perm filename DBL.XGP[NSF,DBL] blob sn#158481 filedate 1975-05-09 generic text, type T, neo UTF8
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␈↓ ↓H␈↓β␈↓ ¬5NSF PROPOSAL
␈↓ ↓H␈↓β␈↓ ∧*Program-Understanding Group

␈↓ ↓H␈↓β␈↓ αtvery (sigh), very ROUGH DRAFT: May 9, 1975
␈↓ ↓H␈↓β␈↓ ¬	(additons from DBL)
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓β␈↓ ∧rTABLE OF CONTENTS

␈↓ ↓H␈↓α␈↓ αλSection␈↓ λxPage
␈↓ ↓H␈↓␈↓ αλ           Initial State of AM's Knowledge␈↓ λx5
␈↓ ↓H␈↓␈↓ αλ           The Goal: AM running␈↓ λx6
␈↓ ↓H␈↓␈↓ αλ           Books and Memos␈↓ λx11
␈↓ ↓H␈↓␈↓ αλ           Articles␈↓ λx18









































␈↓ ↓H␈↓␈↓ ε0␈↓↓page i␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓β␈↓ ¬∀FIXUPS to be made















































␈↓ ↓H␈↓␈↓ ε+␈↓↓page ii␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓␈↓ α_IV.2 AM (p38)


␈↓ ↓H␈↓␈↓ α_How␈α
might␈α∞one␈α
go␈α∞about␈α
automating␈α∞the␈α
acquisition␈α∞of␈α
domain␈α∞knowledge?␈α
 First,␈α∞a␈α
speci≡c

␈↓ ↓H␈↓domain␈α∞must␈α∂be␈α∞chosen.␈α∂The␈α∞character␈α∂of␈α∞its␈α∞knowledge␈α∂must␈α∞be␈α∂studied␈α∞and␈α∂perhaps␈α∞organized.

␈↓ ↓H␈↓Similarly,␈αone␈αwould␈αexamine␈αand␈αcategorize␈αthe␈αmethods␈αtypically␈αused␈αto␈αgain␈αnew␈αinformation␈αin

␈↓ ↓H␈↓this␈αdomain.␈α
 Next␈αwould␈α
come␈αa␈α
di≠cult␈αbut␈αcrucial␈α
step:␈αformulate␈α
a␈αmodel␈α
of␈αhow␈αnew␈α
knowledge

␈↓ ↓H␈↓is␈α∂acquired␈α∞in␈α∂this␈α∞domain.␈α∂ The␈α∂≡nal␈α∞task␈α∂would␈α∞be␈α∂the␈α∂realization␈α∞of␈α∂the␈α∞model␈α∂in␈α∂a␈α∞computer

␈↓ ↓H␈↓program;␈α~this␈α~sticky␈α~problem␈α~would␈α~involve␈α~choosing␈α~representations,␈α~simulating␈α~primitive

␈↓ ↓H␈↓constituents of the model, replacing vague hints in the model with precise algorithms, etc.


␈↓ ↓H␈↓␈↓ α_AM␈α⊃is␈α⊂an␈α⊃attempt␈α⊂at␈α⊃such␈α⊃a␈α⊂knowledge␈α⊃acquisition␈α⊂system,␈α⊃for␈α⊂the␈α⊃domain␈α⊃of␈α⊂elementary

␈↓ ↓H␈↓mathematics.␈α⊃ The␈α⊃problem␈α⊃of␈α⊃implementing␈α⊃the␈α⊃general␈α⊃model␈α⊃as␈α⊃a␈α⊃running␈α⊃program␈α⊃is␈α⊂being

␈↓ ↓H␈↓attacked␈α∃with␈α∀program-understanding␈α∃methods.␈α∃ The␈α∀paragraphs␈α∃below␈α∀describe␈α∃the␈α∃kinds␈α∀of

␈↓ ↓H␈↓concepts↑*␈α∞AM␈α∞must␈α∞deal␈α∞with,␈α
how␈α∞new␈α∞ones␈α∞are␈α∞obtained,␈α
a␈α∞tentative␈α∞model␈α∞for␈α∞enlarging␈α
one's

␈↓ ↓H␈↓repertoire␈α∞of␈α∞mathematical␈α∞concepts,␈α∂and␈α∞a␈α∞description␈α∞of␈α∂how␈α∞AM␈α∞is␈α∞designed.␈α∂ The␈α∞information

␈↓ ↓H␈↓that␈α↔AM␈α↔begins␈α↔with,␈α_and␈α↔how␈α↔knowledge␈α↔is␈α↔represented,␈α_is␈α↔of␈α↔some␈α↔interest.␈α_Since␈α↔that

␈↓ ↓H␈↓representation␈α⊂is␈α∂procedural,␈α⊂acquiring␈α∂a␈α⊂new␈α∂concept␈α⊂really␈α∂means␈α⊂writing␈α∂a␈α⊂new␈α⊂little␈α∂program.

␈↓ ↓H␈↓Although␈αthere␈αis␈αno␈αpredetermined␈α"goal",␈αwe␈αcan␈αsketch␈αsome␈αof␈αthe␈αconcepts␈αwhich␈αwe␈αhope␈αAM

␈↓ ↓H␈↓will␈α∀acquire.␈α∀ "Acquisition"␈α∀has␈α∀connotations␈α∀ranging␈α∀from␈α∀information␈α∀storage␈α∀to␈α∀learning␈α∀to

␈↓ ↓H␈↓independent␈α∩discovery;␈α∩AM␈α∩should␈α⊃in␈α∩fact␈α∩exhibit␈α∩several␈α⊃levels␈α∩of␈α∩sophistication␈α∩in␈α∩how␈α⊃new

␈↓ ↓H␈↓concepts␈αare␈αintroduced.␈α Appendix␈α1␈αprovides␈αdetailed␈αsamples␈αof␈αbehaviors␈αwe␈αwould␈αbe␈αhappy␈αto

␈↓ ↓H␈↓elicit from AM.


␈↓ ↓H␈↓␈↓ α_Mathematical␈α∞knowledge␈α
is␈α∞organized␈α
into␈α∞a␈α
complex␈α∞network␈α
of␈α∞subdomains␈α∞called␈α
␈↓↓theories␈↓.

␈↓ ↓H␈↓Each␈αtheory␈αis␈αbuilt␈αon␈αa␈αunique␈αfoundation␈αof␈αprimitive␈αconcepts␈α(structures␈αand␈αoperations)␈αwhich



␈↓ ↓H␈↓␈↓ ε-␈↓↓page 1␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓are␈α∂implicitly␈α∂de≡ned␈α∂by␈α∂a␈α∂set␈α∂of␈α∂constraints␈α∂(called␈α∂␈↓↓axioms␈↓).␈α∂From␈α∂this␈α∂␈↓↓basis␈↓,␈α∂implications␈α⊂can␈α∂be

␈↓ ↓H␈↓drawn␈α⊃(called␈α⊃␈↓↓theorems␈↓)␈α⊃and␈α⊃new␈α⊃conventions␈α⊃can␈α⊃be␈α⊃proclaimed␈α⊃(called␈α⊃␈↓↓de≡nitions␈↓).␈α⊂ Additional

␈↓ ↓H␈↓mathematical␈αknowledge␈αexists␈αto␈α
enable␈αtheories␈αto␈α␈↓βdevelop␈↓:␈αsome␈α
of␈αthis␈αis␈αquite␈α
rigorous␈α(e.g.,

␈↓ ↓H␈↓techniques␈α∞for␈α∞proof),␈α∞but␈α∞much␈α∂of␈α∞it␈α∞is␈α∞informal␈α∞(e.g.,␈α∂hints␈α∞for␈α∞when␈α∞some␈α∞relationship␈α∂is␈α∞worth

␈↓ ↓H␈↓singling␈α
out␈α
as␈α
a␈α
new␈α
concept,␈α
via␈α
a␈α
new␈α
de≡nition).␈α
 Some␈α
of␈α
these␈α
"mathematical␈α∞techniques"␈α
are

␈↓ ↓H␈↓speci≡c␈α
to␈α∞the␈α
theory␈α∞involved␈α
(e.g.,␈α∞relaxation␈α
methods),␈α
and␈α∞some␈α
are␈α∞quite␈α
general␈α∞(e.g.,␈α
working

␈↓ ↓H␈↓backwards).


␈↓ ↓H␈↓␈↓ α_Any␈α
knowledge␈αacquisition␈α
system␈αfor␈α
any␈αdomain␈α
of␈αmathematics␈α
should␈αhave␈α
the␈α"general"

␈↓ ↓H␈↓mathematical␈α
techniques␈αmentioned␈α
above.␈α
What␈αelse␈α
must␈αbe␈α
present?␈α
 For␈αeach␈α
theory␈α
which␈αthe

␈↓ ↓H␈↓system␈αis␈αsupposed␈αto␈αknow␈αinitially,␈αit␈αmust␈αhave␈αspeci≡c␈αinformation␈αabout␈αthat␈αtheory's␈α
primitives,

␈↓ ↓H␈↓axioms,␈α∪known␈α∪theorems,␈α∩de≡nitions,␈α∪specialized␈α∪methods,␈α∩etc.␈α∪ In␈α∪addition,␈α∪many␈α∩mathematical

␈↓ ↓H␈↓theories␈α∞are␈α∞built␈α∞upon␈α∞others;␈α∞if␈α∞such␈α∞a␈α∞theory␈α
is␈α∞to␈α∞be␈α∞treated,␈α∞all␈α∞the␈α∞prerequisite␈α∞ones␈α∞must␈α
be

␈↓ ↓H␈↓mastered.␈α
 For␈α
example,␈αone␈α
should␈α
learn␈αabout␈α
set␈α
theory␈α
before␈αstudying␈α
topology.␈α
 So␈αthe␈α
tentative

␈↓ ↓H␈↓speci≡cation␈α∂for␈α∂AM␈α∂calls␈α⊂for␈α∂a␈α∂repertoire␈α∂of␈α⊂general␈α∂techniques␈α∂(proof␈α∂strategies,␈α⊂weak␈α∂problem-

␈↓ ↓H␈↓solving␈αmethods),␈αplus␈αdetails␈αof␈αa␈αfew␈αfundamental␈αtheories␈α(naive␈αset␈αtheory,␈αrelations,␈αlogic).␈α AM

␈↓ ↓H␈↓should␈αeventually␈αacquire␈αconcepts␈αin␈α≡elds␈αwhich␈αdepend␈αon␈αthese,␈αsuch␈αas␈αarithmetic,␈αalgebra,␈αand

␈↓ ↓H␈↓theorem-proving.


␈↓ ↓H␈↓␈↓ α_Given␈α⊂a␈α∂description␈α⊂of␈α⊂the␈α∂mathematical␈α⊂concepts␈α∂we␈α⊂want␈α⊂AM␈α∂to␈α⊂deal␈α∂with,␈α⊂and␈α⊂what␈α∂it

␈↓ ↓H␈↓means␈αto␈αacquire␈αand␈αdevelop␈αthose␈αconcepts,␈αwe␈αnow␈αface␈αthe␈αdi≠cult␈αtasks␈αof␈αproposing␈αa␈αsuitable

␈↓ ↓H␈↓model␈α∀for␈α∀acquiring␈α∃new␈α∀mathematical␈α∀concepts,␈α∀and␈α∃implementing␈α∀that␈α∀model␈α∀in␈α∃a␈α∀running

␈↓ ↓H␈↓computer␈α∀system.␈α∃ After␈α∀much␈α∀reading↑*↑*␈α∃and␈α∀introspecting,␈α∀one␈α∃tentative␈α∀model␈α∃was␈α∀pieced

␈↓ ↓H␈↓together:
␈↓ ↓H␈↓¬⊂αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ X⊃


␈↓ ↓H␈↓␈↓ ε-␈↓↓page 2␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓
␈↓ ↓h1.␈αThe␈αorder␈αof␈αevents␈αin␈αa␈αtypical␈αmathematical␈αinvestigation␈αis:␈α(a)␈αOBSERVE:␈αThe␈αobservation␈αis␈αeither␈αof␈αreality␈αor␈αof␈αan
␈↓ ↓H␈↓
␈↓ α_analogous,␈α
already-established␈α
mathematical␈α
theory.␈α
 (b)␈α
NOTICE␈α
REGULARITY:␈α
Perceive␈α
some␈α
patterns,␈α
some␈αinteresting
␈↓ ↓H␈↓
␈↓ α_relationships␈αthat␈αappear␈αto␈αhold␈αsometimes.␈α (c)␈αFORMALIZE:␈αDecide␈αon␈αsome␈αof␈αthe␈αobjects,␈αoperators,␈αde≡nitions,␈αand
␈↓ ↓H␈↓
␈↓ α_statements␈αthat␈αthe␈αtheory␈αwill␈αcontain.␈α (d)␈αFINALIZE:␈αDecide␈αwhich␈αconcepts␈αare␈αto␈αbe␈αprimitve␈αand␈αwhich␈αaren't;␈α
decide
␈↓ ↓H␈↓
␈↓ α_which␈αstatements␈α
will␈αbe␈α
considered␈αaxioms,␈α
and␈αensure␈α
that␈αthe␈α
others␈αcan␈α
in␈αfact␈α
be␈αderived␈α
from␈αthem.␈α (e)␈α
DEVELOP:
␈↓ ↓H␈↓
␈↓ α_What␈αλadditional␈αλtheorems␈αλcan␈α	be␈αλproven␈αλfrom␈αλthis␈α	formal␈αλsystem;␈αλdo␈αλthey␈αλcorrespond␈α	to␈αλobservable␈αλphenomena␈αλin␈α	the␈αλdomain
␈↓ ↓H␈↓
␈↓ α_which␈α	motivated␈αλthis␈α	new␈αλtheory?␈α	 When␈αλnew␈α	observations␈α	are␈αλmade␈α	in␈αλthat␈α	motivating␈αλdomain,␈α	can␈αλthey␈α	be␈α	naturally␈αλphrased
␈↓ ↓H␈↓
␈↓ α_as formal statements in this theory; and if so, are they provable from the existing axioms and thoerems?
␈↓ ↓H␈↓
␈↓ ↓h2.␈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.␈α
 The␈α	key␈α	to␈α
keeping␈α	this
␈↓ ↓H␈↓
␈↓ α_from␈αbecoming␈αa␈α
blind,␈αexplosive␈αsearch␈α
is␈αthe␈αproper␈α
use␈αof␈αevaluation␈α
criteria.␈αThat␈αis,␈α
one␈αmust␈αconstantly␈αchoose␈α
the
␈↓ ↓H␈↓
␈↓ α_most interesting, aesthetically pleasing, useful,... alternative available.
␈↓ ↓H␈↓
␈↓ ↓h3.␈α
But␈α
many␈αof␈α
those␈α
criteria␈αare␈α
quite␈α
opposite␈α(e.g.,␈α
one␈α
often␈αmust␈α
sacri≡ce␈α
elegance␈αfor␈α
utility,␈α
interestingness␈αfor␈α
safety,
␈↓ ↓H␈↓
␈↓ α_etc.).␈α
How␈α
should␈αone␈α
weight␈α
these␈αfeatures␈α
when␈α
deciding␈αwhat␈α
to␈α
do␈αnext␈α
during␈α
an␈αinvestigation?␈α
 We␈α
believe␈α(and␈α
one
␈↓ ↓H␈↓
␈↓ α_goal␈αof␈αAM␈αis␈αto␈αtest)␈αthat␈αthe␈αnon-formal␈αcriteria␈α(aesthetics,␈αinterestingness,␈αintuitive␈αclarity,␈αutility,␈αanalogy,␈αempirical
␈↓ ↓H␈↓
␈↓ α_evidence)␈α
are␈α
much␈α
more␈α
important␈α
than␈α
formal␈α
deductive␈α
methods␈α
in␈α
developing␈α
mathematically␈α
worthwhile␈α
theories,␈αand␈α
in
␈↓ ↓H␈↓
␈↓ α_avoiding␈α
barren␈αdiversions.␈α
 Among␈α
the␈αsubjective␈α
criteria,␈α
the␈αorder␈α
listed␈αabove␈α
is␈α
roughly␈αtheir␈α
order␈α
of␈αimportance.
␈↓ ↓H␈↓
␈↓ α_However,␈αAM␈αshould␈αhave␈αa␈αdynamically␈αvariable␈α"orientation",␈αwhich␈αunder␈αcertain␈αcircumstances␈αmight␈αinduce␈αit␈αto␈αseek
␈↓ ↓H␈↓
␈↓ α_safety (e.g., utility) rather than uncertainty (e.g., completing an analogy).
␈↓ ↓H␈↓
␈↓ ↓h4.␈α	The␈α	above␈αλcriteria␈α	are␈α	virtually␈α	the␈αλsame␈α	in␈α	all␈αλdomains␈α	of␈α	mathematics,␈α	and␈αλat␈α	all␈α	levels.␈αλEach␈α	factor␈α	encourages␈α	some␈αλpursuits
␈↓ ↓H␈↓
␈↓ α_and␈αλdiscourages␈αλothers.␈αλ It␈α	is␈αλhoped␈αλthat␈αλno␈αλmodi≡cations␈α	need␈αλbe␈αλmade␈αλto␈αλAM's␈α	judgmental␈αλscheme,␈αλas␈αλAM␈αλacquires␈α	more␈αλand
␈↓ ↓H␈↓
␈↓ α_more new concepts.
␈↓ ↓H␈↓
␈↓ ↓h5.␈α	For␈α
true␈α	understanding,␈α
AM␈α	should␈α	relate␈α
to␈α	each␈α
concept␈α	in␈α
several␈α	ways:␈α	declarative␈α
(de≡nition),␈α	operational␈α
(how␈α	to␈α
use␈α	it),
␈↓ ↓H␈↓
␈↓ α_demonic␈αλ(recognizing␈αλwhen␈α	it␈αλis␈αλrelevant),␈α	exemplary␈αλ(especially␈αλboundary␈α	examples),␈αλand␈αλintuitive␈α	(simulated␈αλimage␈αλof␈α	a␈αλreal-
␈↓ ↓H␈↓
␈↓ α_world interpretation).
␈↓ ↓H␈↓
␈↓ ↓h6.␈α
Progress␈α	in␈α
␈↓&any␈↓)αβ␈α
≡eld␈α	of␈α
math␈α
demands␈α	much␈α
intuition␈α
(and␈α	some␈α
formal␈α
knowledge)␈α	of␈α
␈↓&many␈↓)αβ␈α
di≥erent␈α	mathematical␈α
≡elds.␈α
So␈α	a
␈↓ ↓H␈↓
␈↓ α_broad,␈α
universal␈α
core␈α
of␈α
intuition␈α
must␈α
be␈α
established␈αbefore␈α
any␈α
single␈α
theory␈α
can␈α
meaningfully␈α
be␈α
developed.␈α Intuition␈α
is
␈↓ ↓H␈↓
␈↓ α_contrasted␈αλwith␈αλmore␈α	formal␈αλrepresentations␈αλby␈αλthe␈α	fact␈αλthat␈αλit␈αλis␈α	opaque␈αλ(AM␈αλcannot␈αλintrospect␈α	to␈αλdetermine␈αλhow␈α	the␈αλresult
␈↓ ↓H␈↓
␈↓ α_is produced) and fallable.
␈↓ ↓H␈↓¬%αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα␈↓ X$


␈↓ ↓H␈↓␈↓ α_The␈α≡nal␈αstep␈α--␈αthe␈αmain␈αone␈α--␈αis␈αto␈αrealize␈αthis␈αmodel␈αin␈αa␈αprogram,␈αAM.␈α AM␈αwould␈αthen

␈↓ ↓H␈↓have␈α
the␈α
ability␈α
to␈α
acquire␈α
to␈α
new␈α
mathematical␈α
concepts,␈α
in␈α
accord␈α
with␈α
that␈α
model.␈α
This␈α
acquisition

␈↓ ↓H␈↓might␈α
be␈αvia␈α
dialogue,␈αwhere␈α
a␈α
human␈αuser␈α
either␈αtells␈α
AM␈α
about␈αnew␈α
concepts,␈αor␈α
else␈α
where␈αthe

␈↓ ↓H␈↓user␈α∂merely␈α⊂guides␈α∂AM␈α∂as␈α⊂it␈α∂compounds␈α∂its␈α⊂existing␈α∂concepts␈α∂into␈α⊂new␈α∂ones.␈α∂ The␈α⊂major␈α∂design

␈↓ ↓H␈↓details␈α⊃have␈α⊃been␈α⊂tentatively␈α⊃decided␈α⊃upon,␈α⊂and␈α⊃they␈α⊃are␈α⊂based␈α⊃on␈α⊃recently-developed␈α⊂program-

␈↓ ↓H␈↓understanding techniques:


␈↓ ↓H␈↓␈↓ α_1.␈αRepresentation␈αof␈αknowledge␈αin␈αthe␈αsystem:␈αAM␈αwill␈αrepresent␈αeach␈αconcept␈αas␈αa␈αbundle␈αof

␈↓ ↓H␈↓facets␈α⊂(DEFINITION,␈α⊂INTUITION,␈α∂RECOGNITION,␈α⊂INTERESTINGNESS,...),␈α⊂and␈α⊂each␈α∂facet

␈↓ ↓H␈↓will␈α∞be␈α∂stored␈α∞internally␈α∞as␈α∂a␈α∞little␈α∞program.␈α∂ Each␈α∞concept␈α∞will␈α∂have␈α∞precisely␈α∞the␈α∂same␈α∞set␈α∂of␈α∞25

␈↓ ↓H␈↓facets.␈α∞ This␈α
enables␈α∞us␈α
to␈α∞assemble,␈α∞in␈α
advance,␈α∞a␈α
body␈α∞of␈α
knowledge␈α∞(called␈α∞␈↓↓strategic␈↓␈α
knowledge)

␈↓ ↓H␈↓about each facet. This is the same as the program-understanding representation scheme PUP6 used.



␈↓ ↓H␈↓␈↓ ε-␈↓↓page 3␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓␈↓ α_2.␈αControl␈αin␈αthe␈αsystem:␈αAs␈αAM␈αruns,␈αat␈αeach␈αmoment,␈αeach␈αconcept␈αwill␈αhave␈αonly␈αa␈αfew␈αof

␈↓ ↓H␈↓its␈α
facet␈α
programs␈α
≡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

␈↓ ↓H␈↓control␈α
structure␈α∞is␈α
to␈α∞repeatedly␈α
choose␈α∞some␈α
facet␈α∞of␈α
some␈α∞concept,␈α
and␈α∞then␈α
use␈α∞the␈α
appropriate

␈↓ ↓H␈↓strategic␈αknowledge␈α
to␈α≡ll␈αin␈α
that␈αmissing␈αprogram.␈α
 The␈αstrategic␈αknowledge␈α
will␈αtypically␈αaccess␈α
and

␈↓ ↓H␈↓run␈αmany␈αother␈αfacet␈αprograms␈αfrom␈αmany␈αother␈αconcepts.␈α In␈αthe␈αcourse␈αof␈αthis,␈αnew␈αconcepts␈αmay

␈↓ ↓H␈↓be␈α∞constructed␈α
and␈α∞deemed␈α
worth␈α∞giving␈α∞names␈α
to.␈α∞ Whenever␈α
this␈α∞happens,␈α
the␈α∞new␈α∞concept␈α
has

␈↓ ↓H␈↓almost␈αall␈αits␈αfacets␈αblank,␈αwhich␈αmeans␈αAM␈αnow␈αhas␈αabout␈α25␈αspeci≡c␈αtasks␈αto␈αeventually␈αattend␈αto

␈↓ ↓H␈↓(if␈α∂they're␈α∂deemed␈α∞interesting␈α∂enough).␈α∂So␈α∞AM␈α∂should␈α∂never␈α∞run␈α∂out␈α∂of␈α∞things␈α∂to␈α∂do;␈α∂rather,␈α∞the

␈↓ ↓H␈↓number␈αof␈αpossible␈αtasks␈αkeeps␈αgrowing␈αrapidly.␈αOne␈αreason␈αfor␈αactually␈αprogramming␈αAM␈αis␈αto␈αsee

␈↓ ↓H␈↓how sophiticated the judgmental criteria must be to control this growth.


␈↓ ↓H␈↓␈↓ α_3.␈α∂Program-writing:␈α⊂Recall␈α∂that␈α∂each␈α⊂facet␈α∂is␈α∂a␈α⊂little␈α∂program.␈α∂So␈α⊂"≡lling␈α∂in␈α∂a␈α⊂facet"␈α∂means

␈↓ ↓H␈↓synthesizing␈αa␈αprogram␈αwhich␈αmeets␈αthe␈αrequirements.␈αFor␈αfacet␈αF␈αof␈αconcept␈αC,␈αthis␈αrequirement␈αis

␈↓ ↓H␈↓that␈αthe␈αprogram␈αbe␈αable␈αto␈αanswer␈αquestions␈αabout␈αF␈αwhich␈αmight␈αbe␈αput␈αto␈αC.␈αThe␈α
know-how␈αto

␈↓ ↓H␈↓write␈α∃this␈α∃program␈α∃is␈α∃contained␈α∃in␈α∃the␈α∃strategic␈α∃knowledge␈α∃associated␈α∃with␈α∃F.␈α⊗The␈α∃strategic

␈↓ ↓H␈↓knowledge␈α∂is␈α∞thus␈α∂also␈α∂a␈α∞program;␈α∂its␈α∂"argument"␈α∞in␈α∂this␈α∂case␈α∞would␈α∂be␈α∂C␈α∞(and␈α∂implicity,␈α∂all␈α∞the

␈↓ ↓H␈↓existing␈αconcepts),␈αand␈αits␈α"output"␈αwould␈αbe␈αa␈αlittle␈αprogram␈αwhich␈αwould␈αbe␈α≡lled␈αin␈αas␈αfacet␈αF␈αof

␈↓ ↓H␈↓concept␈α
C.␈α All␈α
the␈αtechniques␈α
of␈αprogram-␈↓↓understanding␈↓␈α
are␈αrequired␈α
to␈αinterpret␈α
the␈α
argument␈αC

␈↓ ↓H␈↓meaningfully.␈α∩ All␈α∩the␈α∩techniques␈α⊃of␈α∩automatic␈α∩program␈α∩␈↓↓synthesis␈↓␈α⊃are␈α∩required␈α∩to␈α∩assemble␈α⊃the

␈↓ ↓H␈↓output program correctly.


␈↓ ↓H␈↓␈↓ α_4.␈α∂AM␈α∞is␈α∂interactive:␈α∞AM␈α∂informs␈α∞a␈α∂human␈α∞user␈α∂of␈α∞the␈α∂things␈α∞it␈α∂≡nds␈α∞which␈α∂it␈α∂thinks␈α∞are

␈↓ ↓H␈↓interesting.␈α The␈αuser␈αcan␈αinterrupt␈αAM␈αand␈αinterrogate␈αit,␈αredirect␈αits␈αenergies,␈αand␈αso␈αon.␈αSince␈α
the

␈↓ ↓H␈↓user␈α∩has␈α∩several␈α∩roles,␈α∩AM␈α∩should␈α⊃have␈α∩several␈α∩languages:␈α∩traditional␈α∩math␈α∩notation,␈α⊃textbook

␈↓ ↓H␈↓English, formal (e.g. AUTOMATH or predicate calculus), pictorial (simulated visual intuitions), etc.


␈↓ ↓H␈↓␈↓ ε-␈↓↓page 4␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓αInitial State of AM's Knowledge


␈↓ ↓H␈↓␈↓ α_1.␈αRange␈αof␈αconcepts␈αprovided:␈αAM␈αwill␈αbe␈αgiven␈αstrategic␈αinformation␈αfor␈αeach␈αkind␈αof␈αfacet

␈↓ ↓H␈↓that␈α∞a␈α∂concept␈α∞is␈α∞permitted␈α∂to␈α∞have,␈α∂and␈α∞will␈α∞be␈α∂given␈α∞some␈α∞detailed␈α∂information␈α∞about␈α∂the␈α∞most

␈↓ ↓H␈↓universal mathematical concepts.  A few of these speci≡c concepts are: ␈↓↓Objects␈↓

␈↓ ↓H␈↓
(like␈α	Ordered-pair,␈α	Variable,␈α	List-structure,␈α
Axiom);␈α	␈↓↓Actives␈↓
␈α	(like␈α	Compose,␈α
Insert,␈α	Substitute,␈α	Undo,␈α	Negate,␈α
Membership,␈α	Equipollence,
␈↓ ↓H␈↓
Quanti≡cation,␈α
Extreme);␈α
␈↓↓Higher-order␈α
Actives␈↓
␈α
(like␈α
Select,␈α
Analogize,␈α
Assume,␈α
Prove,␈α
Debug,␈α
Communicate,␈α
Select-representation,
␈↓ ↓H␈↓
Categoricity); and ␈↓↓Higher-order Objects␈↓
 (like Conjecture, Contradiction, Analogy, Problem, Mathematical-theory).


␈↓ ↓H␈↓␈↓ α_2.␈α∞Facets␈α∂that␈α∞each␈α∞concept␈α∂might␈α∞have:␈α∞Each␈α∂facet␈α∞program␈α∞can␈α∂be␈α∞viewed␈α∞as␈α∂answering␈α∞a

␈↓ ↓H␈↓certain␈α⊗family␈α↔of␈α⊗questions␈α↔about␈α⊗the␈α↔concept␈α⊗in␈α⊗which␈α↔it␈α⊗is␈α↔embedded.␈α⊗For␈α↔example,␈α⊗the

␈↓ ↓H␈↓"DEFINITION"␈α
facet␈α
of␈αthe␈α
concept␈α
called␈α
"COMPOSE"␈αshould␈α
be␈α
able␈α
to␈αtell␈α
if␈α
any␈α
given␈αentity␈α
is

␈↓ ↓H␈↓a␈α∪composition.␈α∪ The␈α∪tentative␈α∪set␈α∪of␈α∪25␈α∪facets␈α∪that␈α∪concepts␈α∪can␈α∪have␈α∪breaks␈α∪into␈α∪four␈α∪main

␈↓ ↓H␈↓categories:

␈↓ ↓H␈↓␈↓↓RECOGNITION GROUPING␈↓  Notice when this concept, call it B, is relevant.
␈↓ ↓H␈↓␈↓↓ALTER-ONESELF GROUPING␈↓  Know about variations of B, how good B is, etc.
␈↓ ↓H␈↓␈↓↓ACT-ON-ANOTHER GROUPING␈↓  Look at part of another concept, perhaps do something to it.
␈↓ ↓H␈↓␈↓↓INFO GROUPING␈↓ General information about this concept and how it ≡ts in.


␈↓ ↓H␈↓␈↓ α_3.␈αActual␈αinitial␈αstate:␈αAfter␈α
delineating␈αthe␈αconcepts␈αwhich␈αwill␈α
be␈αgiven,␈αand␈αthe␈αfacets␈αthat␈α
a

␈↓ ↓H␈↓concept␈α
might␈α
have,␈α
there␈α
remain␈α
two␈α
additional␈α
knowledge-gathering␈α
tasks:␈α
(i)␈α
Fill␈α
in␈α
some␈α
of␈α
the

␈↓ ↓H␈↓facets␈α
for␈α
each␈αof␈α
the␈α
concepts␈αinitially␈α
to␈α
be␈α
supplied,␈αand␈α
(ii)␈α
Fill␈αin␈α
the␈α
strategic␈α
information␈αfor

␈↓ ↓H␈↓each␈α∂facet.␈α∂For␈α∞example,␈α∂here␈α∂are␈α∂some␈α∞hints␈α∂that␈α∂might␈α∞be␈α∂placed␈α∂in␈α∂the␈α∞INTERESTINGNESS

␈↓ ↓H␈↓facet␈α
of␈α
the␈α
concept␈α
COMPOSE␈α
(that␈α
is,␈α
here␈α
are␈α
some␈α
procedures␈α
for␈α
determining␈α
when␈α
any␈α
speci≡c

␈↓ ↓H␈↓composition is to viewed as interesting):
␈↓ ↓H␈↓
␈↓ ↓xThe result satis≡es some interesting property which is not true of either argument relation.
␈↓ ↓H␈↓
␈↓ ↓xInteresting properties of both argument relations are preserved.
␈↓ ↓H␈↓
␈↓ ↓xUndesirable properties of both argument relations are lost.
␈↓ ↓H␈↓
␈↓ ↓xInteresting␈αsubsets␈α(cases)␈αof␈αdomain␈αof␈α1st␈αargument␈αoperator␈αmap␈αinto␈αinteresting␈αsubsets␈αof␈αrange␈αof␈α
2nd␈αargument
␈↓ ↓H␈↓
␈↓ α(operator.
␈↓ ↓H␈↓
␈↓ ↓xPreimages␈αof␈αinteresting␈αsubsets␈α(cases)␈αof␈αrange␈αof␈α2nd␈αargument␈αare␈αthemselves␈αinteresting␈αsubsets␈αof␈αdomain␈αof␈α1st
␈↓ ↓H␈↓
␈↓ α(argument.
␈↓ ↓H␈↓
␈↓ ↓xThe range of the ≡rst argument operator is equal to, not just a subset of, the domain of the second argument operator.



␈↓ ↓H␈↓␈↓ ε-␈↓↓page 5␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓␈↓ α_Notice␈α∃that␈α∃the␈α∃≡rst␈α∀≡ve␈α∃factors␈α∃are␈α∃all␈α∀recursive,␈α∃depending␈α∃on␈α∃interestingness␈α∃of␈α∀the

␈↓ ↓H␈↓arguments,␈α∂properties,␈α∂etc.␈α∂The␈α∂≡nal␈α∂hint␈α∂is␈α∂␈↓↓not␈↓␈α∂recursive;␈α∂ultimately,␈α∂every␈α∂interstingness␈α∞"search"

␈↓ ↓H␈↓terminates at such primitive factors.





␈↓ ↓H␈↓αThe Goal: AM running


␈↓ ↓H␈↓␈↓ α_1.␈α∂No␈α∂speci≡c␈α∂goal:␈α∞One␈α∂of␈α∂the␈α∂∨aws␈α∞of␈α∂PUP6␈α∂was␈α∂that␈α∞it␈α∂had␈α∂one␈α∂particular␈α∂target.␈α∞Since

␈↓ ↓H␈↓PUP6␈α∞did␈α∞synthesize␈α
that␈α∞target,␈α∞it␈α∞was␈α
never␈α∞clear␈α∞precisely␈α∞how␈α
important␈α∞various␈α∞aspects␈α∞of␈α
its

␈↓ ↓H␈↓design␈αwere,␈αand␈αthere␈αwere␈αno␈α"experiments"␈αone␈αcould␈αrun␈αon␈αit.␈αBy␈αcontrast,␈αAM␈αhas␈αno␈αset␈αgoal,

␈↓ ↓H␈↓no␈α⊂target␈α⊂"≡nal␈α⊃state"␈α⊂of␈α⊂knowledge.␈α⊃Its␈α⊂actions␈α⊂are␈α⊃knowledge␈α⊂acquisition,␈α⊂guided␈α⊃by␈α⊂evaluation

␈↓ ↓H␈↓criteria␈α
and␈α
discovery␈α∞heuristics.␈α
It␈α
will␈α∞be␈α
a␈α
success␈α∞if␈α
it␈α
does␈α∞something␈α
interesting,␈α
if␈α∞it␈α
develops

␈↓ ↓H␈↓some␈α_mathematically␈α_interesting␈α_concepts␈α_(no␈α↔doubt␈α_ones␈α_that␈α_are␈α_well-known␈α_already,␈α↔like

␈↓ ↓H␈↓cardinality).␈αThis␈αlack␈αof␈αspeci≡city␈αis␈αconsidered␈αan␈αadvantage,␈αsince␈αthe␈αsystem␈αcreators␈αcan't␈α(even

␈↓ ↓H␈↓unconsciously) predetermine what is "necessary" for AM to start with, to acheive a ≡xed goal.


␈↓ ↓H␈↓␈↓ α_2.␈α
Some␈α
possible␈α
developments:␈α
To␈α
say␈α
that␈α∞AM␈α
has␈α
no␈α
speci≡c␈α
goal␈α
does␈α
not␈α
mean␈α∞that␈α
we

␈↓ ↓H␈↓can't␈α
evaluate␈αits␈α
performance,␈αor␈α
that␈αwe␈α
have␈αno␈α
idea␈αwhat␈α
it␈αmight␈α
do.␈α In␈α
fact,␈αhere␈α
is␈αa␈α
list␈αof

␈↓ ↓H␈↓early␈α⊃numerical␈α⊃concepts␈α⊃we␈α⊃expect␈α⊃AM␈α⊃to␈α⊃devlop:␈α⊃Count,␈α⊃Inverse,␈α⊃Commutativity,␈α⊃Transitivity,

␈↓ ↓H␈↓Associativity,␈α∪Singleton,␈α∪Function,␈α∪Successor,␈α∪Zero,␈α∪One,␈α∪Two,␈α∪Plus,␈α∪Times.␈α∪ Appendix␈α∀1␈α∪gives

␈↓ ↓H␈↓scenarios for making it plausible that AM might discover some of these.


␈↓ ↓H␈↓␈↓ α_3.␈α⊂A␈α⊂hypothetical␈α⊂dialogue␈α⊂with␈α⊂AM:␈α⊃Assume␈α⊂that␈α⊂AM␈α⊂has␈α⊂just␈α⊂developed␈α⊂all␈α⊃the␈α⊂simple

␈↓ ↓H␈↓numerical␈α
concepts␈αlisted␈α
in␈α(2),␈α
in␈αaddition␈α
to␈αits␈α
universal␈αstore␈α
(item␈α*␈α
on␈αpage␈α
*).␈α Here␈α
is␈αhow␈α
the

␈↓ ↓H␈↓ensuing dialgoue might appear to the user sitting at a teletype.



␈↓ ↓H␈↓␈↓ ε-␈↓↓page 6␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓ε␈↓πSYSTEM␈↓ε:␈αI␈αam␈α
looking␈αat␈αmore␈αexamples␈α
of␈αTIMES␈↓#
-␈↓#␈↓#
1␈↓#,␈αwhere␈α
TIMES␈αis␈αa␈αrelation␈α
taking␈αa␈αpair␈αof␈α
numbers
␈↓ ↓H␈↓ε␈↓ αλand␈α⊂returning␈α⊂a␈α⊂number.␈α⊂So␈α⊂TIMES␈↓#
-␈↓#␈↓#
1␈↓#␈α⊂takes␈α⊂a␈α⊂number␈α⊂and␈α⊂returns␈α⊂a␈α⊂set␈α⊂of␈α⊂pairs␈α⊂of␈α⊂numbers.
␈↓ ↓H␈↓ε␈↓ αλTIMES␈↓#
-␈↓#␈↓#
1␈↓#(n) seems always to include, as one ordered pair, (n, 1). Same for (1, n). So I can state:
␈↓ ↓H␈↓εConjecture: (∀xεN) (x,1) ε TIMES␈↓#
-␈↓#␈↓#
1␈↓#(x), and (1,x) ε TIMES␈↓#
-␈↓#␈↓#
1␈↓#(x).  Proof is simple: TIMES(x,1)=x=TIMES(1,x).
␈↓"⊃␈↓ ↓H␈↓ε␈↓πSYSTEM␈↓ε:␈α∞The␈α∞two␈α∞pairs␈α∞(n,1)␈α∞and␈α∂(1,n)␈α∞are␈α∞distinct␈α∞iff␈α∞the␈α∞number␈α∂n␈α∞is␈α∞not␈α∞equal␈α∞to␈α∞1.␈α∂ Noticing␈α∞the
␈↓ ↓H␈↓ε␈↓ αλnumber 1 as an exception, we can state:
␈↓ ↓H␈↓εCorollary:␈α
COUNT*TIMES␈↓#
-␈↓#␈↓#
1␈↓#(1)␈α
=␈α∞1;␈α
(∀x>1)␈α
COUNT*TIMES␈↓#
-␈↓#␈↓#
1␈↓#(x)␈α∞≥␈α
2.␈α
 COUNT␈α
*␈α∞TIMES␈↓#
-␈↓#␈↓#
1␈↓#␈α
is␈α
the␈α∞how␈α
many
␈↓ ↓H␈↓ε␈↓ αλways one may express a number as a product of an ordered pair of numbers.
␈↓"⊃␈↓ ↓H␈↓ε␈↓πSYSTEM␈↓ε: Interesting to consider the subset P of N, defined as {xεN | COUNT*TIMES␈↓#
-␈↓#␈↓#
1␈↓#(x)=2}.
␈↓"⊃␈↓ ↓H␈↓ε␈↓πUSER␈↓ε: (interrupting) Why is this interesting?
␈↓"⊃␈↓ ↓H␈↓ε␈↓πSYSTEM␈↓ε:␈α
At␈α
the␈α
moment,␈α
we␈α
have␈α
the␈α
known␈α
fact␈α
f(x)≥y;␈α
try␈α
to␈α
pin␈α
down␈α
extreme␈α
cases,␈α
those␈α∞x␈α
for
␈↓ ↓H␈↓ε␈↓ αλwhich␈α
f(x)␈α
is␈α
precisely␈α
equal␈α
to␈α
y.␈α
f␈α
is␈α
COUNT*TIMES␈↓#
-␈↓#␈↓#
1␈↓#,␈α
in␈α
the␈α
current␈α
case,␈α
and␈α
y␈α
is␈α
2.␈α
 Do␈α
you
␈↓ ↓H␈↓ε␈↓ αλhave␈α
any␈α
special␈α
names␈α
you␈α
would␈α
like␈α
me␈α
to␈α
refer␈α
to␈α
COUNT␈α
*␈α
TIMES␈↓#
-␈↓#␈↓#
1␈↓#␈α
by?␈α
Any␈α
special␈α
names␈α
for
␈↓ ↓H␈↓ε␈↓ αλthe subset P?
␈↓"⊃␈↓ ↓H␈↓ε␈↓πUSER␈↓ε: Call that composition FAC.  Call the set P, of extreme numbers under FAC, PRIMES.
␈↓"⊃␈↓ ↓H␈↓ε␈↓πSYSTEM␈↓ε: Syntactic transformation: "PRIME" is the predicate which means "is in PRIMES".
␈↓"⊃␈↓ ↓H␈↓ε␈↓πUSER␈↓ε:␈α⊂TIMES␈α⊂may␈α⊂also␈α⊂be␈α⊂viewed␈α⊂as␈α⊂operating␈α⊂on␈α⊂a␈α⊂BAG;␈α⊂in␈α⊂this␈α⊂sense,␈α⊂TIMES␈↓#
-␈↓#␈↓#
1␈↓#␈α⊂would␈α⊃return␈α⊂all
␈↓ ↓H␈↓ε␈↓ αλpossible␈αbags␈α(of␈αany␈α
length)␈αwhose␈αproduct␈αis␈α
the␈αgiven␈αnumber.␈αWe␈α
do␈αnot␈αallow␈αbags␈α
to␈αhave
␈↓ ↓H␈↓ε␈↓ αλ<2 members, and we do not allow bags longer than 2 to contain any 1's.
␈↓"⊃␈↓ ↓H␈↓ε␈↓πSYSTEM␈↓ε:␈αIn␈α
this␈αcase,␈αthe␈α
corollary␈αabove␈αjust␈α
says:␈αCOUNT(TIMES␈↓#
-␈↓#␈↓#
1␈↓#(x))␈α
≥␈α1.␈αThe␈α
extremes␈αnow␈αare␈α
just
␈↓ ↓H␈↓ε␈↓ αλ1 and PRIMES.  I want a new name for this new interpretation of TIMES␈↓#
-␈↓#␈↓#
1␈↓#.
␈↓"⊃␈↓ ↓H␈↓ε␈↓πUSER␈↓ε: Call it FACTORS.
␈↓"⊃␈↓ ↓H␈↓ε␈↓πSYSTEM␈↓ε:␈αConjecture:␈αFACTORS(x)␈αalways␈αseems␈αto␈αcontain␈αa␈αbag␈αcontaining␈αonly␈αones␈αand␈αprimes,␈αand
␈↓ ↓H␈↓ε␈↓ αλonly one such bag.
␈↓ ↓H␈↓εProof of existence is by contradiction, using the lemma ¬PRIME(x) → ∃(u,v)εTIMES␈↓#
-␈↓#␈↓#
1␈↓#(x) s.t. 1<u<x, 1<v<x.
␈↓ ↓H␈↓εProof of Uniqueness is by contradiction, using the lemma (∀a,bεN, xεPRIMES). x|TIMES(a,b) iff x|a or x|b.
␈↓"⊃␈↓ ↓H␈↓ε␈↓πUSER␈↓ε:␈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
␈↓ ↓H␈↓ε␈↓ αλthe divisors of a number.


␈↓ ↓H␈↓␈↓ α_4.␈α
How␈α
AM's␈α
knowledge␈α
interacts␈α
to␈α
do␈α∞this:␈α
Appendix␈α
1␈α
explains␈α
in␈α
detail␈α
how␈α∞AM␈α
might

␈↓ ↓H␈↓really manage to make such a discovery as the Unique Factorization Theorem.


␈↓ ↓H␈↓␈↓ α_5. Estimates of AM's parameters:

␈↓ ↓H␈↓     NUMBER␈↓ ¬hInitially␈↓ πλUltimately
␈↓ ↓H␈↓␈↓ ¬h␈↓ πλ (recall that there is no set "goal", however)
␈↓ ↓H␈↓Number of Families of concepts␈↓ ¬h  5␈↓ πλ  5
␈↓ ↓H␈↓Number of concepts per family␈↓ ¬h 30␈↓ πλ100
␈↓ ↓H␈↓Number of Parts per concept␈↓ ¬h 25␈↓ πλ 25
␈↓ ↓H␈↓Size of each part (lines of LISP)␈↓ ¬h  5␈↓ πλ  7
␈↓ ↓H␈↓Number of parts ≡lled in␈↓ ¬h  8␈↓ πλ 20
␈↓ ↓H␈↓Size of avg. concept (LISP lines)␈↓ ¬h 40␈↓ πλ140
␈↓ ↓H␈↓Total number of concepts␈↓ ¬h150␈↓ πλ500
␈↓ ↓H␈↓Core used by AM␈↓ ¬h200K␈↓ πλ500K

␈↓ ↓H␈↓␈↓ ε-␈↓↓page 7␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Time per int. result (cpu min)␈↓ ¬h 15␈↓ πλ  5
␈↓ ↓H␈↓CPU time used (hours)␈↓ ¬h  0␈↓ πλ100


␈↓ ↓H␈↓␈↓ α_________________________________________________________________________


␈↓ ↓H␈↓␈↓ α_FOOTNOTE (from ↑*, above):


␈↓ ↓H␈↓␈↓ α_What␈α∞do␈α∞we␈α∞mean␈α∞by␈α∞a␈α∞"mathematical␈α∞concept"?␈α∞ A␈α∞few␈α∞moments␈α∞of␈α∞pondering␈α∞this␈α∞should

␈↓ ↓H␈↓convince␈α∂the␈α∂reader␈α∂that␈α∂this␈α∂cannot␈α∂be␈α∂answered␈α∂cleanly.␈α∂ Later,␈α∂we␈α∂shall␈α∂indicate␈α∂the␈α⊂classes␈α∂of

␈↓ ↓H␈↓concepts␈αinvolved;␈αfor␈αnow,␈αlet␈αus␈αjust␈αmention␈αa␈αfew␈αspeci≡c␈αones␈αto␈αindicate␈αthe␈αbreadth␈αinvolved.

␈↓ ↓H␈↓Each␈αof␈αthe␈α
following␈αis␈αa␈αmathematical␈α
concept:␈αSet,␈αRelation,␈α{1,␈α
frob,␈α{}},␈αZorn's␈α
Lemma,␈αTheory,

␈↓ ↓H␈↓Union,␈αProve,␈α
Proof,␈αTheorem,␈α
The␈αUnique␈α
Factorization␈αTheorem␈α
(UFT),␈αThe␈α
proof␈αof␈αUFT,␈α
The

␈↓ ↓H␈↓methods␈α∞used␈α
to␈α∞prove␈α
UFT,␈α∞Constructively␈α
proving␈α∞existence,␈α
Associativity.␈α∞ A␈α∞circular␈α
de≡nition

␈↓ ↓H␈↓might␈αbe␈α"all␈α
the␈αthings␈αdiscussed␈α
in␈αmath␈αbooks";␈α
an␈αoperational␈αde≡nition␈α
might␈αbe␈α"whatever␈α
AM

␈↓ ↓H␈↓knows about initially, plus whatever new knowledge it acquires."


␈↓ ↓H␈↓␈↓ α_FOOTNOTE (from ↑*↑*, above):


␈↓ ↓H␈↓␈↓ α_Epecially␈α∪useful␈α∀were␈α∪(see␈α∀Bibliography␈α∪for␈α∪full␈α∀references)␈α∪Polya,␈α∀Hadamard,␈α∪Kershner,

␈↓ ↓H␈↓Skemp.


␈↓ ↓H␈↓␈↓ α_APPENDIX 1: AM details [this needs much more stu≥ inseerted, but not much manhours]


␈↓ ↓H␈↓␈↓ α_1. Plausibility of discovering early numerical concepts:


␈↓ ↓H␈↓␈↓ α_2. More complete listing of the names of the concepts AM will start with:


␈↓ ↓H␈↓␈↓ α_3. Complete list of the facets which each concept might have:


␈↓ ↓H␈↓␈↓ α_4. Sample of what a concept looks like: "COMPOSE":

␈↓ ↓H␈↓␈↓ ε-␈↓↓page 8␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓␈↓ α_5. Details of AM running: how it might manage to discover the Unique Fac. Thm.:


␈↓ ↓H␈↓␈↓ α_Let's␈αconsider␈αjust␈αone␈αstep␈αin␈αthe␈αAM␈αdialogue␈αon␈αpage␈α*:␈αthe␈αinteresting␈αleap␈αfrom␈αthe␈αuser's

␈↓ ↓H␈↓declaration␈α
to␈α
call␈α
the␈α
function␈α
FACTORS,␈αand␈α
AM's␈α
statement␈α
of␈α
the␈α
unique␈αfactorization␈α
theorem.

␈↓ ↓H␈↓Recall␈αthat␈αthe␈αcontrol␈αmechanism␈αis␈αjust:␈αrepeatedly␈αselect␈αa␈αconcept␈αand␈αa␈αfacet,␈αthen␈αwork␈αto␈α≡ll␈αin

␈↓ ↓H␈↓the program for that facet of that concept.  Below, C.F is the notation used for facet F of concept C.

␈↓ ↓H␈↓i.␈α∂Choose␈α∞Concept=FACTORS,␈α∂Facet=TIES.␈α∞ Gather␈α∂relevant␈α∞algorithms␈α∂from␈α∞the␈α∂facets␈α∞labelled:
␈↓ ↓H␈↓␈↓ ↓x[FACTORS.Ties].Fillin,␈α?␈α≡[FACTORS.Genl-Info].Fillin,␈α?␈α≡[FACTORS.Any-Part].Fillin,
␈↓ ↓H␈↓␈↓ ↓x[OPERATION.Ties].Fillin,␈α≠[OPERATION.Genl-Info].Fillin,␈α≠[OPERATION.Any-Part].Fillin,
␈↓ ↓H␈↓␈↓ ↓x[ACTIVE.Ties].Fillin,␈α?␈α?␈αλ[ACTIVE.Genl-Info].Fillin,␈α?␈α?␈απ[ACTIVE.Any-Part].Fillin,
␈↓ ↓H␈↓␈↓ ↓x[ANY-concept.Ties].Fillin, [ANY-concept.Genl-Info].Fillin, [ANY-concept.Any-Part].Fillin.

␈↓ ↓H␈↓ii.␈αFirst␈αhint␈α
collected␈αsays:␈α"Let␈αD␈α
be␈αthe␈αknown␈α
concept␈αrepresenting␈αthe␈αkind␈α
of␈αentity␈αin␈αthe␈α
range
␈↓ ↓H␈↓␈↓ ↓xof␈αFACTORS.␈α
Then␈αask␈αD.INTEREST␈α
how␈αto␈α
look␈αfor␈αinteresting␈α
properties␈αor␈αregularities.␈α
 If
␈↓ ↓H␈↓␈↓ ↓xsparse,␈α
ask␈α(generalization↑*␈α
D).INTEREST␈α
also.␈α Apply␈α
these␈αmethods␈α
to␈α
the␈αoutput␈α
of␈αa␈α
typical
␈↓ ↓H␈↓␈↓ ↓xexample␈α∞of␈α∞FACTORS.␈α∞ Check␈α∞interesting␈α∞property␈α∞found,␈α∞by␈α∞seeing␈α∞if␈α∞it␈α∞holds␈α∞for␈α∞the␈α
other
␈↓ ↓H␈↓␈↓ ↓xoutputs from FACTORS and ensuring it isn't simply part of the de≡nition of FACTORS."

␈↓ ↓H␈↓iii.␈α∃Because␈α∃the␈α⊗output␈α∃of␈α∃a␈α∃call␈α⊗on␈α∃FACTORS␈α∃is␈α⊗a␈α∃␈↓↓set␈↓␈α∃of␈α∃bags,␈α⊗we␈α∃are␈α∃directed␈α⊗to␈α∃ask
␈↓ ↓H␈↓␈↓ ↓xSET.INTEREST␈α⊃for␈α⊃aid␈α⊃in␈α⊃perceiving␈α⊃interesting␈α⊂things␈α⊃about␈α⊃a␈α⊃particular␈α⊃output,␈α⊃say␈α⊂the
␈↓ ↓H␈↓␈↓ ↓xoutput␈α∩{(BAG␈α∩3␈α∩5␈α∩5)␈α∪(BAG␈α∩75)␈α∩(BAG␈α∩5␈α∩15)␈α∪(BAG␈α∩3␈α∩25)}␈α∩from␈α∩the␈α∪call␈α∩FACTORS(75).
␈↓ ↓H␈↓␈↓ ↓xSET.INTEREST is not very big, so we ask STRUCTURE.INTEREST as well.

␈↓ ↓H␈↓iv.␈α∀STRUCTURE.INTEREST␈α∪explains␈α∀that␈α∪there␈α∀are␈α∪three␈α∀distinct␈α∪ways␈α∀a␈α∪structure␈α∀can␈α∪be
␈↓ ↓H␈↓␈↓ ↓xinteresting.␈α∂ First,␈α∂check␈α∞whether␈α∂the␈α∂structure␈α∞satis≡es␈α∂any␈α∂known␈α∞interesting␈α∂property␈α∂of␈α∞that
␈↓ ↓H␈↓␈↓ ↓xtype␈α∩of␈α∩structure.␈α∩ If␈α∪not,␈α∩check␈α∩to␈α∩see␈α∩whether␈α∪every␈α∩element␈α∩satis≡es␈α∩the␈α∪same␈α∩interesting
␈↓ ↓H␈↓␈↓ ↓xproperty.␈α⊃If␈α⊂not,␈α⊃check␈α⊂to␈α⊃see␈α⊂if␈α⊃␈↓↓some␈↓␈α⊂element␈α⊃of␈α⊂the␈α⊃structure␈α⊂satis≡es␈α⊃some␈α⊃very␈α⊂interesting
␈↓ ↓H␈↓␈↓ ↓xproperty.␈α∂ The␈α∂criteria␈α∂for␈α∂interestingness␈α∂being␈α∂talked␈α∂about␈α∂here␈α∂is␈α∂the␈α∂one␈α∂speci≡ed␈α∂by␈α∞the
␈↓ ↓H␈↓␈↓ ↓xconcept␈αrepresenting␈αthe␈αtype␈αof␈αthe␈αelements.␈αIn␈αour␈αpresent␈αcase,␈αour␈αset␈αis␈αa␈αset␈αof␈α␈↓↓bags,␈↓␈αso␈α
that
␈↓ ↓H␈↓␈↓ ↓xmeans␈α∞consult␈α∞all␈α∞the␈α∞hints␈α∞and␈α∞factors␈α∞present␈α∞under␈α∞BAG.INTEREST.␈α∞But␈α∞this␈α∞is␈α∞also␈α∞very
␈↓ ↓H␈↓␈↓ ↓xsparse, hence we recursively turn to STRUCTURE.INTEREST for evaluation criteria.

␈↓ ↓H␈↓v.␈α
After␈αa␈α
reasonable␈αtime,␈α
AM␈αcannot␈α
≡nd␈α
any␈αinteresting␈α
property␈αsatis≡ed␈α
by␈αthe␈α
given␈αoutput␈α
set,
␈↓ ↓H␈↓␈↓ ↓x{(BAG␈α3␈α5␈α5)␈α(BAG␈α75)␈α(BAG␈α5␈α15)␈α(BAG␈α3␈α25)}.␈α AM␈αalso␈αfails␈αto␈α≡nd␈αany␈αsingle␈αinteresting
␈↓ ↓H␈↓␈↓ ↓xproperty satis≡ed by all four bags which form the elements of that output set.

␈↓ ↓H␈↓vi.␈αNow␈αAM␈αlooks␈αat␈αeach␈αelement␈αin␈αturn,␈αthat␈αis,␈αeach␈αbag.␈αFirst␈αwe␈αconsider␈α(BAG␈α75),␈αsay.␈αThis
␈↓ ↓H␈↓␈↓ ↓xsatis≡es␈α∞the␈α∂property␈α∞SINGLETON.␈α∂ We␈α∞check␈α∞with␈α∂other␈α∞examples␈α∂of␈α∞FACTORS␈α∂and,␈α∞sure
␈↓ ↓H␈↓␈↓ ↓xenough,␈α∞each␈α∂one␈α∞of␈α∞them␈α∂contains,␈α∞as␈α∞an␈α∂element,␈α∞a␈α∞bag␈α∂having␈α∞the␈α∂property␈α∞SINGLETON.
␈↓ ↓H␈↓␈↓ ↓xComparing␈αthese␈αsingletons␈αto␈αthe␈αinputs␈αto␈α
FACTORS,␈αwe␈αconjecture␈αthat␈α(BAG␈αx)␈αwill␈α
always
␈↓ ↓H␈↓␈↓ ↓xappear in the output set of FACTORS(x).



␈↓ ↓H␈↓␈↓ ε-␈↓↓page 9␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓vii.␈αWe␈αgo␈αback␈αto␈αlooking␈αat␈αthe␈αindividual␈αbags␈αin␈αFACTORS(75).␈α This␈αtime␈αwe␈αlook␈αat␈α(BAG␈α
3
␈↓ ↓H␈↓␈↓ ↓x5␈α∞5).␈α
 Each␈α∞element␈α
␈↓↓does␈↓␈α∞satisfy␈α
an␈α∞interesting␈α
property,␈α∞namely␈α
PRIME.␈α∞We␈α
check␈α∞against␈α
the
␈↓ ↓H␈↓␈↓ ↓xother␈αexamples␈αof␈αFACTORS,␈αand␈αsure␈αenough␈αeach␈αone␈αof␈αthem␈αcontains␈αan␈αelement␈αwhich␈αis
␈↓ ↓H␈↓␈↓ ↓xa␈αbag␈α
of␈αprimes.␈αThere␈α
doesn't␈αseem␈αto␈α
be␈αany␈αobvious␈α
relationship␈αto␈αthe␈α
input␈αargument,␈αso␈α
we
␈↓ ↓H␈↓␈↓ ↓xmerely␈α∂conjecture␈α⊂that␈α∂FACTORS(x)␈α⊂always␈α∂contains␈α∂a␈α⊂bag␈α∂of␈α⊂primes,␈α∂without␈α⊂saying␈α∂which
␈↓ ↓H␈↓␈↓ ↓xprimes␈αor␈αhow␈αto␈αcompute␈αthem.␈αThis␈αis␈αone␈αhalf␈αof␈αthe␈αUnique␈αFactorization␈α
Theorem.␈αNotice
␈↓ ↓H␈↓␈↓ ↓xthat␈αthis␈αis␈α"rough␈αaround␈αthe␈αedges",␈αnamely␈αfor␈αthe␈αcases␈αof␈αfactors␈αof␈αzero␈αand␈αone,␈αbut␈αthese
␈↓ ↓H␈↓␈↓ ↓xwill␈αbe␈αcaught␈αlater␈αby␈αan␈αexpert␈αconcept␈αwho␈αspecializes␈αin␈αchecking␈αconjectures␈αjust␈αbefore␈αwe
␈↓ ↓H␈↓␈↓ ↓xstart to prove them.

␈↓ ↓H␈↓viii.␈α
Each␈α
element␈α
of␈α
(BAG␈α
3␈α
5␈α
5)␈α
also␈αsatis≡es␈α
the␈α
property␈α
ODD.␈α
But␈α
this␈α
is␈α
quickly␈α
rejected␈αby
␈↓ ↓H␈↓␈↓ ↓xlooking at the example FACTORS(2) = {(BAG 2)}.

␈↓ ↓H␈↓ix.␈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
␈↓ ↓H␈↓␈↓ ↓xinteresting is found here or in the next case, (BAG 3 25).

␈↓ ↓H␈↓x.␈α⊃Instead␈α⊃of␈α⊃going␈α⊃on␈α⊃to␈α⊃prove␈α⊃some␈α⊂of␈α⊃these␈α⊃conjectures,␈α⊃let's␈α⊃see␈α⊃how␈α⊃AM␈α⊃might␈α⊃notice␈α⊂the
␈↓ ↓H␈↓␈↓ ↓xuniqueness␈α~aspects␈α≠of␈α~them.␈α~AM␈α≠knows␈α~that␈α~some␈α≠elements␈α~of␈α≠FACTORS(x)␈α~satisfy
␈↓ ↓H␈↓␈↓ ↓xMAPSTRUC(PRIME),␈αbut␈αsome␈αdon't.␈αIt␈αwants␈αto␈α≡nd␈αout␈αhow␈αto␈αcharacterize␈αthose␈αwhich␈αdo;
␈↓ ↓H␈↓␈↓ ↓xnamely,␈αthose␈αbags␈αof␈αprimes␈αfrom␈αthose␈αcontaining␈αa␈αnonprime.␈α So␈αAM␈αwill␈αtemporarily␈αcreate
␈↓ ↓H␈↓␈↓ ↓xa␈αnew␈αconcept,␈αcalled␈αsay␈αPF,␈αde≡ned␈αas␈α␈↓εPF(x)␈α=␈αFACTORS(x)␈α∩␈αMAPSTRUC(PRIME,␈αx)␈α=␈α{b␈α|␈αBAG(b)
␈↓ ↓H␈↓ε␈↓ ↓x∧␈α∞TIMES(b)=x␈α∞∧␈α∞1¬εb␈α∞∧␈α∞∀zεb.␈α∞PRIME(z)}␈↓.␈α∂ Which␈α∞means:␈α∞all␈α∞bags␈α∞of␈α∞primes␈α∞whose␈α∞TIMES␈α∂is␈α∞x;
␈↓ ↓H␈↓␈↓ ↓xwhich also means all factorizations of x into bags containing only primes.

␈↓ ↓H␈↓xi.␈α⊃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
␈↓ ↓H␈↓␈↓ ↓xSINGLETON.␈α∞That␈α∞is,␈α∞there␈α∞is␈α∞only␈α∞one␈α∞bag␈α∞of␈α∞primes␈α∞in␈α∞the␈α∞FACTORS(x)␈α∞collection␈α∂for␈α∞a
␈↓ ↓H␈↓␈↓ ↓xgiven␈α
x.␈α How␈α
does␈α
it␈αdo␈α
this?␈α
 The␈αunique␈α
factorization␈αtheorem␈α
can␈α
be␈αconsisely␈α
be␈α
stated␈αas
␈↓ ↓H␈↓␈↓ ↓x"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␈α∞surprising␈α∞that␈α∞AM␈α∂will␈α∞routinely
␈↓ ↓H␈↓␈↓ ↓xinvestigate it.


␈↓ ↓H␈↓␈↓ α_6. Estimated timetable for AM:


␈↓ ↓H␈↓␈↓ α_(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).

␈↓ ↓H␈↓␈↓
Reality: See Given Knowledge, as presented in a separate volume. Completed in December, 1974.␈↓


␈↓ ↓H␈↓␈↓ α_(ii).␈α
Formulate␈α
a␈α
su≠cient␈α
set␈α
of␈αnew␈α
ideas,␈α
design␈α
decisions,␈α
and␈α
intuitive␈α
assumptions␈αto␈α
make

␈↓ ↓H␈↓the task meaninful and feasable.  ␈↓
Reality: ≡rmed up in January, 1975.␈↓


␈↓ ↓H␈↓␈↓ α_(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




␈↓ ↓H␈↓␈↓ ε%␈↓↓page 10␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓concrete,␈α
simulated␈α
system.␈α	 ␈↓
Reality:␈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-

␈↓ ↓H␈↓
simulations done during February and March, 1975, with this "paper" system.␈↓


␈↓ ↓H␈↓␈↓ α_(iv). Implement a realization of AM as a computer program.  ␈↓
Reality: Under way in April, 1975.␈↓


␈↓ ↓H␈↓␈↓ α_(v).␈αDebug␈α
and␈αrun␈αthe␈α
system.␈αAdd␈αthe␈α
natural␈αlanguage␈αabilities␈α
gradually,␈αas␈αneeded.␈α
 ␈↓
Reality:

␈↓ ↓H␈↓
Scheduled for May to November of 1975. First interesting results expected in late June.␈↓


␈↓ ↓H␈↓␈↓ α_(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

␈↓ ↓H␈↓automating␈αcreative␈αmathematical␈αdiscovery␈αand␈αtheory␈αdevelopment;␈αadequacy␈αof␈αthe␈αinitial␈αcore␈αof

␈↓ ↓H␈↓knowledge;␈α_adequacy␈α_of␈α→the␈α_ideas,␈α_design␈α→decisions,␈α_implementation␈α_details,␈α→and␈α_theoretical

␈↓ ↓H␈↓assumptions.␈α Use␈αthe␈αresults␈αto␈αimprove␈αthe␈αsystem;␈αwhen␈α"adequate,"␈αforge␈αahead␈αas␈αfar␈αas␈α
possible

␈↓ ↓H␈↓into as many domains as possible, then reanalyze. ␈↓
Reality: the (v)↔(vi) cycle will terminate in the Winter of 1976.␈↓


␈↓ ↓H␈↓␈↓ α_(vii). Experiment with AM. ␈↓
Reality: scheduled to begin in December, 1975.␈↓


␈↓ ↓H␈↓␈↓ α_BIBLIOGRAPHY


␈↓ ↓H␈↓␈↓ α_All␈α∃the␈α∃references␈α∃below␈α∃have␈α∃actually␈α⊗been␈α∃read␈α∃as␈α∃background␈α∃for␈α∃AM.␈α⊗ I␈α∃strongly

␈↓ ↓H␈↓recommend those with an "@" sign; the others proved to be merely supplementary.





␈↓ ↓H␈↓αBooks and Memos

␈↓ ↓H␈↓Adams, James L., zz4Conceptual Blockbusting, W.H. Freeman and Co., San Francisco, 1974.

␈↓ ↓H␈↓Allendoerfer,␈α↔Carl␈α↔B.,␈α⊗and␈α↔Oakley,␈α↔Cletis␈α⊗O.,␈α↔␈↓βPrinciples␈α↔of␈α↔Mathematics␈↓,␈α⊗Third
␈↓ ↓H␈↓␈↓ αλEdition, McGraw-Hill, New York, 1969.

␈↓ ↓H␈↓Alexander,␈α⊃Stephen,␈α⊃␈↓βOn␈α⊃the␈α⊂Fundamental␈α⊃Principles␈α⊃of␈α⊃Mathematics␈↓,␈α⊃B.␈α⊂L.
␈↓ ↓H␈↓␈↓ αλHamlen, New Haven, 1849.


␈↓ ↓H␈↓␈↓ ε%␈↓↓page 11␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Aschenbrenner,␈α
Karl,␈α
␈↓βThe␈α∞Concepts␈α
of␈α
Value␈↓,␈α∞D.␈α
Reidel␈α
Publishing␈α∞Company,␈α
Dordrecht,
␈↓ ↓H␈↓␈↓ αλHolland, 1971.

␈↓ ↓H␈↓Atkin,␈α
A.␈α∞O.␈α
L.,␈α∞and␈α
Birch,␈α∞B.␈α
J.,␈α
eds.,␈α∞␈↓βComputers␈α
in␈α∞Number␈α
Theory␈↓,␈α∞Proceedings␈α
of
␈↓ ↓H␈↓␈↓ αλthe 1969 SRCA Oxford Symposium, Academic Press, New York, 1971.

␈↓ ↓H␈↓Avey,␈α∞Albert␈α∂E.,␈α∞␈↓βThe␈α∂Function␈α∞and␈α∞Forms␈α∂of␈α∞Thought␈↓,␈α∂Henry␈α∞Holt␈α∂and␈α∞Company,
␈↓ ↓H␈↓␈↓ αλNew York, 1927.

␈↓ ↓H␈↓@Badre,␈α∪Nagib␈α∩A.,␈α∪␈↓βComputer␈α∩Learning␈α∪From␈α∩English␈α∪Text␈↓,␈α∪Memorandum␈α∩No.
␈↓ ↓H␈↓␈↓ αλERL-M372,␈α∞Electronics␈α∞Research␈α∞Laboratory,␈α∞UCB,␈α∞December␈α∞20,␈α∞1972.␈α∞ Also␈α∂summarized␈α∞in
␈↓ ↓H␈↓␈↓ αλ␈↓βCLET␈α≠--␈α~A␈α≠Computer␈α~Program␈α≠that␈α~Learns␈α≠Arithmetic␈α~from
␈↓ ↓H␈↓β␈↓ αλan Elementary Textbook␈↓, IBM Research Report RC 4235, February 21, 1973.

␈↓ ↓H␈↓Bahm,␈α∩A.␈α∩J.,␈α∩␈↓βTypes␈α∩of␈α∪Intuition␈↓,␈α∩University␈α∩of␈α∩New␈α∩Mexico␈α∩Press,␈α∪Albuquerque,␈α∩New
␈↓ ↓H␈↓␈↓ αλMexico, 1960.

␈↓ ↓H␈↓Banks, J. Houston, ␈↓βElementary-School Mathematics␈↓, Allyn and Bacon, Boston, 1966.

␈↓ ↓H␈↓Berkeley,␈α!Edmund␈α C.,␈α!␈↓βA␈α Guide␈α!to␈α Mathematics␈α!for␈α!the␈α Intelligent
␈↓ ↓H␈↓β␈↓ αλNonmathematician␈↓, Simon and Schuster, New York, 1966.

␈↓ ↓H␈↓Berkeley,␈α∂Hastings,␈α∂␈↓βMysticism␈α∂in␈α∂Modern␈α∂Mathematics␈↓,␈α∂Oxford␈α∂U.␈α∂Press,␈α∂London,
␈↓ ↓H␈↓␈↓ αλ1910.

␈↓ ↓H␈↓Beth,␈α*Evert␈α)W.,␈α*and␈α*Piaget,␈α)Jean,␈α*␈↓βMathematical␈α*Epistemology␈α)and
␈↓ ↓H␈↓β␈↓ αλPsychology␈↓, Gordon and Breach, New York, 1966.

␈↓ ↓H␈↓Black, Max, ␈↓βMargins of Precision␈↓, Cornell University Press, Ithaca, New York, 1970.

␈↓ ↓H␈↓Blackburn,␈α∃Simon,␈α⊗␈↓βReason␈α∃and␈α∃Prediction␈↓,␈α⊗Cambridge␈α∃University␈α⊗Press,␈α∃Cambridge,
␈↓ ↓H␈↓␈↓ αλ1973.

␈↓ ↓H␈↓@Brotz,␈α∩Douglas␈α∩K.,␈α⊃␈↓βEmbedding␈α∩Heuristic␈α∩Problem␈α⊃Solving␈α∩Methods␈α∩in␈α⊃a
␈↓ ↓H␈↓β␈↓ αλMechanical␈α~Theorem␈α~Prover␈↓,␈α≠dissertation␈α~published␈α~as␈α≠Stanford␈α~Computer
␈↓ ↓H␈↓␈↓ αλScience Report STAN-CS-74-443, AUgust, 1974.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓␈↓ αλCognition Project, John Wiley & Sons, New York, 1956.

␈↓ ↓H␈↓Charosh, Mannis, ␈↓βMathematical Challenges␈↓, NCTM, Wahington, D.C., 1965.

␈↓ ↓H␈↓Cohen,␈α∪Paul␈α∀J.,␈α∪␈↓βSet␈α∀Theory␈α∪and␈α∀the␈α∪Continuum␈α∀Hypothesis␈↓,␈α∪W.A.Benjamin,
␈↓ ↓H␈↓␈↓ αλInc., New York, 1966.



␈↓ ↓H␈↓␈↓ ε%␈↓↓page 12␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Copeland,␈α≠Richard␈α≠W.,␈α≠␈↓βHow␈α~Children␈α≠Learn␈α≠Mathematics␈↓,␈α≠The␈α~MacMillan
␈↓ ↓H␈↓␈↓ αλCompany, London, 1970.

␈↓ ↓H␈↓Courant,␈α∞Richard,␈α
and␈α∞Robins,␈α
Herbert,␈α∞␈↓βWhat␈α
is␈α∞Mathematics␈↓,␈α
Oxford␈α∞University␈α
Press,
␈↓ ↓H␈↓␈↓ αλNew York, 1941.

␈↓ ↓H␈↓D'Augustine,␈α∩Charles,␈α∩␈↓βMultiple␈α∩Methods␈α∩of␈α∩Teaching␈α∩Mathematics␈α∩in␈α∩the
␈↓ ↓H␈↓β␈↓ αλElementary School␈↓, Harper & Row, New York, 1968.

␈↓ ↓H␈↓Dornbusch,␈α∃Sanford,␈α∃and␈α∃Scott,␈α⊗␈↓βEvaluation␈α∃and␈α∃the␈α∃Exercise␈α⊗of␈α∃Authority␈↓,
␈↓ ↓H␈↓␈↓ αλJossey-Bass, San Francisco, 1975.

␈↓ ↓H␈↓Douglas, Mary (ed.), ␈↓βRules and Meanings␈↓, Penguin Education, Baltimore, Md., 1973.

␈↓ ↓H␈↓Dowdy, S. M., ␈↓βMathematics: Art and Science␈↓, John Wiley & Sons, NY, 1971.

␈↓ ↓H␈↓Dubin, Robert, ␈↓βTheory Building␈↓, The Free Press, New York, 1969.

␈↓ ↓H␈↓Dubs, Homer H., ␈↓βRational Induction␈↓, U. of Chicago Press, Chicago, 1930.

␈↓ ↓H␈↓Dudley,␈α
Underwood,␈α
␈↓βElementary␈α
Number␈α
Theory␈↓,␈α
W.␈α
H.␈α
Freeman␈α
and␈α∞Company,␈α
San
␈↓ ↓H␈↓␈↓ αλFrancisco, 1969.

␈↓ ↓H␈↓Eynden,␈α!Charles␈α!Vanden,␈α!␈↓βNumber␈α Theory:␈α!An␈α!Introduction␈α!to␈α Proof␈↓,
␈↓ ↓H␈↓␈↓ αλInternational Textbook Comapny, Scranton, Pennsylvania, 1970.

␈↓ ↓H␈↓Fuller, R. Buckminster, ␈↓βIntuition␈↓, Doubleday, Garden City, New York, 1972.

␈↓ ↓H␈↓GCMP, ␈↓βKey Topics in Mathematics␈↓, Science Research Associates, Palo Alto, 1965.

␈↓ ↓H␈↓Goldstein,␈α∀Ira,␈α∀␈↓βElementary␈α∀Geometry␈α∀Theorem␈α∀Proving␈↓,␈α∀MIT␈α∀AI␈α∀Memo␈α∪280,
␈↓ ↓H␈↓␈↓ αλApril, 1973.

␈↓ ↓H␈↓Goodstein,␈α⊂R.␈α⊃L.,␈α⊂␈↓βFundamental␈α⊂Concepts␈α⊃of␈α⊂Mathematics␈↓,␈α⊂Pergamon␈α⊃Press,␈α⊂New
␈↓ ↓H␈↓␈↓ αλYork, 1962.

␈↓ ↓H␈↓Goodstein,␈α∨R.␈α L.,␈α∨␈↓βRecursive␈α∨Number␈α Theory␈↓,␈α∨North-Holland␈α Publishing␈α∨Co.,
␈↓ ↓H␈↓␈↓ αλAmsterdam, 1964.

␈↓ ↓H␈↓@Green,␈α⊗Waldinger,␈α⊗Barstow,␈α⊗Elschlager,␈α⊗Lenat,␈α⊗McCune,␈α⊗Shaw,␈α⊗and␈α↔Steinberg,␈α⊗␈↓βProgress
␈↓ ↓H␈↓β␈↓ αλReport␈α⊂on␈α⊃Program-Understanding␈α⊂Systems␈↓,␈α⊃Memo␈α⊂AIM-240,␈α⊃CS␈α⊂Report
␈↓ ↓H␈↓␈↓ αλSTAN-CS-74-444,Arti≡cial Intelligence Laboratory, Stanford University, August, 1974.

␈↓ ↓H␈↓@Hadamard,␈α⊗Jaques,␈α⊗␈↓βThe␈α⊗Psychology␈α⊗of␈α⊗Invention␈α⊗in␈α↔the␈α⊗Mathematical
␈↓ ↓H␈↓β␈↓ αλField␈↓, Dover Publications, New York, 1945.

␈↓ ↓H␈↓Halmos, Paul R., ␈↓βNaive Set Theory␈↓, D. Van Nostrand Co., Princeton, 1960.

␈↓ ↓H␈↓␈↓ ε%␈↓↓page 13␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Hanson,␈α→Norwood␈α→R.,␈α→␈↓βPerception␈α→and␈α→Discovery␈↓,␈α→Freeman,␈α→Cooper␈α→&␈α~Co.,␈α→San
␈↓ ↓H␈↓␈↓ αλFrancisco, 1969.

␈↓ ↓H␈↓Hartman,␈α↔Robert␈α↔S.,␈α↔␈↓βThe␈α⊗Structure␈α↔of␈α↔Value:␈α↔Foundations␈α↔of␈α⊗Scientific
␈↓ ↓H␈↓β␈↓ αλAxiology␈↓, Southern Illinois University Press, Carbondale, Ill., 1967.

␈↓ ↓H␈↓Hempel,␈α≥Carl␈α≡G.,␈α≥␈↓βFundamentals␈α≥of␈α≡Concept␈α≥Formation␈α≡in␈α≥Empirical
␈↓ ↓H␈↓β␈↓ αλScience␈↓, University of Chicago Press, Chicago, 1952.

␈↓ ↓H␈↓Hibben, John Grier, ␈↓βInductive Logic␈↓, Charles Scribner's Sons, New York, 1896.

␈↓ ↓H␈↓Hilpinen,␈α∪Risto,␈α∀␈↓βRules␈α∪of␈α∀Acceptance␈α∪and␈α∀Inductive␈α∪Logic␈↓,␈α∀Acta␈α∪Philosophica
␈↓ ↓H␈↓␈↓ αλFennica, Fasc. 22, North-Holland Publishing Company, Amsterdam, 1968.

␈↓ ↓H␈↓Hintikka, Jaako, ␈↓βKnowledge and Belief␈↓, Cornell U. Press, Ithaca, NY, 1962.

␈↓ ↓H␈↓Hintikka,␈α
Jaako,␈α
and␈α
Suppes,␈α
Patrick␈α
(eds.),␈α
␈↓βAspects␈α
of␈α
Inductive␈α
Logic␈↓,␈αNorth-Holland
␈↓ ↓H␈↓␈↓ αλPublishing Company, Amsterdam, 1966.

␈↓ ↓H␈↓Jouvenal, Bertrand de, ␈↓βThe Art of Conjecture␈↓, Basic Books, Inc., New York, 1967.

␈↓ ↓H␈↓@Kershner,␈α⊂R.B.,␈α⊂and␈α∂L.R.Wilcox,␈α⊂␈↓βThe␈α⊂Anatomy␈α∂of␈α⊂Mathematics␈↓,␈α⊂The␈α⊂Ronald␈α∂Press
␈↓ ↓H␈↓␈↓ αλCompany, New York, 1950.

␈↓ ↓H␈↓Klauder,␈α↔Francis␈α⊗J.,␈α↔␈↓βThe␈α⊗Wonder␈α↔of␈α⊗Intelligence␈↓,␈α↔Christopher␈α↔Publishing␈α⊗House,
␈↓ ↓H␈↓␈↓ αλNorth QUincy, Mass., 1973.

␈↓ ↓H␈↓Klerner,␈α&M.,␈α&and␈α&J.␈α&Reinfeld,␈α&eds.,␈α&␈↓βInteractive␈α&Systems␈α'for␈α&Applied
␈↓ ↓H␈↓β␈↓ αλMathematics␈↓,␈α∩ACM␈α∩Symposium,␈α⊃held␈α∩in␈α∩Washington,␈α⊃D.C.,␈α∩AUgust,␈α∩1967.␈α⊃Academic
␈↓ ↓H␈↓␈↓ αλPress, NY, 1968.

␈↓ ↓H␈↓Kline,␈α≤M.␈α≤(ed),␈α≤␈↓βMathematics␈α≤in␈α≤the␈α≤Modern␈α≤World:␈α≥Readings␈α≤from
␈↓ ↓H␈↓β␈↓ αλScientific American␈↓, W.H.Freeman and Co., San Francisco, 1968.

␈↓ ↓H␈↓@Kling,␈α"Robert␈α"Elliot,␈α#␈↓βReasoning␈α"by␈α"Analogy␈α"with␈α#Applications␈α"to
␈↓ ↓H␈↓β␈↓ αλHeuristic␈α∩Problem␈α∪Solving:␈α∩A␈α∩Case␈α∪Study␈↓,␈α∩Stanford␈α∪Arti≡cial␈α∩Intelligence
␈↓ ↓H␈↓␈↓ αλProject Memo AIM-147, CS Department report CS-216, August, 1971.

␈↓ ↓H␈↓Koestler, Arthur, ␈↓βThe Act of Creation␈↓, New York, Dell Pub., 1967.

␈↓ ↓H␈↓Korner, Stephan, ␈↓βConceptual Thinking␈↓, Dover Publications, New York, 1959.

␈↓ ↓H␈↓Krivine,␈α∃Jean-Louis,␈α∀␈↓βIntroduction␈α∃to␈α∃Axiomatic␈α∀Set␈α∃Theory␈↓,␈α∃Humanities␈α∀Press,
␈↓ ↓H␈↓␈↓ αλNew York, 1971.



␈↓ ↓H␈↓␈↓ ε%␈↓↓page 14␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Kubinski,␈α(Tadeusz,␈α(␈↓βOn␈α(Structurality␈α'of␈α(Rules␈α(of␈α(Inference␈↓,␈α'Prace
␈↓ ↓H␈↓␈↓ αλWroclawskiego Towarzystwa Naukowego, Seria A, Nr. 107, Worclaw, Poland, 1965.

␈↓ ↓H␈↓Lakatos,␈α∂Imre␈α∞(ed.),␈α∂␈↓βThe␈α∞Problem␈α∂of␈α∞Inductive␈α∂Logic␈↓,␈α∞North-Holland␈α∂Publishing␈α∞Co.,
␈↓ ↓H␈↓␈↓ αλAmsterdam, 1968.

␈↓ ↓H␈↓Lamon,␈α~William␈α~E.,␈α~␈↓βLearning␈α~and␈α~the␈α~Nature␈α~of␈α~Mathematiccs␈↓,␈α→Science
␈↓ ↓H␈↓␈↓ αλResearch Associates, Palo Alto, 1972.

␈↓ ↓H␈↓Lang, Serge, ␈↓βAlgebra␈↓, Addison-Wesley, Menlo Park, 1971.

␈↓ ↓H␈↓Lefrancois, Guy R., ␈↓βPsychological Theories and Human Learning␈↓, 1972.

␈↓ ↓H␈↓Le␈α∞Lionnais,␈α∂F.,␈α∞␈↓βGreat␈α∂Currents␈α∞of␈α∂Mathematical␈α∞Thought␈↓,␈α∂Dover␈α∞Publications,
␈↓ ↓H␈↓␈↓ αλNew York, 1971.

␈↓ ↓H␈↓Margenau,␈α∃Henry,␈α∃␈↓βIntegrative␈α∃Principles␈α∀of␈α∃Modern␈α∃Thought␈↓,␈α∃Gordon␈α∀and
␈↓ ↓H␈↓␈↓ αλBreach, New York, 1972.

␈↓ ↓H␈↓Martin,␈α∂James,␈α∂␈↓βDesign␈α∂of␈α∂Man-Computer␈α∂Dialogues␈↓,␈α∂Prentice-Hall,␈α∂Inc.,␈α∞Englewood
␈↓ ↓H␈↓␈↓ αλCli≥s, N. J., 1973.

␈↓ ↓H␈↓Martin,␈α_R.␈α_M.,␈α_␈↓βToward␈α→a␈α_Systematic␈α_Pragmatics␈↓,␈α_North␈α→Holland␈α_Publishing
␈↓ ↓H␈↓␈↓ αλCompany, Amsterdam, 1959.

␈↓ ↓H␈↓Mendelson,␈α⊃Elliott,␈α⊃␈↓βIntroduction␈α⊃to␈α⊃Mathematical␈α⊃Logic␈↓,␈α⊃Van␈α∩Nostrand␈α⊃Reinhold
␈↓ ↓H␈↓␈↓ αλCompany, New York, 1964.

␈↓ ↓H␈↓Meyer,␈α
Jerome␈α
S.,␈α␈↓βFun␈α
With␈α
Mathematics␈↓,␈αFawcett␈α
Publications,␈α
Greenwich,␈αConnecticut,
␈↓ ↓H␈↓␈↓ αλ1952.

␈↓ ↓H␈↓Mirsky, L., ␈↓βStudies in Pure Mathematics␈↓, Academic Press, New York, 1971.

␈↓ ↓H␈↓Moore,␈α∀Robert␈α∀C.,␈α∀␈↓βD-SCRIPT:␈α∀A␈α∀Computational␈α∀Theory␈α∃of␈α∀Descriptions␈↓,
␈↓ ↓H␈↓␈↓ αλMIT AI Memo 278, February, 1973.

␈↓ ↓H␈↓National␈α~Council␈α~of␈α~Teachers␈α~of␈α~Mathematics,␈α~␈↓βThe␈α~Growth␈α≠of␈α~Mathematical
␈↓ ↓H␈↓β␈↓ αλIdeas␈↓, 24th yearbook, NCTM, Washington, D.C., 1959.

␈↓ ↓H␈↓Newell, Allen, and Simon, Herbert, ␈↓βHuman Problem Solving␈↓, 1972.

␈↓ ↓H␈↓Nevins,␈α≠Arthur␈α≠J.,␈α≠␈↓βA␈α≠Human␈α≠Oriented␈α≠Logic␈α≠for␈α≠Automatic␈α≠Theorem
␈↓ ↓H␈↓β␈↓ αλProving␈↓, MIT AI Memo 268, October, 1972.

␈↓ ↓H␈↓Niven,␈α≥Ivan,␈α≥and␈α≤Zuckerman,␈α≥Herbert,␈α≥␈↓βAn␈α≤Introduction␈α≥to␈α≥the␈α≥Theory␈α≤of
␈↓ ↓H␈↓β␈↓ αλNumbers␈↓, John Wiley & Sons, Inc., New York, 1960.


␈↓ ↓H␈↓␈↓ ε%␈↓↓page 15␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Olson, Robert G., ␈↓βMeaning and Argument␈↓, Harcourt, Brace & World, New York, 1969.

␈↓ ↓H␈↓Ore, Oystein, ␈↓βNumber Theory and its History␈↓, McGraw-Hill, New York, 1948.

␈↓ ↓H␈↓Pietarinen,␈α≤Juhani,␈α≥␈↓βLawlikeness,␈α≤Analogy,␈α≥and␈α≤Inductive␈α≥Logic␈↓,␈α≤North-
␈↓ ↓H␈↓␈↓ αλHolland,␈αAmsterdam,␈α
published␈αas␈αv.␈α
26␈αof␈α
the␈αseries␈αActa␈α
Philosophica␈αFennica␈α
(J.␈αHintikka,
␈↓ ↓H␈↓␈↓ αλed.), 1972.

␈↓ ↓H␈↓@Poincare',␈α∀Henri,␈α∪␈↓βThe␈α∀Foundations␈α∀of␈α∪Science:␈α∀Science␈α∀and␈α∪Hypothesis,
␈↓ ↓H␈↓β␈↓ αλThe␈α∂Value␈α⊂of␈α∂Science,␈α⊂Science␈α∂and␈α⊂Method␈↓,␈α∂The␈α⊂Science␈α∂Press,␈α⊂New␈α∂York,
␈↓ ↓H␈↓␈↓ αλ1929.

␈↓ ↓H␈↓@Polya,␈α↔George,␈α↔␈↓βMathematics␈α↔and␈α↔Plausible␈α↔Reasoning␈↓,␈α↔Princeton␈α⊗University
␈↓ ↓H␈↓␈↓ αλPress, Princeton, Vol. 1, 1954; Vol. 2, 1954.

␈↓ ↓H␈↓@Polya,␈α∂George,␈α∞␈↓βHow␈α∂To␈α∞Solve␈α∂It␈↓,␈α∞Second␈α∂Edition,␈α∞Doubleday␈α∂Anchor␈α∞Books,␈α∂Garden␈α∞City,
␈↓ ↓H␈↓␈↓ αλNew York, 1957.

␈↓ ↓H␈↓@Polya,␈α
George,␈α∞␈↓βMathematical␈α
Discovery␈↓,␈α
John␈α∞Wiley␈α
&␈α
Sons,␈α∞New␈α
York,␈α
Vol.␈α∞1,␈α
1962;
␈↓ ↓H␈↓␈↓ αλVol. 2, 1965.

␈↓ ↓H␈↓Richardson,␈α≤Robert␈α≤P.,␈α≤and␈α≤Edward␈α≤H.␈α≤Landis,␈α≤␈↓βFundamental␈α≤Conceptions␈α≠of
␈↓ ↓H␈↓β␈↓ αλModern Mathematics␈↓, The Open Court Publishing Company, Chicago, 1916.

␈↓ ↓H␈↓Rosskopf,␈α↔Ste≥e,␈α↔Taback␈α↔(eds.),␈α↔␈↓βPiagetian␈α↔Cognitive-Development␈α↔Research
␈↓ ↓H␈↓β␈↓ αλand␈α∀Mathematical␈α∃Education␈↓,␈α∀National␈α∀Council␈α∃of␈α∀Teachers␈α∃of␈α∀Mathematics,
␈↓ ↓H␈↓␈↓ αλNew York, 1971.

␈↓ ↓H␈↓Rulison,␈α∞Je≥,␈α∂and...␈α∞␈↓βQA4,␈α∂A␈α∞Procedural␈α∞Frob...␈↓,␈α∂Technical␈α∞Note...,␈α∂Arti≡cial␈α∞Intelligence
␈↓ ↓H␈↓␈↓ αλCenter, SRI, Menlo Park, California, ..., 1973.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓β␈↓ αλMathematical Sciences␈↓, McGraw-Hill Book Company, New York, 1969.

␈↓ ↓H␈↓Schminke,␈α⊃C.␈α⊃W.,␈α⊂and␈α⊃Arnold,␈α⊃William␈α⊂R.,␈α⊃eds.,␈α⊃␈↓βMathematics␈α⊂is␈α⊃a␈α⊃Verb␈↓,␈α⊃The␈α⊂Dryden
␈↓ ↓H␈↓␈↓ αλPress, Hinsdale, Illinois, 1971.

␈↓ ↓H␈↓Singh,␈α⊗Jagjit,␈α⊗␈↓βGreat␈α⊗Ideas␈α∃of␈α⊗Modern␈α⊗Mathematics␈↓,␈α⊗Dover␈α⊗Publications,␈α∃New
␈↓ ↓H␈↓␈↓ αλYork, 1959.

␈↓ ↓H␈↓@Skemp,␈α→Richard␈α_R.,␈α→␈↓βThe␈α→Psychology␈α_of␈α→Learning␈α→Mathematics␈↓,␈α_Penguin
␈↓ ↓H␈↓␈↓ αλBooks, Ltd., Middlesex, England, 1971.

␈↓ ↓H␈↓Slocum,␈α_Jonathan,␈α_␈↓βThe␈α→Graph-Processing␈α_Language␈α_GROPE␈↓,␈α_U.␈α→Texas␈α_at
␈↓ ↓H␈↓␈↓ αλAustin, Technical Report NL-22, August, 1974.



␈↓ ↓H␈↓␈↓ ε%␈↓↓page 16␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Smith,␈α_Nancy␈α_Woodland,␈α↔␈↓βA␈α_Question-Answering␈α_System␈α_for␈α↔Elementary
␈↓ ↓H␈↓β␈↓ αλMathematics␈↓,␈α_Stanford␈α_Institute␈α_for␈α_Mathematical␈α_Studies␈α_in␈α_the␈α→Social␈α_Sciences,
␈↓ ↓H␈↓␈↓ αλTechnical Report 227, April 19, 1974.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓β␈↓ αλTheory of Meaning␈↓, Stanford IMSSS Technical Report 238, October 25, 1974.

␈↓ ↓H␈↓Stein,␈α-Sherman␈α-K.,␈α-␈↓βMathematics:␈α-The␈α-Man-Made␈α-Universe:␈α,An
␈↓ ↓H␈↓β␈↓ αλIntroduction␈α≠to␈α≠the␈α≠Spirit␈α≠of␈α≠Mathematics␈↓,␈α≠Second␈α≠Edition,␈α≠W.␈α~H.
␈↓ ↓H␈↓␈↓ αλFreeman and Company, San Francisco, 1969.

␈↓ ↓H␈↓Stewart, B. M., ␈↓βTheory of Numbers␈↓, The MacMillan Co., New York, 1952.

␈↓ ↓H␈↓Stokes,␈α∃C.␈α∃Newton,␈α∃␈↓βTeaching␈α∃the␈α∃Meanings␈α∃of␈α⊗Arithmetic␈↓,␈α∃Appleton-Century-
␈↓ ↓H␈↓␈↓ αλCrofts, New York, 1951.

␈↓ ↓H␈↓Suppes,␈α∞Patrick,␈α∂␈↓βA␈α∞Probabilistic␈α∂Theory␈α∞of␈α∂Causality␈↓,␈α∞Acta␈α∂Philosophica␈α∞Fennica,
␈↓ ↓H␈↓␈↓ αλFasc. 24, North-Holland Publishing Company, Amsterdam, 1970.

␈↓ ↓H␈↓Teitelman, Warren, ␈↓βINTERLISP Reference Manual␈↓, XEROX PARC, 1974.

␈↓ ↓H␈↓Venn,␈α∃John,␈α∃␈↓βThe␈α∃Principles␈α∃of␈α∃Empirical␈α∃or␈α∃Inductive␈α⊗Logic␈↓,␈α∃MacMillan
␈↓ ↓H␈↓␈↓ αλand Co., London, 1889.

␈↓ ↓H␈↓Waismann,␈α∂Friedrich,␈α⊂␈↓βIntroduction␈α∂to␈α⊂Mathematical␈α∂Thinking␈↓,␈α⊂Frederick␈α∂Ungar
␈↓ ↓H␈↓␈↓ αλPublishing Co., New York, 1951.

␈↓ ↓H␈↓Wickelgren,␈α∪Wayne␈α∀A.,␈α∪␈↓βHow␈α∀to␈α∪Solve␈α∀Problems:␈α∪Elements␈α∀of␈α∪a␈α∀Theory␈α∪of
␈↓ ↓H␈↓β␈↓ αλProblems and Problem Solving␈↓, W. H. Freeman and Co., Sanf Francisco, 1974.

␈↓ ↓H␈↓Wilder,␈α⊃Raymond␈α⊃L.,␈α∩␈↓βEvolution␈α⊃of␈α⊃Mathematical␈α∩Concepts␈↓,␈α⊃John␈α⊃Wiley␈α∩&␈α⊃Sons,
␈↓ ↓H␈↓␈↓ αλInc., NY, 1968.

␈↓ ↓H␈↓Winston,␈α∂P.,␈α∂(ed.),␈α∞"New␈α∂Progress␈α∂in␈α∞Arti≡cial␈α∂Intelligence",␈α∂␈↓βMIT␈α∞AI␈α∂Lab␈α∂Memo␈α∞AI-TR-
␈↓ ↓H␈↓β␈↓ αλ310␈↓,␈αJune,␈α1974.␈αGood␈αsummaries␈αof␈αwork␈αon␈αFrames,␈αDemons,␈αHacker,␈αHeterarchy,␈αDialogue,
␈↓ ↓H␈↓␈↓ αλand Belief.

␈↓ ↓H␈↓Wittner,␈α↔George␈α_E.,␈α↔␈↓βThe␈α↔Structure␈α_of␈α↔Mathematics␈↓,␈α↔Xerox␈α_College␈α↔Publishing,
␈↓ ↓H␈↓␈↓ αλLexington, Mass, 1972.

␈↓ ↓H␈↓Wright,␈α∀Georg␈α∀H.␈α∀von,␈α∃␈↓βA␈α∀Treatise␈α∀on␈α∀Induction␈α∀and␈α∃Probability␈↓,␈α∀Routledge
␈↓ ↓H␈↓␈↓ αλand Kegan Paul, London, 1951.





␈↓ ↓H␈↓␈↓ ε%␈↓↓page 17␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓αArticles

␈↓ ↓H␈↓Amarel,␈α≥Saul,␈α≥␈↓βOn␈α≥Representations␈α≥of␈α≥Problems␈α≥of␈α≥Reasoning␈α≤about
␈↓ ↓H␈↓β␈↓ αλActions␈↓, Machine Intelligence 3, 1968, pp. 131-171.

␈↓ ↓H␈↓Bledsoe,␈α!W.␈α!W.,␈α!␈↓βSplitting␈α and␈α!Reduction␈α!Heuristics␈α!in␈α Automatic
␈↓ ↓H␈↓β␈↓ αλTheorem Proving␈↓, Arti≡cial Intelligence 2, 1971, pp. 55-77.

␈↓ ↓H␈↓Bledsoe␈α⊂and␈α∂Bruell,␈α⊂Peter,␈α∂␈↓βA␈α⊂Man-Machine␈α∂Theorem-Proving␈α⊂System␈↓,␈α∂Arti≡cial
␈↓ ↓H␈↓␈↓ αλIntelligence 5, 1974, 51-72.

␈↓ ↓H␈↓Bourbaki,␈α∪Nicholas,␈α∪␈↓βThe␈α∀Architechture␈α∪of␈α∪Mathematics␈↓,␈α∀American␈α∪Mathematics
␈↓ ↓H␈↓␈↓ αλMonthly, v. 57, pp. 221-232, Published by the MAA, Albany, NY, 1950.

␈↓ ↓H␈↓@Boyer,␈α⊃Robert␈α⊃S.,␈α⊃and␈α⊃J.␈α⊃S.␈α⊃Moore,␈α⊃␈↓βProving␈α⊃Theorems␈α⊃about␈α⊃LISP␈α⊂Functions␈↓,
␈↓ ↓H␈↓␈↓ αλJACM, V. 22, No. 1, January, 1975, pp. 129-144.

␈↓ ↓H␈↓@Bruijn,␈α⊂N.␈α∂G.␈α⊂de,␈α⊂␈↓βAUTOMATH,␈α∂a␈α⊂language␈α⊂for␈α∂mathematics␈↓,␈α⊂Notes␈α⊂taken␈α∂by
␈↓ ↓H␈↓␈↓ αλBarry␈α
Fawcett,␈α
of␈α
Lecures␈α
given␈α∞at␈α
the␈α
Seminare␈α
de␈α
mathematiques␈α
Superieurs,␈α∞University␈α
de
␈↓ ↓H␈↓␈↓ αλMontreal, June, 1971. Stanford University Computer Science Library report number is 005913.

␈↓ ↓H␈↓@Buchanan,␈α→Feigenbaum,␈α→and␈α_Sridharan,␈α→␈↓βHeuristic␈α→Theory␈α→Formation␈↓,␈α_Machine
␈↓ ↓H␈↓␈↓ αλIntelligence 7, 1972, pp. 267-...

␈↓ ↓H␈↓@Bundy, Alan, ␈↓βDoing Arithmetic with Diagrams␈↓, 3rd IJCAI, 1973, pp. 130-138.

␈↓ ↓H␈↓Daalen,␈α∃D.␈α∃T.␈α∃van,␈α∃␈↓βA␈α∃Description␈α∃of␈α∃AUTOMATH␈α∃and␈α∃some␈α∃aspects␈α∀of
␈↓ ↓H␈↓β␈↓ αλits␈α→language␈α_theory␈↓,␈α→in␈α_the␈α→Proceedings␈α_of␈α→the␈α_SYmposium␈α→on␈α→APL,␈α_Paris,
␈↓ ↓H␈↓␈↓ αλDecember,␈α1973,␈αP.␈αBra≥ort␈α
(ed).␈αThis␈αvolume␈αalso␈αcontains␈α
other,␈αmore␈αdetailed␈αarticles␈αon␈α
this
␈↓ ↓H␈↓␈↓ αλproject, by Bert Jutting and Ids Zanlevan.

␈↓ ↓H␈↓Engelman,␈α!C.,␈α"␈↓βMATHLAB:␈α!A␈α!Program␈α"for␈α!On-Line␈α"Assistance␈α!in
␈↓ ↓H␈↓β␈↓ αλSymbolic Computation␈↓, in Proceedings of the FJCC, Volume 2, Spartan Books, 1965.

␈↓ ↓H␈↓Engelman, C., ␈↓βMATHLAB '68␈↓, in IFIP, Edinburgh, 1968.

␈↓ ↓H␈↓Gardner,␈α
Martin,␈α␈↓βMathematical␈α
Games␈↓,␈α
Scienti≡c␈αAmerican,␈α
numerous␈α
columns,␈αincluding
␈↓ ↓H␈↓␈↓ αλespecially: February, 1975.

␈↓ ↓H␈↓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
␈↓ ↓H␈↓β␈↓ αλComputing␈α↔Machines,␈↓␈α_pages␈α↔1:33␈α↔of␈α_Volumne␈α↔5␈α↔of␈α_A.␈α↔H.␈α↔Taub␈α_(ed),␈α↔␈↓βThe
␈↓ ↓H␈↓β␈↓ αλCollected Works of John von Neumann␈↓, Pergamon Press, NY, 1963.

␈↓ ↓H␈↓Guard,␈α∞J.␈α
R.,␈α∞et␈α
al.,␈α∞␈↓βSemi-Automated␈α∞Mathematics␈↓,␈α
JACM␈α∞16,␈α
January,␈α∞1969,␈α∞pp.␈α
49-
␈↓ ↓H␈↓␈↓ αλ62.


␈↓ ↓H␈↓␈↓ ε%␈↓↓page 18␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Halmos,␈α
Paul␈α
R.,␈α␈↓βInnovation␈α
in␈α
Mathematics␈↓,␈α
in␈αKline,␈α
M.␈α
(ed),␈α
␈↓βMathematics␈αin
␈↓ ↓H␈↓β␈↓ αλthe␈α-Modern␈α-World:␈α-Readings␈α-from␈α.Scientific␈α-American␈↓,
␈↓ ↓H␈↓␈↓ αλW.H.Freeman␈α∀and␈α∃Co.,␈α∀San␈α∀Francisco,␈α∃1968,␈α∀pp.␈α∀6-13.␈α∃Originally␈α∀in␈α∃Scienti≡c␈α∀American,
␈↓ ↓H␈↓␈↓ αλSeptember, 1958.

␈↓ ↓H␈↓Hasse,␈α⊃H.,␈α⊃␈↓βMathemakik␈α⊃als␈α⊂Wissenschaft,␈α⊃Kunst␈α⊃und␈α⊃Macht␈↓,␈α⊂(Mathematics
␈↓ ↓H␈↓␈↓ αλas Science, Art, and Power), Baden-Badeb, 1952.

␈↓ ↓H␈↓@Hewitt,␈α∃Carl,␈α∃␈↓βA␈α∃Universal␈α∃Modular␈α∃ACTOR␈α∃Formalism␈α∃for␈α∃Artificial
␈↓ ↓H␈↓β␈↓ αλIntelligence␈↓,␈α
Third␈α
International␈α
Joint␈αConference␈α
on␈α
Arti≡cial␈α
Intelligence,␈α
1973,␈αpp.␈α
235-
␈↓ ↓H␈↓␈↓ αλ245.

␈↓ ↓H␈↓Menges,␈α_Gunter,␈α_␈↓βInference␈α_and␈α_Decision␈↓,␈α_A␈α_Volume␈α_in␈α→␈↓βSelecta␈α_Statistica
␈↓ ↓H␈↓β␈↓ αλCanadiana␈↓, John Wiley & Sons, New York, 1973, pp. 1-16.

␈↓ ↓H␈↓Kling,␈α⊃Robert␈α⊂E.,␈α⊃␈↓βA␈α⊃Paradigm␈α⊂for␈α⊃Reasoning␈α⊂by␈α⊃Analogy␈↓,␈α⊃Arti≡cial␈α⊂Intelligence
␈↓ ↓H␈↓␈↓ αλ2, 1971, pp. 147-178.

␈↓ ↓H␈↓Knuth,Donald␈α∂E.,␈α∂␈↓βAncient␈α∂Babylonian␈α∂Algorithms␈↓,␈α∂CACM␈α∂15,␈α∂July,␈α∂1972,␈α∂pp.␈α∂671-
␈↓ ↓H␈↓␈↓ αλ677.

␈↓ ↓H␈↓Lee,␈α∀Richard␈α∃C.␈α∀T.,␈α∃␈↓βFuzzy␈α∀Logic␈α∃and␈α∀the␈α∃Resolution␈α∀Principle␈↓,␈α∃JACM␈α∀19,
␈↓ ↓H␈↓␈↓ αλJanuary, 1972, pp. 109-119.

␈↓ ↓H␈↓@Lenat, D., ␈↓βBEINGs: Knowledge as Interacting Experts␈↓, 4th IJCAI, 1975.

␈↓ ↓H␈↓McCarthy,␈α∃John,␈α∀and␈α∃Hayes,␈α∀Patrick,␈α∃␈↓βSome␈α∀Philosophical␈α∃Problems␈α∃from␈α∀the
␈↓ ↓H␈↓β␈↓ αλStandpoint␈α⊂of␈α∂Artificial␈α⊂Intelligence␈↓,␈α∂Machine␈α⊂Intelligence␈α∂4,␈α⊂1969,␈α⊂pp.␈α∂463-
␈↓ ↓H␈↓␈↓ αλ502.

␈↓ ↓H␈↓Martin,␈α∞W.,␈α∞and␈α∞Fateman,␈α∞R.,␈α
␈↓βThe␈α∞MACSYMA␈α∞System␈↓,␈α∞Second␈α∞Symposium␈α∞on␈α
Symbolic
␈↓ ↓H␈↓␈↓ αλand Algebraic Manipulation, 1971, pp. 59-75.

␈↓ ↓H␈↓@Minsky, Marvin, ␈↓βFrames␈↓, in (Winston) ␈↓βPsychology of Computer Vision␈↓, 1974.

␈↓ ↓H␈↓Moore,␈α∩J.,␈α∩and␈α∩Newell,␈α∪␈↓βHow␈α∩Can␈α∩Merlin␈α∩Understand?␈↓,␈α∪Carnegie-Mellon␈α∩University
␈↓ ↓H␈↓␈↓ αλDepartment of Computer Science "preprint", November 15, 1973.

␈↓ ↓H␈↓@Neumann,␈α
J.␈α∞von,␈α
␈↓βThe␈α
Mathematician␈↓,␈α∞in␈α
R.B.␈α
Heywood␈α∞(ed),␈α
␈↓βThe␈α
Works␈α∞of␈α
the
␈↓ ↓H␈↓β␈↓ αλMind␈↓, U. Chicago Press, pp. 180-196, 1947.

␈↓ ↓H␈↓Neumann,␈α∩J.␈α∩von,␈α∩␈↓βThe␈α∩Computer␈α∩and␈α∩the␈α∩Brain␈↓,␈α∩Silliman␈α∩Lectures,␈α∩Yale␈α∩U.␈α∩Press,
␈↓ ↓H␈↓␈↓ αλ1958.

␈↓ ↓H␈↓Pager,␈α∀David,␈α∀␈↓βA␈α∀Proposal␈α∀for␈α∀a␈α∀Computer-based␈α∀Interactive␈α∪Scientific
␈↓ ↓H␈↓β␈↓ αλCommunity␈↓, CACM 15, February, 1972, pp. 71-75.

␈↓ ↓H␈↓␈↓ ε%␈↓↓page 19␈↓␈↓ h
␈↓ ↓H␈↓␈↓αNSF Proposal␈↓ ¬¬Program-Understanding Group␈↓ 	QDRAFT: May 9, 1975␈↓


␈↓ ↓H␈↓Pager,␈α∪David,␈α∪␈↓βOn␈α∀the␈α∪Problem␈α∪of␈α∪Communicating␈α∀Complex␈α∪Information␈↓,
␈↓ ↓H␈↓␈↓ αλCACM 16, May, 1973, pp. 275-281.

␈↓ ↓H␈↓@Sloman,␈α'Aaron,␈α'␈↓βInteractions␈α'Between␈α'Philosophy␈α'and␈α'Artificial
␈↓ ↓H␈↓β␈↓ αλIntelligence:␈α∃The␈α∃Role␈α⊗of␈α∃Intuition␈α∃and␈α⊗Non-Logical␈α∃Reasoning
␈↓ ↓H␈↓β␈↓ αλin Intelligence␈↓, Arti≡cial Intelligence 2, 1971, pp. 209-225.

␈↓ ↓H␈↓Sloman, Aaron, ␈↓βOn Learning about Numbers␈↓,...

␈↓ ↓H␈↓Winston,␈α∩Patrick,␈α∩␈↓βLearning␈α∩Structural␈α∩Descriptions␈α∩from␈α∪Examples␈↓,␈α∩Ph.D.
␈↓ ↓H␈↓␈↓ αλthesis,␈αDept.␈αof␈αElectrical␈αEngineering,␈αTR-76,␈αProject␈αMAC,␈αTR-231,␈αMIT␈αAI␈αLab,␈α
September,
␈↓ ↓H␈↓␈↓ αλ1970.




































␈↓ ↓H␈↓␈↓ ε%␈↓↓page 20␈↓␈↓ h