈α∂subset,␈α∂etc.␈α∂ Notice␈α∂that␈α∂these␈α∂require␈α⊂use␈α∂of

␈↓ ↓*␈↓numbers,␈αordering,␈αsets,␈αetc.,␈αso␈αthe␈αfunctions␈αwhich␈αaccomplish␈αthem␈αmust␈αbe␈αopaque.␈α The␈αinteraction␈αwith


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	46␈↓ε␈α	 This␈α	is␈α
worth␈α	rejustifying:␈α	is␈α
fair␈α	to␈α	write␈α
a␈α	LISP␈α	program␈α
(which␈α	uses␈α	the␈α
function␈α	TIMES)␈α	whose␈α
task␈α	is␈α	to␈α
synthesize␈α	code␈α	for␈α
the␈α	function
␈↓ ↓*␈↓ε␈↓ αJTIMES, so long as the program does not have access to, does not even know about its use of TIMES.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  46␈↓

␈↓ ↓*␈↓the␈αrest␈αof␈αthe␈αsystem␈αwill␈αbe␈αfor␈αthese␈αpictures␈αto␈αsuggest␈αand␈αreinforce␈αand␈αveto␈αvarious␈αconjectures.␈α They

␈↓ ↓*␈↓serve␈α⊃to␈α⊂generate␈α⊃empirical␈α⊂evidence␈α⊃for␈α⊂the␈α⊃rest␈α⊃of␈α⊂the␈α⊃system.␈α⊂ To␈α⊃avoid␈α⊂gerrymandering,␈α⊃it␈α⊃might␈α⊂be

␈↓ ↓*␈↓necessary␈αto␈α
view␈αa␈αset␈α
as␈αa␈αlist␈α
(of␈αarbitrary␈αlength)␈α
of␈αordered␈αpairs;␈α
an␈αabsent␈αpair␈α
can␈αbe␈αassumed␈α
to␈αbe

␈↓ ↓*␈↓some default pair␈↓	47␈↓.


␈↓ ↓*␈↓Still␈αother␈αrepresentations␈αwill␈αbe␈αmaintained:␈αpointer␈αstructures␈α(graphs␈αwhere␈αeach␈αarc␈αrepresents␈αa␈α
relation,

␈↓ ↓*␈↓e.g. ␈↓¬ε, ⊂␈↓), activities (the characteristic function for the set), algebraic equations describing regions of space, etc.


␈↓ ↓*␈↓How should the system choose which intuitive representation(s) of a set to use?  Some considerations are:


␈↓ ↓*␈↓What␈α∞operations␈α∞are␈α∂to␈α∞be␈α∞done␈α∂to␈α∞this␈α∞set␈α∂(e.g.,␈α∞␈↓¬ε␈↓,␈α∞⊂,␈α∂∩,␈α∞∪,␈α∞␈↓¬≡␈↓,␈α∂=,␈α∞',...)?␈α∞The␈α∂representations␈α∞di≥er␈α∞in␈α∂cost␈α∞of

␈↓ ↓*␈↓maintenance and in the ease with which each of these operations can be carried out.


␈↓ ↓*␈↓How␈αarti≡cial␈αis␈α
the␈αrepresentation␈αfor␈α
the␈αgiven␈αset?␈α
 Some␈αwill␈αbe␈α
quite␈αnatural,␈αe.g.,␈α
if␈αthe␈αset␈α
is␈αa␈αnest␈α
then

␈↓ ↓*␈↓use␈α∩the␈α∪pointer␈α∩structure;␈α∪if␈α∩the␈α∪set␈α∩is␈α∪a␈α∩relation␈α∪over␈α∩the␈α∪small␈α∩set␈α∪AxB,␈α∩then␈α∪use␈α∩the␈α∪lattice␈α∩points

␈↓ ↓*␈↓representation.


␈↓ ↓*␈↓How␈αmuch␈α
is␈α"given␈αaway"␈α
by␈αthe␈αmodel?␈α
This␈αis␈αa␈α
question␈αof␈αfairness,␈α
and␈αmeans␈αthat␈α
the␈αsystem-writers

␈↓ ↓*␈↓must build in opacity constraints and/or make the intuitive operations faulty.  We shall do both.


␈↓ ↓*␈↓How␈α⊂compatible␈α⊂is␈α⊂each␈α⊃representation␈α⊂with␈α⊂the␈α⊂computer's␈α⊂physiology?␈α⊃ Thus␈α⊂it␈α⊂is␈α⊂almost␈α⊃impossible␈α⊂to

␈↓ ↓*␈↓represent␈α∂pictures␈α∂or␈α∂blobs␈α∂directly,␈α∂but␈α∂very␈α∞suitable␈α∂to␈α∂store␈α∂algebraic␈α∂equations␈α∂de≡ning␈α∂such␈α∞geometric

␈↓ ↓*␈↓images.


␈↓ ↓*␈↓Does␈α
the␈α
representation␈α
suggest␈αa␈α
set␈α
theory␈α
with␈αbasic␈α
elements␈α
which␈α
are␈αnon-sets;␈α
with␈α
an␈α
in≡nite␈αmodel;

␈↓ ↓*␈↓with␈α∩any␈α⊃special␈α∩desirable␈α⊃or␈α∩undesirable␈α⊃qualities?␈α∩For␈α⊃example,␈α∩the␈α⊃geometric␈α∩representation␈α∩seems␈α⊃to

␈↓ ↓*␈↓demand the concept of continuity, which the system probably won't ever use in any de≡nite way.


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	47␈↓ε  That is, a set is a simplex in Hilbert space; each set has infinite dimension, but differs from any other in only finitely many of them.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  47␈↓

␈↓ ↓*␈↓There␈α
are␈α
about␈α
150␈α
BEINGs␈α
in␈α
the␈α
proposed␈α
core,␈αand␈α
each␈α
one␈α
of␈α
them␈α
should␈α
have␈α
an␈α
intuition␈αalmost␈α
as

␈↓ ↓*␈↓rich␈αas␈αthat␈αfor␈αSETS,␈αabove.␈αSpace␈αprecludes␈αdelving␈αinto␈αeach␈αone;␈αsome␈αfew␈αlines␈αabout␈αeach␈αβ's␈αintuition

␈↓ ↓*␈↓is present in the document "␈↓βGIVEN KNOWLEDGE␈↓".␈↓	48␈↓











































␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	48␈↓ε This  is available as the file named GIVEN, on the directory [AM,AJC], at the Stanford AI Lab, which is ARPA-NET site named SAIL.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  48␈↓

␈↓ ↓*␈↓␈↓ ∧ ␈↓∧␈↓&12. Example: The Modules Interacting␈↓)αβ␈↓


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&12.1. Some Early Numerical Concepts␈↓)αβ␈↓


␈↓ ↓*␈↓In␈α
order␈α
to␈α
be␈α
interesting,␈α
the␈α∞example␈α
session␈α
in␈α
the␈α
next␈α
section␈α∞takes␈α
place␈α
after␈α
AM␈α
has␈α
learned␈α∞a␈α
few

␈↓ ↓*␈↓numerical␈αconcepts.␈αTo␈αmake␈αthis␈αmore␈αplausible,␈αlet␈αme␈αbrie∨y␈αshow␈αhow␈αsome␈αof␈αthese␈αmight␈αbe␈αdiscovered

␈↓ ↓*␈↓"naturally" by AM.


␈↓ ↓*␈↓One␈α_of␈α→the␈α_things␈α→AM␈α_knows␈α_about␈α→relations␈α_is␈α→when␈α_they␈α_are␈α→interesting.␈α_ (The␈α→facet␈α_named

␈↓ ↓*␈↓"Interestingness"␈αon␈αthe␈αconcept␈αnamed␈α"Relation".)␈αThese␈αfeatures␈αinclude:␈α
(i)␈αIf␈αR␈αis␈αa␈αrelation␈αfrom␈αA␈αto␈α
B,

␈↓ ↓*␈↓then␈α
R␈α
is␈α
interesting␈α∞if␈α
the␈α
image␈α
of␈α∞each␈α
element␈α
␈↓¬xεA␈↓␈α
satis≡es␈α
some␈α∞B-interesting␈α
property;␈α
or,␈α
(ii)␈α∞all␈α
(x,y)

␈↓ ↓*␈↓pairs,␈α
for␈α
␈↓¬x,yεA␈↓,␈α
are␈α
in␈α
some␈α
 B-interesting␈α
relation.␈α Suppose␈α
AM␈α
is␈α
considering␈α
only␈α
relations␈α
from␈α
sets␈αto

␈↓ ↓*␈↓sets,␈α⊃so␈α⊃A␈α⊃and␈α⊃B␈α⊃are␈α⊃two␈α⊃sets.␈α⊃ Then␈α⊃SET.INTEREST␈α⊃is␈α⊃accessed,␈α⊃and␈α⊃he␈α⊃tells␈α⊃the␈α⊃system␈α⊃ that␈α⊃some

␈↓ ↓*␈↓interesting␈αset␈αproperties␈αand␈αrelations␈αare␈αknown␈αto␈αbe:␈αsingleton,␈αequal,␈αdisjoint.␈αThese␈αlead,␈αrespectively,␈αto

␈↓ ↓*␈↓the␈α⊂concepts␈α⊂of␈α⊂Function␈α∂(all␈α⊂images␈α⊂are␈α⊂singletons),␈α∂Constant␈α⊂(all␈α⊂images␈α⊂are␈α∂equal),␈α⊂1-1␈α⊂(no␈α⊂two␈α∂images

␈↓ ↓*␈↓intersect).


␈↓ ↓*␈↓As␈α⊂another␈α⊂example,␈α⊂a␈α⊂composition␈α⊂is␈α⊂known␈α⊂to␈α⊃be␈α⊂interesting␈α⊂if␈α⊂its␈α⊂component␈α⊂relations␈α⊂are,␈α⊂and␈α⊃if␈α⊂its

␈↓ ↓*␈↓domain␈α'and␈α(range␈α'are␈α'related␈α(by␈α'an␈α'interesting␈α(relation.␈α' This␈α'leads␈α(to␈α'considering

␈↓ ↓*␈↓MAP-STRUCTURE(REVERSE-ORDERED-PAIR);␈α⊃that␈α⊂is,␈α⊃take␈α⊂a␈α⊃relation,␈α⊂view␈α⊃it␈α⊂as␈α⊃a␈α⊂set␈α⊃of␈α⊂ordered

␈↓ ↓*␈↓pairs,␈α∂then␈α∂reverse␈α⊂each␈α∂one,␈α∂then␈α∂view␈α⊂the␈α∂resultant␈α∂set␈α∂as␈α⊂a␈α∂relation␈α∂again.␈α∂This␈α⊂is␈α∂just␈α∂the␈α⊂concept␈α∂of

␈↓ ↓*␈↓Inverse,␈α⊗and␈α⊗is␈α⊗interesting␈α⊗because␈α↔it␈α⊗takes␈α⊗relations␈α⊗on␈α⊗CROSS-PRODUCT(A,B)␈α⊗into␈α↔relations␈α⊗on

␈↓ ↓*␈↓CROSS-PRODUCT(REVERSE-ORDERED-PAIR((A,B)).


␈↓ ↓*␈↓For␈α
our␈α
example,␈α
all␈α
the␈α
following␈α
concepts␈α
should␈α
be␈α
developed␈α
already:␈α
Count␈↓	49␈↓,␈α
Inverse,␈α
Commutativity,

␈↓ ↓*␈↓Transitivity, Associativity, Singleton, Function, Successor, Zero, Plus, Times, One, Two.


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	49␈↓ε  This operation converts any list (of length n) into canonical form (perhaps a list of n NILs).
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  49␈↓

␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&12.2. A Hypothetical Session␈↓)αβ␈↓


␈↓ ↓*␈↓Assume␈αthat␈αAM␈αhas␈αa␈αgrasp␈αof␈αthe␈αearly␈αnumerical␈αconcepts␈αmentioned,␈αas␈αwell␈αas␈αof␈αpremathematical␈αones.

␈↓ ↓*␈↓We␈αare␈αnow␈αready␈αto␈αexamine␈αthe␈αexample␈αof␈αhow␈αa␈α session␈αwith␈αAM␈αmight␈αappear␈αto␈αthe␈αuser.␈α In␈αa␈αfew

␈↓ ↓*␈↓minutes,␈α∞we'll␈α∞go␈α∞into␈α∞detail␈α
about␈α∞how␈α∞AM␈α∞actually␈α∞does␈α∞this.␈α
 Suppose␈α∞that␈α∞AM␈α∞has␈α∞just␈α∞considered␈α
the

␈↓ ↓*␈↓concept of repeated TIMES-ing, that is, exponentiation.


␈↓ ↓*␈↓<THE␈α"FOLLOWING␈α"IS␈α#THE␈α"COMPLETE␈α"SLIDE;␈α#AFTERWARDS␈α"IS␈α"THE␈α#SET␈α"OF

␈↓ ↓*␈↓ACCOMPANYING PATTER>

␈↓ ↓*␈↓¬1.␈α
SYSTEM:␈αRate␈α
of␈α
drop␈αof␈α
interest␈α
in␈αRepeated␈α
TIMES-ing␈α
(compared␈αto␈α
SUCCESSOR,␈α
PLUS,␈αTIMES)␈α
is␈αhuge.␈α
 Not
␈↓ ↓*␈↓¬␈↓ ↓Zassociative␈αor␈αcommutative;␈αno␈αLeft␈αidentity␈αelement.␈α Dissuades␈αme␈αfrom␈αpursuing␈αhigher␈αorder␈αconstructed
␈↓ ↓*␈↓¬␈↓ ↓Zoperations.

␈↓ ↓*␈↓¬2. SYSTEM: I am now investigating the inverses of Plus and Times.  In general, the inverses are not functions.

␈↓ ↓*␈↓¬3.␈αSYSTEM:␈αLooking␈αat␈αmore␈α
examples␈αof␈αTIMES␈↓	-1␈↓¬.␈αTIMES␈↓	-1␈↓¬(n)␈αseems␈αalways␈α
to␈αinclude,␈αas␈αone␈αordered␈α
pair,␈α(n,
␈↓ ↓*␈↓¬␈↓ ↓Z1). Same for (1, n).  Conjecture: (∀xεN)  (x,1) ε TIMES␈↓	-1␈↓¬(x), and (1,x) ε TIMES␈↓	-1␈↓¬(x).

␈↓ ↓*␈↓¬4. SYSTEM: Proof is simple: TIMES(x,1)=x=TIMES(1,x).

␈↓ ↓*␈↓¬5.␈αSYSTEM:␈αThe␈αtwo␈αpairs␈α(n,1)␈αand␈α(1,n)␈αare␈αdistinct␈αiff␈αthe␈αnumber␈αn␈αis␈αnot␈αequal␈αto␈α1.␈α Noticing␈αthe␈αnumber␈α1
␈↓ ↓*␈↓¬␈↓ ↓Zas␈α
an␈α∞exception,␈α
we␈α∞can␈α
state:␈α∞Corollary:␈α
COUNT*TIMES␈↓	-1␈↓¬(1)␈α∞=␈α
1;␈α∞ (∀x>1)␈α
COUNT*TIMES␈↓	-1␈↓¬(x)␈α∞≥␈α
2.␈α∞ COUNT␈α
*
␈↓ ↓*␈↓¬␈↓ ↓ZTIMES␈↓	-1␈↓¬ is the how many ways one may express a number as a product of an ordered pair of numbers.

␈↓ ↓*␈↓¬6. SYSTEM: Interesting to consider the subset P of N, defined as {xεN | COUNT*TIMES␈↓	-1␈↓¬(x)=2}.

␈↓ ↓*␈↓¬7. USER: (interrupting) Why is this interesting?

␈↓ ↓*␈↓¬8.␈α
SYSTEM:␈α
At␈α
the␈α
moment,␈α
we␈α
have␈α
the␈α
known␈α
fact␈αf(x)≥y;␈α
try␈α
to␈α
pin␈α
down␈α
extreme␈α
cases,␈α
those␈α
x␈α
for␈αwhich␈α
f(x)
␈↓ ↓*␈↓¬␈↓ ↓Zis␈αprecisely␈αequal␈αto␈αy.␈αf␈αis␈α
COUNT*TIMES␈↓	-1␈↓¬,␈αin␈αthe␈αcurrent␈αcase,␈αand␈αy␈α
is␈α2.␈α Do␈αyou␈αhave␈αany␈αspecial␈α
names
␈↓ ↓*␈↓¬␈↓ ↓Zyou would like me to refer to COUNT * TIMES␈↓	-1␈↓¬ by? Any special names for the subset P?

␈↓ ↓*␈↓¬9. USER: Call that composition FAC.  Call the set P, of extreme numbers under FAC, PRIMES.

␈↓ ↓*␈↓¬10. SYSTEM: Syntactic transformation: PRIME is the predicate which means "is in PRIMES".

␈↓ ↓*␈↓¬11.␈αSYSTEM:␈αTIMES␈αmay␈αalso␈αbe␈αviewed␈αas␈αoperating␈αon␈αa␈αBAG;␈αin␈αthis␈αsense,␈αTIMES␈↓	-1␈↓¬␈αwould␈αreturn␈αall␈αpossible
␈↓ ↓*␈↓¬␈↓ ↓Zbags␈α(of␈α
any␈αlength)␈α
whose␈αproduct␈α
is␈αthe␈α
given␈αnumber.␈αWe␈α
do␈αnot␈α
allow␈αbags␈α
to␈αhave␈α
<2␈αmembers,␈αand␈α
we
␈↓ ↓*␈↓¬␈↓ ↓Zdo␈αnot␈αallow␈αbags␈αlonger␈αthan␈α2␈αto␈αhave␈αany␈α1's␈αin␈αthem.␈α In␈αthis␈αcase,␈α(5)␈αjust␈αsays:␈αCOUNT(TIMES␈↓	-1␈↓¬(x))␈α≥␈α1.
␈↓ ↓*␈↓¬␈↓ ↓ZThe extremes now are just 1 and PRIMES.   I want a new name for this new interpretation of TIMES␈↓	-1␈↓¬.

␈↓ ↓*␈↓¬12. USER: Call it FACTORS.

␈↓ ↓*␈↓¬13.␈αSYSTEM:␈αConjecture:␈αFACTORS(x)␈αalways␈αseems␈αto␈αcontain␈αa␈αbag␈αcontaining␈αonly␈αones␈αand␈αprimes,␈αand␈αonly
␈↓ ↓*␈↓¬␈↓ ↓Zone such bag.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  50␈↓

␈↓ ↓*␈↓¬14.␈α
SYSTEM:␈α
Proof␈α∞of␈α
existence␈α
is␈α
by␈α∞contradiction,␈α
using␈α
the␈α
lemma␈α∞ ¬PRIME(x)␈α
→␈α
∃(u,v)εTIMES␈↓	-1␈↓¬(x)␈α∞s.t.␈α
1<u<x,
␈↓ ↓*␈↓¬␈↓ ↓Z1<v<x.␈α
 Proof␈αof␈α
Uniqueness␈α
is␈αby␈α
contradiction,␈αusing␈α
the␈α
lemma␈α(∀a,bεN,␈α
xεPRIMES).␈αx|TIMES(a,b)␈α
iff␈α
x|a␈αor
␈↓ ↓*␈↓¬␈↓ ↓Zx|b.

␈↓ ↓*␈↓¬15.␈α
USER:␈α
Call␈α
this␈α
the␈α
unique␈α
factorization␈α
theorem.␈α
This␈α
is␈α
very␈α
important.␈α
 Consider␈α
now␈α
the␈α
sum␈α
of␈α
all␈α
the
␈↓ ↓*␈↓¬␈↓ ↓Zdivisors of a number.

␈↓ ↓*␈↓¬16.␈α
SYSTEM:␈α
PLUS␈α*␈α
UNION␈α
*␈α
TIMES␈↓	-1␈↓¬␈αis␈α
being␈α
investigated.␈αSeems␈α
to␈α
range␈α
from␈αa␈α
low␈α
extreme␈αof␈α
Successor(the
␈↓ ↓*␈↓¬␈↓ ↓Znumber)␈αup␈αto␈α
a␈αhigh␈αextreme␈αof␈α
almost␈α Times(the␈αnumber,␈α
itself).␈αThe␈αnumbers␈αat␈α
the␈αlow␈αextreme␈αseem␈α
to
␈↓ ↓*␈↓¬␈↓ ↓Zbe␈α
the␈α
primes.␈α
 The␈α
numbers␈α
near␈α
the␈α
high␈αextreme␈α
may␈α
not␈α
be␈α
worth␈α
naming.␈α
An␈α
intermediate␈α
point␈αis␈α
twice
␈↓ ↓*␈↓¬␈↓ ↓Zthe␈α∞number␈α∞itself.␈α
That␈α∞is,␈α∞those␈α
numbers␈α∞n␈α∞s.t.␈α
Times(2,n)=␈α∞PLUS*UNION*TIMES␈↓	-1␈↓¬(n).␈α∞ Another␈α
intermediate
␈↓ ↓*␈↓¬␈↓ ↓Zset␈αof␈αpoints␈αis␈αPRIMES;␈αthat␈αis,␈αthose␈αnumbers␈αn␈αs.t.␈αPRIME(PLUS*UNION*TIMES␈↓	-1␈↓¬(n))␈αholds.␈α This␈αmay␈αlead␈αto
␈↓ ↓*␈↓¬␈↓ ↓Zthe conjecture that P*U*T(N) = PRIMES.

␈↓ ↓*␈↓¬17.␈αUSER:␈αCall␈α
these␈α=2n␈αnumbers␈α"Perfect␈α
Numbers."␈αCall␈αthe␈αothers␈α
nothing.␈α There␈αare␈αnot␈α
many␈αinteresting
␈↓ ↓*␈↓¬␈↓ ↓Zthings to prove involving either idea.

␈↓ ↓*␈↓¬18. SYSTEM: Going on to new topic, then. I shall consider comparing TIMES␈↓	-1␈↓¬ collections from two numbers.

␈↓ ↓*␈↓¬<this leads to GCD and relative primeness>


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&12.3. Comments on the Session␈↓)αβ␈↓

␈↓ ↓*␈↓¬1.␈α∞␈↓	50␈↓¬␈α∞AM␈α∞is␈α∞not␈α∞thrilled␈α∞with␈α
exponentiation,␈α∞so␈α∞it␈α∞doesn't␈α∞even␈α∞consider␈α∞the␈α
hyper-(repeated-)exponentiation
␈↓ ↓*␈↓¬␈↓ ↓Zoperation.

␈↓ ↓*␈↓¬2. SYSTEM: I am now investigating the inverses of Plus and Times.  In general, the inverses are not functions.

␈↓ ↓*␈↓¬3. AM notices that (x,1) and (1,x) are always allowed factorizations of x.

␈↓ ↓*␈↓¬5. A corollary is that TIMES␈↓	-1␈↓¬ has at least two members.

␈↓ ↓*␈↓¬6. AM now decides to consider just those numbers whose TIMES␈↓	-1␈↓¬ has ␈↓βprecisely␈↓¬ 2 members.

␈↓ ↓*␈↓¬7. The user asks why this is interesting.

␈↓ ↓*␈↓¬8.␈α
SYSTEM:␈α
At␈α
the␈α
moment,␈α
we␈α
have␈α
the␈α
known␈α
fact␈αf(x)≥y;␈α
try␈α
to␈α
pin␈α
down␈α
extreme␈α
cases,␈α
those␈α
x␈α
for␈αwhich␈α
f(x)
␈↓ ↓*␈↓¬␈↓ ↓Zis␈αprecisely␈αequal␈αto␈αy.␈αf␈αis␈α
COUNT*TIMES␈↓	-1␈↓¬,␈αin␈αthe␈αcurrent␈αcase,␈αand␈αy␈α
is␈α2.␈α Do␈αyou␈αhave␈αany␈αspecial␈α
names
␈↓ ↓*␈↓¬␈↓ ↓Zyou would like me to refer to COUNT * TIMES␈↓	-1␈↓¬ by? Any special names for the subset P?

␈↓ ↓*␈↓¬9.␈α
USER:␈αCall␈α
that␈αcomposition␈α
FAC,␈αstanding␈α
for␈α"pairwise␈α
factorings".␈α Call␈α
the␈αset␈α
P,␈αof␈α
extreme␈αnumbers␈α
under
␈↓ ↓*␈↓¬␈↓ ↓ZFAC, PRIMES.

␈↓ ↓*␈↓¬11.␈α
AM␈αnotices␈α
that␈αTIMES␈α
may␈αalso␈α
be␈αviewed␈α
as␈αoperating␈α
on␈αa␈α
MULTISET.␈α In␈α
this␈αsense,␈α
TIMES␈↓	-1␈↓¬␈αwould␈α
return
␈↓ ↓*␈↓¬␈↓ ↓Zall␈α∞possible␈α∞multisets␈α∂(of␈α∞any␈α∞length)␈α∞whose␈α∂product␈α∞is␈α∞the␈α∂given␈α∞number.␈α∞In␈α∞this␈α∂case,␈α∞(5)␈α∞just␈α∂says␈α∞that
␈↓ ↓*␈↓¬␈↓ ↓ZTIMES␈↓	-1␈↓¬(x)␈αhas␈αat␈αleast␈αone␈αmember.␈α The␈αextremes␈αnow␈αare␈αjust␈α1␈αand␈αPRIMES.␈α  AM␈αrequests␈αa␈αnew␈αname
␈↓ ↓*␈↓¬␈↓ ↓Zfor this new interpretation of TIMES␈↓	-1␈↓¬.

␈↓ ↓*␈↓¬12. USER: Call it FACTORS.


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	50␈↓ε The numbering here corresponds to that in the last section.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  51␈↓

␈↓ ↓*␈↓¬13.␈αSYSTEM:␈αConjecture:␈αFACTORS(x)␈αalways␈αseems␈αto␈αcontain␈αa␈αbag␈αcontaining␈αonly␈αones␈αand␈αprimes,␈αand␈αonly
␈↓ ↓*␈↓¬␈↓ ↓Zone such bag.

␈↓ ↓*␈↓¬14. AM proves both existence and uniqueness by contradiction, using the two lemmas it devises.

␈↓ ↓*␈↓¬15.␈α
USER:␈α
Call␈α
this␈α
the␈α
unique␈α
factorization␈α
theorem.␈α
This␈α
is␈α
very␈α
important.␈α
 Consider␈α
now␈α
the␈α
sum␈α
of␈α
all␈α
the
␈↓ ↓*␈↓¬␈↓ ↓Zdivisors of a number.

␈↓ ↓*␈↓¬This leads AM to devise PERFECT NUMBERS.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&12.4. The Session Revisited: An In-Depth Example␈↓)αβ␈↓


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.4.1. The control structure reviewed␈↓)αβ␈↓


␈↓ ↓*␈↓Before␈αexamining␈α
how␈αAM␈α
"really"␈αcould␈α
carry␈αon␈α
the␈αhypothetical␈α
dialogue,␈αlet's␈α
review␈αthe␈α
global␈αcontrol

␈↓ ↓*␈↓structure␈α⊂of␈α⊂the␈α⊂system.␈α⊂ The␈α⊂basic␈α∂process␈α⊂is␈α⊂to␈α⊂repeatedly␈α⊂(i)␈α⊂see␈α∂which␈α⊂part␈α⊂of␈α⊂which␈α⊂BEING␈α⊂is␈α∂most

␈↓ ↓*␈↓relevant␈α(w.r.t.␈αCS,␈αthe␈α
current␈αsituation),␈αthen␈α(ii)␈αdeal␈α
with␈αthat␈αpart␈αof␈αthat␈α
BEING␈α(either␈α␈↓βexecute␈↓␈αit,␈αor␈α
try

␈↓ ↓*␈↓to␈α∪≡ll␈α∪it␈α∪in␈α∪more).␈α∩Often,␈α∪new␈α∪entities␈α∪are␈α∪constructed␈↓	51␈↓.␈α∩ When␈α∪this␈α∪happens,␈α∪they␈α∪are␈α∪evaluated␈α∩for

␈↓ ↓*␈↓interestingness.␈α
Any␈αnew,␈α
promising␈αconstruct␈α
is␈α
temporarily␈αgiven␈α
full␈α"BEING"␈α
status:␈α
we␈αassume␈α
that␈αall␈α
25

␈↓ ↓*␈↓parts␈αmight␈αbe␈αworth␈α≡lling␈αin;␈αwe␈αtry␈αto␈αanswer␈αall␈α25␈α"questions"␈αfor␈αthis␈αnew␈αconcept.␈α After␈αa␈αwhile,␈αif␈αit

␈↓ ↓*␈↓doesn't prove interesting, we "forget"  this BEING.␈↓	52␈↓


␈↓ ↓*␈↓Say␈αwe␈αare␈αdealing␈αwith␈αpart␈αP␈αof␈αBEING␈αB,␈αwhich␈αwe␈α abbreviate␈αB.P.␈α In␈αthe␈αcase␈αof␈αextending␈α(or␈α≡lling

␈↓ ↓*␈↓it␈α
in,␈α
if␈α
it␈α
is␈α
currently␈α
empty),␈α
AM␈α
simply␈α
gathers␈α
together␈α
algorithms␈α
suitiable␈α
for␈α
≡lling␈α
in␈α
the␈α
slot␈αnamed␈α
␈↓β<P

␈↓ ↓*␈↓βor␈αany␈αgeneralization␈αof␈α
P>␈↓␈αin␈αthe␈αBEING␈αnamed␈α
␈↓β<B␈αor␈αany␈αgeneralization␈αof␈α
B>␈↓.␈α One␈αextreme␈αcase␈α
of␈αthe

␈↓ ↓*␈↓latter␈αgeneralization␈α
is␈α"algorithms␈αfor␈α
≡lling␈αin␈αslot␈α
P␈αin␈α␈↓βany␈↓␈α
BEING";␈αthough␈αgeneral,␈α
this␈αis␈α
probably␈αthe

␈↓ ↓*␈↓most␈αcommon␈αkind␈αof␈αstrategy␈αinformation.␈α The␈αreason␈αis␈αthat␈αmost␈αof␈αthe␈αinformation␈αfor␈αhow␈αto␈α≡nd␈αthe

␈↓ ↓*␈↓answer␈α
to␈α
some␈αgiven␈α
facet␈α
of␈α
a␈αconcept␈α
does␈α
␈↓βnot␈↓␈αdepend␈α
on␈α
the␈α
kind␈αof␈α
concept␈α
so␈α
much␈αas␈α
the␈α
kind␈αof␈α
facet.

␈↓ ↓*␈↓One of these general packets of knowledge was the ANY-BEING.EXAMPLES BEING we looked at.

␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	51␈↓ε  E.g., when filling in the  EXAMPLES slot of some  BEING
␈↓ ↓*␈↓ε␈↓	52␈↓ε␈α	 Perhaps␈α	we␈αλreplace␈α	it␈α	as␈α	one␈αλsubpart␈α	of␈α	one␈αλslot␈α	of␈α	one␈α	BEING,␈αλwhere␈α	it␈α	came␈α	from.␈αλThis␈α	is␈α	also␈αλwhere␈α	it␈α	would␈α	have␈αλremained,␈α	if␈α	it␈α	had␈αλnever
␈↓ ↓*␈↓ε␈↓ αJseemed interesting.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  52␈↓

␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.4.2. Looking at the new concept named "FACTORS"␈↓)αβ␈↓


␈↓ ↓*␈↓Let␈α
us␈α
now␈α
consider␈α
a␈α
fairly␈α
detailed␈α
example.␈α
We␈αshall␈α
pick␈α
up␈α
near␈α
the␈α
end␈α
of␈α
our␈α
earlier␈α
dialogue,␈αand

␈↓ ↓*␈↓explain␈αhow␈αAM␈αstudies␈αthe␈αnew␈αBEING␈αFACTORS,␈αnotices␈αthe␈αunique␈αfactorization␈αtheorem,␈αand␈αtries␈αto

␈↓ ↓*␈↓prove it.

␈↓ ↓*␈↓After␈α∂the␈α⊂user␈α∂gives␈α⊂this␈α∂Active␈α⊂Operation␈α∂BEING␈α∂the␈α⊂new␈α∂name␈α⊂"FACTORS",␈α∂it␈α⊂is␈α∂so␈α⊂recorded.␈α∂ The
␈↓ ↓*␈↓de≡nition␈α
part␈α∞of␈α
FACTORS␈α
is␈α∞already␈α
≡lled␈α
in,␈α∞namely␈α
␈↓¬"FACTORS(x)␈α
≡␈α∞{b␈α
|␈α
BAG(b)␈α∞∧␈α
TIMES(b)=x␈α∞∧␈α
1¬εb}."␈↓
␈↓ ↓*␈↓The␈αintuition␈α
part␈αalready␈αcontains␈α
the␈αadequate␈α
images␈αof:␈αpartitioning␈α
a␈αset␈αinto␈α
disjoint␈αsubsets␈α
of␈αequal
␈↓ ↓*␈↓size;␈αdissection;␈αlooking␈αmore␈αclosely␈αat␈αan␈αartifact␈αuntil␈αit␈αdivides␈αup␈αinto␈αits␈αconstituent␈αparts;␈αdividing␈αup␈αa
␈↓ ↓*␈↓pile␈α∞of␈α∞marbles␈α
into␈α∞smaller␈α∞piles␈α
which␈α∞all␈α∞balance␈α∞each␈α
other.␈α∞ The␈α∞control␈α
function␈α∞takes␈α∞over.␈α∞Recall␈α
it
␈↓ ↓*␈↓wants to repeatedly choose B and P, then deal with part P of BEING B, abbreviated B.P.

␈↓ ↓*␈↓1.␈α
Neither␈α
P␈α∞nor␈α
B␈α
is␈α∞known.␈α
Ask␈α
each␈α∞BEING␈α
how␈α
relevant␈α
it␈α∞is␈α
to␈α
the␈α∞current␈α
situation,␈α
CS.␈α∞ Since␈α
CS
␈↓ ↓*␈↓␈↓ ↓Zcontains␈αmany␈αreferences␈αto␈αthe␈αrecently-named␈αFACTORS␈αBEING,␈αand␈αsince␈αthat␈αBEING␈αis␈αstill␈αquite
␈↓ ↓*␈↓␈↓ ↓Zincomplete,␈αit␈α
is␈αnot␈α
surprising␈αthat␈α
it␈αwins␈αthis␈α
contest.␈αTo␈α
decide␈αwhich␈α
part␈αof␈α
FACTORS␈αto␈αdeal␈α
with,
␈↓ ↓*␈↓␈↓ ↓Zand␈α∩how,␈α⊃we␈α∩look␈α⊃at␈α∩FACTORS.ORDERING;␈α⊃it␈α∩doesn't␈α⊃exist␈α∩yet.␈α⊃We␈α∩next␈α⊃look␈α∩at␈α⊃(generalization
␈↓ ↓*␈↓␈↓ ↓ZFACTORS).Ordering;␈α~in␈α~this␈α~case,␈α~FACTORS␈α~is␈α≠a␈α~kind␈α~of␈α~OPERATION,␈α~so␈α~we␈α≠look␈α~at
␈↓ ↓*␈↓␈↓ ↓ZOPERATION.ORDERING.␈α
This␈αtoo␈α
is␈αblank.␈α
Ultimately,␈α
we␈αlook␈α
at␈αANY-BEING.ORDERING,␈α
which
␈↓ ↓*␈↓␈↓ ↓Zhas␈αsome␈αquite␈αgeneral␈αhints␈αfor␈αwhich␈αorder␈αthe␈αparts␈αshould␈αbe␈αdealt␈αwith.␈α In␈αparticular,␈α"concentrate
␈↓ ↓*␈↓␈↓ ↓Zon␈α⊃≡lling␈α⊃in␈α⊃the␈α⊃General␈α⊃Information␈α⊃parts␈α∩before␈α⊃doing␈α⊃anything␈α⊃else."␈α⊃These␈α⊃in␈α⊃turn␈α∩are␈α⊃ordered
␈↓ ↓*␈↓␈↓ ↓Z"de≡nition,␈αintuition,␈αexamples,␈αties."␈αThe␈αde≡nition␈αand␈αintuition␈αparts␈αare␈αnicely␈α≡lled␈αout␈αalready.␈α The
␈↓ ↓*␈↓␈↓ ↓ZExamples part is blank, however, and that is where AM chooses to work now.

␈↓ ↓*␈↓2.␈α_So␈α_AM␈α_has␈α_chosen␈α_B=FACTORS,␈α_P=EXAMPLES.␈α↔AM␈α_is␈α_going␈α_to␈α_≡ll␈α_in␈α_the␈α_part␈α↔labelled
␈↓ ↓*␈↓␈↓ ↓ZFACTORS.EXAMPLES.␈α⊗The␈α⊗control␈α∃system␈α⊗now␈α⊗goes␈α∃about␈α⊗collecting␈α⊗relevant␈α⊗algorithms␈α∃(and
␈↓ ↓*␈↓␈↓ ↓Zconstraints)␈αfrom␈αthe␈αparts␈αlabelled␈α[(generalization␈↓	*␈↓␈αFACTORS).(generalization␈↓	*␈↓␈αEXAMPLES)].FILLIN.
␈↓ ↓*␈↓␈↓ ↓ZThese parts are:
␈↓ ↓*␈↓␈↓ ↓J[FACTORS.Examples].Fillin␈↓ ¬~[FACTORS.Genl-Info].Fillin␈↓ λz[FACTORS.Any-Part].Fillin
␈↓ ↓*␈↓␈↓ ↓J[OPERATION.Examples].Fillin␈↓ ¬~[OPERATION.Genl-Info].Fillin␈↓ λz[OPERATION.Any-Part].Fillin
␈↓ ↓*␈↓␈↓ ↓J[ACTIVE.Examples].Fillin␈↓ ¬~[ACTIVE.Genl-Info].Fillin␈↓ λz[ACTIVE.Any-Part].Fillin
␈↓ ↓*␈↓␈↓ ↓J[ANY-BEING.Examples].Fillin␈↓ ¬~[ANY-BEING.Genl-Info].Fillin␈↓ λz[ANY-BEING.Any-Part].Fillin
␈↓ ↓*␈↓␈↓ ↓ZOf␈αthese␈α
9␈αslots,␈αonly␈α
two␈αhave␈αexecutable␈α
code:␈α[ANY-BEING.EXAMPLES].FILLIN␈αcontains␈α
almost␈αa
␈↓ ↓*␈↓␈↓ ↓Zpage␈α∞of␈α∞general␈α∞techniques␈α∞for␈α∂≡lling␈α∞in␈α∞examples.␈α∞[ACTIVE.EXAMPLES].FILLIN␈α∞has␈α∞about␈α∂half␈α∞as
␈↓ ↓*␈↓␈↓ ↓Zmany␈α⊃special-purpose␈α∩hints␈α⊃which␈α⊃apply␈α∩whenever␈α⊃the␈α⊃entity␈α∩is␈α⊃an␈α⊃Activity,␈α∩␈↓βafter␈↓␈α⊃the␈α∩more␈α⊃general
␈↓ ↓*␈↓␈↓ ↓Ztechniques␈α
have␈αbeen␈α
applied.␈α The␈α
control␈αstructure␈α
now␈αsimply␈α
appends␈αthese␈α
speci≡c␈α
algorithms␈αonto
␈↓ ↓*␈↓␈↓ ↓Zthe␈αmore␈αgeneral␈α
algorithms,␈αand␈αcomes␈α
up␈αwith␈αa␈α
little␈αprogram␈αto␈α
≡ll␈αin␈αFACTORS.EXAMPLES.␈α
This
␈↓ ↓*␈↓␈↓ ↓Zprogram␈α∂is␈α∂run,␈α∂with␈α∂(FACTORS,EXAMPLES)␈α∂as␈α⊂its␈α∂arguments,␈α∂and␈α∂then␈α∂control␈α∂will␈α∂revert␈α⊂to␈α∂the
␈↓ ↓*␈↓␈↓ ↓Z"select best B,P" phase.

␈↓ ↓*␈↓<IF␈α
TIME␈α
IS␈α
SHORT:>␈α∞Time␈α
doesn't␈α
permit␈α
me␈α∞to␈α
go␈α
through␈α
the␈α∞whole␈α
assembled␈α
program.␈α
It␈α∞calls␈α
for
␈↓ ↓*␈↓␈↓ α
manipulating␈α
the␈α
de≡nition␈αand␈α
intuition␈α
parts␈αof␈α
FACTORS␈α
to␈α
obtain␈αexamples.␈α
 In␈α
fact,␈αit␈α
manages
␈↓ ↓*␈↓␈↓ α
to␈αcompute␈αFACTORS␈αof␈αthe␈α
numbers␈α0,1,2,␈αand␈α75.␈α <POINT␈α
TO␈α12:>␈αHere,␈αafter␈αthese␈αhave␈α
been
␈↓ ↓*␈↓␈↓ α
computed,␈α~they␈α→are␈α~entered␈α→on␈α~the␈α~FACTORS.EXAMPLES␈α→part.␈α~AM␈α→then␈α~looks␈α~at␈α→the
␈↓ ↓*␈↓␈↓ α
OPERATIONS␈α⊂part␈α⊂of␈α⊂FACTORS,␈α⊂and␈α⊂≡nds␈α⊂various␈α⊂activities␈α⊂listed.␈α⊂For␈α⊂each␈α⊂one␈α⊂of␈α⊃these,␈α⊂it
␈↓ ↓*␈↓␈↓ α
consults␈α⊂its␈α⊂EXAMPLES␈α⊃part.␈α⊂If␈α⊂it's␈α⊂incomplete,␈α⊃it␈α⊂leaves␈α⊂a␈α⊂little␈α⊃note␈α⊂there␈α⊂saying␈α⊂"if␈α⊃you␈α⊂want
␈↓ ↓*␈↓␈↓ α
examples␈αof␈αyour␈αdomain␈α
elements,␈αone␈αsure␈αplace␈αto␈α
≡nd␈αsome␈αis␈αon␈αFACTORS.EXAMPLES."␈α
Here
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  53␈↓

␈↓ ↓*␈↓␈↓ α
we␈α⊂see␈α⊂an␈α⊃instance␈α⊂of␈α⊂how␈α⊂adding␈α⊃knowledge␈α⊂to␈α⊂AM␈α⊂makes␈α⊃later␈α⊂activity␈α⊂run␈α⊂faster,␈α⊃not␈α⊂slower.
␈↓ ↓*␈↓␈↓ α
<SKIP THE REST OF THIS SECTION; GO TO NEXT SECTION>

␈↓ ↓*␈↓<IF TIME IS NOT SHORT:>

␈↓ ↓*␈↓3.␈α⊂Let␈α∂us␈α⊂now␈α∂go␈α⊂through␈α∂this␈α⊂assembled␈α⊂program,␈α∂and␈α⊂see␈α∂how␈α⊂it␈α∂≡lls␈α⊂in␈α∂examples␈α⊂for␈α⊂the␈α∂FACTORS
␈↓ ↓*␈↓␈↓ ↓Zconcept.␈α⊗The␈α∃≡rst␈α⊗thing␈α⊗it␈α∃says␈α⊗to␈α⊗try␈α∃is␈α⊗"for␈α⊗each␈α∃specialization␈α⊗x␈α⊗of␈α∃FACTORS,␈α⊗take␈α⊗all␈α∃of
␈↓ ↓*␈↓␈↓ ↓Zx.EXAMPLES␈α
if␈α
it␈αis␈α
≡lled␈α
in␈α;␈α
if␈α
not,␈αtry␈α
to␈α
instantiate␈α
the␈αdescription␈α
of␈α
x␈αand␈α
thereby␈α
produce␈αsuch␈α
an
␈↓ ↓*␈↓␈↓ ↓Zexample".␈α But␈α
the␈αSpecializations␈αpart␈α
of␈αFACTORS␈αhasn't␈α
been␈α≡lled␈α
in␈αat␈αall,␈α
so␈αthere␈αis␈α
no␈αx␈αto␈α
work
␈↓ ↓*␈↓␈↓ ↓Zwith.␈α
A␈α
note␈α
is␈α
placed␈α
in␈αFACTORS.SPECIALIZATIONS,␈α
which␈α
says:␈α
when␈α
you␈α
get␈α
≡lled␈αin,␈α
instantiate
␈↓ ↓*␈↓␈↓ ↓Zyourself␈α
into␈αan␈α
example␈α
for␈αFACTORS.EXAMPLES".␈α
 This␈α
concludes␈αthe␈α
≡rst␈α
of␈αabout␈α
30␈αtactics␈α
listed
␈↓ ↓*␈↓␈↓ ↓Zunder ANY-BEING.EXAMPLES.FILLIN.

␈↓ ↓*␈↓4.␈αThe␈αsecond␈αthing␈αto␈αtry␈αis␈α"instantiate␈αthe␈αde≡nition␈αof␈αFACTORS".␈α The␈αde≡nition␈αis:␈α␈↓¬"FACTORS(x)␈α≡␈α{b␈α|
␈↓ ↓*␈↓¬␈↓ ↓ZBAG(b)␈α∧␈αTIMES(b)=x␈α∧␈α1¬εb}."␈↓␈αTo␈αinstantiate␈αit,␈αAM␈αmust␈α≡nd␈αa␈αspeci≡c␈αnumber␈αx␈αand␈αa␈αset␈αof␈αbags␈α{b},
␈↓ ↓*␈↓␈↓ ↓Zsatisfying␈αthe␈αconditions.␈αSince␈αwe␈αknow␈αTIMES␈↓	-1␈↓(2)={(1,2),(2,1)},␈αit␈αis␈αclear␈αthat␈αFACTORS(2)␈α=␈α{(BAG
␈↓ ↓*␈↓␈↓ ↓Z2)}.  This is our ≡rst real example.

␈↓ ↓*␈↓5.␈αThe␈αnext␈αhint␈αsays␈α"instantiate␈αthe␈αintuition".␈α Suppose␈αwe␈αlook␈αat␈αthe␈αimage␈α"partition␈αa␈αset␈αinto␈αdisjoint
␈↓ ↓*␈↓␈↓ ↓Zsubsets␈αof␈αequal␈αsize".␈αTo␈αinstantiate␈αit,␈αtake␈αa␈αspeci≡c␈αset,␈αsay␈α{a,b,c,d,e,f}.␈αBy␈αenumeration,␈αthe␈αonly␈αways
␈↓ ↓*␈↓␈↓ ↓Zto␈α
do␈α∞this␈α
are:␈α∞one␈α
subset␈α∞with␈α
6␈α∞elements,␈α
two␈α∞subsets␈α
each␈α∞with␈α
3␈α∞elements,␈α
three␈α∞subsets␈α
each␈α∞with␈α
2
␈↓ ↓*␈↓␈↓ ↓Zelements,␈αand␈α6␈αsubsets␈αeach␈αwith␈α1␈αelement.␈αThe␈αintuition␈αdirects␈αthe␈αtranslation␈αof␈αthese␈αresults␈αinto␈αthe
␈↓ ↓*␈↓␈↓ ↓Zstatement␈α∞that␈α∞the␈α∞only␈α∞bags␈α∂whose␈α∞TIMES␈α∞is␈α∞6␈α∞are:␈α∂(BAG␈α∞1␈α∞6),␈α∞(BAG␈α∞2␈α∂3),␈α∞(BAG␈α∞3␈α∞2),␈α∞(BAG␈α∂6␈α∞1).
␈↓ ↓*␈↓␈↓ ↓ZEliminating␈α1's␈αand␈αmultiple␈αoccurrences␈αof␈αthe␈αsame␈αbag,␈αwe␈αcan␈αsay␈αFACTORS(6)␈α=␈α{(BAG␈α6),(BAG␈α2
␈↓ ↓*␈↓␈↓ ↓Z3)}.  This is our second example.

␈↓ ↓*␈↓6.␈αThe␈α
next␈αthing␈α
to␈αtry␈α
is␈α"≡nd␈α
an␈αanalogy␈α
between␈αFACTORS␈α
and␈αsome␈α
other␈αBEING,␈α
say␈αx,␈α
and␈αthen␈α
try
␈↓ ↓*␈↓␈↓ ↓Zto␈α∞map␈α∞over␈α∞the␈α∞examples␈α∞of␈α∞x␈α∞into␈α∞examples␈α∞of␈α∞FACTORS."␈α∞There␈α∞are␈α∞no␈α∞such␈α∞analogies␈α∞known␈α∞at
␈↓ ↓*␈↓␈↓ ↓Zpresent, so this fails.

␈↓ ↓*␈↓7.␈α∂The␈α⊂next␈α∂type␈α∂of␈α⊂example␈α∂to␈α⊂search␈α∂for␈α∂is␈α⊂coincidental:␈α∂some␈α⊂of␈α∂the␈α∂variables␈α⊂in␈α∂the␈α⊂de≡nition␈α∂might
␈↓ ↓*␈↓␈↓ ↓Zcoinicide.␈αBut␈αthe␈αonly␈αvariables␈αhere␈αare␈αx␈αand␈αb,␈αone␈αof␈αwhich␈αis␈αa␈αnumber␈αand␈αthe␈αother␈αa␈αbag,␈αso␈αno
␈↓ ↓*␈↓␈↓ ↓Zcoincidence is possible.

␈↓ ↓*␈↓8.␈αSearch␈αfor␈αthe␈αsimplest␈αpossible␈αexamples.␈αPlug␈αthe␈αextreme␈αcases␈αof␈αthe␈αdomains␈αinto␈αthe␈αde≡nition.␈αThus
␈↓ ↓*␈↓␈↓ ↓Zwe␈α
ask␈αthe␈α
BEING␈α
named␈αNUMBERS␈α
what␈α
his␈αextreme␈α
examples␈α
are.␈αHe␈α
replies␈α
"0␈αand␈α
1".␈α
AM␈αnow
␈↓ ↓*␈↓␈↓ ↓Ztries␈αto␈αcompute␈αFACTORS(0).␈αThis␈αmeans␈α≡nding␈αa␈αbag␈αsatisfying␈αTIMES(b)=0.␈αAM␈αasks␈αthe␈αBEING
␈↓ ↓*␈↓␈↓ ↓Znamed␈αTIMES␈αif␈αhe␈αknows␈αany␈αsuch␈αb's.␈αTIMES.EXAMPLES␈αreplies␈αYes;␈αin␈αfact,␈α␈↓βany␈↓␈αbag␈αb␈αis␈αallowed
␈↓ ↓*␈↓␈↓ ↓Zif␈α
it␈α
contains␈αa␈α
ZERO.␈α
Thus␈αFACTORS(0)␈α
=␈α
␈↓¬{b␈α
|␈αBAG(b)␈α
∧␈α
0εb␈α∧␈α
1¬εb}␈↓.␈α
 Finding␈α
FACTORS(1)␈αagain
␈↓ ↓*␈↓␈↓ ↓Zinvolves␈α
asking␈αTIMES,␈α
who␈α
says␈αthat␈α
the␈α
only␈αbags␈α
whose␈α
image␈αunder␈α
TIMES␈α
is␈α1␈α
are␈αthose␈α
consisting
␈↓ ↓*␈↓␈↓ ↓Zonly␈α∃of␈α∃ONE's.␈α∃But␈α∃there␈α∃is␈α∃no␈α∃bag␈α∃of␈α∃1's␈α∃not␈α∃containing␈α∃a␈α∃1,␈α∃except␈α∃for␈α∃the␈α∃empty␈α⊗bag.␈α∃So
␈↓ ↓*␈↓␈↓ ↓ZFACTORS(1)={(BAG)}.

␈↓ ↓*␈↓9.␈α
Search␈α
for␈α
a␈αsophisticated,␈α
huge␈α
example.␈α
Plug␈αin␈α
a␈α
sophisticated␈α
case␈αof␈α
the␈α
argument␈α
of␈αFACTORS.␈α
AM
␈↓ ↓*␈↓␈↓ ↓Zasks␈αNUMBER.EXAMPLES␈αfor␈αa␈αbig␈αexample,␈αand␈αgets␈αback␈α(in␈αcanonical␈αform)␈αthe␈αnumber␈α75.␈αAfter
␈↓ ↓*␈↓␈↓ ↓Zmuch␈αtrial␈αand␈α
error,␈αthis␈αis␈α
≡nally␈αconverted␈αinto␈α
the␈αexample:␈αFACTORS(75)={(BAG␈α
3␈α5␈α5),␈α(BAG␈α
75),
␈↓ ↓*␈↓␈↓ ↓Z(BAG 5 15), (BAG 3 25)}.

␈↓ ↓*␈↓10. No prototypical example can be found, because so little is known about FACTORS.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  54␈↓

␈↓ ↓*␈↓11.␈α⊃No␈α⊂boundary␈α⊃examples␈α⊂can␈α⊃be␈α⊃found,␈α⊂since␈α⊃the␈α⊂concepts␈α⊃involved␈α⊂don't␈α⊃have␈α⊃meaningful␈α⊂boundary
␈↓ ↓*␈↓␈↓ ↓Zexamples.

␈↓ ↓*␈↓12.␈α
At␈αthis␈α
point,␈αour␈α
assembled␈αprogram␈α
tells␈αus␈α
to␈αinsert␈α
a␈αmesage␈α
into␈αeach␈α
slot␈αof␈α
the␈αform␈α
<member␈αof
␈↓ ↓*␈↓␈↓ ↓ZFACTORS.OPERATIONS>.EXAMPLES,␈αsaying␈α
"if␈αyou␈α
want␈αexamples␈αof␈α
your␈αdomain␈α
elements,␈αone
␈↓ ↓*␈↓␈↓ ↓Zsure␈α∩place␈α∩to␈α∩≡nd␈α∩some␈α⊃is␈α∩on␈α∩FACTORS.EXAMPLES".␈α∩ Here␈α∩we␈α⊃see␈α∩an␈α∩example␈α∩of␈α∩how␈α⊃adding
␈↓ ↓*␈↓␈↓ ↓Zknowledge␈α
to␈α
AM␈α
will␈α
make␈α
later␈α
processing␈α
␈↓βshorter␈↓,␈α
not␈α
longer:␈α
forestalling␈α
a␈α
future␈α
search␈α
by␈α
leaving␈α
the
␈↓ ↓*␈↓␈↓ ↓Zproper information in a place where it will be found immediately -- but only -- when later needed.

␈↓ ↓*␈↓13.␈α∂That␈α∞was␈α∂the␈α∂≡nal␈α∞hint␈α∂taken␈α∂from␈α∞[ANY-BEING.EXAMPLES].FILLIN;␈α∂AM␈α∂now␈α∞turns␈α∂to␈α∂those␈α∞on
␈↓ ↓*␈↓␈↓ ↓Z[ACTIVE.EXAMPLES].FILLIN.␈α∩ None␈α∩of␈α∩these␈α∩turns␈α∩out␈α∩to␈α∩be␈α∩applicable␈α∩here,␈α∩so␈α∩the␈α∩assembled
␈↓ ↓*␈↓␈↓ ↓Zprogram␈αis␈α≡nished,␈αAM␈αhas␈αdealt␈αwith␈αFACTORS.EXAMPLES,␈αand␈αcontrol␈αreverts␈αto␈αchoosing␈αwhich
␈↓ ↓*␈↓␈↓ ↓ZB,P to deal with next.

␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.4.3. Conjecturing the Unique Factorization Theorem␈↓)αβ␈↓

␈↓ ↓*␈↓THE CONJECTURING PROCESS: abbreviated

␈↓ ↓*␈↓(See commentary immediately following the abbreviation)

␈↓ ↓*␈↓¬1.␈α∀Choose␈α∀B=FACTORS,␈α∀P=TIES.␈α∀ Gather␈α∃relevant␈α∀algorithms␈α∀from␈α∀the␈α∀slots␈α∃labelled:␈α∀[FACTORS.Ties].Fillin,
␈↓ ↓*␈↓¬␈↓ ↓Z[FACTORS.Genl-Info].Fillin,␈α∨[FACTORS.Any-Part].Fillin,␈α∨[OPERATION.Ties].Fillin,␈α∨[OPERATION.Genl-Info].Fillin,
␈↓ ↓*␈↓¬␈↓ ↓Z[OPERATION.Any-Part].Fillin,␈α?␈αα[ACTIVE.Ties].Fillin,␈α?␈αα[ACTIVE.Genl-Info].Fillin,␈α?␈α↓[ACTIVE.Any-Part].Fillin,
␈↓ ↓*␈↓¬␈↓ ↓Z[ANY-BEING.Ties].Fillin, [ANY-BEING.Genl-Info].Fillin, [ANY-BEING.Any-Part].Fillin.

␈↓ ↓*␈↓¬2.␈α
"Let␈α
D␈αbe␈α
the␈α
known␈αBEING␈α
representing␈α
the␈αkind␈α
of␈α
entity␈αin␈α
the␈α
range␈αof␈α
FACTORS.␈α
Then␈α
ask␈αD.INTEREST
␈↓ ↓*␈↓¬␈↓ ↓Zhow␈α⊃to␈α⊃look␈α⊃for␈α⊃interesting␈α⊃properties␈α⊂or␈α⊃regularities.␈α⊃ If␈α⊃sparse,␈α⊃ask␈α⊃(generalization␈↓	*␈↓¬␈α⊃D).INTEREST␈α⊂also.
␈↓ ↓*␈↓¬␈↓ ↓ZApply␈αthese␈αmethods␈αto␈αthe␈αoutput␈αof␈αa␈αtypical␈αexample␈αof␈αFACTORS.␈α Check␈αinteresting␈αproperty␈αfound,␈αby
␈↓ ↓*␈↓¬␈↓ ↓Zseeing␈αif␈αit␈αholds␈αfor␈αthe␈αother␈αoutputs␈αfrom␈αFACTORS␈α and␈αensuring␈αit␈αisn't␈αsimply␈αpart␈αof␈αthe␈αdefinition␈αof
␈↓ ↓*␈↓¬␈↓ ↓ZFACTORS."

␈↓ ↓*␈↓¬3.␈α
Ask␈α
SET.INTEREST␈α
and␈α
STRUCTURE.INTEREST␈α
for␈α
perceptual␈α
guidance␈α
about␈α
a␈α
particular␈α
output,␈α
say␈α
the␈α
output
␈↓ ↓*␈↓¬␈↓ ↓Z{(BAG 3 5 5) (BAG 75) (BAG 5 15) (BAG 3 25)} from the call FACTORS(75).

␈↓ ↓*␈↓¬4.␈α"Three␈αdistinct␈α
ways␈αa␈αstructure␈α
 S␈αcan␈αbe␈αinteresting:␈α
S␈αsatisfies␈αa␈α
known␈αinteresting␈αproperty␈α
of␈αType(S).
␈↓ ↓*␈↓¬␈↓ ↓ZEvery xεS satisfies some single,interesting property.  ␈↓βSome␈↓¬  xεS satisfies some very interesting property."

␈↓ ↓*␈↓¬5.␈αFirst␈αand␈α
second␈αhints␈αfail.␈α
 Now␈αlook␈αat␈α
each␈αelement␈αin␈α
turn,␈αthat␈αis,␈α
each␈αbag.␈αFirst␈α
 consider␈α(BAG␈α75).␈α
This
␈↓ ↓*␈↓¬␈↓ ↓Zsatisfies␈α↔the␈α↔property␈α↔SINGLETON.␈α↔ Checks␈α↔with␈α↔other␈α↔examples␈α↔of␈α↔FACTORS.␈α↔ Conjecture:␈α⊗∀x.(BAG
␈↓ ↓*␈↓¬␈↓ ↓Zx)εFACTORS(x).

␈↓ ↓*␈↓¬6.␈α∩ Look␈α⊃at␈α∩(BAG␈α⊃3␈α∩5␈α∩5).␈α⊃ Each␈α∩element␈α⊃satisfies␈α∩PRIME.␈α∩ All␈α⊃other␈α∩examples␈α⊃of␈α∩FACTORS␈α∩check.␈α⊃ Conjec:
␈↓ ↓*␈↓¬␈↓ ↓ZFACTORS(x) always contains a bag of primes.

␈↓ ↓*␈↓¬7. Look at (BAG 3 5 5) still.  Each element satisfies ODD. Does not check out: FACTORS(2) = {(BAG 2)}.

␈↓ ↓*␈↓¬8. Look at (BAG 5 15) and at (BAG 3 25). Nothing interesting.

␈↓ ↓*␈↓¬9.␈α⊂Jumping␈α⊂ahead:␈α⊂PF(x)␈α⊂=␈α⊂FACTORS(x)␈α⊂∩␈α⊂MAPSTRUC(PRIME,␈α⊂x)␈α∂=␈α⊂{b␈α⊂|␈α⊂BAG(b)␈α⊂∧␈α⊂TIMES(b)=x␈α⊂∧␈α⊂1¬εb␈α⊂∧␈α∂∀zεb.
␈↓ ↓*␈↓¬␈↓ ↓ZPRIME(z)}.  Conjecture: PF is a function.

␈↓ ↓*␈↓COMMENTARY on the Conjecturing abbreviation
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  55␈↓

␈↓ ↓*␈↓0.␈α
A␈αspecial␈α
heuristic,␈α
embedded␈αimmutably␈α
in␈αthe␈α
control␈α
structure,␈αhas␈α
AM␈αmake␈α
a␈α
slight␈αe≥ort␈α
to␈αdeal␈α
with
␈↓ ↓*␈↓␈↓ ↓Zthe␈α⊂same␈α⊂B␈α⊂␈↓βor␈↓␈α⊂P␈α⊂as␈α⊂it␈α⊂did␈α⊂last␈α⊂time,␈α⊂but␈α∂not␈α⊂the␈α⊂same␈α⊂B.P␈α⊂of␈α⊂course.␈α⊂It␈α⊂is␈α⊂not␈α⊂surprising␈α⊂then␈α∂that
␈↓ ↓*␈↓␈↓ ↓ZFACTORS␈α∞is␈α
again␈α∞selected;␈α
this␈α∞time,␈α∞the␈α
TIES␈α∞part␈α
is␈α∞chosen␈α
to␈α∞be␈α∞≡lled␈α
in.␈α∞ So␈α
AM␈α∞is␈α∞looking␈α
for
␈↓ ↓*␈↓␈↓ ↓Zsome conjectures involving the concept FACTORS.

␈↓ ↓*␈↓1. Information is gathered as before; this time, the relevant parts are:
␈↓ ↓*␈↓␈↓ ↓Z[FACTORS.Ties].Fillin␈↓ ¬*[FACTORS.Genl-Info].Fillin␈↓ 	
[FACTORS.Any-Part].Fillin
␈↓ ↓*␈↓␈↓ ↓Z[OPERATION.Ties].Fillin␈↓ ¬*[OPERATION.Genl-Info].Fillin␈↓ 	
[OPERATION.Any-Part].Fillin
␈↓ ↓*␈↓␈↓ ↓Z[ACTIVE.Ties].Fillin␈↓ ¬*[ACTIVE.Genl-Info].Fillin␈↓ 	
[ACTIVE.Any-Part].Fillin
␈↓ ↓*␈↓␈↓ ↓Z[ANY-BEING.Ties].Fillin␈↓ ¬*[ANY-BEING.Genl-Info].Fillin␈↓ 	
[ANY-BEING.Any-Part].Fillin

␈↓ ↓*␈↓2. just read through

␈↓ ↓*␈↓3.␈αBecause␈αthe␈αoutput␈αof␈αa␈α
call␈αon␈αFACTORS␈αis␈αa␈α␈↓βset␈↓␈αof␈α
bags,␈αwe␈αare␈αdirected␈αto␈αask␈α
SET.INTEREST␈αfor
␈↓ ↓*␈↓␈↓ ↓Zaid␈α∞in␈α
perceiving␈α∞interesting␈α∞things␈α
about␈α∞a␈α∞particular␈α
output,␈α∞say␈α∞the␈α
output␈α∞{(BAG␈α∞3␈α
5␈α∞5)␈α∞(BAG␈α
75)
␈↓ ↓*␈↓␈↓ ↓Z(BAG␈α∂5␈α∞15)␈α∂(BAG␈α∞3␈α∂25)}␈α∞from␈α∂the␈α∞call␈α∂FACTORS(75).␈α∞ SET.INTEREST␈α∂is␈α∞not␈α∂very␈α∞big,␈α∂so␈α∂we␈α∞ask
␈↓ ↓*␈↓␈↓ ↓ZSTRUCTURE.INTEREST as well.

␈↓ ↓*␈↓4.␈α
STRUCTURE.INTEREST␈α
explains␈α
that␈α
there␈α
are␈αthree␈α
distinct␈α
ways␈α
a␈α
structure␈α
can␈α
be␈αinteresting.␈α
 First,
␈↓ ↓*␈↓␈↓ ↓Zcheck␈α∂whether␈α∂the␈α∂structure␈α∂satis≡es␈α∂any␈α∂known␈α⊂interesting␈α∂property␈α∂of␈α∂that␈α∂type␈α∂of␈α∂structure.␈α⊂ If␈α∂not,
␈↓ ↓*␈↓␈↓ ↓Zcheck␈α∂to␈α∞see␈α∂whether␈α∂every␈α∞element␈α∂satis≡es␈α∂the␈α∞same␈α∂interesting␈α∂property.␈α∞If␈α∂not,␈α∂check␈α∞to␈α∂see␈α∂if␈α∞␈↓βsome␈↓
␈↓ ↓*␈↓␈↓ ↓Zelement␈α∂of␈α∞the␈α∂structure␈α∞satis≡es␈α∂some␈α∞very␈α∂interesting␈α∞property.␈α∂ The␈α∞criteria␈α∂for␈α∂interestingness␈α∞being
␈↓ ↓*␈↓␈↓ ↓Ztalked␈αabout␈α
here␈αis␈α
the␈αone␈α
speci≡ed␈αby␈αthe␈α
BEING␈αrepresenting␈α
the␈αtype␈α
of␈αthe␈α
elements.␈αIn␈αour␈α
present
␈↓ ↓*␈↓␈↓ ↓Zcase,␈α⊗our␈α↔set␈α⊗is␈α↔a␈α⊗set␈α↔of␈α⊗␈↓βbags,␈↓␈α↔so␈α⊗that␈α⊗means␈α↔consult␈α⊗all␈α↔the␈α⊗hints␈α↔and␈α⊗factors␈α↔present␈α⊗under
␈↓ ↓*␈↓␈↓ ↓ZBAG.INTEREST.␈αBut␈αthis␈αis␈αalso␈αvery␈αsparse,␈αhence␈αwe␈αrecursively␈αturn␈αto␈αSTRUCTURE.INTEREST
␈↓ ↓*␈↓␈↓ ↓Zfor evaluation criteria.

␈↓ ↓*␈↓5.␈αAfter␈αa␈α
reasonable␈αtime,␈αAM␈α
cannot␈α≡nd␈αany␈α
interesting␈αproperty␈αsatis≡ed␈α
by␈αthe␈αgiven␈α
output␈αset.␈α  It␈α
also
␈↓ ↓*␈↓␈↓ ↓Zfails␈α∂to␈α∞≡nd␈α∂any␈α∞single␈α∂interesting␈α∞property␈α∂satis≡ed␈α∞by␈α∂all␈α∞four␈α∂bags␈α∞which␈α∂form␈α∞the␈α∂elements␈α∂of␈α∞that
␈↓ ↓*␈↓␈↓ ↓Zoutput set.

␈↓ ↓*␈↓Now␈αAM␈αlooks␈αat␈αeach␈αelement␈αin␈αturn,␈αthat␈αis,␈αeach␈αbag.␈αFirst␈αwe␈αconsider␈α(BAG␈α75),␈αsay.␈αThis␈αsatis≡es␈αthe
␈↓ ↓*␈↓␈↓ ↓Zproperty␈α∞SINGLETON.␈α
 We␈α∞check␈α
with␈α∞other␈α
examples␈α∞of␈α
FACTORS␈α∞and,␈α
sure␈α∞enough,␈α
each␈α∞one␈α
of
␈↓ ↓*␈↓␈↓ ↓Zthem␈α
contains,␈α
as␈α
an␈α
element,␈α
a␈α∞bag␈α
having␈α
the␈α
property␈α
SINGLETON.␈α
Comparing␈α
these␈α∞singletons␈α
to
␈↓ ↓*␈↓␈↓ ↓Zthe␈α∀inputs␈α∀to␈α∃FACTORS,␈α∀we␈α∀conjecture␈α∃that␈α∀(BAG␈α∀x)␈α∃will␈α∀always␈α∀appear␈α∃in␈α∀the␈α∀output␈α∃set␈α∀of
␈↓ ↓*␈↓␈↓ ↓ZFACTORS(x).

␈↓ ↓*␈↓6.␈α∞We␈α∞go␈α∞back␈α∂to␈α∞looking␈α∞at␈α∞the␈α∞individual␈α∂bags␈α∞in␈α∞FACTORS(75).␈α∞ This␈α∞time␈α∂we␈α∞look␈α∞at␈α∞(BAG␈α∞3␈α∂5␈α∞5).
␈↓ ↓*␈↓␈↓ ↓ZEach␈αelement␈α␈↓βdoes␈↓␈αsatisfy␈α
an␈αinteresting␈αproperty,␈αnamely␈αPRIME.␈α
We␈αcheck␈αagainst␈αthe␈α
other␈αexamples
␈↓ ↓*␈↓␈↓ ↓Zof␈αFACTORS,␈αand␈αsure␈αenough␈αeach␈αone␈αof␈αthem␈αcontains␈αan␈αelement␈αwhich␈αis␈αa␈αbag␈αof␈αprimes.␈αThere
␈↓ ↓*␈↓␈↓ ↓Zdoesn't␈α∪seem␈α∪to␈α∪be␈α∪any␈α∪obvious␈α∪relationship␈α∪to␈α∪the␈α∪input␈α∪argument,␈α∪so␈α∪we␈α∪merely␈α∀conjecture␈α∪that
␈↓ ↓*␈↓␈↓ ↓ZFACTORS(x)␈α
always␈α
contains␈α
a␈α
bag␈α
of␈α
primes,␈α
without␈α
saying␈α
which␈α
primes␈α
or␈α
how␈α
to␈α
compute␈α
them.
␈↓ ↓*␈↓␈↓ ↓ZThis␈α∂is␈α∂one␈α∂half␈α∞of␈α∂the␈α∂Unique␈α∂Factorization␈α∞Theorem.␈α∂Notice␈α∂that␈α∂this␈α∞is␈α∂"rough␈α∂around␈α∂the␈α∞edges",
␈↓ ↓*␈↓␈↓ ↓Znamely␈αfor␈αthe␈αcases␈αof␈αfactors␈αof␈αzero␈αand␈αone,␈αbut␈αthese␈αwill␈αbe␈αcaught␈αlater␈αby␈αan␈αexpert␈αBEING␈αwho
␈↓ ↓*␈↓␈↓ ↓Zspecializes in checking conjectures just before we start to prove them.

␈↓ ↓*␈↓7.␈α
Each␈α
element␈α
of␈α
(BAG␈α3␈α
5␈α
5)␈α
also␈α
satis≡es␈αthe␈α
property␈α
ODD.␈α
But␈α
this␈αis␈α
quickly␈α
rejected␈α
by␈α
looking␈αat␈α
the
␈↓ ↓*␈↓␈↓ ↓Zexample FACTORS(2) = {(BAG 2)}.

␈↓ ↓*␈↓8.␈α
We␈α
now␈α
look␈αat␈α
the␈α
next␈α
individual␈αbag␈α
in␈α
FACTORS(75),␈α
namely␈α(BAG␈α
5␈α
15).␈α
 Nothing␈α
interesting␈αis
␈↓ ↓*␈↓␈↓ ↓Zfound here or in the next case, (BAG 3 25).
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  56␈↓

␈↓ ↓*␈↓9.␈αBefore␈αgoing␈αon␈αto␈αprove␈αsome␈αof␈αthese␈αconjectures,␈αlet's␈αsee␈αhow␈αAM␈αmight␈αnotice␈αthe␈αuniqueness␈αaspects
␈↓ ↓*␈↓␈↓ ↓Zof␈αthem.␈αAM␈αknows␈αthat␈αsome␈αelements␈αof␈αFACTORS(x)␈αsatisfy␈αMAPSTRUC(PRIME),␈αbut␈αsome␈αdon't.
␈↓ ↓*␈↓␈↓ ↓ZIt␈α⊃wants␈α∩to␈α⊃≡nd␈α⊃out␈α∩how␈α⊃to␈α⊃characterize␈α∩those␈α⊃which␈α⊃do;␈α∩namely,␈α⊃those␈α⊃bags␈α∩of␈α⊃primes␈α∩from␈α⊃those
␈↓ ↓*␈↓␈↓ ↓Zcontaining␈αa␈αnonprime.␈α So␈αAM␈αwill␈αtemporarily␈αcreate␈α
a␈αnew␈αBEING,␈αcalled␈αsay␈αPF,␈αde≡ned␈αas␈α
␈↓¬PF(x)␈α=
␈↓ ↓*␈↓¬␈↓ ↓ZFACTORS(x)␈α∩␈α
MAPSTRUC(PRIME,␈αx)␈α=␈α
{b␈α|␈αBAG(b)␈α
∧␈αTIMES(b)=x␈α∧␈α
1¬εb␈α∧␈α∀zεb.␈α
PRIME(z)}␈↓.␈α Which␈αmeans:␈α
all
␈↓ ↓*␈↓␈↓ ↓Zbags␈α∞of␈α∞primes␈α∞whose␈α∞TIMES␈α
is␈α∞x;␈α∞which␈α∞also␈α∞means␈α∞all␈α
factorizations␈α∞of␈α∞x␈α∞into␈α∞bags␈α∞containing␈α
only
␈↓ ↓*␈↓␈↓ ↓Zprimes.

␈↓ ↓*␈↓In␈α
a␈α
manner␈α
similar␈α
to␈α
the␈α
above,␈α
AM␈α
will␈α
notice␈α
that␈α
PF␈α
of␈α
each␈α
number␈α
seems␈α
to␈α
be␈α
a␈α
SINGLETON.␈α
That
␈↓ ↓*␈↓␈↓ ↓Zis,␈α
there␈α
is␈α
only␈αone␈α
bag␈α
of␈α
primes␈αin␈α
the␈α
FACTORS(x)␈α
collection␈α
for␈αa␈α
given␈α
x.␈α
The␈αunique␈α
factorization
␈↓ ↓*␈↓␈↓ ↓Ztheorem␈α∞can␈α∞now␈α∞be␈α∞consisely␈α∞be␈α∞stated␈α
as␈α∞"PF␈α∞is␈α∞a␈α∞function␈α∞de≡ned␈α∞on␈α
N".␈α∞ In␈α∞such␈α∞a␈α∞form,␈α∞it␈α∞is␈α
not
␈↓ ↓*␈↓␈↓ ↓Zsurprising that AM will routinely investigate it.

␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.4.4. Proving Existence␈↓)αβ␈↓

␈↓ ↓*␈↓THE PROVING PROCESS abbreviated

␈↓ ↓*␈↓(See commentary immediately following the abbreviation)

␈↓ ↓*␈↓¬1. B = UFT, P= Justification.

␈↓ ↓*␈↓¬2. [ANY-BEING.JUSTIFICATION].FILLIN says to execute GUESS.ALGORITHMS and PROVE.ALGORITHMS.

␈↓ ↓*␈↓¬3.␈α"Supporting␈αContacts".␈α Test␈αUFT␈αon␈αextreme␈αcases.␈αMeans␈αselect␈αa␈αnumber␈αx,␈αthen␈αcomput␈αFACTORS(x),␈αthen
␈↓ ↓*␈↓¬␈↓ ↓Zcheck␈α
that␈α
one␈α
of␈αthe␈α
bags␈α
so␈α
produced␈α
consists␈αpurely␈α
of␈α
primes.␈α
 NUMBER␈α
says␈αthat␈α
the␈α
extremes␈α
are␈α0,␈α
1,
␈↓ ↓*␈↓¬␈↓ ↓Zand␈α⊃sometimes␈α⊃2.␈α⊃Failure␈α⊃of␈α⊃0,1.␈α⊃ Modify␈α⊃the␈α⊃wording␈α⊃of␈α⊃UFT:␈α⊃∀numbers␈α⊃x>1,␈α⊃∃␈α⊃bag␈α⊃bεFACTORS(x)␈α⊃s.t.
␈↓ ↓*␈↓¬␈↓ ↓Z∀zεb.PRIME(z).

␈↓ ↓*␈↓¬4.␈α
PROVE.ALGORITHMS␈αpasses␈α
off␈αto␈α
Math-Induction.ALGORITHMS.␈αBase␈α
case␈αx=2;␈α
constructor␈α
SUCCESSOR.␈αBase
␈↓ ↓*␈↓¬␈↓ ↓Zcase already solved.

␈↓ ↓*␈↓¬5.␈αAssume␈αUFT␈α
for␈αall␈αnumber␈αbetween␈α
2␈αand␈αx.␈α Problem:␈α
prove␈αUFT(SUCCESSOR(x)).␈α Not␈αtrivial␈α
transformation.
␈↓ ↓*␈↓¬␈↓ ↓ZIntuition indicates: proof by cases, where cases are: SUCCESSOR(x) is/isn't prime.

␈↓ ↓*␈↓¬6. Assume x+1 is a prime. Lemma: ∀x>2,(BAG x)εFACTORS(x).

␈↓ ↓*␈↓¬7.␈α∞Assume␈α
x+1␈α∞is␈α∞not␈α
prime.␈α∞ Write␈α∞ x+1␈α
 as␈α∞TIMES(u,v),␈α∞where␈α
both␈α∞ u␈α∞and␈α
v␈α∞are␈α∞between␈α
1␈α∞and␈α∞x+1.␈α
 Work
␈↓ ↓*␈↓¬␈↓ ↓Zbackwards;␈α∞Lemma:␈α∞For␈α∞any␈α∞number␈α∞a,␈α∞bag␈α∞b␈α∞of␈α∞numbers,␈α∞bag␈α∞d␈α∞of␈α∞numbers,␈α∞if␈α∞a=TIMES(b)␈α∞and␈α∞aεd,␈α∞then
␈↓ ↓*␈↓¬␈↓ ↓ZTIMES(d) = TIMES( SUBSTITUTE(b for a in d)).

␈↓ ↓*␈↓¬8. Proofs of the two lemmas...

␈↓ ↓*␈↓COMMENTARY on the Proving abbreviation

␈↓ ↓*␈↓There␈α
are␈αmany␈α
good␈αautomated␈α
theorem␈αprovers␈α
around,␈αmany␈α
of␈αthem␈α
embodying␈αsome␈α
of␈α
the␈αheuristics
␈↓ ↓*␈↓␈↓ ↓Zthat␈α
AM␈α
does.␈α
So␈α
it's␈α
probably␈α
not␈α
worth␈α∞laboring␈α
on␈α
through␈α
the␈α
proof␈α
of␈α
the␈α
UFT.␈α∞ The␈α
interesting
␈↓ ↓*␈↓␈↓ ↓Zaspect of AM is that it ␈↓βproposed␈↓ the conjecture.

␈↓ ↓*␈↓Let's␈αjust␈α
go␈αa␈αstep␈α
into␈αthis,␈αto␈α
see␈αhow␈α
AM␈αcleans␈αup␈α
the␈α(slightly-incorrect)␈αstatement␈α
of␈αUFT␈α
which␈αwas
␈↓ ↓*␈↓␈↓ ↓Zproposed␈αas␈αa␈αconjecture.␈α This␈αconjecture␈αis␈αmade␈αinto␈αa␈αtemporary␈αBEING,call␈αit␈αUFT,␈αand␈αeventaully
␈↓ ↓*␈↓␈↓ ↓ZAM gets around to ≡lling in its JUSTIFICATION part.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  57␈↓

␈↓ ↓*␈↓1.␈α∀Skipping␈α∪some␈α∀of␈α∀the␈α∪details,␈α∀assume␈α∀that␈α∪the␈α∀control␈α∀structure␈α∪chooses␈α∀to␈α∀work␈α∪on␈α∀≡lling␈α∀in␈α∪the
␈↓ ↓*␈↓␈↓ ↓ZJUSTIFICATION part of that new conjecture.

␈↓ ↓*␈↓2.␈α+The␈α+plan␈α+for␈α,action␈α+is␈α+assembled␈α+out␈α,of␈α+the␈α+nonblank␈α+slot␈α,labelled:␈α+[ANY-
␈↓ ↓*␈↓␈↓ ↓ZBEING.JUSTIFICATION].FILLIN,␈α∃which␈α∃in␈α∃turn␈α∃directs␈α∃our␈α∃energies␈α∃to␈α∃following␈α∃the␈α∀activities
␈↓ ↓*␈↓␈↓ ↓Zspeci≡ed by GUESS.ALGORITHMS and PROVE.ALGORITHMS.

␈↓ ↓*␈↓3.␈αThe␈αonly␈αhint␈αon␈αGUESS.ALGORITHMS␈αwhich␈αa≥ects␈αUFT␈αis␈αthe␈αone␈αcalled␈α"Supporting␈αContacts".␈αIt
␈↓ ↓*␈↓␈↓ ↓Zsays␈α
that␈α
we␈α
should␈αtry␈α
to␈α
test␈α
the␈αconjecture␈α
on␈α
extreme␈α
cases.␈αBy␈α
asking␈α
UFT,␈α
we␈α≡nd␈α
that␈α
testing␈α
it␈αin␈α
a
␈↓ ↓*␈↓␈↓ ↓Zspeci≡c␈α
case␈α
means␈α
selecting␈α
a␈α
number␈α
x,␈α
then␈α
computing␈α
FACTORS(x),␈α
then␈α
checking␈α
that␈α
one␈α
of␈α
the
␈↓ ↓*␈↓␈↓ ↓Zbags␈α⊃so␈α⊂produced␈α⊃consists␈α⊂purely␈α⊃of␈α⊂primes.␈α⊃ To␈α⊂≡nd␈α⊃extreme␈α⊂cases␈α⊃means␈α⊂therefore␈α⊃to␈α⊃≡nd␈α⊂extreme
␈↓ ↓*␈↓␈↓ ↓Znumbers.␈αNUMBER␈αsays␈α
that␈αthe␈αextremes␈αare␈α
0,␈α1,␈αand␈α
sometimes␈α2.␈αSure␈αenough,␈α
zero␈αand␈αone␈αfail␈α
the
␈↓ ↓*␈↓␈↓ ↓ZUFT␈α∂conjecture!␈α∂The␈α∂hint␈α∂on␈α∂GUESS.ALGORITHMS␈α∂says␈α∂that␈α∂if␈α∂only␈α∂extreme␈α∂cases␈α∂fail,␈α⊂to␈α∂simply
␈↓ ↓*␈↓␈↓ ↓Zmodify␈α
the␈αstatement␈α
of␈αthe␈α
conjecture␈αto␈α
allow␈α
for␈αthese␈α
exceptions.␈αThe␈α
de≡nition␈αof␈α
UFT␈αnow␈α
becomes:
␈↓ ↓*␈↓␈↓ ↓Z␈↓¬∀numbers x>1, ∃ bag b of prime factors␈↓

␈↓ ↓*␈↓4. We now execute the code found under PROVE.ALGORITHMS, with UFT as our argument.

␈↓ ↓*␈↓This is where we shall leave our proof example.

␈↓ ↓*␈↓<IF TIME IS SHORT, SKIP TO NEXT SECTION!!>

␈↓ ↓*␈↓This␈α
sets␈α
up␈αsome␈α
demons␈α
to␈α
watch␈αfor␈α
places␈α
to␈α
declare␈αlemmas,␈α
use␈α
intuition,␈α
etc.␈αThen␈α
each␈α
type␈αof␈α
proof␈↓	53␈↓
␈↓ ↓*␈↓␈↓ ↓Z(which␈α
are␈α
themselves␈α
BEINGs)␈α
examine␈α
UFT␈α
to␈α
see␈α
how␈α
relevant␈α
they␈α
are.␈α
 The␈α
winner␈α
is,␈α
say,␈α
Proving-
␈↓ ↓*␈↓␈↓ ↓ZUniversal-Statments␈α∃(abbreviated␈α∃Univ).␈α∀ His␈α∃ALGORITHMS␈α∃part␈α∀says␈α∃to␈α∃consider␈α∀Mathematical
␈↓ ↓*␈↓␈↓ ↓ZInduction␈α
≡rst.␈α
 The␈α∞Math-Induction␈α
BEING's␈α
ALGORITHMS␈α∞part␈α
says␈α
to␈α∞clearly␈α
decide␈α
on␈α∞the␈α
base
␈↓ ↓*␈↓␈↓ ↓Zcase␈α⊂and␈α∂the␈α⊂constructor␈α⊂function.␈α∂In␈α⊂the␈α∂case␈α⊂of␈α⊂UFT,␈α∂this␈α⊂is␈α∂simply␈α⊂the␈α⊂case␈α∂x=2␈α⊂and␈α⊂the␈α∂function
␈↓ ↓*␈↓␈↓ ↓ZSUCCESSOR.␈αA␈αbig␈αboost␈α
in␈αcon≡dence␈αoccurs␈αwhen␈αMath-Induction␈α
≡nds␈αthe␈αbase␈αcase␈αhas␈α
just␈αbeen
␈↓ ↓*␈↓␈↓ ↓Zsolved elsewhere (UFT was veri≡ed for the case x=2 when we looked at the extremes 0,1,2).

␈↓ ↓*␈↓5.␈αThe␈α
assumption␈αnow␈αmade␈α
is␈αthat␈α
UFT␈αis␈αtrue␈α
for␈αx␈αand␈α
all␈αits␈α
predessors␈α(down␈αto␈α
the␈αnumber␈α2,␈α
though
␈↓ ↓*␈↓␈↓ ↓Znot␈αbelow).␈αThe␈αproblem␈αis␈αnow␈αto␈αprove␈αthat␈αUFT␈αholds␈αwhen␈αx␈αis␈αreplaced␈αby␈αSUCCESSOR(x),␈αcall␈α
it
␈↓ ↓*␈↓␈↓ ↓Zy.␈α∞ After␈α∞some␈α
e≥ort,␈α∞AM␈α∞gives␈α
up␈α∞trying␈α∞to␈α
establish␈α∞this␈α∞in␈α
any␈α∞trivial␈α∞way.␈α
 The␈α∞intuition␈α∞for␈α
UFT
␈↓ ↓*␈↓␈↓ ↓Zshould␈αprovide␈αthe␈αhint␈αthat␈αUFT␈αwould␈αbe␈αtrue␈αfor␈αy␈αeither␈αbecuase␈αy␈αis␈αa␈αfundamental␈αbuilding␈αblock,
␈↓ ↓*␈↓␈↓ ↓Zor␈α
if␈αnot␈α
then␈αy␈α
must␈α
be␈αbuilt␈α
out␈αof␈α
blocks␈α
much␈αsmaller␈α
than␈αy.␈α
 This␈α
translates␈αas␈α
a␈αproof␈α
by␈αcases,␈α
and
␈↓ ↓*␈↓␈↓ ↓ZPROOF-BY-CASES␈α⊂tries␈α⊂to␈α∂run.␈α⊂ The␈α⊂two␈α∂cases␈α⊂suggested␈α⊂by␈α∂intuition␈α⊂are:␈α⊂SUCCESSOR(x)␈α∂is/isn't
␈↓ ↓*␈↓␈↓ ↓Zprime.

␈↓ ↓*␈↓6.␈α
The␈α
≡rst␈αof␈α
these␈α
subtasks␈α
is␈αstarted␈α
by␈α
assuming␈αthat␈α
UFT␈α
is␈α
true␈αfor␈α
2,3,...,x,␈α
and␈α
that␈αx+1␈α
is␈α
a␈αprime.␈α
But
␈↓ ↓*␈↓␈↓ ↓Zthe␈α⊂only␈α⊂conjecture␈α⊂that␈α⊂≡lling␈α⊂in␈α⊂FACTORS.TIES␈α⊂provided,␈α⊂besides␈α⊂UFT,␈α⊂was␈α⊂the␈α⊂one␈α⊂saying␈α⊂that
␈↓ ↓*␈↓␈↓ ↓Z(BAG␈α
x)␈α
is␈α
an␈α
element␈α
of␈α
FACTORS(x).␈α
So␈α
in␈α
our␈α
present␈α
case␈α
(BAG␈α
x+1)␈α
would␈α
be␈α
a␈α∞singleton␈α
bag,
␈↓ ↓*␈↓␈↓ ↓Zwhose␈αonly␈α
element␈αwas␈α
a␈αprime,␈α
and␈αwhich␈α
was␈αfound␈α
in␈αFACTORS(x+1).␈α
 Thus␈αwe␈α
set␈αup,␈α
as␈αa␈α
lemma,
␈↓ ↓*␈↓␈↓ ↓Zthe fact that ␈↓¬∀x>2,(BAG x)εFACTORS(x).␈↓

␈↓ ↓*␈↓7.␈αThe␈α
second␈αsubproblem␈αis␈α
started␈αby␈αassuming␈α
that␈αUFT␈αis␈α
true␈αfor␈α2,3,..,x,␈α
and␈αthat␈αx+1␈α
is␈αnot␈α
a␈αprime.
␈↓ ↓*␈↓␈↓ ↓ZIntuitively,␈α
that␈α
means␈α
that␈α
x+1␈α
can␈α
be␈α
split␈α
into␈α
two␈α
very␈α
small␈α
pieces,␈α
each␈α
much␈α
smaller␈α
than␈α
x,␈αand
␈↓ ↓*␈↓␈↓ ↓Zhence␈α
for␈α∞whom␈α
UFT␈α
holds.␈α∞One␈α
of␈α
the␈α∞known␈α
properties␈α
of␈α∞PRIME␈α
is␈α
that␈α∞␈↓βnot␈↓␈α
being␈α
prime␈α∞lets␈α
you

␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	53␈↓ε  Universal, Existential, Forward, Backward, Direct, Indirect, Constructive, Nonconstructive, Mathematical Induction,...
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  58␈↓

␈↓ ↓*␈↓␈↓ ↓Zconclude␈α
that␈αthe␈α
number␈α
is␈αthe␈α
product␈αof␈α
two␈α
which␈αare␈α
smaller␈αbut␈α
not␈α
as␈αsmall␈α
as␈α1.␈α
In␈α
the␈αpresent
␈↓ ↓*␈↓␈↓ ↓Zcase,␈α∞noting␈α∞that␈α∞we␈α
again␈α∞need␈α∞to␈α∞ensure␈α∞x>1,␈α
we␈α∞are␈α∞thus␈α∞able␈α
to␈α∞ write␈α∞ x+1␈α∞ as␈α∞TIMES(u,v),␈α
where
␈↓ ↓*␈↓␈↓ ↓Zboth␈α u␈αand␈αv␈αare␈αbetween␈α1␈αand␈αx+1.␈α To␈αuse␈αthis␈αfact,␈αwe␈αmust␈αwork␈αbackwards␈αand␈αobserve␈αthat␈αwhat
␈↓ ↓*␈↓␈↓ ↓Zwe␈αneed␈αis␈αthe␈αlemma:␈αFor␈α
any␈αnumber␈αa,␈αbag␈αb␈αof␈αnumbers,␈α
bag␈αd␈αof␈αnumbers,␈αif␈αa=TIMES(b)␈αand␈α
aεd,
␈↓ ↓*␈↓␈↓ ↓Zthen␈αTIMES(d)␈α=␈αTIMES(␈αSUBSTITUTE(b␈αfor␈αa␈αin␈αd)).␈α Thus␈αwe␈αcan␈αexpress␈αx+1␈αas␈αTIMES((BAG␈αu
␈↓ ↓*␈↓␈↓ ↓Zv)),␈αbut␈α
by␈αthe␈α
inductive␈αhypothesis␈α
u=TIMES(r)␈αand␈αv=TIMES(s),␈α
for␈αsome␈α
two␈αbags␈α
of␈αprimes␈α
r,s.␈αBy
␈↓ ↓*␈↓␈↓ ↓Zthe␈α
lemma,␈α
we␈α
can␈α
replace␈α
u␈α
and␈α
v␈α
by␈α
the␈α
elements␈αof␈α
r␈α
and␈α
s,␈α
get␈α
a␈α
new␈α
large␈α
bag␈α
of␈α
primes,␈αand␈α
TIMES
␈↓ ↓*␈↓␈↓ ↓Zapplied to this large bag is still x+1.

␈↓ ↓*␈↓8.␈αThe␈αtwo␈αlemmas␈αused␈αin␈αthis␈αproof␈αare␈αset␈αup␈αas␈αnew␈αBEINGs,␈αand␈αtheir␈αjusti≡cations␈αwill␈α
eventually␈αbe
␈↓ ↓*␈↓␈↓ ↓Zattended␈αto.␈αThe␈α≡rst␈αone␈αis␈αproved␈αso␈αeasily␈αthat␈αit␈αis␈αnot␈αeven␈αmentioned␈αto␈αthe␈αuser;␈αthe␈αsecond␈αone␈αis
␈↓ ↓*␈↓␈↓ ↓Zinteresting␈αin␈αits␈α
own␈αright␈αas␈α
well␈αas␈αlonger␈α
to␈αprove,␈αhence␈α
it␈αis␈αmentioned␈α
and␈αremembered.␈αThe␈α
proofs
␈↓ ↓*␈↓␈↓ ↓Zof␈αthese␈αlemmas,␈αand␈αthe␈αproof␈αof␈αthe␈αuniqueness␈αhalf␈αof␈αthe␈αUFT,␈αwill␈αbe␈αomitted␈αhere.␈α Notice␈αthat␈αin
␈↓ ↓*␈↓␈↓ ↓Zall␈α
this␈α
processing,␈α
just␈α
a␈α
few␈α
lines␈α
have␈α
been␈α
printed␈α
out␈α
to␈α
the␈α
user,␈α
and␈α
there␈α
has␈α
been␈α∞no␈α
guidance
␈↓ ↓*␈↓␈↓ ↓Zfrom him whatsoever.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&12.5. A Few Other Examples␈↓)αβ␈↓


␈↓ ↓*␈↓Let␈αus␈αskim␈αover␈αa␈αfew␈αsituations␈αand␈αsee␈αhow␈αAM␈αwould␈αhandle␈αthem,␈αhow␈αit␈αwould␈αdiscover␈αsome␈α
of␈αthe

␈↓ ↓*␈↓interesting information which we won't supply it with initially.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.5.1. E≥ort of Noticing New Analogies␈↓)αβ␈↓


␈↓ ↓*␈↓AM␈α∂will␈α⊂have␈α∂a␈α⊂general␈α∂strategy,␈α∂located␈α⊂in␈α∂the␈α⊂ORDERING␈α∂part␈α∂of␈α⊂the␈α∂BEING␈α⊂named␈α∂ANY-BEING,

␈↓ ↓*␈↓which␈α
says␈α
that␈α
if␈αno␈α
e≥ort␈α
has␈α
been␈α
expended␈αwhatsoever␈α
to␈α
try␈α
to␈α≡nd␈α
analogues␈α
of␈α
the␈α
current␈αBEING,

␈↓ ↓*␈↓then␈α
there␈α
is␈α
a␈α
high␈α
interest␈α
in␈α
doing␈α
this␈α
activity␈α
(about␈α
the␈α
same␈α
motivation␈α
as␈α
≡nding␈α
examples␈α
of␈α
it,␈αif

␈↓ ↓*␈↓there are none yet).


␈↓ ↓*␈↓This␈αwill␈α
lead␈αto␈α
two␈αdistinct␈α
types␈αof␈αbehavior.␈α
When␈αthe␈α
system␈αis␈α
≡rst␈αstarted,␈α
whichever␈αBEING␈αis␈α
chosen

␈↓ ↓*␈↓to␈α
be␈αcompleted␈α
by␈α
COMPLETE,␈αits␈α
Analogy␈α
subpart␈αof␈α
its␈αTies␈α
part␈α
will␈αbe␈α
blank,␈α
hence␈αearly␈α
on␈α
it␈αwill

␈↓ ↓*␈↓want␈αto␈α≡nd␈α
analogies␈αto␈αitself␈α(it␈α
will␈αtrigger␈αAnalogize,␈α
with␈αitself␈αas␈αthe␈α
only␈αknown␈αargument).␈αThe␈α
second

␈↓ ↓*␈↓type␈α∞of␈α∞activity␈α∂occurs␈α∞when␈α∞a␈α∂new␈α∞BEING␈α∞is␈α∂created␈α∞by␈α∞the␈α∂system.␈α∞It␈α∞will␈α∂usually␈α∞be␈α∞worked␈α∂on␈α∞almost

␈↓ ↓*␈↓immediately,␈α
and␈α
after␈α
the␈α
highest-priority␈α
parts␈α
are␈α
≡lled␈α
in␈α
(like␈α
intuition,␈α
de≡nition,␈α
perhaps␈αsome␈α
examples

␈↓ ↓*␈↓and family ties) the above-mentioned strategy will direct attention to ≡nding analogies to it.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  59␈↓

␈↓ ↓*␈↓When␈α⊃a␈α⊃BEING␈α⊃wants␈α⊃to␈α⊃≡nd␈α⊃its␈α⊃analogues,␈α⊃Analogize␈α⊃will␈α⊃look␈α⊃within␈α⊃BEING's␈α⊃family,␈α⊃scanning␈α⊂for

␈↓ ↓*␈↓another␈αBEING␈αwhich␈αhas␈αone␈αof:␈α(i)␈αsyntactically␈αsimilar␈αde≡nition,␈α(ii)␈αintuition␈αview␈αwhich␈αapplies␈αto␈αthe

␈↓ ↓*␈↓same␈αreal-world␈αsituation,␈α(iii)␈α
syntactically␈αsimilar␈αexamples,␈α(iv)␈α
similarities␈αbetween␈αthe␈αtwo␈α
BEINGs'␈αTies

␈↓ ↓*␈↓parts.


␈↓ ↓*␈↓The␈α∞initial␈α
∨urry␈α∞of␈α
analogy␈α∞quests␈α
will␈α∞number␈α∞about␈α
 18,000␈α∞(␈α
=␈α∞5␈α
families␈α∞ x␈α
 30␈α∞BEINGs/family␈α∞ x␈α
30

␈↓ ↓*␈↓BEINGs␈α
to␈αinteract␈α
with␈α
 x␈α4␈α
part-pairings␈α
to␈αexamine)␈α
Some␈α
of␈αthese␈α
will␈α
be␈αprecluded␈α
almost␈α
instantly,␈αso␈α
a

␈↓ ↓*␈↓reasonable␈α≡gure␈αis␈αabout␈αthree␈αCPU␈αhours␈αof␈αtime␈αexpended,␈α≡nding␈αabout␈α1,000␈αpossible␈αanalogies␈αin␈αtoto,

␈↓ ↓*␈↓of␈αwhich␈α
only␈αabout␈α
100␈αwill␈αprove␈α
intersting␈αupon␈α
careful␈αexamination␈αand␈α
will␈αbe␈α
made␈αinto␈αnew␈α
BEINGs.

␈↓ ↓*␈↓Another␈αspeedup␈αwill␈αoccur␈αbecause␈αmany␈αof␈αthe␈αinitially␈αsupplied␈αBEINGs␈αwill␈αnot␈αhave␈αanything␈α
in␈αtheir

␈↓ ↓*␈↓Examples or Ties parts to begin with, so those matches will fail trivially.


␈↓ ↓*␈↓The␈αsecondary␈αprocess␈αof␈αanalogizing,␈αwhen␈αa␈αnew␈αTies,␈αEx,␈αDefn,␈αor␈αIntu␈αpart␈αis␈αadded,␈αis␈αof␈αcourse␈αonly␈αa

␈↓ ↓*␈↓matter of  30 ( = 1 same family  x  30 BEING's to consider x 1 same part as the new one) things to look at.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.5.2. Filling in the Examples parts of Objects␈↓)αβ␈↓


␈↓ ↓*␈↓Let␈αus␈αnow␈αconsider␈αa␈αfairly␈αdetailed␈αexample.␈αWhat␈αhappens␈αwhen␈αthe␈αsystem␈α≡rst␈αstarts␈αup?␈α Each␈αBEING

␈↓ ↓*␈↓will␈α∞probably␈α∞only␈α∞have␈α∞a␈α∞few␈α∞parts,␈α∞hence␈α∞all␈α∞will␈α∞clamor␈α∞for␈α∞attention.␈α∞The␈α∞ORDERING␈α∞part␈α∂of␈α∞ANY-

␈↓ ↓*␈↓BEING␈αindicates␈αthat␈αafter␈αthe␈αde≡nition␈αand␈αintutition,␈αthe␈αnext␈αmost␈αimportant␈αpart␈αto␈α≡ll␈αin␈αis␈αExamples.

␈↓ ↓*␈↓The reason is, both for motivation and for later empirical evidence.


␈↓ ↓*␈↓The␈α∞environment␈α∞function␈α∞Complete␈α∞takes␈α∂over;␈α∞see␈α∞page␈α∞33.␈α∞The␈α∂numbers␈α∞below␈α∞refer␈α∞to␈α∞the␈α∂steps␈α∞listed

␈↓ ↓*␈↓there. The details of each access are omitted, for brevity.


␈↓ ↓*␈↓1.␈αNeither␈αP␈αnor␈αB␈αis␈αknown.␈αAsk␈αeach␈αBEING␈αhow␈αrelevant␈αit␈αis␈αto␈αthe␈αcurrent␈αsituation,␈αCS,␈αwhich␈αat␈αthe

␈↓ ↓*␈↓moment␈αis␈α
almost␈αtotally␈αnull.␈α
 Since␈αmost␈αBEING's␈α
require␈α␈↓βsome␈↓␈α
constrained␈αstate␈αof␈α
CS,␈αin␈αwhich␈α
something

␈↓ ↓*␈↓is␈αtrue␈αor␈αwithin␈αcertain␈αbounds,␈αthere␈αare␈αonly␈αabout␈αthirty␈αresponders␈α(out␈αof␈αabout␈α125␈αBEING's).␈α These
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  60␈↓

␈↓ ↓*␈↓include␈αStructures␈α(who␈αwant␈αexamples),␈αActives␈α(who␈αwant␈αexamples␈αand␈αties),␈αStatic-Metas␈α(examples),␈αand

␈↓ ↓*␈↓a␈α∃couple␈α⊗Active-Metas␈α∃(Guess,␈α∃Analogize)␈α⊗who␈α∃also␈α⊗want␈α∃some␈α∃examples.␈α⊗ The␈α∃Time␈α⊗component␈α∃of

␈↓ ↓*␈↓Analogize's␈α⊃Worth␈α⊃part␈α⊃is␈α∩incredibly␈α⊃low␈α⊃(since␈α⊃no␈α⊃arguments␈α∩--␈α⊃suggesions␈α⊃for␈α⊃either␈α⊃candidate␈α∩in␈α⊃the

␈↓ ↓*␈↓analogy␈α
--␈α
have␈α∞been␈α
proposed).␈α
Any␈α∞of␈α
the␈α
Actives,␈α
and␈α∞also␈α
Guess,␈α
would␈α∞≡rst␈α
ensure␈α
that␈α∞the␈α
Examples

␈↓ ↓*␈↓parts␈αof␈αthe␈αthings␈αit␈α
deals␈αwith␈αbe␈α≡lled␈αin,␈αhence␈α
we␈αmay␈αas␈αwell␈αassume␈α
that␈αwe␈αare␈α≡lling␈αin␈αthe␈α
Examples

␈↓ ↓*␈↓part␈αof␈αa␈αStructure␈αor␈αa␈αStatic␈αMeta.␈α The␈αlatter␈αcategory␈αare␈αnot␈αas␈αeasy,␈αand␈αare␈αgenerally␈αproduced␈αby␈αan

␈↓ ↓*␈↓Active␈α∞Meta's␈α∞direct␈α∞command.␈α
 Decide␈α∞to␈α∞work␈α∞on␈α
structures.␈α∞ Of␈α∞all␈α∞the␈α
structures,␈α∞Set␈α∞and␈α∞Bag␈α∞are␈α
tied

␈↓ ↓*␈↓with␈αthe␈αhigest␈α
ORD␈αvalue␈α(based␈α
on␈αtheir␈αWorth␈αparts).␈α
 The␈αlist␈α(Set␈α
Bag)␈αis␈αprinted␈α
to␈αthe␈αuser,␈αwho␈α
then

␈↓ ↓*␈↓has␈α
a␈αfew␈α
seconds␈α
to␈αrespond␈α
before␈α
the␈αsystem␈α
begins␈αworking␈α
on␈α
one␈αof␈α
them.␈α
 Say␈αthe␈α
user␈α
doesn't␈αcare,

␈↓ ↓*␈↓and␈α
B␈α
is␈α
now␈αdetermined␈α
to␈α
be␈α
Set.␈α
The␈αnext␈α
step␈α
is␈α
to␈α
choose␈αP,␈α
the␈α
part␈α
of␈α
the␈αSet␈α
BEING␈α
to␈α
work␈αon␈α
now.

␈↓ ↓*␈↓We␈α⊃collect␈α⊃all␈α⊃the␈α⊃facts␈α⊃on␈α⊃up␈↓	*␈↓(Set).ORDERING,␈α⊃which␈α⊃means␈α⊃Set.Ordering,␈α∩Structure.Ord,␈α⊃Object.Ord,

␈↓ ↓*␈↓AnyBEING.Ord.␈α∞In␈α∞this␈α∞case,␈α∞only␈α∂the␈α∞last␈α∞of␈α∞these␈α∞is␈α∂nonempty.␈α∞Each␈α∞factor␈α∞is␈α∞evaluated,␈α∂and␈α∞Examples

␈↓ ↓*␈↓wins with a factor of .6 on a 0 - 1 scale.   So P is chosen to be Examples.


␈↓ ↓*␈↓2.␈α∂Create␈α∂a␈α⊂plan␈α∂for␈α∂≡lling␈α∂in␈α⊂Set.Examples.␈α∂ Collect␈α∂any␈α∂helpful␈α⊂information␈α∂from␈α∂the␈α⊂following␈α∂sources:

␈↓ ↓*␈↓Examples.Fillin␈α(which␈αcontains␈αmany␈αthings␈αto␈αtry␈αto␈αget␈αnew␈αexamples),␈αSet.Examples.Fillin␈α(nothing␈αthere),

␈↓ ↓*␈↓Structure.Examples.Fillin␈α∞(which␈α∞contains␈α∞some␈α∞specialized␈α
hints:␈α∞Convert␈α∞other␈α∞structures'␈α∞Examples;␈α
make

␈↓ ↓*␈↓some␈α∩of␈α⊃the␈α∩interestingness␈α⊃features␈α∩present␈α⊃in␈α∩up␈↓	*␈↓(Set).Interest␈α⊃not␈α∩just␈α⊃desirable␈α∩but␈α∩actually␈α⊃required,

␈↓ ↓*␈↓thereby␈α⊂guaranteeing␈α⊂an␈α⊃␈↓βinteresting␈↓␈α⊂example),␈α⊂Object.Examples.Fillin␈α⊃(empty),␈α⊂AnyBEING.Examples.Fillin

␈↓ ↓*␈↓(empty).␈α∂ Finally,␈α∞some␈α∂additonal␈α∞parts␈α∂of␈α∞the␈α∂Examples␈α∞BEING␈α∂might␈α∞be␈α∂relevant␈α∞later␈α∂on.␈α∞ The␈α∂plan␈α∞is

␈↓ ↓*␈↓simple:␈α∞try␈α∞the␈α∞Examples.Fillin␈α∞activities,␈α∞then␈α∞the␈α∞second␈α∞Structure.Examples.Fillin␈α∞activity,␈α∞then␈α∞check␈α∞the

␈↓ ↓*␈↓results with the Examples.Check part.


␈↓ ↓*␈↓3/4. Try to instantiate specializations of Set.  There are none.  Fail.


␈↓ ↓*␈↓3/4.␈α
Try␈αto␈α
instantiate␈α
de≡nition(s)␈αof␈α
Set.␈α
 A␈αsimple␈α
linear␈α
analysis␈αof␈α
the␈α
base␈αstep␈α
of␈α
the␈αrecursive␈α
de≡nition

␈↓ ↓*␈↓yields the fact that (CLASS ), called PHI, is an example of a set. Return (CLASS ).
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  61␈↓

␈↓ ↓*␈↓3/4.␈αTry␈αto␈αinstantiate␈αand␈αapply␈αthe␈αintuition(s).␈αThe␈αset␈αintuition␈αrequires␈αa␈αpurpose␈α(and␈αoften␈αother␈αsets);

␈↓ ↓*␈↓the␈α∞intuition␈α∂is␈α∞not␈α∂designed␈α∞to␈α∂create␈α∞sets␈α∂from␈α∞nothing␈α∂for␈α∞no␈α∂purpose␈α∞(if␈α∂it␈α∞were,␈α∂this␈α∞would␈α∂be␈α∞highly

␈↓ ↓*␈↓suspect!). Thus this fails.


␈↓ ↓*␈↓3/4.␈α
Try␈α
to␈α
invert␈αthe␈α
recursive␈α
de≡nition,␈α
so␈αit␈α
produces␈α
more␈α
complicated␈αexamples.␈α
 In␈α
the␈α
current␈αcase,␈α
this

␈↓ ↓*␈↓is␈α∞trivial.␈α
We␈α∞simply␈α∞apply␈α
the␈α∞algorithm:␈α
Take␈α∞known␈α∞sets␈α
and␈α∞apply␈α
Set-insert␈α∞to␈α∞them.␈α
To␈α∞start␈α∞out,␈α
we

␈↓ ↓*␈↓plug␈αin␈αthe␈α
speci≡c␈αbase-step␈αset,␈α
namely␈αPhi.␈αThe␈α
result␈αis␈α(CLASS␈α(CLASS␈α
)␈α),␈αusually␈α
written␈α{␈α{}␈α
}.␈α We

␈↓ ↓*␈↓reapply the algorithm with one argument PHI, and get
␈↓ ↓*␈↓[CLASS (CLASS (CLASS)) (CLASS )]; with both arguments equal to (CLASS (CLASS)), we obtain
␈↓ ↓*␈↓[CLASS (CLASS (CLASS)) (CLASS (CLASS)) ]  =  { {{}}, {{}} }.

␈↓ ↓*␈↓There␈αis␈αno␈αreason␈αjust␈αnow␈αto␈αgo␈αon,␈αsince␈αwe␈αhave␈αthe␈αalgorithm,␈αso␈αwe␈αreturn␈αthese␈αfew␈αexamples␈αplus␈αwe

␈↓ ↓*␈↓also return the inverted recursive de≡nition. ␈↓	54␈↓


␈↓ ↓*␈↓3/4.␈α∂Tag␈α∞the␈α∂Examples␈α∞parts␈α∂of␈α∞Set.Ops␈α∂(namely␈α∞Member,␈α∂Containment,␈α∞Get-Some-Member,␈α∂Equality,␈α∞Set-

␈↓ ↓*␈↓insert,␈α
and␈α
Set-delete)␈α
as␈α∞follows:␈α
Put␈α
a␈α
little␈α∞note␈α
in␈α
each␈α
of␈α∞these␈α
parts,␈α
saying␈α
that␈α∞Set.Examples␈α
contains

␈↓ ↓*␈↓some␈α⊂examples␈α⊂of␈α⊂Domain␈α∂elements␈α⊂for␈α⊂these␈α⊂operations.␈α∂This␈α⊂will␈α⊂raise␈α⊂the␈α∂level␈α⊂of␈α⊂estimated␈α⊂worth␈α∂of

␈↓ ↓*␈↓working on ␈↓βtheir␈↓ examples parts. ␈↓	55␈↓


␈↓ ↓*␈↓3/4.␈αNow␈αwe␈αpass␈αfrom␈αthe␈αgeneral␈αExamples.Fillin␈αstrategies␈αto␈αthe␈αStructure.Examples.Fillin␈αstrategies.␈α We

␈↓ ↓*␈↓must␈α
conjoin␈α
the␈αInterest␈α
properties␈α
as␈α
if␈αthey␈α
were␈α
requirements.␈α Set.Interest␈α
asks␈α
for␈α
the␈αelements␈α
of␈α
a␈αset␈α
to

␈↓ ↓*␈↓be␈αrelated␈αby␈αsome␈αother␈αknown,␈αinteresting␈αrelation␈α(besides␈αbeing␈αmembers␈αof␈αthe␈αsame␈αset).␈α Things␈αwhich

␈↓ ↓*␈↓are␈α∂so␈α∞related␈α∂are␈α∂located␈α∞via␈α∂their␈α∞Ties␈α∂parts,␈α∂so␈α∞the␈α∂task␈α∞is␈α∂≡nd␈α∂some␈α∞Tied␈α∂entities␈α∞and␈α∂make␈α∂them␈α∞the

␈↓ ↓*␈↓elements␈αof␈α
a␈αset.␈α
The␈α≡rst␈αpart␈α
is␈αdone,␈α
say,␈αby␈α
noticing␈αthe␈αTies␈α
part␈αof␈α
Set␈αitself,␈α
namely␈αto␈αother␈α
structures,


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	54␈↓ε␈α
Aside:␈α
We␈α
have␈α
no␈α
way␈α
of␈α
knowing,␈α
though,␈α
whether␈α
this␈α
process␈α
always␈α
gives␈α
new␈α
sets.␈α
That␈α
is,␈α
Phi␈α
might␈α
equal␈α
Setinsert((Class␈α
Phi),(class
␈↓ ↓*␈↓ε␈↓ αJPhi))=(Class␈α(Class␈αPhi)␈αPhi).␈αOne␈αwould␈αhave␈αto␈αdo␈αinference,␈αusing␈αsome␈αfoundation␈αaxiom,␈αto␈αprove␈αthat␈αx␈αcan␈αnever␈αbe␈αan
␈↓ ↓*␈↓ε␈↓ αJelement␈α	of␈αλx,␈α	just␈α	to␈αλprove␈α	that␈α	we␈αλhave␈α	three␈α	distinct␈αλsets␈α	here;␈α	the␈αλactual␈α	proof␈α	that␈αλthe␈α	chain␈α	[x␈↓#vn␈↓#␈↓#v+␈↓#␈↓#v1␈↓#␈αλ=␈α	Set-insert(x␈↓#vn␈↓#,␈α	x␈↓#vn␈↓#)]␈αλnever
␈↓ ↓*␈↓ε␈↓ αJrepeats␈αis␈αnot␈αtrivial␈αunless␈αthe␈αaxiom␈αis␈αphrased␈αin␈αjust␈αthe␈αright␈αway.␈αThe␈αfact␈αthat␈αthis␈αarose␈αout␈αof␈αinverting␈αa␈αrecursive
␈↓ ↓*␈↓ε␈↓ αJdefinition,␈α	however,␈α	strongly␈α	suggests␈α	that␈α	this␈αλalgorithm␈α	will␈α	in␈α	fact␈α	yield␈α	an␈α	infinite␈αλnumber␈α	of␈α	distinct␈α	sets␈α	if␈α	there␈α	are␈αλinfinitely
␈↓ ↓*␈↓ε␈↓ αJmany.␈α
An␈α
indirect␈α
proof␈αcould␈α
now␈α
be␈α
proposed,␈αnamely␈α
assume␈α
that␈α
A,B,...,Z␈αare␈α
the␈α
only␈α
distinct␈αsets␈α
which␈α
can␈α
exist,␈αand␈α
then
␈↓ ↓*␈↓ε␈↓ αJderive a contradiction.
␈↓ ↓*␈↓ε␈↓	55␈↓ε  This is one small example of how tangential knowledge speeds up, rather than slows down, the acquisition of new information.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  62␈↓

␈↓ ↓*␈↓and␈αthe␈αsecond␈αpart␈αis␈αdone␈αwhen␈αMake-a-Set␈αrecognizes␈αits␈αown␈αrelevance.␈αThe␈αresult␈αis␈αthus␈α(CLASS␈αHist

␈↓ ↓*␈↓List␈αBag␈αOset␈αSet␈αOrdered␈αPair).␈αThe␈αease␈αwith␈αwhich␈αthis␈αwas␈αdone␈αsignals␈αthat␈αit␈αmay␈αbe␈αexplosive,␈αso␈αwe

␈↓ ↓*␈↓don't pursue this method of set construction any further right now.


␈↓ ↓*␈↓3/4.␈αStructure.Interest␈αsays␈αthat␈αthe␈αstructure␈αshould␈αbe␈αsuch␈αthat␈αcertain␈αinteresting␈αoperations␈αare␈αdoable␈αto

␈↓ ↓*␈↓it␈α∞e≠cently,␈α∞without␈α∞going␈α∞into␈α∞detail␈α∂about␈α∞which␈α∞operations.␈α∞ The␈α∞part␈α∞Set.Operations␈α∂contains␈α∞Member,

␈↓ ↓*␈↓Get-some-member,␈α∀Subset,␈α∀Equal,␈α∀Setinsert,␈α∀Setdelete,␈α∀and␈α∀some␈α∀invariance␈α∀data.␈α∀ After␈α∀studying␈α∪these

␈↓ ↓*␈↓operations,␈α∞it␈α∞decides␈α∞that␈α∞PHI␈α∞is␈α∞the␈α∞most␈α∞e≠cient␈α∞argument␈α∞to␈α∞each␈α∞and␈α∞every␈α∞one␈α∞of␈α∞them.␈α∞This␈α
result,

␈↓ ↓*␈↓while␈αtrivial,␈αis␈αnoticed␈αas␈αa␈αsurprising␈α(to␈αthe␈αsystem)␈αdiscovery,␈αand␈αmay␈αbe␈αsu≠cent␈αto␈αensure␈αthat␈αPHI␈αis

␈↓ ↓*␈↓made into a BEING itself, and its properties studied.


␈↓ ↓*␈↓5.␈αThe␈αpercentage␈αof␈αsuccess␈αfactor␈αin␈αSet.Worth␈αvector␈αis␈αincremented␈α(say␈αfrom␈α.9␈αto␈α.91),␈αits␈αanalogic␈αutility

␈↓ ↓*␈↓factor␈α
likewise␈α(from␈α
.8␈α
to␈α.9).␈α
 There␈α
is␈αnot␈α
enough␈αactivation␈α
energy␈α
left␈αto␈α
pursue␈α
any␈αmore␈α
examples␈αof␈α
Set

␈↓ ↓*␈↓just␈αnow.␈αA␈αmarker␈αis␈αleft␈αhere␈αindicating␈αhow␈αmuch␈αe≥ort␈αwas␈αspent,␈αhow␈αthe␈αinverted␈α
recursive␈αde≡nition

␈↓ ↓*␈↓can be used, and the hint of a conjecture about the diversity of its results.


␈↓ ↓*␈↓1.␈α
Return␈α
and␈α
decide␈αif␈α
Set␈α
is␈α
still␈αthe␈α
best␈α
BEING,␈α
and/or␈αif␈α
Examples␈α
is␈α
still␈αthe␈α
best␈α
part␈α
to␈αconcentrate

␈↓ ↓*␈↓upon.␈αProbably,␈αthe␈α
Examples␈αpart␈αof␈αBag␈α
will␈αbe␈αthe␈αhighest␈α
priority␈αpart␈αot␈α≡ll␈α
in␈αat␈αthe␈αmoment.␈α
In␈αthis

␈↓ ↓*␈↓manner,␈αwe␈αmay␈αsuppose␈αin␈αlater␈αexamples␈αthat␈αthe␈αsystem␈αhas␈αspent␈αsome␈αtime␈αto␈αcollect␈αexamples␈αof␈αall␈αthe

␈↓ ↓*␈↓structures.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.5.3. Considering New Compositions of Operations␈↓)αβ␈↓


␈↓ ↓*␈↓The␈α_activity␈α_for␈α↔≡nding␈α_examples␈α_of␈α↔Actives␈α_is␈α_similar␈α↔to␈α_≡nding␈α_examples␈α↔of␈α_Sets,␈α_except␈α↔that

␈↓ ↓*␈↓Structure.Examples.Fillin␈α
is␈α
not␈α
relevant␈α∞to␈α
Actives,␈α
and␈α
its␈α∞place␈α
is␈α
taken␈α
by␈α∞Active.Examples.Fillin,␈α
which

␈↓ ↓*␈↓contains␈αone␈αspeci≡c␈αactivity:␈αafter␈αa␈αnew␈αspecialized␈αoperator␈αis␈αfound,␈αgo␈αto␈αthe␈αtrouble␈αof␈αapplying␈αit␈αto␈α a

␈↓ ↓*␈↓few␈α∂examples␈α⊂of␈α∂its␈α∂domain.␈α⊂This␈α∂gets␈α∂a␈α⊂bit␈α∂intricate␈α∂if,␈α⊂e.g.,␈α∂its␈α∂domain␈α⊂is␈α∂itself␈α∂a␈α⊂set␈α∂of␈α⊂operators.␈α∂The
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  63␈↓

␈↓ ↓*␈↓trickiest␈α
case,␈α
when␈α
the␈αActive␈α
is␈α
the␈α
BEING␈α
named␈αCompose,␈α
is␈α
presented␈α
now␈αto␈α
help␈α
clarify␈α
(??)␈α
all␈αthis

␈↓ ↓*␈↓"examples of examples of..."  confusion.


␈↓ ↓*␈↓1.␈α
Complete␈α
wants␈α
to␈α
determine␈α
P␈α
and␈α
B.␈α
Again␈α
the␈α
same␈α
thirty␈α
or␈α
so␈α
BEING's␈α
respond,␈α
as␈α
in␈α∞example␈α
3.

␈↓ ↓*␈↓This␈α∞time,␈α∞we␈α∞assume␈α∂that␈α∞the␈α∞static␈α∞BEING's␈α∂have␈α∞their␈α∞examples␈α∞≡lled␈α∂in,␈α∞likewise␈α∞most␈α∞of␈α∂the␈α∞Actives.

␈↓ ↓*␈↓Guess␈α∩and␈α∩Analogize␈α∩still␈α∩are␈α∩low␈α∩in␈α⊃their␈α∩run-time␈α∩worth␈α∩components␈α∩(due␈α∩to␈α∩unspeci≡ed␈α⊃arguments).

␈↓ ↓*␈↓Suppose␈αthat␈αno␈αexamples␈αare␈αknown␈αfor␈αthe␈αActive␈αBEING␈αnamed␈αCompose,␈αand␈αits␈αWorth␈αpart␈αlets␈αit␈αbe

␈↓ ↓*␈↓chosen.  Ordering (of Any-BEING, actually) speci≡es that Examples should be ≡lled in.


␈↓ ↓*␈↓2.␈α∞Must␈α
devise␈α∞a␈α∞plan␈α
for␈α∞≡lling␈α∞in␈α
the␈α∞Examples␈α∞part␈α
of␈α∞the␈α∞Compose␈α
BEING.␈α∞ Much␈α∞information␈α
exists

␈↓ ↓*␈↓under␈αExamples.Fillin,␈αand␈αsome␈α
also␈αunder␈αActive.Examples.Fillin.␈αThe␈α
latter␈αindicates␈α"Afterwards",␈αso␈αit␈α
is

␈↓ ↓*␈↓done␈αonly␈αafter␈αthe␈αExamples.Fillin␈αstrategies␈αare␈αexhausted.␈αThe␈α≡nal␈αinformation␈αrecognized␈αto␈αbe␈αrelevant

␈↓ ↓*␈↓is␈α⊂present␈α⊂in␈α⊂Compose.Algorithms,␈α⊂and␈α⊂is␈α⊂used␈α⊂when␈α⊂the␈α⊂terms␈α⊂of␈α⊂the␈α⊂de≡nition␈α⊂get␈α⊂replaced␈α⊂by␈α⊂speci≡c

␈↓ ↓*␈↓examples of themselves.


␈↓ ↓*␈↓3/4.␈α∂ Specialize␈α∂the␈α∂de≡nition␈α∂of␈α∂Compose.␈α∂ Must␈α∂≡nd␈α∂an␈α∂ordered␈α∂pair␈α∂of␈α∂operations␈α∂(f,g),␈α∂ with␈α∂dom(f)␈α∞⊃

␈↓ ↓*␈↓ran(g).␈α Access␈αthe␈αFind.Algorithms␈αpart.␈α This␈αsays␈αto␈αconsider␈αK={set␈αof␈αknown␈αoperators}.␈α Form␈αthe␈α
cross-

␈↓ ↓*␈↓product␈α∞C=KxK.␈α∞If␈α∂there␈α∞is␈α∞some␈α∂reasonable␈α∞ordering␈α∞on␈α∂C,␈α∞order␈α∞it.␈α∂ Pick␈α∞an␈α∞e␈α∂in␈α∞C,␈α∞say␈α∂e=(j,k).␈α∞Check

␈↓ ↓*␈↓whether␈α⊃ran(k)␈α⊃⊂␈α⊃dom(j).␈α⊃ If␈α⊂so,␈α⊃apply␈α⊃Compose.Algorithm.␈α⊃In␈α⊃either␈α⊂case,␈α⊃you␈α⊃can␈α⊃continue␈α⊃by␈α⊂picking

␈↓ ↓*␈↓another element of C, etc.


␈↓ ↓*␈↓By␈α
applying␈αthe␈α
above␈αalgorithm,␈α
the␈αsystem␈α
uncovers␈α
a␈αwealth␈α
of␈αpossible␈α
compositions.␈α For␈α
example,␈α
α␈α␈↓εo␈↓

␈↓ ↓*␈↓Delete,␈α
where␈α
α␈α∞can␈α
be␈α
any␈α
one␈α∞of:␈α
Insert,␈α
Delete,␈α
Convert,␈α∞Subst,␈α
Unite,␈α
Common-parts,␈α∞Member,␈α
Contain.

␈↓ ↓*␈↓Some␈αsecond-order␈αof␈αworth␈αcompositions␈αinclude␈αcompose*(delete,delete)␈α and␈αalso␈αequal*(delete,delete).␈αSome

␈↓ ↓*␈↓third-order ones are compose*(delete,insert) and equal*(delete,insert).


␈↓ ↓*␈↓Each␈α⊂example␈α⊂found␈α⊂(and␈α⊂there␈α⊂will␈α⊂be␈α⊂about␈α⊂300␈α⊂of␈α⊂them)␈α⊂is␈α⊂made␈α⊂into␈α⊂a␈α⊂BEING␈α⊂whose␈α⊂Worth␈α⊂part
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  64␈↓

␈↓ ↓*␈↓indicates␈α⊂high␈α⊃interest␈α⊂but␈α⊃a␈α⊂short␈α⊃lifetime␈α⊂if␈α⊃nothing␈α⊂new␈α⊂and␈α⊃interesting␈α⊂is␈α⊃found␈α⊂out␈α⊃about␈α⊂it␈α⊃(if␈α⊂no

␈↓ ↓*␈↓interesting␈αTie␈α
is␈αdiscovered␈αrapidly).␈α
 This␈αis␈αdone␈α
by␈αActive.Examples.Fillin,␈αwhich␈α
also␈αspeci≡cally␈αcalls␈α
for

␈↓ ↓*␈↓the␈αinvestigation␈αof␈αthe␈αspeci≡c␈α
Ties␈αof␈αthe␈αform:␈α(input,output)␈α
satisfy␈αsome␈αknown␈αinteresting␈αrelation;␈αsee␈α
if

␈↓ ↓*␈↓the␈α
new␈α
operation␈α
is␈α
related␈αto␈α
another␈α
by␈α
yet␈α
a␈αthird.␈α
 The␈α
300␈α
new␈α
Active␈αBEING's␈α
are␈α
chosen␈α
one␈α
at␈αa

␈↓ ↓*␈↓time, and their intuition and examples parts are ≡lled out.


␈↓ ↓*␈↓Let␈α
us␈α
take␈α
a␈α
few␈α
examples␈α(not␈α
all␈α
300!!).␈α
Consider␈α
Insert␈α
o␈α
Delete.␈αThis␈α
means␈α
take␈α
a␈α
structure␈α
S,␈αproduce␈α
all

␈↓ ↓*␈↓the␈α
pairs␈α
(e,␈αresult␈α
of␈α
deleting␈α
e␈αfrom␈α
S),␈α
then␈αcall␈α
Insert␈α
on␈α
each␈αof␈α
these␈α
pairs,␈α
getting␈αa␈α
list␈α
of␈αnew␈α
structures

␈↓ ↓*␈↓characterized␈α
by␈α
 {S'␈α
|␈α
∃e.␈α
S'␈α
=␈α
result␈α
of␈α
inserting␈α
e␈α
into␈α
the␈α
result␈α
of␈α
deleting␈α
e␈α
from␈α
S}.␈α
 Another␈α
view␈α
is␈αto␈α
be

␈↓ ↓*␈↓given␈α⊂a␈α∂structure␈α⊂S␈α∂and␈α⊂an␈α∂element␈α⊂e,␈α∂then␈α⊂perform␈α∂Delete(e,S),␈α⊂obtaining␈α∂structure␈α⊂R,␈α∂then␈α⊂perform␈α∂the

␈↓ ↓*␈↓(explosive) operation Insert(R), yielding all the pairs (f, result of inserting f into R).
␈↓ ↓*␈↓Some examples are found, say PHI → {(e,PHI)} → { {e} };
␈↓ ↓*␈↓{f} → { (e, {f}), (f, PHI) } → { {e,f}, {f} };
␈↓ ↓*␈↓(f) → { (e, (f)), (f, NIL) } → { (e,f), (f) };
␈↓ ↓*␈↓(f,g) → { (e, (f,g)), (f, (g)), (g, (f)) }  →  { (e,f,g), (f,g), (g,f) }.


␈↓ ↓*␈↓Filling␈α
in␈α
the␈α
intuition␈αpart␈α
of␈α
Insert␈α
o␈αDelete,␈α
we␈α
get␈α
the␈α
tight␈αmeshing␈α
of:␈α
pull␈α
x␈αout␈α
of␈α
container␈α
S␈αand␈α
then

␈↓ ↓*␈↓drop␈α
it␈α∞back␈α
in.␈α
This␈α∞should␈α
indicate␈α
that␈α∞Insert*Delete␈α
may␈α
leave␈α∞the␈α
original␈α
container␈α∞unchanged.␈α
That

␈↓ ↓*␈↓would␈α
mean␈α
that␈α
Insert␈α
o␈α
Delete␈α
(S)␈α
=␈α
{S};␈α
a␈α
slight␈α
weakening␈α
would␈α
be␈α
the␈α
statement␈α
"S␈α
is␈α
a␈α
member␈α
of␈α
Insert

␈↓ ↓*␈↓o␈αDelete␈α
(S)".␈αOne␈α
potential␈αexception␈αis␈α
when␈αyou␈α
can't␈αpull␈αthe␈α
thing␈αx␈α
out␈αof␈αS;␈α
when␈αS␈α
is␈αnull␈α
or␈α(more

␈↓ ↓*␈↓intelligently)␈α
when␈α
the␈α
entity␈α
x␈α
is␈α
not␈α
already␈α
a␈α
member␈α
of␈α
S.␈α
 Either␈α
the␈α
intuition␈α
or␈α
the␈α
examples␈α
should

␈↓ ↓*␈↓indicate a conjecture of the form
␈↓ ↓*␈↓S non-null  ↔  S ␈↓¬ε␈↓ Insert o Delete(S), or perhaps even the sophisticated one:
␈↓ ↓*␈↓x is in set S  ↔  Insert(x,Delete(x,S))=S.


␈↓ ↓*␈↓Consider␈αnow␈αthe␈αcomposition␈αMember␈αo␈αDelete,␈αwhich␈αtakes␈αa␈αstructure␈αS,␈αdeletes␈αsome␈αelement␈αx,␈αthen␈αsees

␈↓ ↓*␈↓if␈α
x␈α
is␈α
a␈α
member␈α
of␈α
S.␈α
Notice␈α
that␈α
the␈α
range␈α
is␈α
therefore␈α
{T,F}.␈α
From␈α
examples␈α
alone,␈α
the␈α
system␈αshould␈α
notice

␈↓ ↓*␈↓that␈αall␈αsets␈αand␈αosets␈αmap␈αinto␈αFalse,␈αand␈αsome-but-not-all␈αlists␈αand␈αbags␈αmap␈αthe␈αsame␈αway.␈α The␈αintuition

␈↓ ↓*␈↓should␈αanswer␈αthe␈αriddle␈α"how/when␈αdoes␈αa␈αlist/bag␈α␈↓βnot␈↓␈αmap␈αinto␈αFalse?"␈αThe␈αanswer␈αis␈αthat␈αthere␈αwas␈αmore
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  65␈↓

␈↓ ↓*␈↓than␈αone␈αx␈αin␈αthe␈αoriginal␈αstructure,␈αso␈αthere␈αis␈αstill␈αat␈αleast␈αone␈αleft␈αwhen␈αyou␈αpluck␈αout␈αone␈αx.␈αOne␈αway␈αto

␈↓ ↓*␈↓ensure␈αthat␈αthis␈α
␈↓βwill␈↓␈αoccur␈αis␈α
to␈αinsert␈αx␈αtwice␈α
into␈αthe␈α list␈α
or␈αbag␈αbefore␈αyou␈α
try␈αto␈αapply␈α
this␈αcomposition.

␈↓ ↓*␈↓The␈α∂conjecture␈α∞thus␈α∂arrived␈α∞at␈α∂is:␈α∞Member*Delete*Insert*Insert(S)␈α∂contains␈α∞T␈α∂i≥␈α∞S␈α∂is␈α∞a␈α∂bag␈α∞or␈α∂a␈α∂list,␈α∞and

␈↓ ↓*␈↓otherwise (for sets and osets) it is {False}.  The user may name this property "duplicity".


␈↓ ↓*␈↓The␈αthird␈αexample␈αof␈αa␈αcomposition␈αwe␈αshall␈αdeal␈αwith␈αexplicitly␈αhere␈αis␈αthe␈αawful␈αone␈αAssign*Never.␈α First

␈↓ ↓*␈↓we␈α∂take␈α∞an␈α∂unquanti≡ed␈α∂proposition␈α∞P,␈α∂then␈α∂turn␈α∞out␈α∂ordered␈α∞pairs␈α∂(x,␈α∂␈↓¬∀x.¬␈↓P(x))␈α∞for␈α∂all␈α∂possible␈α∞variable

␈↓ ↓*␈↓names␈α
x.␈α∞For␈α
each␈α
such␈α∞pair␈α
(x,Q),␈α
we␈α∞then␈α
assign␈α∞to␈α
the␈α
variable␈α∞x␈α
the␈α
value␈α∞Q.␈α
 This␈α∞bizarre␈α
operation

␈↓ ↓*␈↓thus␈αcould␈αmap␈α
P="x␈αis␈αin␈α
S"␈αto␈αthe␈α
situation␈αwhere␈αy␈α
had␈αthe␈αvalue␈α
"for␈αall␈αy,␈α
x␈αis␈αnever␈α
in␈αS",␈αwhere␈αx␈α
had

␈↓ ↓*␈↓the␈α
value␈α
"␈↓¬∀x.¬xεS␈↓",␈α
etc.␈α∞After␈α
much␈α
groping,␈α
this␈α
might␈α∞lead␈α
to␈α
distinguishing␈α
the␈α
positive␈α∞quanti≡ers␈α
(␈↓¬∀,∃␈↓)

␈↓ ↓*␈↓from␈α
the␈α
negative␈α
quanti≡ers␈α
(never␈α
and␈α
not␈α
always).␈α
 The␈α
intuitions␈α
will␈α
positively␈α
rebel␈α
at␈α
this␈αunnatural

␈↓ ↓*␈↓composition,␈αand␈α
the␈αBEING␈αwill␈α
be␈αallowed␈α
to␈αdie␈αsoon.␈α
All␈αthe␈α
other␈αcompositions␈αof␈α
the␈αform␈α
"Assign␈αo

␈↓ ↓*␈↓<quantify>"␈αwill␈αbe␈αnoticed␈αas␈αsimilar␈αto␈αthis␈αdismal␈αfailure,␈αhence␈αthe␈αsystem␈αwill␈αnot␈αeven␈αwaste␈αthis␈αmuch

␈↓ ↓*␈↓time␈αon␈α
them.␈α Notice␈α
that␈αthere␈α
is␈αnothing␈αwrong␈α
with␈αglancing␈α
at␈αthis␈α
horrible␈αcomposition.␈α
 It␈αis␈αonly␈α
upon

␈↓ ↓*␈↓examination␈α
that␈αthe␈α
intuitions␈αare␈α
asked␈α
to␈αripple␈α
outward,␈αhopefully␈α
towards␈α
each␈αother,␈α
until␈αthey␈α
intersect

␈↓ ↓*␈↓in␈αsome␈αimage.␈αIn␈α
this␈αcase,␈αthey␈αdon't␈αmeet␈α
in␈αany␈αreasonable␈αtime,␈α
which␈αis␈αthe␈αcomputational␈αequivalent␈α
of

␈↓ ↓*␈↓saying␈α
that␈α
their␈α
composition␈α
is␈α
aesthetically␈α
disgusting.␈α
The␈α
only␈α
"stupidity"␈α
would␈α
be␈α
to␈α
notice␈α
this␈α
mismatch

␈↓ ↓*␈↓and ignore it;  the common brand of "ignorance" would be never to uncover this intuitive incompatibility.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.5.4. Proving a Conjecture␈↓)αβ␈↓


␈↓ ↓*␈↓Let␈α⊃us␈α⊃discuss␈α⊃the␈α⊃examination␈α⊃of␈α⊃the␈α⊃intuitively␈α⊃and␈α⊃empirically␈α⊃justi≡ed␈α⊃conjecture␈α⊃≡rst␈α∩mentioned␈α⊃in

␈↓ ↓*␈↓example 4, on the last page: For any non-null structure S, S is one  result of Insert*Delete(S).


␈↓ ↓*␈↓The␈αTest␈αBEING␈αgrabs␈αcontrol,␈αand␈αsince␈αthe␈αconjecture␈αis␈αbelieved␈α(intuitively␈αand␈αempirically)␈αhe␈αcalls␈αon

␈↓ ↓*␈↓Prove.␈αThe␈α≡rst␈αaction␈αunder␈αProve.Algorithms␈αis␈αto␈αclarify␈αthe␈αexisting␈αinformal␈αjusti≡cation.␈α The␈α
intuition

␈↓ ↓*␈↓was␈α
to␈α
break␈α
o≥␈α
a␈α
piece␈α
x␈α
from␈αS␈α
and␈α
then␈α
glue␈α
it␈α
back␈α
into␈α
the␈αsame␈α
place;␈α
to␈α
pull␈α
a␈α
thing␈α
out␈α
of␈αa␈α
container
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  66␈↓

␈↓ ↓*␈↓and␈αthen␈αthrow␈αit␈αback␈αin.␈α The␈αexamples␈α include␈αmany␈αof␈αeach␈αknown␈αtype␈αof␈αstructure.␈α Prove␈αthen␈αasks

␈↓ ↓*␈↓which␈αtype␈αof␈αproving␈αseems␈αmost␈αrelevant.␈αProving-␈↓¬∀␈↓'s␈αis␈αeager,␈αand␈αnarrowly␈αwins␈αover␈αCases␈αand␈αNatural

␈↓ ↓*␈↓Deduction.␈α∃This␈α∃is␈α∀to␈α∃no␈α∃avail,␈α∃for␈α∀Proving-␈↓¬∀␈↓'s␈α∃immediately␈α∃asks␈α∀about␈α∃Cases␈α∃of␈α∃structures␈α∀anyway.

␈↓ ↓*␈↓Structures.Specializations informs him that the types are Hist, List, Oset, Bag, Set.


␈↓ ↓*␈↓Before␈αworking␈α
on␈αseparate␈α
case␈αproofs,␈α
as␈αmuch␈αgeneral-purpose␈α
proof␈αas␈α
possible␈αshould␈α
be␈α≡rmed␈αup.␈α
The

␈↓ ↓*␈↓intuition␈αis␈α
asked␈αto␈αnotice␈α
features␈αabout␈αthe␈α
element␈αwhich␈αis␈α
deleted␈αin␈αthe␈α
successful␈αcases.␈αIt␈α
infers␈αthat

␈↓ ↓*␈↓the␈αelement␈αwas␈αalways␈αa␈αmember␈αof␈αthe␈αstructure;␈αit␈αprobably␈αdoesn't␈αnotice␈αthat␈αit␈αwas␈αthe␈α≡rst␈αmember␈αin

␈↓ ↓*␈↓the␈α
ordered␈α
case␈α
(lists␈α
and␈α
osets␈α
and␈α
hists)␈α
and␈α
any␈α
member␈α
in␈α
the␈α
unordered␈α
case␈α
(sets␈α
and␈α
bags).␈αThe␈α
second

␈↓ ↓*␈↓step,␈αthat␈αof␈αinsertion,␈αbrings␈α
you␈αback␈αto␈αthe␈αoriginal␈α
structure␈αif␈αyou␈αthen␈αglue␈α
it␈αback␈αin␈αprecisely␈αthe␈α
place

␈↓ ↓*␈↓you␈α∂took␈α∂it␈α∞from␈α∂(ordered)␈α∂or␈α∞anywhere␈α∂(non-ordered).␈α∂ The␈α∞Insertion␈α∂BEING␈α∂is␈α∞asked␈α∂where␈α∂it␈α∂puts␈α∞the

␈↓ ↓*␈↓element,␈αand␈α
its␈αintuition␈αreplies␈α
that␈αit␈αgoes␈α
at␈αthe␈α
front␈αof␈αthe␈α
structure,␈αso␈αthat␈α
Some-member␈αcan␈α
grab␈αit

␈↓ ↓*␈↓next.␈α
 The␈αreasoning␈α
now␈α
is␈αthat␈α
if␈αx␈α
is␈α
placed␈αsuch␈α
that␈α
Some-member␈αgrabs␈α
it␈αnext,␈α
and␈α
it␈αhad␈α
to␈αbe␈α
placed

␈↓ ↓*␈↓where␈αit␈αcame␈αfrom,␈αthen␈αit␈αhad␈αto␈αcome␈αfrom␈αthe␈αplace␈αwhich␈αSome-member␈αwould␈αgrab.␈αThat␈αis,␈αx␈αhad␈αto

␈↓ ↓*␈↓be Some-member of S. The suggested proof is:
␈↓ ↓*␈↓x is Assigned the value Some-member(S).
␈↓ ↓*␈↓S' is Assigned the value Delete(x,S).
␈↓ ↓*␈↓S'' is Assigned the value Insert(x,S').
␈↓ ↓*␈↓Claim S'' = S.
␈↓ ↓*␈↓That is, S = Insert(Some-member(S), Delete(Some-member(S), S)).


␈↓ ↓*␈↓The␈α
general-structure␈α
axioms␈α∞are␈α
insu≠cent␈α
to␈α∞prove␈α
this,␈α
so␈α∞we␈α
≡nally␈α
do␈α∞break␈α
the␈α
conjecture␈α∞into␈α
cases,

␈↓ ↓*␈↓hopefully␈αusing␈αthe␈αsuggested␈αproof␈αas␈αour␈αmodel.␈αFor␈αthe␈αcases␈αof␈αSets␈αand␈αOsets,␈αthis␈αproof␈αworks␈α≡ne.␈αFor

␈↓ ↓*␈↓the␈α∂rest,␈α∂however,␈α⊂the␈α∂duplicity␈α∂simply␈α⊂confuses␈α∂the␈α∂issues␈α⊂and␈α∂leads,␈α∂e.g.,␈α⊂to␈α∂in≡nite␈α∂chains␈α⊂of␈α∂inductions

␈↓ ↓*␈↓which␈αsimply␈αdon't␈αget␈αany␈αeasier.␈αThe␈αcore␈αof␈αthis␈αdilemma␈αis␈αthe␈αneed␈αto␈αcount␈αthe␈αnumber␈αof␈αoccurrences

␈↓ ↓*␈↓of␈αeach␈αelement␈αin␈αa␈αbag␈αor␈αlist,␈αwhich␈αof␈α
course␈αthe␈αsystem␈αcan't␈αdo␈αnow.␈αThe␈αtwo␈αalteraatives␈αare␈α
to␈αdefer

␈↓ ↓*␈↓this␈α∂until␈α⊂later,␈α∂but␈α⊂rely␈α∂on␈α⊂it␈α∂unless␈α⊂proven␈α∂otherwise,␈α∂or␈α⊂to␈α∂add␈α⊂new␈α∂list/bag␈α⊂axiom(s)␈α∂from␈α⊂which␈α∂this

␈↓ ↓*␈↓conjecture␈α∞could␈α
be␈α∞deduced.␈α
 The␈α∞former␈α
is␈α∞preferred,␈α∞since␈α
it␈α∞avoids␈α
interacting␈α∞with␈α
the␈α∞user,␈α∞and␈α
since
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  67␈↓

␈↓ ↓*␈↓Assuming␈α⊂is␈α⊂a␈α⊂way␈α⊂of␈α⊂"copping␈α⊂out".␈α⊂ So␈α⊂we␈α⊂postpone␈α⊂further␈α⊂attempts␈α⊂at␈α⊂proving␈α⊂this␈α⊂until␈α⊂some␈α⊂new,

␈↓ ↓*␈↓powerful␈αknowledge␈αis␈αgained␈αrelevant␈αto␈αbags␈αand␈αlists.␈α A␈αnote␈αis␈αadded␈αin␈αcase␈αany␈αnew␈αbag␈αor␈αlist␈αaxiom

␈↓ ↓*␈↓is␈α∞later␈α∞considered:␈α∞its␈α∞value␈α∞is␈α∞boosted␈α∞if␈α∞it␈α∞also␈α∞helps␈α∞prove␈α∞this␈α∞conjecture␈α∞(in␈α∞addition␈α∞to␈α∞its␈α∞reason␈α∞for

␈↓ ↓*␈↓existing).


␈↓ ↓*␈↓The next example goes into detail about a much more trivial conjecture.


␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.5.5. Formally Investigating an intuitively believed conjecture␈↓)αβ␈↓


␈↓ ↓*␈↓Note:␈αIt␈αis␈αdi≠cult␈αto␈α≡nd␈αhard␈αproofs␈αat␈αthis␈αlow␈αlevel.␈α PHI=0={}=(CLASS␈α)=empty␈αset.␈α Below␈αis␈αwhat␈αthe

␈↓ ↓*␈↓user might see at near-maximal verbosity level.

␈↓ ↓*␈↓(1) Conjecture: The only relation from 0 to any set X is 0.

␈↓ ↓*␈↓Test recognizes: conjecture
␈↓ ↓*␈↓     Intuitive Justi≡cation: Cannot seem to ≡nd any place for any arrow of the reln. to come
␈↓ ↓*␈↓     from (i.e. can't draw arrow because can't choose an ele. from domain because there aren't any)
␈↓ ↓*␈↓     Conclusion: since this is believed, we shall try to prove it, not disprove it.
␈↓ ↓*␈↓     Access: A relation between A and B is a subset of A X B.
␈↓ ↓*␈↓     Access: A X B is the set of all ordered pairs <a,b> such that a ␈↓¬ε␈↓ A and b ␈↓¬ε␈↓ B

␈↓ ↓*␈↓Containment.Iden: To prove Any α is BEING, consider any α, show it's BEING.
␈↓ ↓*␈↓     Consider any relation R: 0 → X.  Show it is 0.  Ask the BEING named PHI how to prove R=PHI.
␈↓ ↓*␈↓     Answer: Show all subsets of 0 x X are 0; alternative: assume z is an ele, derive contradiction.
␈↓ ↓*␈↓     Intuition: All subsets of a set are empty i≥ the set is empty. (Becomes a lemma.)
␈↓ ↓*␈↓     Must show 0 x X = 0 for all sets X. This is intuitive.  (Becomes a lemma.) Done.

␈↓ ↓*␈↓Prove: To prove p i≥ q, prove p implies q and q implies p.  To prove
␈↓ ↓*␈↓     p implies q, assume p and the negation of q, and derive a contradiction.

␈↓ ↓*␈↓     Now must prove two lemmas, by contradiction:
␈↓ ↓*␈↓     (1) Say X is not empty but all its subsets are.  If X is not empty,
␈↓ ↓*␈↓         there is some x ␈↓¬ε␈↓ X.  If x ␈↓¬ε␈↓ X then {x} ⊂ X. But {x} is not empty. Contradiction.
␈↓ ↓*␈↓         Say X is empty but is has a non-empty subset Y.  If Y is non-
␈↓ ↓*␈↓         empty, there is some y ␈↓¬ε␈↓ Y.  By de≡nition of subset, y ␈↓¬ε␈↓ X.  Contradiction.

␈↓ ↓*␈↓     (2) Access the de≡nition of Cross-product to interpret 0xX. Any member z is of the
␈↓ ↓*␈↓         form (a,b), with a in 0 and b in X. But a can never be in 0. Contradiction.

␈↓ ↓*␈↓Popping up, we discover that (1) is now proved.

␈↓ ↓*␈↓Try to prove the converse of (1). Analogy with last proof (this will actually work)
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  68␈↓

␈↓ ↓*␈↓␈↓ ↓:␈↓↓␈↓&12.5.6. Other potentially productive examples␈↓)αβ␈↓

␈↓ ↓*␈↓The following might be interesting to actually simulate by hand before programming:

␈↓ ↓*␈↓Discovering and developing a particular analogy.
␈↓ ↓*␈↓Discovering and developing the idea of the transpose of a relation.
␈↓ ↓*␈↓Working out a complicated inductive proof.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  69␈↓

␈↓ ↓*␈↓␈↓ βC␈↓∧␈↓&13. Parameters: Size / Timetable / Results so far␈↓)αβ␈↓

␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&13.1. Parameters Characterizing the Magnitude of AM␈↓)αβ␈↓

␈↓ ↓*␈↓     NUMBER␈↓ ¬ZInitially␈↓ εjUltimately (but recall: there is no set goal)
␈↓ ↓*␈↓Number of Families of BEINGs␈↓ ¬Z  5␈↓ εj  5
␈↓ ↓*␈↓Number of BEINGs per family␈↓ ¬Z 30␈↓ εj100
␈↓ ↓*␈↓Number of Parts per BEING␈↓ ¬Z 25␈↓ εj 25
␈↓ ↓*␈↓Size of each part (lines of LISP)␈↓ ¬Z  5␈↓ εj  7
␈↓ ↓*␈↓Number of parts ≡lled in␈↓ ¬Z  8␈↓ εj 20
␈↓ ↓*␈↓Size of avg. BEING (LISP lines)␈↓ ¬Z 40␈↓ εj140
␈↓ ↓*␈↓Total number of BEINGs␈↓ ¬Z150␈↓ εj500
␈↓ ↓*␈↓Core used by AM␈↓ ¬Z150K␈↓ εj400K
␈↓ ↓*␈↓Time per int. result (cpu min)␈↓ ¬Z 15␈↓ εj 15
␈↓ ↓*␈↓CPU time used (hours)␈↓ ¬Z  0␈↓ εj 50
␈↓ ↓*␈↓Human time used (person-hours)␈↓ ¬Z300␈↓ εj1500
␈↓ ↓*␈↓Dissertations completed␈↓ ¬Z  0␈↓ εj  1


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&13.2. Timetable for the AM Project␈↓)αβ␈↓


␈↓ ↓*␈↓(i).␈α∞Codify␈α∂the␈α∞necessary␈α∞core␈α∂of␈α∞initial␈α∞knowledge␈α∂(facts␈α∞and␈α∞the␈α∂wisdom␈α∞to␈α∞employ␈α∂them).␈α∞ ␈↓εReality:␈α∂See␈α∞Given

␈↓ ↓*␈↓εKnowledge, as presented in a separate volume. Completed in December, 1974.␈↓


␈↓ ↓*␈↓(ii).␈α∂Formulate␈α∂a␈α∂su≠cient␈α∂set␈α⊂of␈α∂new␈α∂ideas,␈α∂design␈α∂decisions,␈α⊂and␈α∂intuitive␈α∂assumptions␈α∂to␈α∂make␈α⊂the␈α∂task

␈↓ ↓*␈↓meaninful and feasable.  ␈↓εReality: firmed up in January, 1975.␈↓


␈↓ ↓*␈↓(iii).␈αUse␈αthese␈αideas␈αto␈αrepresent␈αthe␈αcore␈αknowledge␈αof␈αmathematics␈αcollected␈αin␈α(i),␈αin␈αa␈αconcrete,␈αsimulated

␈↓ ↓*␈↓system.␈α ␈↓εReality:␈αthe␈αcurrent␈αversion␈αof␈αGiven␈αKnowledge␈αcasts␈αthis␈αinto␈αthe␈αconcept-family␈αformat.␈α Hand-simulations␈αdone␈αduring␈αFebruary,

␈↓ ↓*␈↓εMarch, and April, 1975, with this "paper" system.␈↓


␈↓ ↓*␈↓(iv). Implement a realization of AM as a  computer program.  ␈↓εReality: Under way May, 1975.␈↓


␈↓ ↓*␈↓(v).␈αDebug␈αand␈αrun␈αthe␈αsystem.␈αAdd␈αthe␈αnatural␈αlanguage␈αabilities␈αgradually,␈αas␈αneeded.␈α ␈↓εReality:␈αJune␈αto␈α
November

␈↓ ↓*␈↓εof 1975. First interesting results obtained in November.␈↓


␈↓ ↓*␈↓(vi).␈α
Analyze␈α∞the␈α
results␈α∞obtained␈α
from␈α∞AM,␈α
with␈α∞an␈α
eye␈α∞toward:␈α
overall␈α∞feasability␈α
of␈α∞automating␈α
creative
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  70␈↓

␈↓ ↓*␈↓mathematical␈αdiscovery␈αand␈αtheory␈αdevelopment;␈αadequacy␈αof␈αthe␈αinitial␈αcore␈αof␈αknowledge;␈αadequacy␈αof␈αthe

␈↓ ↓*␈↓ideas,␈α
design␈α
decisions,␈α
implementation␈α
details,␈α
and␈α
theoretical␈α
assumptions.␈α
 Use␈α
the␈α
results␈α
to␈α∞improve␈α
the

␈↓ ↓*␈↓system;␈α
when␈α
"adequate,"␈α
 forge␈α
ahead␈α
as␈α
far␈α
as␈α
possible␈α
into␈α
as␈α
many␈α
domains␈α
as␈α
possible,␈α
then␈α
reanalyze.

␈↓ ↓*␈↓␈↓εReality: the (v)↔(vi) cycle will terminate about April, 1976.␈↓


␈↓ ↓*␈↓(vii). Experiment with AM. ␈↓εReality: scheduled to begin in January, 1975.␈↓


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&13.3. Results as of 5/14/75␈↓)αβ␈↓


␈↓ ↓*␈↓The␈α
control␈α∞structure␈α
has␈α
been␈α∞written␈α
and␈α∞debugged;␈α
it␈α
involves␈α∞a␈α
priority␈α
queue␈α∞of␈α
candidates␈α∞for␈α
AM's

␈↓ ↓*␈↓attention.␈α∞The␈α
size␈α∞is␈α
about␈α∞5␈α
pages␈α∞of␈α∞top-level␈α
routines(␈α∞e.g.,␈α
Find-new-candidates),␈α∞and␈α
5␈α∞pages␈α∞of␈α
utility

␈↓ ↓*␈↓function␈α∂(e.g.,␈α∂Dump-new-Beings-≡le).␈α∂ A␈α∂few␈α∂BEINGs␈α∂have␈α∂been␈α∂partially␈α∂coded␈α∂and␈α∂introduced␈α∞(notably

␈↓ ↓*␈↓AnyBeing.Examples␈α∪and␈α∪also␈α∪Set-structure).␈α∩The␈α∪system␈α∪decided␈α∪that␈α∪it␈α∩would␈α∪be␈α∪interesting␈α∪to␈α∪≡ll␈α∩in

␈↓ ↓*␈↓examples␈α
of␈α
sets,␈αand␈α
came␈α
up␈α
with␈αthe␈α
sets:␈α
(CLASS),␈α
(CLASS␈αDOUG␈α
AVRA␈α
CORDELL␈α
ED),␈α(CLASS

␈↓ ↓*␈↓(CLASS)),␈α⊂and␈α⊃(CLASS␈α⊂R3-5␈α⊃R3-6␈α⊂R4-5␈α⊂R4-6␈α⊃R5-5␈α⊂R5-6).␈α⊃These␈α⊂correspond␈α⊂to␈α⊃the␈α⊂empty␈α⊃set,␈α⊂a␈α⊃set␈α⊂of

␈↓ ↓*␈↓usernames,␈αthe␈α
set␈αcontaining␈α
the␈αempty␈α
set,␈αand␈αa␈α
2x3␈αrectangle␈α
of␈αintegral␈α
lattice␈αpoints.␈α
 This␈αtook␈αa␈α
couple

␈↓ ↓*␈↓cpu␈αseconds.␈αAfterwards,␈α
AM␈αcouldn't␈α≡nd␈α
anything␈αinteresting␈αto␈α
do,␈αand␈αcollapsed.␈α
The␈αcurrent␈αtask␈αis␈α
now

␈↓ ↓*␈↓to␈α
encode␈α
and␈α
feed␈α
in␈α
the␈α
BEINGs␈α
involving␈α
Relation,␈α
instances␈α
of␈α
Relations,␈α
Composition;␈α
 that␈α
might␈αbe

␈↓ ↓*␈↓enough␈αto␈αallow␈αAM␈αto␈α"take␈αo≥"␈α--␈αto␈αnever␈αrun␈αout␈αof␈αthings␈αto␈αdo,␈αalthough␈αthe␈αinterest␈αlevel␈αof␈αconcepts

␈↓ ↓*␈↓built just on these few basic ones cannot be very good.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&13.4. Results as of 11/7/75␈↓)αβ␈↓


␈↓ ↓*␈↓The␈α⊃control␈α⊃structure␈α∩and␈α⊃the␈α⊃Beings␈α⊃themselves␈α∩have␈α⊃been␈α⊃rewritten.␈α⊃ The␈α∩system␈α⊃now␈α⊃is␈α∩much␈α⊃more

␈↓ ↓*␈↓uniform.␈α⊗The␈α⊗concepts␈α⊗of␈α⊗Interesting-Set,␈α⊗Interesting-Composition,␈α⊗Coalesed␈α⊗operations,␈α⊗etc.␈α⊗are␈α∃being

␈↓ ↓*␈↓explored␈αby␈α
AM.␈αExamples␈α
of␈αstructures␈α
and␈αof␈α
each␈αnewly␈α
de≡ned␈αoperation␈α
are␈α≡lled␈α
in␈αroutinely.␈αNo␈α
large

␈↓ ↓*␈↓e≥ort␈α
vis-a-vis␈αintuition␈α
has␈αbeen␈α
begun␈αyet,␈α
nor␈αhas␈α
proving␈αeven␈α
been␈αconsidered.␈α
The␈αonly␈α
proposing␈αin
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  71␈↓

␈↓ ↓*␈↓fact␈α
have␈α
been␈α
trivial␈α
aspects␈α
of␈α
new␈α
concepts.␈α
However,␈α
as␈α
we␈α
saw,␈α
the␈α
UFT␈α
can␈α
be␈α
developed␈α
in␈α
this␈αway,␈α
to

␈↓ ↓*␈↓the␈α∞point␈α∞of␈α∞stating␈α∞as␈α∞"this␈α∞certain␈α∞Being␈α∞is␈α∞a␈α∞function".␈α∞The␈α∞concept␈α∞of␈α∞Cardinality␈α∞has␈α∞been␈α∞discovered

␈↓ ↓*␈↓through␈αthe␈αgeneralization␈αof␈αa␈αrecursive␈αde≡nition␈αfor␈αequality␈αof␈αtwo␈αsets,␈αbut␈αAM␈αhas␈αnot␈αfully␈α
recognized

␈↓ ↓*␈↓its lasting importance.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  72␈↓

␈↓ ↓*␈↓␈↓ ε∧␈↓∧␈↓&Bibliography␈↓)αβ␈↓


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&Comparison to Other Systems␈↓)αβ␈↓


␈↓ ↓*␈↓One␈α∞popular␈α∞way␈α∞to␈α∞explicate␈α∂a␈α∞system's␈α∞design␈α∞ideas␈α∞is␈α∞to␈α∂compare␈α∞it␈α∞to␈α∞other,␈α∞similar␈α∞systems,␈α∂and/or␈α∞to

␈↓ ↓*␈↓others'␈αproposed␈αcriteria␈αfor␈αsuch␈αsystems.␈αThere␈αis␈αvirtually␈αno␈αsimilar␈αproject␈αknown␈αto␈αthe␈αauthor,␈αdespite

␈↓ ↓*␈↓an␈α∪exhaustive␈α∩search␈α∪(see␈α∩Bibliography).␈α∪A␈α∩couple␈α∪tangential␈α∩e≥orts␈α∪will␈α∩be␈α∪mentioned,␈α∩followed␈α∪by␈α∩a

␈↓ ↓*␈↓discussion␈α
of␈α∞how␈α
AM␈α∞will␈α
measure␈α
up␈α∞to␈α
the␈α∞understanding␈α
standards␈α
set␈α∞forth␈α
by␈α∞Moore␈α
and␈α∞Newell␈α
in

␈↓ ↓*␈↓their␈α∞MERLIN␈α∞paper.␈α∂ Next␈α∞comes␈α∞a␈α∂listing␈α∞of␈α∞the␈α∞books␈α∂which␈α∞were␈α∞read,␈α∂and␈α∞≡nally␈α∞a␈α∂bibliography␈α∞of

␈↓ ↓*␈↓relevant articles.


␈↓ ↓*␈↓Several␈α⊂projects␈α⊂have␈α⊂been␈α⊂undertaken␈α⊂which␈α⊂comprise␈α⊂a␈α⊂small␈α⊂piece␈α⊂of␈α⊂the␈α⊂proposed␈α⊂system,␈α⊂plus␈α⊂deep

␈↓ ↓*␈↓concentration␈αon␈αsome␈αarea␈α␈↓βnot␈↓␈αunder␈αstudy␈αhere.␈αFor␈αexample,␈αBoyer␈αand␈αMoore's␈αtheorem-prover␈αembodies

␈↓ ↓*␈↓some␈α
of␈α
the␈α
spirit␈αof␈α
this␈α
e≥ort,␈α
but␈α
its␈αknowledge␈α
base␈α
is␈α
minimal␈αand␈α
its␈α
methods␈α
purely␈α
formal.␈α Badre's

␈↓ ↓*␈↓CLET␈α
system␈αworked␈α
on␈αlearning␈α
 the␈αdecimal␈α
addition␈αalgorithm␈α
␈↓	56␈↓␈αbut␈α
the␈α␈↓βmathematics␈α
discovery␈↓␈αaspects␈α
of

␈↓ ↓*␈↓the␈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

␈↓ ↓*␈↓communication␈α⊃study.␈α⊃The␈α⊃same␈α⊂comment␈α⊃applies␈α⊃to␈α⊃several␈α⊂related␈α⊃studies␈α⊃by␈α⊃IMSSS␈↓	57␈↓.␈α⊃ Gelernter␈α⊂has

␈↓ ↓*␈↓worked␈αon␈αusing␈αprototypical␈αexamples␈αas␈αanalogic␈αmodels␈αto␈αguide␈αsearch␈αin␈αgeometry,␈αand␈αBundy␈αhas␈αused

␈↓ ↓*␈↓"sticks"␈α
to␈αhelp␈α
his␈αprogram␈α
work␈αwith␈α
natural␈αnumbers.␈α
 Kling␈αhas␈α
studied␈αthe␈α
single␈αheuristic␈α
of␈αanalogy,

␈↓ ↓*␈↓and␈α⊂Brotz␈α⊂has␈α∂written␈α⊂a␈α⊂system␈α⊂which␈α∂uses␈α⊂this␈α⊂to␈α⊂propose␈α∂useful␈α⊂lemmata;␈α⊂both␈α⊂of␈α∂these␈α⊂are␈α⊂set␈α⊂up␈α∂as

␈↓ ↓*␈↓theorem␈α∞provers,␈α
again␈α∞not␈α
as␈α∞discoverers.␈α∞ One␈α
aspect␈α∞that␈α
each␈α∞of␈α∞these␈α
systems␈α∞lacked␈α
was␈α∞size:␈α∞they␈α
all

␈↓ ↓*␈↓worked␈α∪in␈α∪tiny␈α∪toy␈α∩domains,␈α∪with␈α∪miniscule,␈α∪carefully␈α∩prearranged␈α∪knowledge␈α∪bases,␈α∪with␈α∪just␈α∩enough

␈↓ ↓*␈↓information␈α
to␈αdo␈α
the␈αjob␈α
well,␈αbut␈α
not␈αso␈α
much␈αthat␈α
the␈αsystem␈α
might␈αbe␈α
swamped.␈αAM␈α
is␈αopen␈α
to␈α
all␈αthe

␈↓ ↓*␈↓advantages␈α∞and␈α
all␈α∞the␈α
dangers␈α∞of␈α
a␈α∞non-toy␈α
system␈α∞with␈α
a␈α∞massive␈α
corpus␈α∞of␈α
data␈α∞to␈α
manage.␈α∞ The␈α
other


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	56␈↓ε  Given the addition table up to 10 + 10, plus an English text description of what it means to carry, how and when to carry, etc.
␈↓ ↓*␈↓ε␈↓	57␈↓ε See [Smith], for example.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  73␈↓

␈↓ ↓*␈↓systems␈α∞did␈α∂not␈α∞deal␈α∞with␈α∂intuition,␈α∞or␈α∞indeed␈α∂any␈α∞multiple␈α∞knowlege␈α∂source␈α∞(except␈α∞examples␈α∂or␈α∞syntactic

␈↓ ↓*␈↓analogy).␈α∞Certainly␈α∞none␈α∞has␈α∞considered␈α∞the␈α∞paradigm␈α∞of␈α∞␈↓βdiscovery␈α∞and␈α∞evaluation␈α∞of␈α∞the␈α∞interestingness␈α∞of

␈↓ ↓*␈↓βstructure␈↓;␈αthe␈αothers␈αhave␈αbeen␈α"here␈αis␈αyour␈αtask,␈αtry␈αand␈αprove␈αit,"␈α or,␈αin␈αBadre's␈αcase,␈α"here␈αis␈αthe␈αanswer,

␈↓ ↓*␈↓try and translate/use it."


␈↓ ↓*␈↓There␈α
is␈α∞very␈α
little␈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

␈↓ ↓*␈↓thinkers␈α∞like␈α∞Polya␈α∂and␈α∞Poincare'␈α∞treat␈α∞mathematical␈α∂ability␈α∞as␈α∞a␈α∞sacred,␈α∂almost␈α∞mystic␈α∞quality,␈α∞tied␈α∂to␈α∞the

␈↓ ↓*␈↓unconscious.␈α⊗ The␈α⊗writings␈α⊗of␈α⊗philosophers␈α⊗and␈α⊗psychologists␈α⊗invariably␈α⊗attempt␈α⊗to␈α⊗examine␈α⊗human

␈↓ ↓*␈↓performance␈α∩and␈α∩belief,␈α∪which␈α∩are␈α∩far␈α∪ more␈α∩manageable␈α∩than␈α∩creativity␈α∪in␈α∩vitro.␈α∩ Belief␈α∪formulae␈α∩in

␈↓ ↓*␈↓inductive␈α∞logic␈α∞(eg.,␈α∞Carnap,␈α∞Pietarinin)␈α∞invariably␈α∞fall␈α
back␈α∞upon␈α∞how␈α∞well␈α∞they␈α∞≡t␈α∞human␈α
measurements.

␈↓ ↓*␈↓The␈α∞abilities␈α∞of␈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

␈↓ ↓*␈↓algorithms!) one possesses which match those of the other.


␈↓ ↓*␈↓In␈αan␈αearlier␈αsection␈αwe␈α
discussed␈αcriteria␈αfor␈αthe␈αsystem.␈α
 Two␈αimportant␈αcriteria␈αare␈α≡nal␈α
performance␈αand

␈↓ ↓*␈↓initial␈αstarting␈αpoint.␈α
 That␈αis,␈αwhat␈α
is␈αit␈αgiven␈α
(including␈αthe␈αknowledge␈α
in␈αthe␈αprogram␈α
environment),␈αand

␈↓ ↓*␈↓what␈αdoes␈αAM␈αdo␈αwith␈αthat␈αinformation?␈α Moore␈αand␈αNewell␈αhave␈αpublished␈αsome␈αreasonable␈αdesign␈αissues

␈↓ ↓*␈↓for any proposed understanding system, and we shall now see how our system answers their questions␈↓	58␈↓.

␈↓ ↓*␈↓¬Representation: Families of BEINGs, simple situation/rules, opaque functions.
␈↓ ↓*␈↓¬        Scope: Each family of BEINGs characterizes one type of knowledge.
␈↓ ↓*␈↓¬                        Each BEING represents one very specialized expert.
␈↓ ↓*␈↓¬                        The opaque functions can represent intuition and the real world.
␈↓ ↓*␈↓¬        Grain: Partial knowledge about a topic X is naturally expressed as an incomplete BEING X.
␈↓ ↓*␈↓¬        Multiple representations: Each differently-named part has its own format, so, e.g.,
␈↓ ↓*␈↓¬                examples of an operation can be stored as i/o pairs, the intuition points to an
␈↓ ↓*␈↓¬                opaque function, the recognition section is sit/action productions, the
␈↓ ↓*␈↓¬                algorithms part is a quasi-executable partially-ordered list of things to try.
␈↓ ↓*␈↓¬Action: Most knowledge is stored in BEING-parts in a nearly-executable way; the remainder is
␈↓ ↓*␈↓¬        stored so that the "active" segment can easily use it as it runs.  The place that
␈↓ ↓*␈↓¬        a piece of information is stored is carefully chosen so that it will be evoked
␈↓ ↓*␈↓¬        in almost all the situations in which it is relevant.  The only real action in the
␈↓ ↓*␈↓¬        system is the selective completion of BEINGs parts (occasionally creating a new BEING).
␈↓ ↓*␈↓¬Assimilation: There is no sharp distinction between the internal knowledge and the
␈↓ ↓*␈↓¬        task; the task is really nothing more than to extend the given knowledge while
␈↓ ↓*␈↓¬        maintaining interest and asethetic worth.  The only external entities are the
␈↓ ↓*␈↓¬        user and the simulated physical world. Contact with the first is through a
␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	58␈↓ε Each point of the taxonomy which they provide before these questions is covered by the proposed system.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  74␈↓

␈↓ ↓*␈↓¬        simpleminded translation scheme, with the latter through evaluation of opaque
␈↓ ↓*␈↓¬        functions on observable data and examination of the results.
␈↓ ↓*␈↓¬Accomodation: translation of alien messages; inference from (simulated) real-world examples data.
␈↓ ↓*␈↓¬Directionality: The Environment gathers up the relevant knowledge at each step to fill
␈↓ ↓*␈↓¬        in the currently worked-on part of the current BEING, simply by asking that part
␈↓ ↓*␈↓¬        (its archetypical representative), that BEING, and its Tied BEINGs what to do.
␈↓ ↓*␈↓¬        Keep-progressing: at each stage, there will be hundreds or thousands of unfilled-in
␈↓ ↓*␈↓¬                parts, and the system simply chooses the most interesting one to work on.
␈↓ ↓*␈↓¬Efficiency:
␈↓ ↓*␈↓¬        Interpreter: Will the contents of BEING's parts be compilable, or must they remain
␈↓ ↓*␈↓¬                completely inspectable? One alternative is to provide two versions, one
␈↓ ↓*␈↓¬                fast one for executing and one transparent one for examining.
␈↓ ↓*␈↓¬                Also provide access to a compiler, to recompile any changed (or new) part.
␈↓ ↓*␈↓¬        Immediacy: There need not be close, rapidifire comunication with a human,
␈↓ ↓*␈↓¬                but whenever communicating with him, time ␈↓βwill␈↓¬ be important; thus the
␈↓ ↓*␈↓¬                only requirement on speed is placed upon the translation modules, and
␈↓ ↓*␈↓¬                they are fairly simple (due to the clean nature of the mathematical domain).
␈↓ ↓*␈↓¬        Formality: There is a probabilistic belief rating for everything, and a descriptive
␈↓ ↓*␈↓¬                "Justifications" component for all BEINGs for which it is meaningful.
␈↓ ↓*␈↓¬                There are experts who know about Bugs, Debugging, Contradiction, etc.
␈↓ ↓*␈↓¬                Frame problem: when the world changes, make no effort to update everything.
␈↓ ↓*␈↓¬                        Whenever a contradiction is encountered, study its origins and
␈↓ ↓*␈↓¬                        recompute belief values until it goes away.
␈↓ ↓*␈↓¬Depth of Understanding:  Each BEING is an expert, one of whose duties is to announce his
␈↓ ↓*␈↓¬        own relevance whenever he recognizes it. The specific desire will generally
␈↓ ↓*␈↓¬        indicate which part of the relevant BEING is the one to examine. In case this loses,
␈↓ ↓*␈↓¬        each BEING has a part which (on the basis of how it failed) points to alternatives.
␈↓ ↓*␈↓¬        Access to all implications: The intuitive functions must simulate this ability,
␈↓ ↓*␈↓¬                since they are to be analogic. The BEINGs certainly don't have such access.



␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&Books and Memos␈↓)αβ␈↓


␈↓ ↓*␈↓All␈α
the␈α
references␈α
below␈α
have␈α
actually␈α
been␈α
read␈α
as␈α
background␈α
for␈α
AM.␈α
 They␈α
form␈α
a␈α
large␈α
yet␈α∞far␈α
from

␈↓ ↓*␈↓comprehensive␈α
list␈α∞of␈α
publications␈α∞dealing␈α
with␈α
automated␈α∞theory␈α
formation␈α∞and␈α
with␈α∞how␈α
mathematicians

␈↓ ↓*␈↓do research.␈↓	59␈↓ I actually relied upon those with an "@" sign; the others proved to be merely supplementary.


␈↓ ↓*␈↓@Adams, James L., ␈↓βConceptual Blockbusting␈↓, W.H. Freeman and Co., San Francisco, 1974.

␈↓ ↓*␈↓Allendoerfer,␈α
Carl␈α
B.,␈α
and␈α
Oakley,␈α
Cletis␈α
O.,␈α
␈↓βPrinciples␈α
of␈α
Mathematics␈↓,␈α
Third␈α
Edition,␈α
McGraw-Hill,␈αNew
␈↓ ↓*␈↓␈↓ ↓jYork, 1969.

␈↓ ↓*␈↓Alexander, Stephen, ␈↓βOn the Fundamental Principles of Mathematics␈↓, B. L. Hamlen, New Haven, 1849.

␈↓ ↓*␈↓Aschenbrenner, Karl, ␈↓βThe Concepts of Value␈↓, D. Reidel Publishing Company, Dordrecht, Holland, 1971.


␈↓ ↓*␈↓________________________________________________________________________________________
␈↓ ↓*␈↓ε␈↓	59␈↓ε From a single author or project (e.g., DENDRAL), only one or two recent papers will be listed.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  75␈↓

␈↓ ↓*␈↓Atkin,␈αA.␈αO.␈αL.,␈αand␈αBirch,␈αB.␈αJ.,␈αeds.,␈α␈↓βComputers␈αin␈αNumber␈αTheory␈↓,␈αProceedings␈αof␈αthe␈α1969␈αSRCA␈α
Oxford
␈↓ ↓*␈↓␈↓ ↓jSymposium, Academic Press, New York, 1971.

␈↓ ↓*␈↓Avey, Albert E., ␈↓βThe Function and Forms of Thought␈↓, Henry Holt and Company, New York, 1927.

␈↓ ↓*␈↓@Badre,␈α∀Nagib␈α∀A.,␈α∀␈↓βComputer␈α∀Learning␈α∀From␈α∀English␈α∀Text␈↓,␈α∀Memorandum␈α∀No.␈α∀ERL-M372,␈α∪Electronics
␈↓ ↓*␈↓␈↓ ↓jResearch␈α
Laboratory,␈α
UCB,␈α
December␈α
20,␈α
1972.␈α
 Also␈α
summarized␈α
in␈α
␈↓βCLET␈α
--␈α
A␈α
Computer␈α
Program␈α
that
␈↓ ↓*␈↓β␈↓ ↓jLearns Arithmetic from an Elementary Textbook␈↓, IBM Research Report RC 4235, February 21, 1973.

␈↓ ↓*␈↓Bahm, A. J., ␈↓βTypes of Intuition␈↓, University of New Mexico Press, Albuquerque, New Mexico, 1960.

␈↓ ↓*␈↓Banks, J. Houston, ␈↓βElementary-School Mathematics␈↓, Allyn and Bacon, Boston, 1966.

␈↓ ↓*␈↓Berkeley,␈α∞Edmund␈α∂C.,␈α∞␈↓βA␈α∞Guide␈α∂to␈α∞Mathematics␈α∂for␈α∞the␈α∞Intelligent␈α∂Nonmathematician␈↓,␈α∞Simon␈α∂and␈α∞Schuster,
␈↓ ↓*␈↓␈↓ ↓jNew York, 1966.

␈↓ ↓*␈↓Berkeley, Hastings, ␈↓βMysticism in Modern Mathematics␈↓, Oxford U. Press, London, 1910.

␈↓ ↓*␈↓Beth,␈α⊂Evert␈α⊂W.,␈α⊂and␈α⊂Piaget,␈α⊂Jean,␈α⊂␈↓βMathematical␈α⊂Epistemology␈α⊂and␈α⊂Psychology␈↓,␈α⊂Gordon␈α⊂and␈α⊂Breach,␈α∂New
␈↓ ↓*␈↓␈↓ ↓jYork, 1966.

␈↓ ↓*␈↓Black, Max, ␈↓βMargins of Precision␈↓, Cornell University Press, Ithaca, New York, 1970.

␈↓ ↓*␈↓Blackburn, Simon, ␈↓βReason and Prediction␈↓, Cambridge University Press, Cambridge, 1973.

␈↓ ↓*␈↓Bongard, ␈↓βPattern Recognition␈↓, USSR

␈↓ ↓*␈↓@Brotz,␈α∃Douglas␈α∀K.,␈α∃␈↓βEmbedding␈α∀Heuristic␈α∃Problem␈α∀Solving␈α∃Methods␈α∀in␈α∃a␈α∀Mechanical␈α∃Theorem␈α∀Prover␈↓,
␈↓ ↓*␈↓␈↓ ↓jdissertation published as Stanford Computer Science Report STAN-CS-74-443, AUgust, 1974.

␈↓ ↓*␈↓Bruner,␈α
Jerome␈α
S.,␈α
Goodnow,␈α
J.␈α
J.,␈α
and␈α
Austin,␈α
G.␈α
A.,␈α
␈↓βA␈α
Study␈α
of␈α
Thinking␈↓,␈α
Harvard␈α
Cognition␈α
Project,␈α
John
␈↓ ↓*␈↓␈↓ ↓jWiley & Sons, New York, 1956.

␈↓ ↓*␈↓Charosh, Mannis, ␈↓βMathematical Challenges␈↓, NCTM, Wahington, D.C., 1965.

␈↓ ↓*␈↓Cohen, Paul J., ␈↓βSet Theory and the Continuum Hypothesis␈↓,  W.A.Benjamin, Inc., New York, 1966.

␈↓ ↓*␈↓Copeland, Richard W., ␈↓βHow Children Learn Mathematics␈↓, The MacMillan Company, London, 1970.

␈↓ ↓*␈↓Courant, Richard, and Robins, Herbert, ␈↓βWhat is Mathematics␈↓, Oxford University Press, New York, 1941.

␈↓ ↓*␈↓D'Augustine,␈αCharles,␈α
␈↓βMultiple␈αMethods␈α
of␈αTeaching␈α
Mathematics␈αin␈α
the␈αElementary␈α
School␈↓,␈αHarper␈α
&␈αRow,
␈↓ ↓*␈↓␈↓ ↓jNew York, 1968.

␈↓ ↓*␈↓Dodge, Clayton W., ␈↓βSets, Logic, and Numbers␈↓, Prindle, Weber & Schmidt, Inc., Boston, 1969.

␈↓ ↓*␈↓Dornbusch, Sanford, and Scott, ␈↓βEvaluation and the Exercise of Authority␈↓, Jossey-Bass, San Francisco, 1975.

␈↓ ↓*␈↓Douglas, Mary (ed.), ␈↓βRules and Meanings␈↓, Penguin Education, Baltimore, Md., 1973.

␈↓ ↓*␈↓Dowdy, S. M., ␈↓βMathematics: Art and Science␈↓, John Wiley & Sons, NY, 1971.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  76␈↓

␈↓ ↓*␈↓Dubin, Robert, ␈↓βTheory Building␈↓, The Free Press, New York,  1969.

␈↓ ↓*␈↓Dubs, Homer H., ␈↓βRational Induction␈↓, U. of Chicago Press, Chicago, 1930.

␈↓ ↓*␈↓Dudley, Underwood, ␈↓βElementary Number Theory␈↓, W. H. Freeman and Company, San Francisco, 1969.

␈↓ ↓*␈↓Eynden,␈α∪Charles␈α∩Vanden,␈α∪␈↓βNumber␈α∪Theory:␈α∩An␈α∪Introduction␈α∪to␈α∩Proof␈↓,␈α∪International␈α∪Textbook␈α∩Comapny,
␈↓ ↓*␈↓␈↓ ↓jScranton, Pennsylvania, 1970.

␈↓ ↓*␈↓Fuller, R. Buckminster, ␈↓βIntuition␈↓, Doubleday, Garden City, New York, 1972.

␈↓ ↓*␈↓Fuller, R. Buckminster, ␈↓βSynergetics␈↓, ...

␈↓ ↓*␈↓GCMP, ␈↓βKey Topics in Mathematics␈↓, Science Research Associates, Palo Alto, 1965.

␈↓ ↓*␈↓George, F. H., ␈↓βModels of Thinking␈↓, Schenkman Publishing Co., Inc., Cambridge, Mass., 1972.

␈↓ ↓*␈↓Goldstein, Ira, ␈↓βElementary Geometry Theorem Proving␈↓, MIT AI Memo 280, April, 1973.

␈↓ ↓*␈↓Goodstein, R. L., ␈↓βFundamental Concepts of Mathematics␈↓, Pergamon Press, New York, 1962.

␈↓ ↓*␈↓Goodstein, R. L., ␈↓βRecursive Number Theory␈↓, North-Holland Publishing Co., Amsterdam, 1964.

␈↓ ↓*␈↓@Green,␈αWaldinger,␈αBarstow,␈αElschlager,␈αLenat,␈αMcCune,␈αShaw,␈αand␈αSteinberg,␈α␈↓βProgress␈αReport␈αon␈αProgram-
␈↓ ↓*␈↓β␈↓ ↓jUnderstanding␈αSystems␈↓,␈αMemo␈αAIM-240,␈αCS␈αReport␈αSTAN-CS-74-444,Arti≡cial␈αIntelligence␈αLaboratory,
␈↓ ↓*␈↓␈↓ ↓jStanford University, August, 1974.

␈↓ ↓*␈↓@Hadamard,␈αJaques,␈α␈↓βThe␈αPsychology␈αof␈αInvention␈αin␈αthe␈αMathematical␈αField␈↓,␈αDover␈αPublications,␈α
New␈αYork,
␈↓ ↓*␈↓␈↓ ↓j1945.

␈↓ ↓*␈↓Halmos, Paul R., ␈↓βNaive Set Theory␈↓, D. Van Nostrand Co., Princeton, 1960.

␈↓ ↓*␈↓Hanson, Norwood R., ␈↓βPerception and Discovery␈↓, Freeman, Cooper & Co., San Francisco, 1969.

␈↓ ↓*␈↓Hartman,␈α
Robert␈α
S.,␈α␈↓βThe␈α
Structure␈α
of␈αValue:␈α
Foundations␈α
of␈αScienti≡c␈α
Axiology␈↓,␈α
Southern␈α
Illinois␈αUniversity
␈↓ ↓*␈↓␈↓ ↓jPress, Carbondale, Ill., 1967.

␈↓ ↓*␈↓Hempel,␈α∂Carl␈α∂G.,␈α⊂␈↓βFundamentals␈α∂of␈α∂Concept␈α⊂Formation␈α∂in␈α∂Empirical␈α⊂Science␈↓,␈α∂University␈α∂of␈α⊂Chicago␈α∂Press,
␈↓ ↓*␈↓␈↓ ↓jChicago, 1952.

␈↓ ↓*␈↓Hibben, John Grier, ␈↓βInductive Logic␈↓, Charles Scribner's Sons, New York, 1896.

␈↓ ↓*␈↓Hilpinen,␈α
Risto,␈α
␈↓βRules␈α
of␈αAcceptance␈α
and␈α
Inductive␈α
Logic␈↓,␈αActa␈α
Philosophica␈α
Fennica,␈α
Fasc.␈α22,␈α
North-Holland
␈↓ ↓*␈↓␈↓ ↓jPublishing Company, Amsterdam, 1968.

␈↓ ↓*␈↓Hintikka, Jaako, ␈↓βKnowledge and Belief␈↓, Cornell U. Press, Ithaca, NY, 1962.

␈↓ ↓*␈↓Hintikka,␈αJaako,␈αand␈αSuppes,␈αPatrick␈α(eds.),␈α␈↓βAspects␈αof␈αInductive␈αLogic␈↓,␈αNorth-Holland␈αPublishing␈αCompany,
␈↓ ↓*␈↓␈↓ ↓jAmsterdam, 1966.

␈↓ ↓*␈↓Jouvenal, Bertrand de, ␈↓βThe Art of Conjecture␈↓, Basic Books, Inc., New York, 1967.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  77␈↓

␈↓ ↓*␈↓@Kershner,␈α∂R.B.,␈α∂and␈α∂L.R.Wilcox,␈α∂␈↓βThe␈α∂Anatomy␈α∂of␈α∂Mathematics␈↓,␈α∂The␈α∂Ronald␈α∂Press␈α∂Company,␈α⊂New␈α∂York,
␈↓ ↓*␈↓␈↓ ↓j1950.

␈↓ ↓*␈↓Klauder, Francis J., ␈↓βThe Wonder of Intelligence␈↓, Christopher Publishing House, North QUincy, Mass., 1973.

␈↓ ↓*␈↓Klerner,␈α∞M.,␈α
and␈α∞J.␈α
Reinfeld,␈α∞eds.,␈α
␈↓βInteractive␈α∞Systems␈α
for␈α∞Applied␈α
Mathematics␈↓,␈α∞ACM␈α
Symposium,␈α∞held␈α
in
␈↓ ↓*␈↓␈↓ ↓jWashington, D.C., August, 1967. Academic Press, NY, 1968.

␈↓ ↓*␈↓Kline,␈αM.␈α(ed),␈α␈↓βMathematics␈αin␈αthe␈αModern␈αWorld:␈αReadings␈αfrom␈αScienti≡c␈αAmerican␈↓,␈αW.H.Freeman␈αand␈αCo.,
␈↓ ↓*␈↓␈↓ ↓jSan Francisco, 1968.

␈↓ ↓*␈↓@Kling,␈α∞Robert␈α
Elliot,␈α∞␈↓βReasoning␈α
by␈α∞Analogy␈α
with␈α∞Applications␈α
to␈α∞Heuristic␈α
Problem␈α∞Solving:␈α
A␈α∞Case␈α
Study␈↓,
␈↓ ↓*␈↓␈↓ ↓jStanford Arti≡cial Intelligence Project Memo AIM-147, CS Department report CS-216, August, 1971.

␈↓ ↓*␈↓Koestler, Arthur, ␈↓βThe Act of Creation␈↓,  New York, Dell Pub., 1967.

␈↓ ↓*␈↓Korner, Stephan, ␈↓βConceptual Thinking␈↓, Dover Publications, New York, 1959.

␈↓ ↓*␈↓Krivine, Jean-Louis, ␈↓βIntroduction to Axiomatic Set Theory␈↓, Humanities Press, New York, 1971.

␈↓ ↓*␈↓Kubinski,␈α
Tadeusz,␈α∞␈↓βOn␈α
Structurality␈α
of␈α∞Rules␈α
of␈α
Inference␈↓,␈α∞Prace␈α
Wroclawskiego␈α∞Towarzystwa␈α
Naukowego,
␈↓ ↓*␈↓␈↓ ↓jSeria A, Nr. 107, Worclaw, Poland, 1965.

␈↓ ↓*␈↓Lakatos, Imre (ed.), ␈↓βThe Problem of Inductive Logic␈↓, North-Holland Publishing Co., Amsterdam, 1968.

␈↓ ↓*␈↓Lamon, William E., ␈↓βLearning and the Nature of Mathematiccs␈↓, Science Research Associates, Palo Alto, 1972.

␈↓ ↓*␈↓Lang, Serge, ␈↓βAlgebra␈↓, Addison-Wesley, Menlo Park, 1971.

␈↓ ↓*␈↓Lefrancois, Guy R., ␈↓βPsychological Theories and Human Learning␈↓, 1972.

␈↓ ↓*␈↓Le Lionnais, F., ␈↓βGreat Currents of Mathematical Thought␈↓, Dover Publications, New York, 1971.

␈↓ ↓*␈↓Margenau, Henry, ␈↓βIntegrative Principles of Modern Thought␈↓, Gordon and Breach, New York, 1972.

␈↓ ↓*␈↓Martin, James, ␈↓βDesign of Man-Computer Dialogues␈↓, Prentice-Hall, Inc., Englewood Cli≥s, N. J., 1973.

␈↓ ↓*␈↓Martin, R. M., ␈↓βToward a Systematic Pragmatics␈↓, North Holland Publishing Company, Amsterdam, 1959.

␈↓ ↓*␈↓Mendelson, Elliott, ␈↓βIntroduction to Mathematical Logic␈↓, Van Nostrand Reinhold Company, New York, 1964.

␈↓ ↓*␈↓Meyer, Jerome S., ␈↓βFun With Mathematics␈↓, Fawcett Publications, Greenwich, Connecticut, 1952.

␈↓ ↓*␈↓Mirsky, L., ␈↓βStudies in Pure Mathematics␈↓, Academic Press, New York, 1971.

␈↓ ↓*␈↓Moore, Robert C., ␈↓βD-SCRIPT: A Computational Theory of Descriptions␈↓, MIT AI Memo 278, February, 1973.

␈↓ ↓*␈↓Nagel, Ernst, ␈↓βThe Structure of Science␈↓, Harcourt, Brace, & World, Inc., N. Y., 1961.

␈↓ ↓*␈↓National␈α∞Council␈α∞of␈α∞Teachers␈α∞of␈α∞Mathematics,␈α∞␈↓βThe␈α∞Growth␈α∞of␈α∞Mathematical␈α∞Ideas␈↓,␈α∞24th␈α∞yearbook,␈α
NCTM,
␈↓ ↓*␈↓␈↓ ↓jWashington, D.C., 1959.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  78␈↓

␈↓ ↓*␈↓Newell, Allen, and Simon, Herbert, ␈↓βHuman Problem Solving␈↓, 1972.

␈↓ ↓*␈↓Nevins,␈α
Arthur␈α
J.,␈α
␈↓βA␈α∞Human␈α
Oriented␈α
Logic␈α
for␈α
Automatic␈α∞Theorem␈α
Proving␈↓,␈α
MIT␈α
AI␈α
Memo␈α∞268,␈α
October,
␈↓ ↓*␈↓␈↓ ↓j1972.

␈↓ ↓*␈↓Niven,␈α
Ivan,␈α
and␈α
Zuckerman,␈α
Herbert,␈α
␈↓βAn␈α
Introduction␈α
to␈α
the␈α
Theory␈α
of␈α
Numbers␈↓,␈α
John␈α
Wiley␈α
&␈α
Sons,␈αInc.,
␈↓ ↓*␈↓␈↓ ↓jNew York, 1960.

␈↓ ↓*␈↓Olson, Robert G., ␈↓βMeaning and Argument␈↓, Harcourt, Brace & World, New York, 1969.

␈↓ ↓*␈↓Ore, Oystein, ␈↓βNumber Theory and its History␈↓, McGraw-Hill, New York, 1948.

␈↓ ↓*␈↓Parish,␈αCharles,␈α
and␈αRoy␈αMcCormick,␈α
␈↓βA␈αStructral␈αApproach␈α
to␈αArithmetic␈↓,␈αVan␈α
Nostrand␈αReinhold␈αCo.,␈α
N.Y.,
␈↓ ↓*␈↓␈↓ ↓j1970.

␈↓ ↓*␈↓Parker, Francis D., ␈↓βThe Structure of Number Systems␈↓, Prentice-Hall, Inc., Englewood Cli≥s, N.J.,  1966.

␈↓ ↓*␈↓Pietarinen,␈α
Juhani,␈α
␈↓βLawlikeness,␈α
Analogy,␈α
and␈α
Inductive␈α
Logic␈↓,␈α
North-Holland,␈α
Amsterdam,␈α
published␈α
as␈α
v.␈α
26
␈↓ ↓*␈↓␈↓ ↓jof the series Acta Philosophica Fennica (J. Hintikka, ed.), 1972.

␈↓ ↓*␈↓@Poincare',␈α∂Henri,␈α∂␈↓βThe␈α∂Foundations␈α∂of␈α∂Science:␈α⊂Science␈α∂and␈α∂Hypothesis,␈α∂The␈α∂Value␈α∂of␈α∂Science,␈α⊂Science␈α∂and
␈↓ ↓*␈↓β␈↓ ↓jMethod␈↓, The Science Press, New York, 1929.

␈↓ ↓*␈↓@Polya,␈α∞George,␈α∞␈↓βMathematics␈α
and␈α∞Plausible␈α∞Reasoning␈↓,␈α
Princeton␈α∞University␈α∞Press,␈α
Princeton,␈α∞Vol.␈α∞1,␈α
1954;
␈↓ ↓*␈↓␈↓ ↓jVol. 2, 1954.

␈↓ ↓*␈↓@Polya, George, ␈↓βHow To Solve It␈↓, Second Edition, Doubleday Anchor Books, Garden City, New York, 1957.

␈↓ ↓*␈↓@Polya, George, ␈↓βMathematical Discovery␈↓, John Wiley & Sons, New York, Vol. 1, 1962; Vol. 2, 1965.

␈↓ ↓*␈↓Richardson,␈α
Robert␈αP.,␈α
and␈α
Edward␈αH.␈α
Landis,␈α␈↓βFundamental␈α
Conceptions␈α
of␈αModern␈α
Mathematics␈↓,␈αThe␈α
Open
␈↓ ↓*␈↓␈↓ ↓jCourt Publishing Company, Chicago, 1916.

␈↓ ↓*␈↓Rosskopf,␈α∩Ste≥e,␈α∪Taback␈α∩ (eds.),␈α∩␈↓βPiagetian␈α∪Cognitive-Development␈α∩Research␈α∩and␈α∪Mathematical␈α∩Education␈↓,
␈↓ ↓*␈↓␈↓ ↓jNational Council of Teachers of Mathematics, New York, 1971.

␈↓ ↓*␈↓Rulison,␈α∂Je≥,␈α⊂and...␈α∂␈↓βQA4,␈α∂A␈α⊂Procedural␈α∂Frob...␈↓,␈α⊂Technical␈α∂Note...,␈α∂Arti≡cial␈α⊂Intelligence␈α∂Center,␈α⊂SRI,␈α∂Menlo
␈↓ ↓*␈↓␈↓ ↓jPark, California, ..., 1973.

␈↓ ↓*␈↓Saaty,␈α⊃Thomas␈α⊃L.,␈α∩and␈α⊃Weyl,␈α⊃F.␈α∩Joachim␈α⊃(eds.),␈α⊃␈↓βThe␈α∩Spirit␈α⊃and␈α⊃the␈α∩Uses␈α⊃of␈α⊃the␈α∩Mathematical␈α⊃Sciences␈↓,
␈↓ ↓*␈↓␈↓ ↓jMcGraw-Hill Book Company, New York, 1969.

␈↓ ↓*␈↓Schminke,␈αC.␈αW.,␈αand␈αArnold,␈αWilliam␈αR.,␈αeds.,␈α␈↓βMathematics␈αis␈αa␈αVerb␈↓,␈αThe␈αDryden␈αPress,␈αHinsdale,␈αIllinois,
␈↓ ↓*␈↓␈↓ ↓j1971.

␈↓ ↓*␈↓Singh, Jagjit, ␈↓βGreat Ideas of Modern Mathematics␈↓, Dover Publications, New York, 1959.

␈↓ ↓*␈↓@Skemp,␈α∂Richard␈α∂R.,␈α∂␈↓βThe␈α∂Psychology␈α∂of␈α∞Learning␈α∂Mathematics␈↓,␈α∂Penguin␈α∂Books,␈α∂Ltd.,␈α∂Middlesex,␈α∞England,
␈↓ ↓*␈↓␈↓ ↓j1971.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  79␈↓

␈↓ ↓*␈↓Slocum,␈α
Jonathan,␈α
␈↓βThe␈α
Graph-Processing␈α
Language␈α
GROPE␈↓,␈αU.␈α
Texas␈α
at␈α
Austin,␈α
Technical␈α
Report␈αNL-22,
␈↓ ↓*␈↓␈↓ ↓jAugust, 1974.

␈↓ ↓*␈↓Smith,␈α∞Nancy␈α∞Woodland,␈α∂␈↓βA␈α∞Question-Answering␈α∞System␈α∞for␈α∂Elementary␈α∞Mathematics␈↓,␈α∞Stanford␈α∂Institute␈α∞for
␈↓ ↓*␈↓␈↓ ↓jMathematical Studies in the Social Sciences, Technical Report 227, April 19, 1974.

␈↓ ↓*␈↓Smith,␈α∂R.L.,␈α∞Nancy␈α∂Smith,␈α∞and␈α∂F.L.␈α∞Rawson,␈α∂␈↓βCONSTRUCT:␈α∞In␈α∂Search␈α∞of␈α∂a␈α∞Theory␈α∂of␈α∂Meaning␈↓,␈α∞Stanford
␈↓ ↓*␈↓␈↓ ↓jIMSSS Technical Report 238, October 25, 1974.

␈↓ ↓*␈↓Stein,␈α⊂Sherman␈α⊂K.,␈α⊂␈↓βMathematics:␈α∂The␈α⊂Man-Made␈α⊂Universe:␈α⊂An␈α∂Introduction␈α⊂to␈α⊂the␈α⊂Spirit␈α⊂of␈α∂Mathematics␈↓,
␈↓ ↓*␈↓␈↓ ↓jSecond Edition, W. H. Freeman and Company, San Francisco,  1969.

␈↓ ↓*␈↓Stewart, B. M., ␈↓βTheory of Numbers␈↓, The MacMillan Co., New York, 1952.

␈↓ ↓*␈↓Stokes, C. Newton, ␈↓βTeaching the Meanings of Arithmetic␈↓, Appleton-Century-Crofts, New York, 1951.

␈↓ ↓*␈↓Suppes,␈α⊂Patrick,␈α⊂␈↓βA␈α⊃Probabilistic␈α⊂Theory␈α⊂of␈α⊂Causality␈↓,␈α⊃Acta␈α⊂Philosophica␈α⊂Fennica,␈α⊂Fasc.␈α⊃24,␈α⊂North-Holland
␈↓ ↓*␈↓␈↓ ↓jPublishing Company, Amsterdam, 1970.

␈↓ ↓*␈↓Teitelman, Warren, ␈↓βINTERLISP Reference Manual␈↓, XEROX PARC, 1974.

␈↓ ↓*␈↓Tullock, Gordon,  ␈↓βThe Organization of Inquiry␈↓, Duke U. Press, Durham, N. C., 1966.

␈↓ ↓*␈↓Venn, John, ␈↓βThe Principles of Empirical or Inductive Logic␈↓, MacMillan and Co., London, 1889.

␈↓ ↓*␈↓Waismann,␈α∞Friedrich,␈α∞␈↓βIntroduction␈α∞to␈α∞Mathematical␈α∞Thinking␈↓,␈α∞Frederick␈α∞Ungar␈α∞Publishing␈α∞Co.,␈α∞New␈α
York,
␈↓ ↓*␈↓␈↓ ↓j1951.

␈↓ ↓*␈↓Watzlawick,␈αP.,␈αJohn␈αWeakland,␈αand␈αRichard␈αFisch,␈α␈↓βChange:␈αPrinciples␈αof␈αProblem␈αFormulation␈αand␈αProblem
␈↓ ↓*␈↓β␈↓ ↓jResolution␈↓, W.W.Norton & Co., Inc., N.Y. 1974.

␈↓ ↓*␈↓Wickelgren,␈αWayne␈αA.,␈α␈↓βHow␈αto␈αSolve␈αProblems:␈αElements␈αof␈αa␈αTheory␈αof␈αProblems␈αand␈αProblem␈αSolving␈↓,␈αW.␈α
H.
␈↓ ↓*␈↓␈↓ ↓jFreeman and Co., Sanf Francisco, 1974.

␈↓ ↓*␈↓Wilder, Raymond L., ␈↓βEvolution of Mathematical Concepts␈↓, John Wiley & Sons, Inc., NY, 1968.

␈↓ ↓*␈↓Winston, P., (ed.), "New Progress in Arti≡cial Intelligence", ␈↓βMIT AI Lab Memo AI-TR-310␈↓, June, 1974.

␈↓ ↓*␈↓Wittner, George E., ␈↓βThe Structure of Mathematics␈↓, Xerox College Publishing, Lexington, Mass, 1972.

␈↓ ↓*␈↓Wright, Georg H. von, ␈↓βA Treatise on Induction and Probability␈↓, Routledge and Kegan Paul, London, 1951.




␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&Articles␈↓)αβ␈↓

␈↓ ↓*␈↓Amarel,␈α
Saul,␈α␈↓βOn␈α
Representations␈α
of␈αProblems␈α
of␈α
Reasoning␈αabout␈α
Actions␈↓,␈α
Machine␈αIntelligence␈α
3,␈α
1968,␈αpp.
␈↓ ↓*␈↓␈↓ ↓j131-171.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  80␈↓

␈↓ ↓*␈↓Bledsoe,␈α
W.␈α
W.,␈α
␈↓βSplitting␈α∞and␈α
Reduction␈α
Heuristics␈α
in␈α∞Automatic␈α
Theorem␈α
Proving␈↓,␈α
Arti≡cial␈α∞Intelligence␈α
2,
␈↓ ↓*␈↓␈↓ ↓j1971, pp. 55-77.

␈↓ ↓*␈↓Bledsoe and Bruell, Peter, ␈↓βA Man-Machine Theorem-Proving System␈↓, Arti≡cial Intelligence 5, 1974, 51-72.

␈↓ ↓*␈↓Bourbaki,␈α∞Nicholas,␈α
␈↓βThe␈α∞Architechture␈α
of␈α∞Mathematics␈↓,␈α
American␈α∞Mathematics␈α
Monthly,␈α∞v.␈α
57,␈α∞pp.␈α
221-232,
␈↓ ↓*␈↓␈↓ ↓jPublished by the MAA, Albany, NY, 1950.

␈↓ ↓*␈↓@Boyer,␈αRobert␈αS.,␈αand␈αJ.␈αS.␈αMoore,␈α␈↓βProving␈αTheorems␈αabout␈αLISP␈αFunctions␈↓,␈αJACM,␈αV.␈α22,␈αNo.␈α1,␈αJanuary,
␈↓ ↓*␈↓␈↓ ↓j1975, pp. 129-144.

␈↓ ↓*␈↓Bruijn,␈αN.␈αG.␈αde,␈α␈↓βAUTOMATH,␈αa␈αlanguage␈αfor␈α
mathematics␈↓,␈αNotes␈αtaken␈αby␈αBarry␈αFawcett,␈αof␈αLecures␈α
given
␈↓ ↓*␈↓␈↓ ↓jat␈αthe␈αSeminare␈αde␈αmathematiques␈αSuperieurs,␈αUniversity␈αde␈αMontreal,␈αJune,␈α1971.␈αStanford␈αUniversity
␈↓ ↓*␈↓␈↓ ↓jComputer Science Library report number is 005913.

␈↓ ↓*␈↓@Buchanan, Feigenbaum, and Sridharan, ␈↓βHeuristic Theory Formation␈↓, Machine Intelligence 7, 1972, pp. 267-...

␈↓ ↓*␈↓@Bundy, Alan, ␈↓βDoing Arithmetic with Diagrams␈↓, 3rd IJCAI, 1973, pp. 130-138.

␈↓ ↓*␈↓Daalen,␈α
D.␈αT.␈α
van,␈α
␈↓βA␈αDescription␈α
of␈α
AUTOMATH␈αand␈α
some␈α
aspects␈αof␈α
its␈α
language␈αtheory␈↓,␈α
in␈αthe␈α
Proceedings
␈↓ ↓*␈↓␈↓ ↓jof␈α
the␈α
SYmposium␈α∞on␈α
APL,␈α
Paris,␈α
December,␈α∞1973,␈α
P.␈α
Bra≥ort␈α
(ed).␈α∞This␈α
volume␈α
also␈α∞contains␈α
other,
␈↓ ↓*␈↓␈↓ ↓jmore detailed articles on this project, by  Bert Jutting and Ids Zanlevan.

␈↓ ↓*␈↓Engelman,␈α
C.,␈α
␈↓βMATHLAB:␈αA␈α
Program␈α
for␈αOn-Line␈α
Assistance␈α
in␈α
Symbolic␈αComputation␈↓,␈α
in␈α
Proceedings␈αof␈α
the
␈↓ ↓*␈↓␈↓ ↓jFJCC, Volume 2, Spartan Books, 1965.

␈↓ ↓*␈↓Engelman, C., ␈↓βMATHLAB '68␈↓, in IFIP, Edinburgh, 1968.

␈↓ ↓*␈↓Gardner,␈α→Martin,␈α→␈↓βMathematical␈α_Games␈↓,␈α→Scienti≡c␈α→American,␈α_numerous␈α→columns,␈α→including␈α_especially:
␈↓ ↓*␈↓␈↓ ↓jFebruary, 1975.

␈↓ ↓*␈↓@Gelernter,␈α⊃H.,␈α⊂␈↓βRealization␈α⊃of␈α⊂a␈α⊃Geometry-Theorem␈α⊃Proving␈α⊂Machine␈↓,␈α⊃in␈α⊂(Feigenbaum␈α⊃and␈α⊃Feldman,␈α⊂eds.)
␈↓ ↓*␈↓␈↓ ↓j␈↓βComputers and Thought␈↓, Part 1, Section 3, pages 134-152, McGraw-Hill Book Co., New York, 1963.

␈↓ ↓*␈↓Goldstine,␈α
Herman␈α
H.,␈α
and␈α
J.␈α
von␈α
Neumann,␈α␈↓βOn␈α
the␈α
Principles␈α
of␈α
Large␈α
Scale␈α
Computing␈α
Machines,␈↓␈αpages
␈↓ ↓*␈↓␈↓ ↓j1:33␈α
of␈α∞Volumne␈α
5␈α
of␈α∞A.␈α
H.␈α
Taub␈α∞(ed),␈α
␈↓βThe␈α
Collected␈α∞Works␈α
of␈α
John␈α∞von␈α
Neumann␈↓,␈α∞Pergamon␈α
Press,
␈↓ ↓*␈↓␈↓ ↓jNY, 1963.

␈↓ ↓*␈↓Guard, J. R., et al., ␈↓βSemi-Automated Mathematics␈↓, JACM 16, January, 1969, pp. 49-62.

␈↓ ↓*␈↓Halmos,␈αPaul␈αR.,␈α␈↓βInnovation␈αin␈αMathematics␈↓,␈αin␈αKline,␈αM.␈α(ed),␈α␈↓βMathematics␈αin␈αthe␈αModern␈αWorld:␈αReadings
␈↓ ↓*␈↓β␈↓ ↓jfrom␈α
Scienti≡c␈α
American␈↓,␈α
W.H.Freeman␈α∞and␈α
Co.,␈α
San␈α
Francisco,␈α∞1968,␈α
pp.␈α
6-13.␈α
Originally␈α∞in␈α
Scienti≡c
␈↓ ↓*␈↓␈↓ ↓jAmerican, September, 1958.

␈↓ ↓*␈↓Hasse,␈α⊃H.,␈α∩␈↓βMathemakik␈α⊃als␈α⊃Wissenschaft,␈α∩Kunst␈α⊃und␈α∩Macht␈↓,␈α⊃(Mathematics␈α⊃as␈α∩Science,␈α⊃Art,␈α∩and␈α⊃Power),
␈↓ ↓*␈↓␈↓ ↓jBaden-Badeb, 1952.

␈↓ ↓*␈↓@Hewitt,␈α
Carl,␈α␈↓βA␈α
Universal␈αModular␈α
ACTOR␈α
Formalism␈αfor␈α
Arti≡cial␈αIntelligence␈↓,␈α
Third␈α
International␈αJoint
␈↓ ↓*␈↓␈↓ ↓jConference on Arti≡cial Intelligence, 1973, pp. 235-245.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  81␈↓

␈↓ ↓*␈↓Menges,␈αGunter,␈α
␈↓βInference␈αand␈αDecision␈↓,␈α
A␈αVolume␈α
in␈α␈↓βSelecta␈αStatistica␈α
Canadiana␈↓,␈αJohn␈α
Wiley␈α&␈αSons,␈α
New
␈↓ ↓*␈↓␈↓ ↓jYork,  1973, pp. 1-16.

␈↓ ↓*␈↓Kling, Robert E., ␈↓βA Paradigm for Reasoning by Analogy␈↓, Arti≡cial Intelligence 2, 1971, pp. 147-178.

␈↓ ↓*␈↓Knuth,Donald E., ␈↓βAncient Babylonian Algorithms␈↓, CACM 15, July, 1972, pp. 671-677.

␈↓ ↓*␈↓Lee, Richard C. T., ␈↓βFuzzy Logic and the Resolution Principle␈↓, JACM 19, January, 1972, pp. 109-119.

␈↓ ↓*␈↓@Lenat, D., ␈↓βBEINGs: Knowledge as Interacting Experts␈↓, 4th IJCAI, 1975.

␈↓ ↓*␈↓McCarthy,␈α⊗John,␈α⊗and␈α⊗Hayes,␈α⊗Patrick,␈α↔␈↓βSome␈α⊗Philosophical␈α⊗Problems␈α⊗from␈α⊗the␈α⊗Standpoint␈α↔of␈α⊗Arti≡cial
␈↓ ↓*␈↓β␈↓ ↓jIntelligence␈↓, Machine Intelligence 4, 1969, pp. 463-502.

␈↓ ↓*␈↓Martin,␈α⊃W.,␈α∩and␈α⊃Fateman,␈α⊃R.,␈α∩␈↓βThe␈α⊃MACSYMA␈α∩System␈↓,␈α⊃Second␈α⊃Symposium␈α∩on␈α⊃Symbolic␈α∩and␈α⊃Algebraic
␈↓ ↓*␈↓␈↓ ↓jManipulation, 1971, pp. 59-75.

␈↓ ↓*␈↓Minsky, Marvin, ␈↓βFrames␈↓, in (Winston) ␈↓βPsychology of Computer Vision␈↓, 1974.

␈↓ ↓*␈↓Moore,␈αJ.,␈αand␈αNewell,␈α␈↓βHow␈αCan␈αMerlin␈αUnderstand?␈↓,␈αCarnegie-Mellon␈αUniversity␈αDepartment␈αof␈αComputer
␈↓ ↓*␈↓␈↓ ↓jScience "preprint", November 15, 1973.

␈↓ ↓*␈↓Nevins,␈αArthur␈α
J.,␈α␈↓βPlane␈α
Geometry␈αTheorem␈α
Proving␈αUsing␈α
Forward␈αChaining␈↓,␈α
Arti≡cial␈αIntelligence␈α6,␈α
Spring
␈↓ ↓*␈↓␈↓ ↓j1975, pp. 1-23.

␈↓ ↓*␈↓Newell, A., ␈↓βProduction Systems␈↓, ...

␈↓ ↓*␈↓Neumann,␈αJ.␈αvon,␈α␈↓βThe␈αMathematician␈↓,␈αin␈αR.B.␈αHeywood␈α(ed),␈α␈↓βThe␈αWorks␈αof␈αthe␈αMind␈↓,␈αU.␈αChicago␈αPress,␈αpp.
␈↓ ↓*␈↓␈↓ ↓j180-196, 1947.

␈↓ ↓*␈↓Neumann, J. von, ␈↓βThe Computer and the Brain␈↓, Silliman Lectures, Yale U. Press, 1958.

␈↓ ↓*␈↓Pager,␈αDavid,␈α␈↓βA␈αProposal␈αfor␈αa␈αComputer-based␈αInteractive␈αScienti≡c␈αCommunity␈↓,␈αCACM␈α15,␈α
February,␈α1972,
␈↓ ↓*␈↓␈↓ ↓jpp. 71-75.

␈↓ ↓*␈↓Pager, David, ␈↓βOn the Problem of Communicating Complex Information␈↓, CACM 16, May, 1973, pp. 275-281.

␈↓ ↓*␈↓@Sloman,␈αAaron,␈α
␈↓βInteractions␈αBetween␈αPhilosophy␈α
and␈αArti≡cial␈α
Intelligence:␈αThe␈αRole␈α
of␈αIntuition␈α
and␈αNon-
␈↓ ↓*␈↓β␈↓ ↓jLogical Reasoning in Intelligence␈↓, Arti≡cial Intelligence 2, 1971, pp. 209-225.

␈↓ ↓*␈↓Sloman, Aaron, ␈↓βOn Learning about Numbers␈↓,...

␈↓ ↓*␈↓Winston,␈α↔Patrick,␈α↔␈↓βLearning␈α↔Structural␈α↔Descriptions␈α↔from␈α↔Examples␈↓,␈α↔Ph.D.␈α↔thesis,␈α↔Dept.␈α_of␈α↔Electrical
␈↓ ↓*␈↓␈↓ ↓jEngineering, TR-76, Project MAC, TR-231, MIT AI Lab, September, 1970.


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&Acknowledgements␈↓)αβ␈↓
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  82␈↓

␈↓ ↓*␈↓I␈α⊂owe␈α⊂a␈α⊂great␈α⊂debt␈α⊂of␈α⊂thanks␈α⊂to␈α⊂many␈α⊂people,␈α∂both␈α⊂for␈α⊂the␈α⊂input␈α⊂of␈α⊂new␈α⊂ideas␈α⊂and␈α⊂for␈α⊂the␈α∂evaluation,

␈↓ ↓*␈↓channelling,␈α
and␈α
pruning␈α
of␈α
my␈α
own.␈α
Let␈α∞me␈α
mention,␈α
alphabetically:␈α
B.␈α
Buchanan,␈α
A.␈α
Cohn,␈α
R.␈α∞Davis,␈α
E.

␈↓ ↓*␈↓Feigenbaum,␈α∂R.␈α∞Floyd,␈α∂C.␈α∞Green,␈α∂D.␈α∞Knuth,␈α∂M.␈α∂Lenat,␈α∞E.␈α∂Sacerdoti,␈α∞R.␈α∂Waldinger,␈α∞R.␈α∂Weyrauch,␈α∂and␈α∞M.

␈↓ ↓*␈↓Wilber.  Let me also thank SRI, for providing some of the computer time for this research.


␈↓ ↓*␈↓The␈αapplication␈αof␈αBEINGs␈αto␈αan␈αAutomatic␈αProgramming␈αtask␈αis␈αdescribed␈αin␈α[Lenat].␈α The␈αproblems␈αwith

␈↓ ↓*␈↓the␈αdomain␈αof␈αconcept-formation-program-writing,␈αstudied␈αtherein␈αand␈αsummarized␈αhere␈αin␈αAppendix␈α1,␈αled

␈↓ ↓*␈↓to␈α
the␈α
AM␈α
project.␈α
A␈αmore␈α
complete␈α
description␈α
of␈α
AM␈αcan␈α
be␈α
perused␈α
as␈α
SYS4[TLK,DBL]@SU-AI.␈α The

␈↓ ↓*␈↓full␈α⊂body␈α⊂of␈α⊂knowledge␈α⊂we␈α⊂expect␈α⊂to␈α⊂provide␈α⊂to␈α⊂AM␈α⊂ is␈α⊂found␈α⊂on␈α⊂≡le␈α⊃GIVEN[TLK,DBL]@SU-AI.␈α⊂The

␈↓ ↓*␈↓running␈α⊗AM␈α⊗program␈α⊗is␈α⊗stored␈α⊗at␈α⊗SUMEX.␈α⊗ From␈α⊗Interlisp,␈α⊗one␈α⊗need␈α⊗only␈α⊗load␈α⊗in␈α⊗the␈α⊗two␈α∃≡les

␈↓ ↓*␈↓<LENAT>TOP5 and <LENAT>CON5.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  83␈↓

␈↓ ↓*␈↓␈↓ αa␈↓∧␈↓&Appendix 1: Background for readers unfamiliar with Beings␈↓)αβ␈↓


␈↓ ↓*␈↓This␈αappendix␈αintroduces␈αthe␈αreader␈αto␈αthe␈αconcept␈αof␈αorganizing␈αknowledge␈αas␈αsimilarly-structured␈αmodules,

␈↓ ↓*␈↓called␈α∪BEINGs.␈α∪It␈α∪should␈α∪be␈α∪skipped␈α∪by␈α∪those␈α∪familiar␈α∪with␈α∪that␈α∪construction.␈α∪For␈α∪a␈α∪more␈α∪thorough

␈↓ ↓*␈↓treatment,␈αread␈α
either:␈αSection␈α
4.6␈αof␈α
Green␈αet␈α
al.,␈α␈↓βProgress␈α
Report␈αon␈α
Program-Understanding␈αSystems␈↓,␈α
Memo

␈↓ ↓*␈↓AIM-240,␈αCS␈αReport␈αSTAN-CS-74-444,␈αArti≡cial␈αIntelligence␈αLaboratory,␈αStanford␈αUniversity,␈αAugust,␈α1974;

␈↓ ↓*␈↓or␈αLenat,␈α␈↓βBEINGS:␈αKnowledge␈αas␈αInteracting␈αExperts␈↓,␈α4th␈αIJCAI,␈α1975␈α(preprint␈αavailable␈αfrom␈αthe␈α
author),

␈↓ ↓*␈↓or␈α∞Lenat,␈α
␈↓βSynthesis␈α∞of␈α
Large␈α∞Programs␈α∞from␈α
Speci≡c␈α∞Dialogues␈↓,␈α
3rd␈α∞World␈α
Conference␈α∞on␈α∞Cybernetics␈α
and

␈↓ ↓*␈↓Systems, Rumania, 1975 (preprint).


␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&BEINGs and Experts␈↓)αβ␈↓


␈↓ ↓*␈↓Consider␈αan␈αinterdisciplinary␈αenterprise,␈αattempted␈αby␈αa␈αcommunity␈αof␈αhuman␈αexperts␈αwho␈αare␈αspecialists␈αin

␈↓ ↓*␈↓--␈αand␈αonly␈α
in␈α--␈αtheir␈α
own␈α≡elds.␈α What␈αmodes␈α
of␈αi}teractions␈αwill␈α
be␈αproductive?␈α The␈α
dominant␈αparadigm

␈↓ ↓*␈↓might␈α
well␈α
settle␈α
into␈α
␈↓βquestioning␈α
and␈α
answering␈↓␈α
each␈α
other.␈α
 Instead␈α
of␈α
a␈α
chairman,␈α
suppose␈α
the␈α
group␈α
adopts

␈↓ ↓*␈↓rules␈α∞for␈α∞gaining␈α∞the␈α∞∨oor,␈α∞what␈α∞a␈α∞speaker␈α∞may␈α∞do,␈α∞ and␈α∞how␈α∞to␈α∞resolve␈α∞disputes.␈α∞ When␈α∞a␈α∞topic␈α∞is␈α∞being

␈↓ ↓*␈↓considered,␈αone␈αor␈αtwo␈αexperts␈αmight␈αrecognize␈αit␈αand␈αspeak␈αup.␈αIn␈αthe␈αcourse␈αof␈αtheir␈αexposition␈αthey␈αmight

␈↓ ↓*␈↓need␈α
to␈αcall␈α
on␈α
other␈αspecialists.␈α
This␈α
might␈αbe␈α
by␈αname,␈α
by␈α
specialty,␈αor␈α
simply␈α
by␈αposing␈α
a␈αnew␈α
sub-question

␈↓ ↓*␈↓and␈α
hoping␈α
someone␈α
could␈α
recognize␈αhis␈α
own␈α
relevance␈α
and␈α
volunteer␈α
a␈αsuggestion.␈α
 If␈α
the␈α
task␈α
is␈αto␈α
construct

␈↓ ↓*␈↓something,␈α⊂then␈α∂the␈α⊂activities␈α⊂of␈α∂the␈α⊂experts␈α⊂should␈α∂not␈α⊂be␈α∂strictly␈α⊂verbal.␈α⊂ Often,␈α∂one␈α⊂will␈α⊂recognize␈α∂his

␈↓ ↓*␈↓relevance to the current situation and ask to ␈↓βdo␈↓ something: clarify or modify or (rarely) create.


␈↓ ↓*␈↓What␈α
would␈α
it␈α
mean␈αto␈α
␈↓βsimulate␈↓␈α
the␈α
above␈αactivity?␈α
 Imagine␈α
several␈α
little␈αprograms,␈α
each␈α
one␈α
modelling␈αa

␈↓ ↓*␈↓di≥erent␈α∞expert.␈α
What␈α∞should␈α∞each␈α
program,␈α∞called␈α∞a␈α
␈↓βBEING␈↓,␈α∞be␈α∞capable␈α
of?␈α∞ It␈α∞must␈α
possess␈α∞a␈α∞corpus␈α
of

␈↓ ↓*␈↓speci≡c␈αfacts␈αand␈αstrategies␈αfor␈αits␈αdesignated␈αspeciality.␈αIt␈αmust␈αinteract␈αvia␈αquestioning␈αand␈αanswering␈αother

␈↓ ↓*␈↓BEINGs.␈αEach␈αBEING␈αshould␈αbe␈αable␈αto␈αrecognize␈α
when␈αit␈αis␈αrelevant.␈α It␈αmust␈αset␈αup␈αand␈α
alter␈αstructures,

␈↓ ↓*␈↓just as the human specialists do.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  84␈↓

␈↓ ↓*␈↓Let␈α
us␈α
return␈α∞to␈α
our␈α
meeting␈α
of␈α∞human␈α
experts.␈α
 To␈α
be␈α∞more␈α
concrete,␈α
suppose␈α
their␈α∞task␈α
is␈α
to␈α∞design␈α
and

␈↓ ↓*␈↓code␈α∂a␈α∂large␈α∂computer␈α∂program:␈α∂a␈α∂concept␈α∂formation␈α∂system[Hempel].␈α∂Experts␈α∂who␈α∂will␈α∂be␈α∂useful␈α∂include

␈↓ ↓*␈↓scienti≡c␈αprogrammers,␈αnon-programming␈αpsychologists,␈αsystem␈αhackers,␈αand␈αmanagement␈α
personnel.␈α Within

␈↓ ↓*␈↓each␈αof␈αthese␈αfour␈αmajor␈αfamilies␈αwill␈αbe␈αmany␈αindividual␈αspecialists.␈α What␈αhappens␈αin␈αthe␈αensuing␈αsession?

␈↓ ↓*␈↓When␈αan␈α
expert␈αparticipates,␈α
he␈αwill␈α
either␈αbe␈α
aiding␈αa␈α
colleague␈αin␈α
some␈αdi≠culty␈α
or␈αelse␈α
transferring␈αa␈α
tiny,

␈↓ ↓*␈↓customized␈α
bit␈α
of␈α
his␈αexpertise␈α
(facts␈α
about␈α
his␈α≡eld)␈α
into␈α
a␈α
programmed␈αfunction␈α
which␈α
can␈α
do␈αsomething.

␈↓ ↓*␈↓The␈α≡nal␈αcode␈αre∨ects␈αthe␈αmembers'␈αknowledge,␈αin␈αthat␈αsense.␈α Of␈αcourse,␈αexperts␈αwithin␈αthe␈αsame␈αfamily␈α
will

␈↓ ↓*␈↓be␈αable␈αto␈αcommunicate␈αthings␈α
among␈αthemselves␈αwhich␈αare␈αunintelligible␈α
to␈αoutsiders␈α(e.g.,␈αwhen␈αthe␈α
hackers

␈↓ ↓*␈↓start␈α
arguing␈α
about␈α
how␈αto␈α
patch␈α
the␈α
system␈α
bugs␈αthat␈α
appear).␈α
Nevertheless,␈α
if␈αwe␈α
press␈α
him,␈α
any␈α
of␈αthese

␈↓ ↓*␈↓specialists␈α∞could␈α∞transform␈α∂his␈α∞compressed␈α∞jargon␈α∂into␈α∞a␈α∞more␈α∂universal␈α∞message␈α∞(losing␈α∂some␈α∞information

␈↓ ↓*␈↓and some e≠cency), by giving examples and analogies, perhaps.


␈↓ ↓*␈↓Suppose␈α∞the␈α∞project␈α∂sponsor␈α∞is␈α∞quasi-active,␈α∂submitting␈α∞an␈α∞initial␈α∞speci≡cation␈α∂order␈α∞for␈α∞the␈α∂program,␈α∞and

␈↓ ↓*␈↓then␈α
participating␈α
in␈αthe␈α
work␈α
as␈αa␈α
(somewhat␈α
priveleged)␈α
member␈αof␈α
the␈α
team.␈αThis␈α
individual␈α
is␈α
the␈αone

␈↓ ↓*␈↓who wants the ≡nal product, hence will be called the ␈↓βuser␈↓.


␈↓ ↓*␈↓How␈αcould␈αBEINGs␈αdo␈αall␈αthis?␈αThere␈αwould␈αbe␈αsome␈αlittle␈αprogram␈αcontaining␈αinformation␈α
about␈α␈↓εCONCEPT-

␈↓ ↓*␈↓εFORMATION␈↓␈α(much␈α
more␈αthan␈α
would␈αbe␈α
used␈αin␈α
writing␈αany␈α
single␈αconcept␈α
formation␈αprogram),␈αanother␈α
BEING

␈↓ ↓*␈↓who␈α∂knows␈α∂how␈α∂to␈α∂manage␈α∂a␈α∂group␈α⊂to␈α∂␈↓εWRITE-PROGRAMS␈↓,␈α∂and␈α∂many␈α∂lower-level␈α∂specialists,␈α∂for␈α⊂example␈α∂␈↓εINFO-

␈↓ ↓*␈↓εOBTAINER,␈α
TEST,␈α∞MODIFY-DATA-STRUCTURE,␈α
UNTIL-LOOP,␈α∞VISUAL-PERCEPTION,␈α
AVOID-CONTRADICTION,␈α∞PROPOSE-PLAUSIBLE-NAME␈↓.␈α
 Like

␈↓ ↓*␈↓the␈αhuman␈αspecialists,␈αthe␈αBEINGs␈αwould␈αcontain␈αfar␈αtoo␈αmuch␈αinformation,␈αfar␈αtoo␈αine≠ciently␈αrepresented,

␈↓ ↓*␈↓to␈α∞be␈α
able␈α∞to␈α∞say␈α
"we␈α∞ourselves␈α∞constitute␈α
the␈α∞desired␈α∞program!"␈α
They␈α∞would␈α∞have␈α
to␈α∞discuss,␈α∞and␈α
perhaps

␈↓ ↓*␈↓carry␈αout,␈α
the␈αconcept␈αformation␈α
task.␈αThey␈αwould␈α
write␈αspecialized␈αversions␈α
of␈αthemselves,␈α
programs␈αwhich

␈↓ ↓*␈↓could␈α
do␈α
exactly␈α
what␈α
the␈α
BEINGs␈α
did␈α
to␈α
carry␈α
out␈α
the␈α
task,␈α
no␈α
more␈α
nor␈α
less␈α
(although␈α
they␈α
would␈α
hopefully

␈↓ ↓*␈↓take␈α
much␈α
less␈α
time,␈α
be␈αmore␈α
customized).␈α
 Some␈α
BEINGs␈α
(e.g.,␈α␈↓εTEST␈↓)␈α
may␈α
have␈α
several␈α
distinct,␈αstreamlined

␈↓ ↓*␈↓fractions␈αof␈α
themselves␈αin␈α
the␈α≡nal␈αprogram.␈α
BEINGs␈αwhich␈α
only␈αaided␈α
other␈αBEINGs␈α(e.g.,␈α
␈↓εPROPOSE-PLAUSIBLE-

␈↓ ↓*␈↓εNAME␈↓) may not have ␈↓βany␈↓ new correlates in the synthesized code.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  85␈↓

␈↓ ↓*␈↓An␈α∪experimental␈α∪system,␈α∩PUP6,␈α∪was␈α∪designed␈α∩and␈α∪partially␈α∪implemented.␈α∩PUP6␈α∪synthesized␈α∪a␈α∩concept

␈↓ ↓*␈↓formation␈α
program␈α
(similar␈α
to␈α
Winston's),␈α
but␈α
the␈α∞user,␈α
who␈α
is␈α
human,␈α
 must␈α
come␈α
up␈α
with␈α∞certain␈α
speci≡c

␈↓ ↓*␈↓answers␈αto␈α
some␈αof␈αthe␈α
BEINGs'␈αcritical␈αqueries.␈α
 A␈αgrammatical␈αinference␈α
program␈αand␈αa␈α
 simple␈αproperty

␈↓ ↓*␈↓list␈α
maintenance␈αroutine␈α
were␈αalso␈α
generated.␈α
Only␈αa␈α
few␈αnew␈α
BEINGs␈α
had␈αto␈α
be␈αadded␈α
to␈α
PUP6's␈αorginal

␈↓ ↓*␈↓pool␈α
of␈α100␈α
BEINGs␈αin␈α
order␈α
to␈αsynthesize␈α
them,␈αbut␈α
communication␈α
∨exibility␈αproblems␈α
existed.␈α The␈α
choice

␈↓ ↓*␈↓of mathematics as the domain for the proposed system was made partially to alleviate this problem.




␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&Internal Design of BEINGs␈↓)αβ␈↓


␈↓ ↓*␈↓Now␈α
that␈α
we␈α
have␈α
developed␈α∞our␈α
"external␈α
speci≡cations"␈α
for␈α
what␈α∞the␈α
BEINGs␈α
must␈α
do,␈α
how␈α∞exactly␈α
will

␈↓ ↓*␈↓they␈α⊂do␈α⊂it?␈α⊃ Have␈α⊂we␈α⊂merely␈α⊃pushed␈α⊂the␈α⊂problem␈α⊂of␈α⊃Arti≡cial␈α⊂Intelligence␈α⊂down␈α⊃into␈α⊂the␈α⊂coding␈α⊃of␈α⊂the

␈↓ ↓*␈↓BEINGs?␈α⊂ Perhaps␈α∂not,␈α⊂for␈α∂we␈α⊂still␈α⊂have␈α∂our␈α⊂analogy␈α∂to␈α⊂the␈α⊂interacting␈α∂experts.␈α⊂Let␈α∂us␈α⊂carry␈α⊂it␈α∂further,

␈↓ ↓*␈↓analyze synergetic cooperation among the humans, then try to model that in our internal design of BEINGs.


␈↓ ↓*␈↓Viewing␈α⊂the␈α∂group␈α⊂of␈α⊂experts␈α∂as␈α⊂a␈α⊂single␈α∂entity,␈α⊂what␈α⊂makes␈α∂it␈α⊂productive?␈α⊂The␈α∂members␈α⊂must␈α⊂be␈α∂very

␈↓ ↓*␈↓di≥erent␈α⊃in␈α⊂abilities,␈α⊃in␈α⊂order␈α⊃to␈α⊃handle␈α⊂a␈α⊃complex␈α⊂task,␈α⊃yet␈α⊃similar␈α⊂in␈α⊃basic␈α⊂cognitive␈α⊃structure␈α⊃(in␈α⊂the

␈↓ ↓*␈↓anatomy␈αof␈αtheir␈αminds)␈αto␈αpermit␈αfacile␈αcommunications␈αto␈α∨ow.␈α For␈αexample,␈αeach␈αpsychologist␈αknows␈α
how

␈↓ ↓*␈↓to␈αdirect␈αa␈αprogrammer␈αto␈αdo␈αsome␈αof␈αthe␈αthings␈αhe␈αcan␈αdo,␈αbut␈αthe␈αspeci≡c␈αfacts␈αhe␈αhas␈αtucked␈αaway␈αunder

␈↓ ↓*␈↓this␈α
category␈α∞must␈α
be␈α∞quite␈α
unique.␈α∞Similarly,␈α
each␈α∞expert␈α
may␈α∞have␈α
a␈α∞set␈α
of␈α∞strategies␈α
for␈α∞recognizing␈α
his

␈↓ ↓*␈↓own␈α∪relevance␈α∪to␈α∪a␈α∪proposed␈α∪question,␈α∪but␈α∀the␈α∪␈↓βcontents␈↓␈α∪of␈α∪that␈α∪knowledge␈α∪varies␈α∪from␈α∀individual␈α∪to

␈↓ ↓*␈↓individual.␈α The␈αproposed␈α
hypothesis␈αis␈αthat␈α
all␈αthe␈αexperts␈α
can␈αbe␈αsaid␈α
to␈αconsist␈αof␈αcategorized␈α
information,

␈↓ ↓*␈↓where␈αthe␈αset␈α
of␈αcategories␈αis␈αfairly␈α
standard,␈αand␈αindicates␈α
the␈α␈↓βtypes␈↓␈αof␈αquestions␈α
any␈αexpert␈αcan␈αbe␈α
expected

␈↓ ↓*␈↓to␈αanswer.␈αAn␈αexpert␈αis␈αconsidered␈α␈↓βequivalent␈↓␈αto␈αhis␈αanswers␈αto␈αseveral␈αstandard␈αquestions.␈α Each␈αexpert␈αhas

␈↓ ↓*␈↓the␈α
same␈α
mental␈α
"parts",␈αit␈α
is␈α
only␈α
the␈α
values␈αstored␈α
in␈α
these␈α
parts,␈α
their␈αcontents,␈α
which␈α
distinguish␈α
him␈αas␈α
an

␈↓ ↓*␈↓individual.␈αThe␈αparticular␈αset␈αof␈αquestions␈αhe␈αcan␈αdeal␈αwith␈αis␈α≡xed,␈αdepending␈αon␈αwhich␈αfamily␈α
the␈αexpert
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  86␈↓

␈↓ ↓*␈↓belongs␈α
to.␈αThere␈α
is␈α
much␈α--␈α
but␈α
not␈αtotal␈α
--␈α
overlap␈αbetween␈α
what␈α
two␈αhumans␈α
from␈α
di≥erent␈αprofessions␈α
can

␈↓ ↓*␈↓meaningly answer.


␈↓ ↓*␈↓Armed␈α∞with␈α∞this␈α∞dubious␈α∞view␈α∞of␈α∞intelligence,␈α∞let␈α∞us␈α∞return␈α∞to␈α∞the␈α∞design␈α∞of␈α∞BEINGs.␈α∞Each␈α∞BEING␈α∞shall

␈↓ ↓*␈↓have␈αmany␈αparts,␈αeach␈αpossessing␈αa␈αname␈α(a␈αquestion␈αit␈αcan␈αdeal␈αwith)␈αand␈αa␈αvalue␈α(a␈αprocedure␈αcapable␈αof

␈↓ ↓*␈↓answering␈α∞that␈α
question).␈α∞ Henceforth,␈α∞"␈↓βpart␈↓"␈α
will␈α∞be␈α∞used␈α
in␈α∞this␈α
technical␈α∞sense.␈α∞ When␈α
a␈α∞BEING␈α∞asks␈α
a

␈↓ ↓*␈↓question,␈αit␈αis␈αreally␈αjust␈αone␈αpart␈αwho␈αis␈αasking.␈αIn␈αfact,␈αit␈αmust␈αbe␈αthat␈αthe␈α␈↓βvalue␈↓␈αsubpart␈αof␈αsome␈αpart␈αcan't

␈↓ ↓*␈↓answer␈α
␈↓βhis␈↓␈α
question␈α
without␈αfurther␈α
assistance.␈α
He␈α
may␈αnot␈α
know␈α
enough␈α
to␈αcall␈α
on␈α
speci≡c␈α
other␈αBEINGs␈α
(in

␈↓ ↓*␈↓which␈αcase␈αhe␈αbroadcasts␈αhis␈αplea,␈αlettting␈αanyone␈αrespond␈αwho␈αfeels␈αrelevant),␈αbut␈αhe␈αshould␈α␈↓βalways␈↓␈αspecify

␈↓ ↓*␈↓what␈α
BEING␈α
␈↓βpart␈↓␈α
the␈α
question␈α
should␈α
be␈α
answered␈α
by.␈α
 By␈α
analogy␈α
with␈α
the␈α
experts,␈α
each␈α
BEING␈α
in␈αthe

␈↓ ↓*␈↓same␈αfamily␈α
will␈αhave␈αthe␈α
same␈α≡xed␈αset␈α
of␈αtypes␈αof␈α
parts␈α(will␈αanswer␈α
the␈αsame␈αkinds␈α
of␈αqueries),␈α
and␈αthis

␈↓ ↓*␈↓uniformity␈α
should␈α
permit␈α
painless␈α
intercommunication␈α
between␈α
specialists␈α
in␈α
the␈α
same␈α
profession.␈α
 Many␈α
of

␈↓ ↓*␈↓these parts will be common to more than one family (e.g., "How long-winded are you").


␈↓ ↓*␈↓Since␈α
the␈αparadigm␈α
of␈α
the␈αmeeting␈α
is␈α
questioning␈αand␈α
anwwering,␈α
the␈αnames␈α
of␈α
the␈αparts␈α
should␈α
cover␈αall␈α
the

␈↓ ↓*␈↓types␈α
of␈α
questions␈α
one␈α
expert␈α
wants␈α
to␈α
ask␈α
another.␈α
Each␈α
part␈α
of␈α
each␈α
BEING␈α
will␈α
have␈α
implicit␈α
access␈αto␈α
this

␈↓ ↓*␈↓list:␈α
it␈α
may␈α
ask␈α
only␈α
these␈α
types␈α
of␈α
questions.␈α
Each␈α
BEING␈α
should␈α
␈↓βnot␈↓␈α
have␈α
access␈α
to␈α
the␈α
list␈α
of␈α
all␈αBEINGs␈α
in

␈↓ ↓*␈↓the␈α⊂system:␈α⊂requests␈α⊂should␈α⊂be␈α⊂phrased␈α⊂in␈α⊂terms␈α⊂of␈α⊂what␈α⊂is␈α⊂wanted;␈α⊂rarely␈α⊂is␈α⊂the␈α⊂name␈α⊂of␈α⊃the␈α⊂answerer

␈↓ ↓*␈↓speci≡ed␈αin␈αadvance.␈α (By␈αanalogy:␈αthe␈αhuman␈αspeaker␈αis␈α
not␈αaware␈αof␈αprecisely␈αwho␈αis␈αin␈αthe␈αroom;␈αwhen␈α
he

␈↓ ↓*␈↓feels inadequate, he asks for help and hopes someone responds).


␈↓ ↓*␈↓Once␈α∂again:␈α∂the␈α∞concept␈α∂of␈α∂a␈α∂system␈α∞of␈α∂BEINGs␈α∂is␈α∞that␈α∂many␈α∂entities␈α∂coexist,␈α∞clumped␈α∂into␈α∂a␈α∂few␈α∞major

␈↓ ↓*␈↓family␈α
groupings.␈α
Each␈α
individual␈α
BEING␈α
has␈α
a␈αcomplex␈α
structure,␈α
but␈α
that␈α
structure␈α
does␈α
not␈α
vary␈αmuch

␈↓ ↓*␈↓from␈α
BEING␈α
to␈α
BEING;␈α
it␈α
does␈αnot␈α
vary␈α
at␈α
all␈α
among␈α
BEINGs␈αof␈α
the␈α
same␈α
family.␈α
 This␈α
idea␈αhas␈α
analogues

␈↓ ↓*␈↓in many ≡elds: transactional analysis in psychology, anatomy in medicine, modular design in architechture.


␈↓ ↓*␈↓To␈α∂carry␈α⊂out␈α∂these␈α∂ideas,␈α⊂we␈α∂build␈α⊂a␈α∂system␈α∂of␈α⊂BEINGs,␈α∂a␈α⊂modular␈α∂program␈α∂which␈α⊂will␈α∂interact␈α⊂with␈α∂a
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  87␈↓

␈↓ ↓*␈↓human␈αuser␈α
and␈αgo␈α
through␈αthe␈α
same␈αconversations,␈α
and␈αarrive␈α
at␈αthe␈α
same␈αend␈α
products,␈αthat␈α
our␈αhuman

␈↓ ↓*␈↓experts␈α
would.␈α Recasting␈α
the␈αidea␈α
into␈α
operational␈αterms,␈α
we␈αarrive␈α
at␈α
this␈αprocedure␈α
for␈αwriting␈α
a␈α
pool␈αof

␈↓ ↓*␈↓BEINGs:


␈↓ ↓*␈↓(1)␈αStudy␈αthe␈αtask␈αwhich␈αthe␈αpool␈αis␈αto␈αdo.␈αSee␈αwhat␈αkinds␈αof␈αquestions␈αare␈αasked␈αby␈αsimulated␈αexperts,␈αand

␈↓ ↓*␈↓notice␈α
how␈α
the␈αexperts␈α
divide␈α
into␈αa␈α
few␈α
major␈αfamilies␈α
{f␈↓#vi␈↓#}.␈α
 The␈αtotal␈α
number␈α
of␈αfamilies␈α
is␈α
important:␈αif

␈↓ ↓*␈↓there␈αare␈αtoo␈αmany,␈αit␈αis␈αhard␈αfor␈αspecialized␈αcommunication␈αto␈αoccur;␈αif␈αtoo␈αfew␈αfamilies,␈αmany␈αBEINGs␈αwill

␈↓ ↓*␈↓be forced to answer questions they consider irrelevant.


␈↓ ↓*␈↓(2)␈α
Distill␈α
the␈α∞corpus␈α
of␈α
collected␈α
communications␈α∞into␈α
a␈α
core␈α∞of␈α
simple␈α
questions,␈α
Q␈↓#vf␈↓#,␈α∞for␈α
each␈α
family␈α∞f,␈α
in

␈↓ ↓*␈↓such␈α
a␈α
way␈α
that␈αeach␈α
inter-expert␈α
question␈α
or␈α
transfer␈αof␈α
control␈α
can␈α
be␈α
rephrased␈αin␈α
terms␈α
of␈α
these␈αQ's.␈α
 The

␈↓ ↓*␈↓sizes␈α
of␈α
the␈α
sets␈α
Q␈α
are␈α
very␈α
important.␈α
 If␈α
a␈αQ␈α
is␈α
huge,␈α
addition␈α
of␈α
new␈α
BEINGs␈α
will␈α
demand␈α
either␈αgreat

␈↓ ↓*␈↓e≥ort␈α
or␈α
great␈α
intelligence␈α(an␈α
example␈α
of␈α
a␈αsystem␈α
like␈α
this␈α
is␈αACTORS).␈α
If␈α
a␈α
Q␈αis␈α
too␈α
small,␈α
all␈α
the␈αnon-

␈↓ ↓*␈↓uniformity␈α∂is␈α∞simply␈α∂pushed␈α∞down␈α∂into␈α∞the␈α∂values␈α∂of␈α∞one␈α∂or␈α∞two␈α∂general␈α∞catchall␈α∂questions␈α∂(all␈α∞≡rst-order

␈↓ ↓*␈↓logical languages do this).


␈↓ ↓*␈↓(3)␈αList␈αall␈α
the␈αBEINGs␈αwho␈α
will␈αbe␈αpresent␈α
in␈αthe␈αpool,␈α
by␈αfamily,␈αand␈α
≡ll␈αin␈αtheir␈α
parts.␈αThe␈αtime␈αto␈α
encode

␈↓ ↓*␈↓knowledge␈αinto␈α
many␈αsimple␈α
representation␈αschemes␈α
is␈αproportional␈α
to␈αthe␈α
square␈αof␈α(occasionally␈α
exponential

␈↓ ↓*␈↓in)␈αthe␈αamount␈αof␈αinterrelated␈αknowledge␈α(e.g.,␈αconsider␈αthe␈αframe␈αproblem).␈α The␈α≡lling␈αin␈αof␈αa␈αnew␈αBEING

␈↓ ↓*␈↓is␈α⊃ ␈↓βindependent␈↓␈α⊃of␈α⊂the␈α⊃number␈α⊃of␈α⊃BEINGs␈α⊂already␈α⊃in␈α⊃the␈α⊃pool,␈α⊂because␈α⊃BEINGs␈α⊃can␈α⊃communicate␈α⊂via

␈↓ ↓*␈↓nondeterministic␈α
goal␈α
mechanisms,␈α
and␈αnot␈α
have␈α
to␈α
know␈αthe␈α
names␈α
of␈α
the␈αBEINGs␈α
who␈α
will␈α
answer␈αtheir

␈↓ ↓*␈↓queries.␈α⊂This␈α⊂≡lling␈α⊂in␈α⊂is␈α⊂ of␈α∂course␈α⊂linear␈α⊂in␈α⊂the␈α⊂number␈α⊂of␈α∂questions␈α⊂a␈α⊂BEING␈α⊂must␈α⊂answer␈α⊂(e.g.,␈α∂the

␈↓ ↓*␈↓maximum size of any Q␈↓#vf␈↓#).


␈↓ ↓*␈↓(4) The human user interacts with the completed BEING community, until the desired task is complete.




␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&BEINGs Interacting␈↓)αβ␈↓
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  88␈↓

␈↓ ↓*␈↓We␈αnow␈α
have␈αsome␈α
idea␈αof␈αwhat␈α
the␈αinternal␈α
structure␈αof␈α
BEINGs␈αare,␈αbut␈α
how␈αdo␈α
they␈αmeet␈α
the␈αexternal

␈↓ ↓*␈↓speci≡cations␈α
we␈α
set␈α
for␈α
them:␈α
the␈α
reproduction␈α
of␈α
the␈α
human␈α
experts'␈α
conversations␈α
and␈α
≡nished␈αproducts?

␈↓ ↓*␈↓The␈αquestion␈αis␈αthat␈αof␈αcontrol␈αin␈αthe␈αsystem,␈αand␈αit␈αsplits␈αinto␈αtwo␈αparts:␈αhow␈αdoes␈αthe␈α"right"␈αBEING␈αgain

␈↓ ↓*␈↓control, and what does he "do" when he gets control?


␈↓ ↓*␈↓The␈α
scenario,␈α
in␈α
PUP6,␈α
runs␈α
as␈α
follows.␈α
When␈α
control␈α
is␈α
about␈α
to␈α
be␈α
passed,␈α
the␈α
relinquisher␈α
will␈α
specify␈α
a␈α
set

␈↓ ↓*␈↓of␈αpossible␈αrecipients␈α(successors).␈αTwo␈α
extreme␈αbut␈αcommon␈αcases␈αare␈α
a␈αsingleton␈α(he␈αkno≥s␈αwho␈α
should␈αgo

␈↓ ↓*␈↓next)␈αand␈α
the␈αset␈α
of␈αall␈αBEINGs␈α
(he␈αhas␈α
no␈αidea␈α
whatsoever).␈α Each␈αpossible␈α
candidate␈αfor␈α
control␈αis␈αasked␈α
if

␈↓ ↓*␈↓he␈αis␈αrelevant␈αto␈α
the␈αcurrent␈αsituation.(If␈αa␈α
goal␈αis␈αcurrently␈αset␈α
forth,␈αhis␈αEFFECTS␈αpart␈α
will␈αbe␈αasked␈αif␈α
this

␈↓ ↓*␈↓guy␈α∂can␈α∂bring␈α∂about␈α∂that␈α∂goal;␈α∂if␈α∂an␈α∂unintelligible␈α∂piece␈α∂of␈α∂information␈α∂is␈α∂sitting␈α∂around␈α⊂somewhere,␈α∂his

␈↓ ↓*␈↓IDENTIFY␈αpart␈αwill␈αbe␈αasked␈αif␈αthis␈αguy␈αcan␈αrecognize␈αthat␈αpiece␈αof␈αinfo.)␈αIf␈αmore␈αthan␈αone␈αBEING␈αreplies

␈↓ ↓*␈↓that␈αit␈αfeels␈αrelevant,␈αthen␈αtheir␈αWHEN␈αcomponents␈αare␈αasked␈α␈↓βhow␈↓␈αrelevant␈αthey␈αare␈αright␈αnow.␈α If␈αa␈αtie␈αstill

␈↓ ↓*␈↓exists,␈α
their␈α
COMPLEXITY␈α∞components␈α
are␈α
asked␈α
to␈α∞decide␈α
which␈α
will␈α
be␈α∞faster,␈α
surer,␈α
lead␈α∞to␈α
auxilliary

␈↓ ↓*␈↓desired␈αe≥ects,␈αetc.␈α There␈αwill␈αalways␈αbe␈α␈↓βsome␈↓␈αBEING␈αwho␈αwill␈αtake␈αover;␈αthe␈αgeneral␈αmanagement␈αtypes␈αof

␈↓ ↓*␈↓BEINGs are always able  -- but reluctant  -- to do so.


␈↓ ↓*␈↓Once␈α
in␈αcontrol,␈α
a␈αBEING␈α
B␈αpicks␈α
one␈αof␈α
its␈αparts,␈α
evaluates␈αit,␈α
and␈αrepeats␈α
this␈αprocess␈α
until␈αit␈α
decides␈αto

␈↓ ↓*␈↓relinquish␈α
control.␈α
At␈α
that␈αtime,␈α
it␈α
puts␈α
forth␈αa␈α
list␈α
of␈α
possible␈α
successors.␈α For␈α
example,␈α
the␈α
ARGS␈αpart␈α
might

␈↓ ↓*␈↓be␈α
≡rst;␈α
if␈α
it␈α
asks␈α
for␈αsome␈α
arguments␈α
which␈α
no␈α
BEING␈α
has␈αsupplied,␈α
then␈α
the␈α
whole␈α
BEING␈α
might␈αdecide␈α
to

␈↓ ↓*␈↓fail.␈αSome␈α parts,␈αwhen␈αevaluated,␈αmight␈αcreate␈αa␈αnew␈αBEING,␈αmight␈αask␈αquestions␈αwhich␈αrequire␈αthis␈αwhole

␈↓ ↓*␈↓process␈α
to␈α∞repeat␈α
recursively,␈α∞etc.␈α
This␈α
"asking"␈α∞really␈α
means␈α∞broadcasting␈α
a␈α
request␈α∞to␈α
one␈α∞or␈α
two␈α∞parts␈α
of

␈↓ ↓*␈↓some␈αother␈αBEINGs␈α(often␈αevery␈αother␈αBEING);␈αfor␈αexample␈α"Is␈αthere␈αa␈αknown␈αfast␈αway␈αof␈αgronking␈αtoves?"

␈↓ ↓*␈↓would␈α∞be␈α∞asked␈α∞as␈α∞a␈α∞search␈α∞for␈α∂a␈α∞BEING␈α∞whose␈α∞COMPLEXITY␈α∞indicated␈α∞speed,␈α∞and␈α∂whose␈α∞EFFECTS

␈↓ ↓*␈↓part␈α
contained␈αa␈α
production␈αwith␈α
a␈αtemplate␈α
matching␈α
"gronking␈αtoves".␈α
 A␈αlist␈α
of␈αthe␈α
responders␈α
would␈αbe

␈↓ ↓*␈↓returned.␈αThe␈αquestioner␈αmight␈αpose␈αsome␈αnew␈α
questions␈αdirectly␈αto␈αthese␈αBEINGs,␈αmight␈αturn␈α
control␈αover

␈↓ ↓*␈↓to them directly, might simply want to know that some exist, etc.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  89␈↓

␈↓ ↓*␈↓How␈αdoes␈αeach␈αBEING␈α
decide␈αwhich␈αparts␈αto␈α
evaluate,␈αand␈αin␈αwhich␈αorder,␈α
once␈αit␈αgains␈αcontrol?␈α
 For␈αour

␈↓ ↓*␈↓humans,␈αthe␈αanswer␈αis:␈α
a␈αcombination␈αof␈αindividual␈α
intelligence,␈αthe␈αtraining␈αinherent␈α
in␈αthe␈αfamily␈αof␈α
expert

␈↓ ↓*␈↓you␈α∞are,␈α∞and␈α
universally␈α∞accepted␈α∞constraints␈α
(common␈α∞sense).␈α∞For␈α
our␈α∞BEINGs,␈α∞we␈α
postulate␈α∞a␈α∞part␈α
called

␈↓ ↓*␈↓ORDERING;␈α⊃each␈α∩BEING␈α⊃consults␈α∩its␈α⊃own␈α⊃Ordering␈α∩part,␈α⊃the␈α∩Ordering␈α⊃part␈α⊃for␈α∩its␈α⊃Family,␈α∩and␈α⊃the

␈↓ ↓*␈↓universal␈αOrdering␈αArchetypical␈αBEING.␈αThey␈αpartially␈αconstrain␈αwhat␈αpart␈αmust␈αbe␈αevaluated␈αbefore␈αwhat

␈↓ ↓*␈↓other␈α∞part.␈α∞ This␈α∂appears␈α∞to␈α∞be␈α∂di≠cult␈α∞or␈α∞tedious␈α∞for␈α∂whoever␈α∞writes␈α∞BEINGs,␈α∂since␈α∞it␈α∞might␈α∂vary␈α∞from

␈↓ ↓*␈↓BEING␈αto␈αBEING.␈αIn␈αfact,␈αit␈αrarely␈αdoes␈αvary,␈α
and␈αmost␈αof␈αthe␈αnecessary␈αconstraints␈αcan␈αbe␈αlearned␈α
by␈αthe

␈↓ ↓*␈↓system as it runs, and inserted into the proper slots.


␈↓ ↓*␈↓Reexamine␈α∂the␈α∞question:␈α∂"What␈α∞parts␈α∂are␈α∞evaluated,␈α∂and␈α∞in␈α∂what␈α∞order,␈α∂when␈α∞a␈α∂particular␈α∂BEING␈α∞gains

␈↓ ↓*␈↓control?"␈α
This␈αdecision␈α
depends␈αprimarily␈α
on␈α
the␈α␈↓βtypes␈↓␈α
of␈αparts␈α
present␈α
in␈αthe␈α
BEING,␈αnot␈α
on␈α
their␈α␈↓βvalues␈↓.

␈↓ ↓*␈↓But␈α⊃every␈α⊂BEING␈α⊃in␈α⊃a␈α⊂family␈α⊃has␈α⊃the␈α⊂same␈α⊃anatomy,␈α⊃so␈α⊂one␈α⊃single␈α⊃algorithm,␈α⊂located␈α⊃in␈α⊃that␈α⊂family's

␈↓ ↓*␈↓Ordering␈α
part,␈α
can␈α
assemble␈αany␈α
BEING's␈α
parts␈α
into␈α
an␈αexecutable␈α
LISP␈α
function.␈α
Moreover,␈α
this␈αassemby

␈↓ ↓*␈↓can␈α
be␈αdone␈α
when␈αthe␈α
system␈αis␈α
≡rst␈αloaded␈α
(or␈αwhen␈α
a␈αnew␈α
BEING␈αis␈α
≡rst␈αcreated),␈α
and␈αneed␈α
only␈αbe␈α
redone

␈↓ ↓*␈↓for␈αa␈α
BEING␈αwhen␈α
the␈αvalues␈α
of␈αits␈αparts␈α
change.␈αSuch␈α
changes␈αare␈α
rare:␈αexperts␈α
are␈αnot␈αoften␈α
open-minded.

␈↓ ↓*␈↓Thus the family's BEINGs can be compiled into executable LISP functions.




␈↓ ↓*␈↓␈↓ α
␈↓↓␈↓&Aspects of BEINGs Systems␈↓)αβ␈↓


␈↓ ↓*␈↓It␈α
would␈αbe␈α
aesthetically␈α
pleasing␈αto␈α
postulate␈α
that␈αthe␈α
only␈α
entities␈αwhich␈α
exist␈α
are␈αBEINGs.␈α
Since␈αthis␈α
would

␈↓ ↓*␈↓require␈α∞BEINGs'␈α∞parts␈α∞to␈α∞be␈α∞BEINGs,␈α∞hence␈α∞have␈α
parts␈α∞of␈α∞their␈α∞own,␈α∞etc.,␈α∞an␈α∞explosive␈α∞recurrence␈α
would

␈↓ ↓*␈↓occur.␈α To␈αavoid␈αthis,␈αwe␈αset␈αa␈αslightly␈αdi≥erent␈αtack.␈αSuppose␈αthat␈αeach␈αpart␈αwhich␈αhas␈αthe␈αsame␈αname␈αmust

␈↓ ↓*␈↓also␈α∂have␈α∂the␈α∂same␈α∞internal␈α∂structure.␈α∂The␈α∂format␈α∂for␈α∞part␈α∂P␈α∂is␈α∂stored␈α∞in␈α∂the␈α∂Representation␈α∂part␈α∂of␈α∞the

␈↓ ↓*␈↓archetypical␈α∃BEING␈α∀named␈α∃P.␈α∀The␈α∃only␈α∀allowable␈α∃formats␈α∀are␈α∃the␈α∀following:␈α∃an␈α∃opaque␈α∀executable

␈↓ ↓*␈↓expression,␈α∂a␈α∂pointer␈α∂to␈α⊂some␈α∂BEING␈α∂or␈α∂some␈α⊂speci≡c␈α∂part␈α∂of␈α∂a␈α∂BEING,␈α⊂a␈α∂list␈α∂of␈α∂executable␈α⊂forms␈α∂and

␈↓ ↓*␈↓pointers to BEINGs. Notice that there are only three, and that they are all quite simple.
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  90␈↓

␈↓ ↓*␈↓We␈α
shall␈α
demand␈α
that␈α
BEINGs␈α
write␈α
only␈α
new␈α
BEINGs,␈α
never␈α
any␈α
new␈α
functions,␈α
production␈α
systems,␈α
etc.

␈↓ ↓*␈↓The␈αhumans␈αoften␈α
succeeded␈αby␈αdistilling␈α
a␈αtiny␈αspecialization␈α
of␈αtheir␈αexpertise;␈α
the␈αBEINGs␈αwork␈α
similarly.

␈↓ ↓*␈↓In␈α∂the␈α⊂process␈α∂of␈α∂discovery,␈α⊂this␈α∂splitting␈α⊂occurs␈α∂usually␈α∂when␈α⊂some␈α∂subpart␈α∂is␈α⊂more␈α∂interesting␈α⊂than␈α∂the

␈↓ ↓*␈↓whole.␈α In␈αthe␈αprocess␈αof␈αautomatic␈αcode-writing,␈αthis␈αcreation␈αoccurred␈αwhen␈αa␈αBEING␈αknew␈αhow␈αto␈αwrite␈αa

␈↓ ↓*␈↓fast,␈αshort,␈αspecialized,␈αstreamlined␈α
version␈αof␈αitself␈αwhich␈αwas␈α
capable␈αof␈αexecuting␈αsome␈α
speci≡c␈αsubprocess

␈↓ ↓*␈↓used in the ≡nal "target" concept formation program.


␈↓ ↓*␈↓To␈α
clarify␈α
what␈α
BEINGs␈α
are␈α
and␈α
are␈α
not,␈α
they␈α
are␈α
contrasted␈α
with␈α
some␈α
other␈α
ideas.␈α
BEINGs␈α
linearly␈α
but

␈↓ ↓*␈↓ine≠cently␈α
subsume␈α∞such␈α
constructions␈α
as␈α∞demons,␈α
functions,␈α
and␈α∞assertions␈α
in␈α
an␈α∞associative␈α
data␈α∞base␈α
(in

␈↓ ↓*␈↓the␈α∞earlier␈α∞papers,␈α∂brief␈α∞demonstrations␈α∞were␈α∞provided).␈α∂ FRAMES␈α∞are␈α∞su≠ciently␈α∞amorphous␈α∂to␈α∞subsume

␈↓ ↓*␈↓BEINGs.␈αIn␈αphilosophy,␈αFRAMES␈αare␈αmeant␈αto␈αmodel␈αperception,␈αand␈αintentionally␈αrely␈αon␈αimplicit␈αdefault

␈↓ ↓*␈↓values;␈α
BEINGs␈αavoid␈α
making␈αdecisions␈α
without␈αfull␈α
awareness␈αof␈α
the␈αjusti≡cation.␈α
This␈αis␈α
also␈αthe␈α
di≥erence

␈↓ ↓*␈↓between␈α∪HACKER␈α∪and␈α∪PUP6,␈α∀the␈α∪≡rst␈α∪experimental␈α∪pool␈α∀of␈α∪BEINGs.␈α∪Since␈α∪PUP6␈α∀wrote␈α∪structured

␈↓ ↓*␈↓programs,␈α∀it␈α∃should␈α∀be␈α∃distinguished␈α∀from␈α∃macro␈α∀expansion.␈α∃Macro␈α∀procedures␈α∃expand␈α∀mechanically:

␈↓ ↓*␈↓␈↓↓expand(sequence␈α  m␈↓#v1␈↓#␈α
 m␈↓#v2␈↓#)␈α=␈α
(sequence␈α expand(m␈↓#v1␈↓#)␈α
 expand(m␈↓#v2␈↓#)))␈↓.␈αBEINGs␈α
could␈αuse␈αinformation␈α
gleaned

␈↓ ↓*␈↓during␈α∞expansion␈α∞of␈α∞m␈↓#v1␈↓#␈α∞to␈α∞improve␈α∞the␈α∞way␈α∞m␈↓#v2␈↓#␈α∞was␈α∞handled.␈α∞ ACTORs,␈α∞unlike␈α∞BEINGs,␈α∞have␈α∞no␈α∞≡xed

␈↓ ↓*␈↓structure␈αimposed,␈αand␈αdo␈α
not␈αbroadcast␈αtheir␈αmessages␈α
(they␈α specify␈αwho␈αgets␈α
each␈αmessage,␈αby␈αname,␈α
to␈αa

␈↓ ↓*␈↓bureaucracy).


␈↓ ↓*␈↓The␈αperformance␈α
of␈αthe␈α
BEINGs␈αrepresentation␈α
itself␈αin␈α
PUP6␈αis␈α
mixed.␈α Two␈α
advantages␈αwere␈α
hoped␈αfor

␈↓ ↓*␈↓by␈αusing␈αa␈αuniform␈αset␈αof␈αBEING␈αparts.␈α Addition␈αof␈αnew␈αBEINGs␈αto␈αthe␈αpool␈αwas␈αnot␈αeasy␈α(for␈αuntrained

␈↓ ↓*␈↓users)␈α⊃but␈α⊃communication␈α⊃among␈α⊂BEINGs␈α⊃␈↓βwas␈↓␈α⊃easy␈α⊃(fast,␈α⊃natural).␈α⊂Two␈α⊃advantages␈α⊃were␈α⊃hoped␈α⊃for␈α⊂by

␈↓ ↓*␈↓keeping␈α⊂the␈α⊃BEINGs␈α⊂highly␈α⊃structured.␈α⊂ The␈α⊃interactions␈α⊂(especially␈α⊂with␈α⊃the␈α⊂user)␈α⊃were␈α⊂brittle,␈α⊃but␈α⊂the

␈↓ ↓*␈↓complex tasks put to the system ␈↓βwere␈↓ successfully completed.


␈↓ ↓*␈↓The␈α∪crippling␈α∀problems␈α∪are␈α∀seen␈α∪to␈α∀be␈α∪with␈α∪user-system␈α∀communication,␈α∪not␈α∀with␈α∪the␈α∀BEINGs␈α∪ideas
␈↓ ↓*␈↓␈↓εAutomated Math Theory Formation␈↓ ε<Doug Lenat␈↓ 
_January 7, 1976     page  91␈↓

␈↓ ↓*␈↓themselves.␈α
 Sophisticated,␈α
bug-free␈α
programs␈α
␈↓βwere␈↓␈α
generated,␈αafter␈α
hours␈α
of␈α
fairly␈α
high␈α
level␈α
dialogue␈αwith

␈↓ ↓*␈↓an␈α⊂active␈α⊃user,␈α⊂after␈α⊃tens␈α⊂of␈α⊂thousands␈α⊃of␈α⊂messages␈α⊃passed␈α⊂among␈α⊂the␈α⊃BEINGs.␈α⊂ Part␈α⊃of␈α⊂this␈α⊃success␈α⊂is

␈↓ ↓*␈↓attributed␈α∂to␈α∂distributing␈α⊂the␈α∂responsibility␈α∂for␈α⊂writing␈α∂code␈α∂and␈α∂for␈α⊂recognizing␈α∂relevance,␈α∂to␈α⊂a␈α∂hundred

␈↓ ↓*␈↓entities,␈α∞rather␈α∞than␈α∞having␈α∞a␈α∞few␈α∞central␈α∞monitors␈α∞worry␈α∞about␈α∞everything.␈α∞ The␈α∞standardization␈α∂of␈α∞parts

␈↓ ↓*␈↓made␈α≡lling␈αin␈αthe␈αBEINGs'␈αcontents␈αfairly␈αpainless,␈αboth␈αfor␈αthe␈αauthor␈αand␈αfor␈αBEINGs␈αwho␈αhad␈αto␈αwrite

␈↓ ↓*␈↓new BEINGs.


␈↓ ↓*␈↓All␈α
this␈α
suggests␈α
two␈α
possible␈α
continuations,␈α
both␈α
of␈α
which␈α
are␈α
underway␈α
here␈α
at␈α
the␈α
Stanford␈α
AI␈α
Lab.␈α
One␈α
is

␈↓ ↓*␈↓to␈α⊃rethink␈α⊃the␈α⊃communication␈α∩problems,␈α⊃and␈α⊃develop␈α⊃a␈α⊃new␈α∩system␈α⊃for␈α⊃the␈α⊃concept␈α∩formation␈α⊃program

␈↓ ↓*␈↓synthesis␈α∂task.␈α∞The␈α∂earliest␈α∞programs␈α∂by␈α∂our␈α∞Automatic␈α∂Programming␈α∞group␈α∂had␈α∞the␈α∂goal␈α∂"synthesize␈α∞the

␈↓ ↓*␈↓target␈α∩program␈α∩somehow";␈α∩the␈α∩later␈α∩PUP1-6␈α∩research␈α⊃insisted␈α∩on␈α∩"getting␈α∩the␈α∩target␈α∩program␈α∩by␈α⊃going

␈↓ ↓*␈↓through␈αone␈α␈↓βproper␈↓␈αsequence␈αof␈αreasoning␈αsteps";␈αthe␈αgroup's␈αproposed␈αcontinuation␈αwants␈αseveral␈αuntrained

␈↓ ↓*␈↓users to  succeed by many di≥erent "proper" routes.


␈↓ ↓*␈↓This␈α
document␈α∞is␈α
proposes␈α
an␈α∞alternative␈α
direction␈α∞for␈α
research␈α
e≥ort.␈α∞ This␈α
other␈α
way␈α∞of␈α
continuing␈α∞is␈α
to

␈↓ ↓*␈↓≡nd␈αa␈αtask␈αwhere␈αBEINGs␈αare␈αwell-suited,␈αwhere␈αthe␈αproblems␈αencountered␈αin␈αPUP6␈αwon't␈αrecur.␈α
What␈α␈↓βare␈↓

␈↓ ↓*␈↓BEINGs␈αgood␈α
for?␈α The␈α
idea␈αof␈α
a␈α≡xed␈α
set␈αof␈αparts␈α
(which␈αdistinguishes␈α
them␈αfrom␈α
ACTORs)␈αis␈α
useful␈αif

␈↓ ↓*␈↓the␈αmass␈αof␈αknowledge␈αis␈αtoo␈αhuge␈αfor␈αone␈αindividual␈αto␈αkeep␈α"on␈αtop"␈αof.␈α It␈αthen␈αshould␈αbe␈αorganized␈αin␈αa

␈↓ ↓*␈↓very␈α∞uniform␈α∞way␈α
(to␈α∞simplify␈α∞preparing␈α
it␈α∞for␈α∞storage),␈α
yet␈α∞it␈α∞must␈α
also␈α∞be␈α∞highly␈α
structured␈α∞(to␈α∞speed␈α
up

␈↓ ↓*␈↓retrieval).␈αA␈α≡nal␈αideal␈αwould␈αbe␈αto␈α≡nd␈αa␈αdomain␈αwhere␈αslightly-trained␈αusers␈αcould␈αwork␈αnaturally,␈αwithout

␈↓ ↓*␈↓(them␈α␈↓βor␈↓␈αus)␈α
having␈αto␈αbattle␈α
the␈αstaggering␈αcomplexities␈α
of␈αnatural␈αlanguage␈α
handling.␈αFor␈αthese␈αreasons,␈α
the

␈↓ ↓*␈↓author␈αhas␈αchosen␈α␈↓βfundamental␈αmathematics␈↓␈αas␈αa␈α
task␈αdomain.␈α BEINGs␈αare␈αbig␈αand␈αslow,␈αbut␈α
valuable␈αfor

␈↓ ↓*␈↓organizing␈α
knowledge␈αin␈α
ways␈α
meaningful␈αto␈α
how␈α
it␈αwill␈α
be␈α
used.␈αIn␈α
this␈α
proposed␈αsystem,␈α
BEINGs␈α
will␈αbe

␈↓ ↓*␈↓one -- but not the only -- internal mechanism for representing and manipulating knowledge.