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␈↓ ↓H␈↓␈↓ βx␈↓∧␈↓&AUTOMATED   MATHEMATICIAN␈↓)αβ␈↓

␈↓ ↓H␈↓␈↓ ∧l␈↓∧Supplementary Materials␈↓

␈↓ ↓H␈↓␈↓ β:for the Stanford ␈↓↓Heuristic Programming Project␈↓ Workshop

␈↓ ↓H␈↓␈↓ ¬Q␈↓βJanuary 5-8, 1976␈↓


␈↓ ↓H␈↓␈↓ ¬WDouglas B. Lenat




␈↓ ↓H␈↓␈↓∧␈↓&Contents␈↓)αβ␈↓
␈↓ ↓H␈↓1. Brief global description of the AM project
␈↓ ↓H␈↓2. The tutorial talk
␈↓ ↓H␈↓3. The details talk
␈↓ ↓H␈↓4. Glossary of concepts and terms
␈↓ ↓H␈↓5. Existing documentation




␈↓ ↓H␈↓␈↓∧␈↓&1. Overview␈↓)αβ␈↓

␈↓ ↓H␈↓Researchers␈αin␈αmost␈αbranches␈αof␈αscience␈αfrequently␈αface␈αthe␈αdi≠cult␈αtask␈α of␈αformulating␈αresearch
␈↓ ↓H␈↓problems␈αwhich␈α must␈αbe␈αsoluble␈αyet␈αnontrivial.␈α In␈αany␈αgiven␈α≡eld,␈αit␈αis␈αusually␈αeasier␈αto␈αtackle␈αa
␈↓ ↓H␈↓speci≡c␈α
given␈α
problem␈α
than␈α
to␈α
 propose␈α
interesting␈α
yet␈α
managable␈α
new␈α
questions␈α
 to␈α investigate.
␈↓ ↓H␈↓For␈α∂ example,␈α∂contrast␈α∂ ␈↓βsolving␈↓␈α∂the␈α∂Missionaries␈α∂and␈α∂ Cannibals␈α∂ problem␈α∂ with␈α∂the␈α⊂ more␈α∂ ill-
␈↓ ↓H␈↓de≡ned reasoning which led to ␈↓βinventing␈↓ it.

␈↓ ↓H␈↓Let's␈αrestrict␈αour␈αattention␈αto␈αcreative␈αtheory␈αformation␈αin␈α␈↓βmathematics␈↓:␈αhow␈αto␈αpropose␈αinteresting
␈↓ ↓H␈↓new␈α∩concepts␈α∩ and␈α⊃plausible␈α∩hypotheses␈α∩connecting␈α∩them.␈α⊃ Although␈α∩many␈α∩great␈α∩minds␈α⊃have
␈↓ ↓H␈↓introspected␈α⊃on␈α⊃this␈α⊃problem␈α⊂[Poincare',␈α⊃Hadamard,␈α⊃Polya],␈α⊃we␈α⊃in␈α⊂AI␈α⊃all␈α⊃know␈α⊃the␈α⊃gulf␈α⊂that
␈↓ ↓H␈↓separates smooth prose from smooth code.

␈↓ ↓H␈↓The␈α∃experimental␈α∀ vehicle␈α∃of␈α∀my␈α∃research␈α∀is␈α∃  a␈α∀   computer␈α∃  program␈α∀  called␈α∃  ␈↓↓AM␈↓␈α∀(for
␈↓ ↓H␈↓␈↓↓␈↓&A␈↓)αβ␈↓utomated␈α∞␈↓↓␈↓&M␈↓)αβ␈↓athematician),␈α∞  which␈α∂carries␈α∞out␈α∞some␈α∞of␈α∂the␈α∞activities␈α∞involved␈α∂in␈α∞mathematical
␈↓ ↓H␈↓research:␈α∂noticing␈α⊂simple␈α∂relationships␈α⊂in␈α∂empirical␈α⊂data,␈α∂formulating␈α⊂ new␈α∂ de≡nitions␈α⊂ out␈α∂ of
␈↓ ↓H␈↓existing␈α∂ ones,␈α∂ proposing␈α∞some␈α∂plausible␈α∂conjectures␈α∞(and,␈α∂less␈α∂importantly,␈α∂sometimes␈α∞ proving
␈↓ ↓H␈↓them), and evaluating the aesthetic "interestingness" of new concepts.

␈↓ ↓H␈↓Before␈α∂discussing␈α∂how␈α∂to␈α∂␈↓βsynthesize␈↓␈α∞a␈α∂new␈α∂theory,␈α∂consider␈α∂brie∨y␈α∞how␈α∂to␈α∂␈↓βanalyze␈↓␈α∂one,␈α∂how␈α∞to
␈↓ ↓H␈↓construct␈α
a␈αplausible␈α
chain␈αof␈α
reasoning␈αwhich␈α
terminates␈αin␈α
a␈αgiven␈α
discovery.␈αOne␈α
can␈α
do␈αthis
␈↓ ↓H␈↓by␈α∂working␈α∂backwards,␈α⊂by␈α∂reducing␈α∂the␈α⊂creative␈α∂act␈α∂to␈α∂simpler␈α⊂and␈α∂simpler␈α∂creative␈α⊂acts.␈α∂ For
␈↓ ↓H␈↓example,␈α∞consider␈α∞ the␈α∞concept␈α∞ of␈α
prime␈α∞ numbers.␈α∞ How␈α∞might␈α∞ one␈α
 be␈α∞led␈α∞ to␈α∞de≡ne␈α∞ such␈α
a
␈↓ ↓H␈↓notion? Notice  the following plausible strategy:
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	K␈↓εSUPPLEMENT     page   2␈↓


␈↓ ↓H␈↓␈↓ α("If␈α∞f␈α∞is␈α∞a␈α∞function␈α∞which␈α∞transforms␈α∂elements␈α∞of␈α∞A␈α∞into␈α∞elements␈α∞of␈α∞B,␈α∞and␈α∂B␈α∞is
␈↓ ↓H␈↓␈↓ α(ordered,␈α∀then␈α∀consider␈α∀just␈α∀those␈α∪members␈α∀of␈α∀A␈α∀which␈α∀are␈α∀transformed␈α∪into
␈↓ ↓H␈↓␈↓ α(␈↓βextremal␈↓ elements of B.  This  set is an interesting subset of A."

␈↓ ↓H␈↓When␈α∞f(x)␈α∞means␈α∞"factors␈α∞of␈α∞x",␈α∞and␈α∞the␈α∞ordering␈α∞is␈α∞"by␈α∞length",␈α∞this␈α∞heuristic␈α∞says␈α∞to␈α
consider
␈↓ ↓H␈↓those␈α⊂numbers␈α∂which␈α⊂have␈α∂a␈α⊂minimal␈↓#
1␈↓#␈α∂ number␈α⊂of␈α⊂factors␈α∂--␈α⊂that␈α∂is,␈α⊂the␈α∂primes.␈α⊂So␈α⊂this␈α∂rule
␈↓ ↓H␈↓actually␈α␈↓βreduces␈↓␈α our␈αtask␈αfrom␈α"proposing␈αthe␈αconcept␈αof␈αprime␈αnumbers"␈αto␈αthe␈αmore␈αelementary
␈↓ ↓H␈↓problems of "inventing factorization" and "discovering cardinality".

␈↓ ↓H␈↓But␈αsuppose␈αwe␈αknow␈αthis␈αgeneral␈αrule:␈α"If␈α f␈α is␈αan␈α interesting␈α function,␈α consider␈αits␈αinverse."␈α
It
␈↓ ↓H␈↓reduces␈α the␈αtask␈αof␈α discovering␈αfactorization␈α to␈αthe␈α simpler␈αtask␈α of␈α discovering␈αmultiplication.
␈↓ ↓H␈↓Eventually,␈α
this␈α
task␈α
reduces␈α
to␈α
the␈α
discovery␈α
of␈α
very␈α
basic␈α
notions,␈α
like␈α
substitution,␈α
set-union,␈α
and
␈↓ ↓H␈↓equality.␈α∩ To␈α∩explain␈α⊃how␈α∩a␈α∩given␈α∩researcher␈α⊃might␈α∩have␈α∩made␈α⊃a␈α∩given␈α∩discovery,␈α∩such␈α⊃an
␈↓ ↓H␈↓analysis␈α⊃is␈α∩continued␈α⊃until␈α∩that␈α⊃inductive␈α∩task␈α⊃is␈α∩reduced␈α⊃to␈α∩"discovering"␈α⊃notions␈α∩which␈α⊃the
␈↓ ↓H␈↓researcher already knew.

␈↓ ↓H␈↓Suppose␈α
a␈αlarge␈α
collection␈α
of␈αthese␈α
 heuristics␈α
has␈αbeen␈α
assembled␈α
(e.g.,␈αby␈α
 analyzing␈α
a␈αgreat␈α
many
␈↓ ↓H␈↓discoveries,␈αand␈αwriting␈α
down␈αnew␈αheuristic␈α
rules␈αwhenever␈α necessary.)␈α
 Instead␈α of␈αusing␈αthem␈α
to
␈↓ ↓H␈↓␈↓βexplain␈↓␈α∂how␈α∂a␈α∂given␈α∂idea␈α∂might␈α∂have␈α∂evolved,␈α∂one␈α∂can␈α∂imagine␈α∂starting␈α∂from␈α∂a␈α∂basic␈α∂core␈α∂of
␈↓ ↓H␈↓knowledge and "running"  the heuristics to ␈↓βgenerate␈↓ new concepts.

␈↓ ↓H␈↓Such␈α⊃syntheses␈α⊂are␈α⊃precisely␈α⊃what␈α⊂AM␈α⊃does.␈α⊃ The␈α⊂program␈α⊃consists␈α⊃of␈α⊂a␈α⊃corpus␈α⊃of␈α⊂primitive
␈↓ ↓H␈↓mathematical␈αconcepts␈αand␈α
a␈αcollection␈αof␈α
guiding␈αheuristics.␈α AM's␈α
activities␈α all␈αserve␈α to␈α
expand
␈↓ ↓H␈↓AM␈α⊂ itself,␈↓#
2␈↓#␈α⊂to␈α⊂ enlarge␈α⊂upon␈α⊂ a␈α⊂given␈α⊂body␈α⊂ of␈α⊂mathematical␈α⊂knowledge.␈α⊂ To␈α⊂cope␈α⊂with␈α∂ the
␈↓ ↓H␈↓enormity␈αof␈αthe␈αpotential␈α"search␈αspace"␈α involved,␈αAM␈α uses␈αits␈αheuristics␈αas␈αjudgmental␈α criteria
␈↓ ↓H␈↓to␈α∩ guide␈α∩development␈α∩ in␈α⊃ the␈α∩ most␈α∩ promising␈α∩  direction.␈α⊃ It␈α∩appears␈α∩that␈α∩ the␈α∩process␈α⊃of
␈↓ ↓H␈↓inventing␈αvaluable␈α
new␈α concepts␈α
can␈αbe␈α
guided␈αsuccessfully␈αusing␈α
a␈αcollection␈α
of␈αa␈α
few␈αhundred
␈↓ ↓H␈↓such heuristics.

␈↓ ↓H␈↓Each␈α concept␈α is␈αrepresented␈α as␈α
 a␈α␈↓¬BEING␈↓␈↓#
3␈↓#,␈αa␈αframe-like␈αdata␈α
structure␈αwith␈α 30␈αdi≥erent␈αfacets␈α
or
␈↓ ↓H␈↓slots.␈α→The␈α~types␈α→of␈α~slots␈α→include:␈α→␈↓¬Examples,␈α~Definitions,␈α→ Generalizations,␈α~Utility,␈α→Analogies,
␈↓ ↓H␈↓¬Interestingness,␈α↔Uninterestingness,␈↓␈α_and␈α↔a␈α↔couple␈α_dozen␈α↔others.␈α↔ The␈α_ ␈↓¬BEINGs␈↓␈α↔representation
␈↓ ↓H␈↓provides␈αa␈αconvenient␈α
scheme␈αfor␈αorganizing␈α
the␈αheuristics;␈αfor␈α
example,␈αthe␈αfollowing␈αstrategy␈α
≡ts
␈↓ ↓H␈↓into␈α∞the␈α∞␈↓βExamples␈↓␈α
slot␈α∞of␈α∞the␈α∞␈↓βPredicate␈↓␈α
concept:␈α∞"If,␈α∞empirically,␈α∞10␈α
times␈α∞as␈α∞many␈α∞elements␈α
␈↓βfail␈↓
␈↓ ↓H␈↓some␈α
predicate␈α
P,␈αas␈α
␈↓βsatisfy␈↓␈α
it,␈α
then␈αsome␈α
␈↓βgeneralization␈↓␈α
(weakened␈α
version)␈αof␈α
P␈α
might␈α
be␈αmore
␈↓ ↓H␈↓interesting␈α∀than␈α∀P".␈α∃ AM␈α∀considers␈α∀this␈α∃suggestion␈α∀after␈α∀trying␈α∀to␈α∃≡ll␈α∀in␈α∀examples␈α∃of␈α∀any
␈↓ ↓H␈↓predicate.␈↓#
4␈↓#

␈↓ ↓H␈↓AM␈αis␈αinitially␈α given␈αa␈αlarge␈αcollection␈αof␈α
core␈αconcepts,␈αwith␈αonly␈α a␈αfew␈αslots␈α≡lled␈αin␈α
for␈αeach.
␈↓ ↓H␈↓Its␈αsole␈α activity␈αis␈αto␈α choose␈αsome␈αfacet␈α of␈αsome␈αconcept,␈α and␈α≡ll␈α in␈αthat␈α particular␈αslot.␈α In␈αso
␈↓ ↓H␈↓doing,␈α
 new␈α
notions␈α
will␈α
often␈α
 emerge.␈α
 Uninteresting␈α
ones␈α
are␈α
forgotten,␈α
mildly␈α∞interesting␈α
ones
␈↓ ↓H␈↓are␈α⊃kept␈α⊃ as␈α⊃parts␈α⊃of␈α⊃one␈α⊃ slot␈α⊃of␈α⊃one␈α⊃concept,␈α⊃ and␈α⊃very␈α⊃ interesting␈α⊃ones␈α⊃ are␈α⊃ granted␈α⊃full
␈↓ ↓H␈↓concept␈α∩ status.␈α∩Such␈α∩ new␈α⊃␈↓¬Beings␈↓␈α∩ will␈α∩ have␈α∩dozens␈α∩ of␈α⊃ blank␈α∩ parts,␈α∩hence␈α∩ the␈α∩ space␈α⊃of
␈↓ ↓H␈↓possible␈αactions␈α(slots␈αto␈α≡ll␈αin)␈αgrows␈αrapidly.␈α The␈αsame␈α heuristics␈αare␈αused␈αboth␈αto␈αsuggest␈αnew
␈↓ ↓H␈↓directions for investigation, and to limit attention: both to grow and to prune.

␈↓ ↓H␈↓The␈α⊂particular␈α∂mathematical␈α⊂domains␈α∂in␈α⊂which␈α∂AM␈α⊂operates␈α∂depend␈α⊂ on␈α∂the␈α⊂choice␈α⊂of␈α∂initial
␈↓ ↓H␈↓concepts.␈αCurrently,␈α AM␈αis␈αgiven␈αabout␈αa␈αhundred␈αconcepts,␈αall␈αof␈αwhich␈αare␈αwhat␈αPiaget␈αmight
␈↓ ↓H␈↓describe␈α
as␈α
␈↓βprenumerical␈↓:␈α
Sets,␈α
substitution,␈α
 equality,␈αrelations,␈α
and␈α
so␈α
on.␈α
 In␈α
particular,␈α
 AM␈αis
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	K␈↓εSUPPLEMENT     page   3␈↓


␈↓ ↓H␈↓not␈α
told␈α
 anything␈α∞about␈α
 proof,␈α
 single-valued␈α
functions,␈α∞or␈α
numbers.␈α
  Although␈α
 it␈α∞ was␈α
never
␈↓ ↓H␈↓able␈α∞to␈α∞ ␈↓βprove␈↓␈α∞ the␈α∞ unique␈α∞ factorization␈α
theorem,␈α∞AM␈α∞actually␈α∞ did␈α∞␈↓βconjecture␈↓␈α∞it.␈↓#
5␈↓#␈α∞ Before␈α
this,
␈↓ ↓H␈↓AM␈α⊂had␈α⊂to␈α⊂de≡ne␈α⊂and␈α⊂investigate␈α⊃concepts␈α⊂corresponding␈α⊂to␈α⊂those␈α⊂we␈α⊂refer␈α⊂to␈α⊃as␈α⊂cardinality,
␈↓ ↓H␈↓multiplication, factors, and primes, based on reasoning similar to that in the example above.

␈↓ ↓H␈↓The␈α∂main␈α∂di≠culty␈α∂with␈α⊂AM␈α∂is␈α∂getting␈α∂it␈α⊂to␈α∂accurately␈α∂judge␈α∂␈↓βa␈α⊂priori␈↓␈α∂the␈α∂value␈α∂of␈α⊂each␈α∂new
␈↓ ↓H␈↓concept,␈αto␈αquickly␈αlose␈αinterest␈αin␈αconcepts␈αwhich␈αaren't␈αgoing␈αto␈αdevelop␈αinto␈αanything.␈α As␈αwith
␈↓ ↓H␈↓many␈α∞AI␈α∞programs,␈α∂one␈α∞aspect␈α∞of␈α∂working␈α∞on␈α∞AM␈α∂is␈α∞the␈α∞degree␈α∂of␈α∞precision␈α∞with␈α∂which␈α∞one's
␈↓ ↓H␈↓ideas␈αmust␈α
be␈αformulated.␈α
 The␈αresultant␈α
body␈αof␈αdetailed␈α
heuristics␈αmay␈α
be␈αthe␈α
germ␈αof␈α
a␈αmore
␈↓ ↓H␈↓e≠cient␈αprogramme␈αfor␈αeducating␈αmath␈αstudents␈αthan␈αthe␈αcurrent␈αdogma.␈↓#
6␈↓#␈αBut␈αperhaps␈αthe␈αmost
␈↓ ↓H␈↓exciting␈αprospect␈αopened␈αup␈αby␈αAM␈αis␈αthat␈αof␈αexperimentation:␈αone␈αcould␈αvary␈αthe␈αconcepts␈αAM
␈↓ ↓H␈↓starts␈αwith,␈α
vary␈αthe␈αheurisitics␈α
available,␈αetc.,␈α
and␈αstudy␈αthe␈α
e≥ects␈αon␈αAM's␈α
behavior.␈α AM␈α
is␈αa
␈↓ ↓H␈↓dissertation project ␈↓βin progress␈↓; few conclusions have been drawn yet.

␈↓ ↓H␈↓The issues to be elaborated upon include:
␈↓ ↓H␈↓(1) What are these heuristics? Where do they come from, what is their justi≡cation, their power?
␈↓ ↓H␈↓(2)␈αWhat␈αis␈α
the␈αAM␈αprogram␈αlike?␈α
What␈αis␈αits␈αcontrol␈α
structure,␈αits␈αrepresentation␈αfor␈α
a␈αconcept?
␈↓ ↓H␈↓How do the heuristics ≡t in?
␈↓ ↓H␈↓(3)␈αHow␈αdoes␈αthe␈αAM␈αprogram␈αwork?␈αWhat␈α
does␈αit␈αstart␈αwith,␈αwhat␈αdoes␈αit␈αdo␈αfrom␈α
there?␈αHow
␈↓ ↓H␈↓and why?
␈↓ ↓H␈↓(4)␈αWhat␈αcan␈αwe␈αall␈αlearn␈αfrom␈αAM?␈αAbstracted␈αout,␈αwhat␈αare␈αthe␈αnew␈αideas,␈αthe␈αtraps␈αthat␈αwere
␈↓ ↓H␈↓fallen into?


␈↓ ↓H␈↓ε*****************************************************************************************************************

␈↓ ↓H␈↓ε␈↓#
1␈↓#␈α	The␈α	other␈α	extreme,␈α	numbers␈α	with␈α	a␈α	MAXIMAL␈α	number␈α	of␈α	factors,␈α	will␈α	also␈α	be␈α	proposed␈α	as␈α	worth␈α	investigating.␈α	 This␈α	leads␈α	to␈α	many
␈↓ ↓H␈↓ε␈↓ α(interesting␈αλquestions;␈α	the␈αλonly␈αλ"new-to-Mankind"␈α	mathematical␈αλresult␈αλso␈α	far␈αλis␈αλin␈α	fact␈αλthat␈αλsuch␈α	maximally-divisible␈αλnumbers
␈↓ ↓H␈↓ε␈↓ α(must␈αλhave␈α	the␈αλform␈αλp␈↓
1␈↓ε␈↓#
a␈↓#␈↓#
1␈↓#p␈↓
2␈↓ε␈↓#
a␈↓#␈↓#
2␈↓#p␈↓
3␈↓ε␈↓#
a␈↓#␈↓#
3␈↓#...p␈↓
k␈↓ε␈↓#
a␈↓#␈↓#
k␈↓#,␈α	where␈αλthe␈α	p␈↓
i␈↓ε's␈αλare␈αλthe␈α	first␈αλk␈α	consecutive␈αλprimes,␈αλand␈α	the␈αλexponents␈α	a␈↓
i␈↓ε␈αλdecrease
␈↓ ↓H␈↓ε␈↓ α(with␈α	i,␈α	and␈α	the␈αλratio␈α	of␈α	(a␈↓
i␈↓ε+1)/(a␈↓
j␈↓ε+1)␈α	is␈α	approximately␈αλ(as␈α	closely␈α	as␈α	is␈α	possibe␈αλfor␈α	integers)␈α	log(p␈↓
j␈↓ε)/log(p␈↓
i␈↓ε).␈α	For␈α	example,␈αλa
␈↓ ↓H␈↓ε␈↓ α(typical␈α⊃divisor-rich␈α⊃number␈α⊃is␈α⊂n=2␈↓#
8␈↓#3␈↓#
5␈↓#5␈↓#
3␈↓#7␈↓#
2␈↓#11␈↓#
2␈↓#13␈↓#
1␈↓#17␈↓#
1␈↓#19␈↓#
1␈↓#23␈↓#
1␈↓#29␈↓#
1␈↓#31␈↓#
1␈↓#37␈↓#
1␈↓#41␈↓#
1␈↓#43␈↓#
1␈↓#47␈↓#
1␈↓#53␈↓#
1␈↓#.␈α⊃ The␈α⊃progression␈α⊃of␈α⊂its
␈↓ ↓H␈↓ε␈↓ α(exponents+1␈αλ(9␈α	6␈αλ4␈αλ3␈α	3␈αλ2␈α	2␈αλ2␈αλ2␈α	2␈αλ2␈αλ2␈α	2␈αλ2␈α	2␈αλ2)␈αλis␈α	about␈αλas␈αλclose␈α	as␈αλone␈α	can␈αλget␈αλto␈α	satisfying␈αλthe␈α	"logarithm"␈αλconstraint.
␈↓ ↓H␈↓ε␈↓ α(This␈α
number␈α	n␈α
has␈α	3,981,312␈α
divisors.␈α	The␈α
"AM␈α	Conjecture"␈α
is␈α
that␈α	no␈α
number␈α	smaller␈α
than␈α	n␈α
has␈α	that␈α
many␈α
divisors.␈α	By
␈↓ ↓H␈↓ε␈↓ α(the way, this n equals 25,608,675,584.

␈↓ ↓H␈↓ε␈↓#
2␈↓#␈αIncidentally,␈αthese␈αbasic␈αconcepts␈αinclude␈αthe␈αoperators␈αwhich␈αenlarge␈αthe␈αspace␈α(e.g.,␈α␈↓¬COMPOSE-2-RELATIONS␈↓ε␈αis␈αboth␈αa
␈↓ ↓H␈↓ε␈↓ α(concept in its own right and a way to generate new ones).

␈↓ ↓H␈↓ε␈↓#
3␈↓# Lenat, Douglas, ␈↓&BEINGS: Knowledge as Interacting Experts␈↓)αβ, 4th IJCAI, 1975, pp. 126-133.

␈↓ ↓H␈↓ε␈↓#
4␈↓#␈αλIn␈α	fact,␈αλafter␈αλAM␈α	attempts␈αλto␈αλfind␈α	examples␈αλof␈α	SET-EQUALITY,␈αλso␈αλfew␈α	are␈αλfound␈αλthat␈α	AM␈αλdecides␈αλto␈α	generalize␈αλthat␈α	predicate.␈αλ The
␈↓ ↓H␈↓ε␈↓ α(result is the predicate which means "Has-the-same-length-as" -- i.e., the rudiments of Cardinality.

␈↓ ↓H␈↓ε␈↓#
5␈↓#␈αλDue␈αλto␈αλthe␈αλfirm␈αλbase␈αλof␈αλpreliminary␈αλconcepts␈α	which␈αλAM␈αλdeveloped,␈αλthis␈αλrelationship␈αλwas␈αλalmost␈αλobvious.␈αλ AM␈αλsought␈α	some␈αλpredicate
␈↓ ↓H␈↓ε␈↓ α(P␈α	which,␈α	for␈α	each␈α	n,␈α
some␈α	member␈α	of␈α	FACTORS-OF(n)␈α	satisfied.␈α	 ALL-PRIMES␈α
was␈α	such␈α	a␈α	predicate.␈α	 AM␈α
next␈α	constructed
␈↓ ↓H␈↓ε␈↓ α(the␈αλrelation␈αλwhich␈αλassociates,␈α	to␈αλeach␈αλnumber␈αλn,␈α	all␈αλfactorizations␈αλof␈αλn␈α	into␈αλprimes.␈αλ The␈αλfull␈α	statement␈αλof␈αλthe␈αλUFT␈α	is␈αλsimply
␈↓ ↓H␈↓ε␈↓ α(that this relation is a function; i.e., it is defined and single-valued for all numbers n.

␈↓ ↓H␈↓ε␈↓#
6␈↓#␈α
Currently,␈α
the␈α
educator␈α
takes␈α
the␈α
very␈α	best␈α
work␈α
any␈α
mathematician␈α
has␈α
ever␈α
done,␈α	polishes␈α
it␈α
until␈α
its␈α
brilliance␈α
is␈α
blinding,␈α	and
␈↓ ↓H␈↓ε␈↓ α(presents␈αit␈αto␈αthe␈αstudent␈αto␈αinduce␈αupon.␈αA␈αfew␈αindividuals␈α(e.g.,␈αMinsky␈αand␈αPapert␈αat␈αMIT,␈αAdams␈αat␈αStanford)␈αare
␈↓ ↓H␈↓ε␈↓ α(experimenting with more realistic strategies for "teaching" creativity.
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	K␈↓εSUPPLEMENT     page   4␈↓


␈↓ ↓H␈↓␈↓∧␈↓&2. Tutorial Talk␈↓)αβ␈↓

␈↓ ↓H␈↓␈↓&Objective␈↓)αβ

␈↓ ↓H␈↓Provide␈αa␈αsolid␈αintroduction␈αto␈αthe␈αsystem␈αfor␈αthose␈αunacquainted␈αwith␈αit.␈αAlso,␈αtry␈α   to␈αconvey␈αa
␈↓ ↓H␈↓feeling for what it is like to actually ␈↓βdo␈↓ simple mathematical research.

␈↓ ↓H␈↓␈↓&Outline␈↓)αβ
␈↓ ↓H␈↓The␈α⊂task:␈α⊂proposing␈α⊂interesting␈α⊂new␈α⊂problems,␈α∂concepts␈α⊂to␈α⊂investigate␈α⊂How␈α⊂one␈α⊂can␈α⊂analyze␈α∂a
␈↓ ↓H␈↓given␈α_discovery,␈α_"explain"␈α_it.␈α_ How␈α_to␈α↔invert␈α_this␈α_process␈α_and␈α_␈↓βgenerate␈↓␈α_new␈α↔discoveries
␈↓ ↓H␈↓Representing␈αmath␈αknowledge␈αas␈αcooperating␈αexperts␈αComments␈αon␈α"theory␈αformation"␈αin␈αgeneral
␈↓ ↓H␈↓Some problems that ␈↓βyou␈↓ might encounter.

␈↓ ↓H␈↓␈↓∧␈↓&3. Details Talk␈↓)αβ␈↓

␈↓ ↓H␈↓␈↓&Objective␈↓)αβ

␈↓ ↓H␈↓After␈α∞a␈α∞brief␈α∞review␈α∞of␈α∂the␈α∞project,␈α∞a␈α∞few␈α∞detailed␈α∂issues␈α∞(which␈α∞have␈α∞analogues␈α∞in␈α∂most␈α∞other
␈↓ ↓H␈↓projects)␈α∞will␈α∂be␈α∞discussed.␈α∂ The␈α∞rest␈α∞of␈α∂the␈α∞time␈α∂will␈α∞be␈α∞spent␈α∂in␈α∞group␈α∂planning/discussion␈α∞of
␈↓ ↓H␈↓AM. Some of these "planning" topics are listed below.

␈↓ ↓H␈↓␈↓&Questions to discuss␈↓)αβ
␈↓ ↓H␈↓Using multiple representations and languages
␈↓ ↓H␈↓Discovering ␈↓βvs␈↓ being led by the nose.
␈↓ ↓H␈↓Philosophical justi≡cations and implications of AM's foundation.
␈↓ ↓H␈↓The name of a computer program: pretentious ␈↓βvs␈↓ meaningless
␈↓ ↓H␈↓Choosing a domain for an AI program, or, "reality was never like this"
␈↓ ↓H␈↓Several directions for future work. What should be the priorities?
␈↓ ↓H␈↓AM next year at Stanford.

␈↓ ↓H␈↓␈↓&Prerequisite concepts␈↓)αβ

␈↓ ↓H␈↓Integers,␈α
Natural␈α
Numbers,␈α
Relations,␈α
Functions,␈α
Ordering:␈α
Just␈α
glance␈α
at␈α
the␈α
Glossary␈α
for␈α
those
␈↓ ↓H␈↓items you feel uncomfortable about.

␈↓ ↓H␈↓Prime␈α
numbers:␈α
What␈α
are␈α
they?␈α
Why␈α
are␈α
they␈α
interesting?␈α
Are␈α
they␈α
useful?␈α
 (If␈α
you␈α
are␈αunsure,
␈↓ ↓H␈↓read the Primes de≡nition in the Glossary).

␈↓ ↓H␈↓Cardinality:␈αIf␈αour␈αconcept␈αof␈α"number"␈αis␈αnot␈αprimitive,␈αwhat␈αis␈αit␈α"built"␈αout␈αof?␈αHow␈αdo␈αyoung
␈↓ ↓H␈↓children␈α
learn␈α
it?␈α∞Is␈α
it␈α
related␈α
to␈α∞the␈α
world␈α
around␈α
us,␈α∞to␈α
our␈α
anatomy,␈α
to␈α∞the␈α
size␈α
of␈α∞our␈α
short-
␈↓ ↓H␈↓term-memory?

␈↓ ↓H␈↓Mathematical␈α∂Research␈α∞␈↓β␈↓&vs␈↓)αβ␈↓␈α∂Finished␈α∞Mathematics:␈α∂If␈α∂you␈α∞don't␈α∂perceive␈α∞the␈α∂di≥erence,␈α∂read␈α∞the
␈↓ ↓H␈↓glossary␈α∪entries␈α∀for␈α∪Mathematical␈α∀concept,␈α∪Mathematical␈α∀theory,␈α∪Mathematical␈α∀intuition,␈α∪and
␈↓ ↓H␈↓Mathematical research.

␈↓ ↓H␈↓Modular␈α⊃Representations␈α⊃of␈α∩Knowledge,␈α⊃Cooperating␈α⊃Knowledge␈α⊃Sources,␈α∩Heterarchy:␈α⊃Despite
␈↓ ↓H␈↓their␈αrecent␈αpopularity␈αin␈αAI␈αcircles,␈αmany␈αpeople␈αhave␈αvery␈αshaky␈αinterpretations␈αof␈αthese␈αterms.
␈↓ ↓H␈↓To see ␈↓βmy␈↓ shaky interpretations, read the Glossary entries for those 3 notions.
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	K␈↓εSUPPLEMENT     page   5␈↓


␈↓ ↓H␈↓␈↓∧␈↓&4a. Glossary of Math Terms␈↓)αβ␈↓

␈↓ ↓H␈↓Cardinality:␈α the␈αconcept␈α of␈α"number".␈α   Two␈αsets␈α
 are␈αof␈α the␈αsame␈αcardinality␈αif␈αthey␈α
have␈αthe
␈↓ ↓H␈↓same number of elements.

␈↓ ↓H␈↓Composition␈α∂of␈α∞two␈α∂relations␈α∂R␈α∞and␈α∂S:␈α∂This␈α∞is␈α∂a␈α∞new␈α∂ relation␈α∂denoted␈α∞R␈↓εo␈↓S,␈α∂ and␈α∂ de≡ned␈α∞ as
␈↓ ↓H␈↓R␈↓εo␈↓S(x)␈α=␈α R(S(x)).␈α So␈α R␈↓εo␈↓S␈α maps␈αelements␈αof␈α
 the␈αdomain␈αof␈αS␈αinto␈αelements␈αof␈αthe␈αrange␈α
of␈αR.
␈↓ ↓H␈↓Notice␈αthat␈αif␈αR␈αand␈αS␈αare␈αboth␈αfunctions,␈αthen␈αso␈αis␈αR␈↓εo␈↓S.␈αThe␈αintuitive␈αpicture␈αof␈αthis␈α process␈αis
␈↓ ↓H␈↓to operate on x  with the relation S, and ␈↓βthen␈↓ apply R to the results.

␈↓ ↓H␈↓Function:␈αan␈α
operation␈αf␈α
which␈αassociates,␈α
to␈αeach␈α
element␈αx␈α
of␈α some␈α
set␈αD,␈α
 an␈αelement␈α
f(x)␈α of
␈↓ ↓H␈↓some␈αset␈α R.␈αD␈αand␈α R␈αare␈αthe␈α domain␈α
and␈αrange␈αof␈αf.␈α Notice␈αthat␈αa␈αfunction␈αmay␈αbe␈α
considered
␈↓ ↓H␈↓a single-valued relation.

␈↓ ↓H␈↓Integers: positive and negative whole numbers; i.e. ...,-2, -1, 0, 1, 2,...

␈↓ ↓H␈↓Map:␈αused␈α as␈αa␈αverb,␈α this␈αword␈αindicates␈α
 the␈αaction␈αof␈α applying␈αa␈αfunction␈α or␈αa␈α
relation;␈αe.g.,
␈↓ ↓H␈↓we say that  ␈↓βsquaring␈↓ maps 7 into 49.  Used as a noun, it is a synonym for function.

␈↓ ↓H␈↓Mathematical␈α∪concept:␈α∩this␈α∪ is␈α∩taken␈α∪to␈α∩ mean␈α∪all␈α∩the␈α∪ constructions,␈α∪de≡nitions,␈α∩ conjectures,
␈↓ ↓H␈↓operations,␈α∞ structures,␈α∞ etc.␈α
   that␈α∞ a␈α∞mathematician␈α
deals␈α∞with.␈α∞Some␈α∞examples:␈α
Set-intersection,
␈↓ ↓H␈↓Sets,  The unique factorization theorem, every entry listed in this glossary.

␈↓ ↓H␈↓Mathematical␈α↔ intuition:␈α⊗this␈α↔ is␈α⊗ the␈α↔mental␈α⊗ imagery␈α↔ which␈α⊗can␈α↔ be␈α⊗brought␈α↔ to␈α⊗ bear.
␈↓ ↓H␈↓Typically,␈α
 we␈α
 transform␈α the␈α
 situation␈α
to␈α an␈α
abstract,␈α
simpli≡ed␈αone,␈α
manipulate␈α
 it␈α
there,␈αand
␈↓ ↓H␈↓re-translate␈α∂ the␈α∂results␈α∞into␈α∂the␈α∂original␈α∞notation.␈α∂ For␈α∂example,␈α∞our␈α∂intuition␈α∂about␈α∞"ordering"
␈↓ ↓H␈↓may␈α⊂involve␈α⊂the␈α⊂image␈α⊂of␈α⊂marks␈α⊃on␈α⊂a␈α⊂yardstick.␈α⊂We␈α⊂can␈α⊂then␈α⊂answer␈α⊃  questions␈α⊂  involving
␈↓ ↓H␈↓ordering␈α  rapidly,␈α  using␈α   this␈αrepresentation.␈α  Three␈α features␈αof␈α the␈αintuitive␈α image␈α should
␈↓ ↓H␈↓be␈αnoted:␈α(i)␈αit␈αis␈α typically␈αfast␈αand␈αsimple,␈α (ii)␈αit␈αis␈αopaque,␈α one␈αcannot␈αintrospect␈α too␈α easily␈αon
␈↓ ↓H␈↓"why it  works", and  (iii) it  is fallible, occasionally leading to wrong results.

␈↓ ↓H␈↓Mathematical␈α
research:␈α
 The␈α
fundamental␈αidea␈α
here␈α
is␈α
that␈αmathematics␈α
is␈α
an␈α
␈↓βempirical␈↓␈α science,
␈↓ ↓H␈↓just␈α
as␈α
much␈α as␈α
chemistry␈α
or␈α physics.␈α
 In␈α
 doing␈α research,␈α
 the␈α
 ultimate␈α goal␈α
is␈α
 the␈α creation
␈↓ ↓H␈↓of␈α new,␈αinteresting␈α theories,␈αbut␈α the␈αtechniques␈α used␈αinclude␈α looking␈αfor␈αpatterns␈αin␈αempirical
␈↓ ↓H␈↓data,␈αinducing␈αnew␈α conjectures,␈αmodelling␈αsome␈αaspects␈αof␈αthe␈αreal␈αworld,␈αetc.␈αAlthough␈αthe␈α≡nal
␈↓ ↓H␈↓product␈α
looks␈α
like␈α
a␈α
smooth,␈α
formal␈α
 development,␈α
magically␈α
∨owing␈α
 from␈α
postulates␈α
 to␈αlemmas␈α
to
␈↓ ↓H␈↓theorems,␈α⊂the␈α⊂actual␈α⊂research␈α⊂process␈α⊂involved␈α⊂untold␈α⊂blind␈α⊂alleys,␈α⊂ rough␈α⊂guesses,␈α⊂ and␈α⊂ hard
␈↓ ↓H␈↓work.   (analogy:  The  process of painting is rarely itself artistic.)

␈↓ ↓H␈↓Mathematical␈α
theory:␈α
to␈αqualify␈α
as␈α
a␈αtheory,␈α
we␈α
must␈αhave␈α
(i)␈α
a␈αbasis␈α
of␈α
unde≡ned␈αprimitive␈α
terms,
␈↓ ↓H␈↓(ii)␈αde≡nitions␈αinvolving␈α
these,␈α(iii)␈αaxioms␈α  involving␈α
 all␈α  the␈α primitives␈α and␈α
  de≡ned␈α terms
␈↓ ↓H␈↓(iv)␈α
conjectures␈α
 and␈α theorems␈α
relating␈α
 these␈α terms.␈α
   To␈α
 be␈α at␈α
 all␈α
worthwhile,␈α
however,␈αthe
␈↓ ↓H␈↓theory␈α
must␈α
also␈α
meet␈α
the␈α
fuzzy␈α
requirements␈α
that␈α
(v)␈α
there␈α
is␈α
some␈α
correspondence␈α
between␈α
the
␈↓ ↓H␈↓primitives␈α_and␈α_some␈α_"real-world"␈α_  concepts,␈α↔ between␈α_  the␈α_  axioms␈α_ and␈α_  some␈α↔  "real"
␈↓ ↓H␈↓relationships,␈α
and␈α
(vi)␈α
 some␈α
of␈α
the␈α
theorems␈α
are␈α
unexpected,␈α
hard␈α
to␈α
prove,␈α∞elegant,␈α
interesting,
␈↓ ↓H␈↓etc.

␈↓ ↓H␈↓Natural numbers: non-negative integers; i.e., 0, 1, 2, 3,...
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	K␈↓εSUPPLEMENT     page   6␈↓


␈↓ ↓H␈↓Ordering:␈α∞the␈α∞ concept␈α∞of␈α∞"before"␈α∞and␈α∞"after".␈α∞ This␈α∞distinguishes␈α∞a␈α∞list␈α∞from␈α∞a␈α∞bag␈α
 (multiset).
␈↓ ↓H␈↓The␈αformal␈α axioms␈αfor␈αordering␈α simply␈αstate␈αthe␈αobvious␈αproperties␈αof␈αthe␈αintuitive␈αimage␈α
of␈αa
␈↓ ↓H␈↓list.

␈↓ ↓H␈↓Prime␈α∞numbers:␈α∞natural␈α
 numbers␈α∞which␈α∞have␈α∞ no␈α
divisors␈α∞other␈α∞ than␈α
1␈α∞and␈α∞themself;␈α∞e.g.,␈α
17,
␈↓ ↓H␈↓but␈α∂␈↓βnot␈↓␈α∂15␈α∞(=3x5).␈α∂Primes␈α∂are␈α∞interesting␈α∂because␈α∂of␈α∞the␈α∂myriad␈α∂times␈α∞they␈α∂crop␈α∂up␈α∂in␈α∞diverse
␈↓ ↓H␈↓theorems␈α∀ --␈α∪from␈α∀the␈α∪Chinese␈α∀ Remainder␈α∪Theorem␈α∀(solving␈α∪systems␈α∀ of␈α∀linear␈α∪congruence
␈↓ ↓H␈↓equations),␈αto␈α the␈αLaw␈αof␈αQuadratic␈αReciprocity,␈αto␈αFermat's␈αTheorem␈α(for␈αall␈αintegers␈α n,␈αfor␈αall
␈↓ ↓H␈↓primes␈α∞p,␈α∞n␈↓#
p␈↓#␈α∞ is␈α∞congruent␈α∞to␈α∞n␈α∞ (mod␈α∞p)).␈α
 The␈α∞"secret"␈α∞of␈α∞ their␈α∞value␈α∞lies␈α∞in␈α∞the␈α∞fact␈α∞that␈α
all
␈↓ ↓H␈↓integers␈α∞can␈α∞be␈α∞factored␈α∞ ␈↓βuniquely␈↓␈α∞into␈α∞ a␈α∞set␈α∞ of␈α∞prime␈α∞ divisors.␈α∞  This␈α∂"Unique␈α∞ Factorization
␈↓ ↓H␈↓Theorem"  lets  us   reduce  questions  about integers to questions about primes.

␈↓ ↓H␈↓Relation:␈αan␈α
operation␈αwhich␈α
associates,␈αfor␈α
each␈αelement␈αof␈α
some␈αset␈α
D,␈αa␈α
set␈αof␈α
elements␈α E␈α=␈α
{e␈↓#v1␈↓#,
␈↓ ↓H␈↓e␈↓#v2␈↓#,...}␈α
of␈α
 some␈α
set␈α∞R.␈α
D␈α
and␈α
R␈α∞ are␈α
the␈α
 domain␈α
and␈α
 range␈α∞of␈α
 the␈α
 relation.␈α
For␈α∞ example,␈α
the
␈↓ ↓H␈↓realtion␈α"␈↓¬≤␈↓"␈αassociates␈α to␈α5␈αthe␈αset␈αof␈αnumbers␈α{5,␈α6,␈α7,␈α8,...}␈α--␈αi.e.,␈αall␈αintegers␈αwhich␈α5␈αis␈αless␈αthan
␈↓ ↓H␈↓or equal to.   The domain and range of this relation are the integers.

␈↓ ↓H␈↓␈↓∧␈↓&4b. Glossary of AI Terms␈↓)αβ␈↓

␈↓ ↓H␈↓ACTORs:␈α⊂A␈α⊂modular␈α⊂form␈α⊂of␈α⊂representation,␈α∂ useful␈α⊂for␈α⊂distributing␈α⊂of␈α⊂the␈α⊂ task␈α⊂ of␈α∂ ␈↓βcontrol␈↓
␈↓ ↓H␈↓among␈α∂ several␈α∂ components␈α∂ in␈α∂a␈α∂ computer␈α∂program.␈α∂Each␈α∂ACTOR␈α∂is␈α∂a␈α∂black␈α∂box,␈α∂with␈α∂no
␈↓ ↓H␈↓parts␈α∂or␈α∂slots,␈α∂but␈α∂which␈α∂does␈α∂have␈α∂ some␈α∂assertions␈α∂(a␈α∂ "contract")␈α∂which␈α∂ he␈α∂must␈α∂honor.␈α∞  It
␈↓ ↓H␈↓merely␈α⊃ responds␈α∩to␈α⊃a␈α⊃≡xed␈α∩ set␈α⊃of␈α⊃messages,␈α∩ by␈α⊃sending␈α⊃out␈α∩certain␈α⊃messages␈α⊃ of␈α∩ his␈α⊃own.
␈↓ ↓H␈↓These  are  delivered  via  a  bureaucracy.  Recursive sending is permitted.

␈↓ ↓H␈↓BEINGs:␈α∀A␈α∃modular␈α∀form␈α∃of␈α∀representation␈α∀of␈α∃knowledge␈α∀as␈α∃a␈α∀collection␈α∃of␈α∀  cooperating
␈↓ ↓H␈↓experts.␈α
     Each␈α   module␈α
  is␈α  a␈α
  list␈α
   of␈αQuestion/Answering-program␈α
pairs,␈αwhere␈α
the␈αset␈α
of
␈↓ ↓H␈↓questions␈α
is␈α
≡xed␈α
for␈α
all␈α
 the␈α
Beings␈α
in␈α
the␈α
system.␈α
When␈α
any␈α
 Being␈α
has␈α
a␈α
question,␈α
he␈α
broadcasts
␈↓ ↓H␈↓it␈α∞to␈α∂the␈α∞entire␈α∂system,␈α∞and␈α∞some␈α∂Being␈α∞who␈α∂ recognizes␈α∞it␈α∞will␈α∂ take␈α∞over␈α∂ control␈α∞ and␈α∂try␈α∞ to
␈↓ ↓H␈↓answer␈α
 it␈α
 by␈α
running␈α
 ␈↓βhis␈↓␈α
appropriate␈α Answering-program.␈α
In␈α
the␈α
process␈α
 of␈α
running␈αthis,␈α
some
␈↓ ↓H␈↓new␈α∂questions␈α∂may␈α⊂arise.␈α∂Notice␈α∂that␈α⊂Beings␈α∂distribute␈α∂responsibility␈α⊂for␈α∂control␈α∂and␈α⊂for␈α∂static
␈↓ ↓H␈↓knowledge.␈α∞ The␈α∞advantages␈α∞of␈α∞having␈α∞each␈α∞BEING␈α∞composed␈α∞of␈α∞the␈α∞same␈α∞structure,␈α∞the␈α
same
␈↓ ↓H␈↓names␈αfor␈α
its␈α"slots",␈αare␈α
 (i)␈α e≠cient␈α communication␈α
 between␈α Beings,␈α and␈α
 (ii)␈α easy␈αcreation␈α
of
␈↓ ↓H␈↓and "≡lling out" of brand new Beings.

␈↓ ↓H␈↓Cooperating␈α
Knowledge␈αSources:␈α
Very␈α
often,␈αin␈α
tackling␈αa␈α
problem,␈α
one␈αreceives␈α
some␈α
hints␈αand
␈↓ ↓H␈↓some␈α∀constraints␈α∀from␈α∪very␈α∀di≥erent␈α∀sources,␈α∀phrased␈α∪ in␈α∀ very␈α∀ di≥erent␈α∀ languages,␈α∪ often
␈↓ ↓H␈↓addressing␈α
 di≥erent␈α
representations␈αof␈α
the␈α
problem.␈α
 For␈αexample,␈α
in␈α
trying␈α
understand␈α a␈α
human
␈↓ ↓H␈↓speaker,␈αour␈αmemory␈αof␈αthe␈αprevious␈αdiscussion␈αand␈αknowledge␈αof␈αthe␈α speaker␈αmay␈αnarrow␈α
down
␈↓ ↓H␈↓the␈α
possible␈α
 ␈↓βmeanings␈↓␈α
of␈α
what␈α
he␈α
is␈αsaying.␈α
Our␈α
ears,␈α
of␈α
course,␈α
register␈α
the␈α
precise␈αacoustic␈α
wave-
␈↓ ↓H␈↓forms␈αhe␈α is␈α uttering.␈α Our␈α English␈α vocabulary␈αforces␈α us␈α to␈α interpret␈αimperfect␈αsignals␈αas␈αreal
␈↓ ↓H␈↓words.␈α  Our␈αeyes␈αsee␈αhis␈αgestures␈αand␈α
 his␈αlip␈αmovements,␈α and␈α give␈αus␈α more␈α
information.␈α All
␈↓ ↓H␈↓these␈α∂ di≥erent␈α⊂sources␈α∂of␈α∂information␈α⊂ must␈α∂be␈α∂used,␈α⊂and␈α∂yet␈α∂they␈α⊂all␈α∂are␈α∂talking␈α⊂in␈α∂di≥erent
␈↓ ↓H␈↓"languages"␈α
to␈α
us.␈α
  The␈α
most␈α
 trivial␈α
solution␈αis␈α
to␈α
 keep␈α
all␈α
the␈α
sources␈α
 independent,␈α
and␈αkeep
␈↓ ↓H␈↓working␈αuntil␈αone␈α of␈αthem␈αcan␈αsolve␈α the␈α problem␈αall␈α by␈αitself.␈α  A␈α much␈α better␈αsolution␈α is␈αto
␈↓ ↓H␈↓transform␈αall␈αtheir␈α
babblings␈αinto␈αone␈α
 canonical␈αrepresentation,␈αone␈α
single␈αlanguage.␈αThere␈αare␈α
in
␈↓ ↓H␈↓fact no more profound ideas around yet on this "interfacing" problem.

␈↓ ↓H␈↓FRAMEs:␈αA␈αmodular␈αrepresentation␈αof␈αknowledge.␈α Each␈αmodule␈αis␈αa␈αlist␈αof␈αFeature/Value␈αpairs.
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	K␈↓εSUPPLEMENT     page   7␈↓


␈↓ ↓H␈↓The␈α␈↓βvalue␈↓␈αrepresents␈αa␈αdefault␈αassumption␈αwhich␈α can␈αbe␈αrelied␈α on␈αuntil/unless␈α new␈αinformation
␈↓ ↓H␈↓comes␈α in␈αabut␈αthat␈αfeature.␈α Each␈αframe␈αhas␈αwhatever␈α␈↓βfeatures␈↓␈α (called␈α"slots")␈αseem␈α appropriate.
␈↓ ↓H␈↓Whenever␈α∀ a␈α∃situation␈α∀ S␈α∀ is␈α∃ encountered,␈α∀ the␈α∃frame(s)␈α∀ for␈α∀ S␈α∃are␈α∀ activated.␈α∃  As␈α∀ new
␈↓ ↓H␈↓information␈α∀ rolls␈α∪ in,␈α∀it␈α∪replaces␈α∀ the␈α∀default␈α∪ information␈α∀ in␈α∪ various␈α∀slots.␈α∀   Notice␈α∪ the
␈↓ ↓H␈↓emphasis on distributing static knowledge (␈↓βdata␈↓), not necessarily control, in such a system.

␈↓ ↓H␈↓Heterarchy:␈α∀This␈α∀term␈α∀refers␈α∀to␈α∀the␈α∃control␈α∀structure␈α∀of␈α∀a␈α∀computer␈α∀program.␈α∃The␈α∀typical
␈↓ ↓H␈↓hierarchical␈α∂structure␈α∂is␈α∂one␈α⊂in␈α∂which␈α∂a␈α∂function␈α∂calls␈α⊂a␈α∂subroutine,␈α∂which␈α∂processes␈α⊂and␈α∂then
␈↓ ↓H␈↓returns␈α
a␈α
value␈α∞to␈α
that␈α
function.␈α
 A␈α∞program␈α
is␈α
viewed␈α
as␈α∞a␈α
tree␈α
structure,␈α
with␈α∞lines␈α
indicating
␈↓ ↓H␈↓"calling".␈α Heterarchical␈αstructuring␈αviews␈αthe␈αwhole␈αprogram␈αas␈αa␈αcollection␈αof␈αequal␈αpartners,␈αan
␈↓ ↓H␈↓unstructured␈α∞set␈α∞of␈α
functions.␈α∞"Control"␈α∞is␈α
viewed␈α∞as␈α∞a␈α
spotlight,␈α∞which␈α∞can␈α
be␈α∞∨icked␈α∞from␈α
one
␈↓ ↓H␈↓function␈αto␈αanother.␈αThe␈αfunctions␈αcan␈αa≥ect␈αwho␈αdoes␈αor␈αdoesn't␈αget␈αcontrol␈αnext,␈αbut␈αthere␈αis␈αno
␈↓ ↓H␈↓guarantee␈α
who␈α
will␈α
get␈α
control,␈α
or␈α
that␈α
control␈αwill␈α
revert␈α
back␈α
to␈α
some␈α
function␈α
which␈α
once␈αhad␈α
it.
␈↓ ↓H␈↓Aside␈α
from␈α
the␈α∞lure␈α
of␈α
its␈α∞democratic␈α
∨avor,␈α
it␈α∞is␈α
clearly␈α
a␈α∞natural␈α
way␈α
to␈α∞represent␈α
cooperating
␈↓ ↓H␈↓knowledge modules.

␈↓ ↓H␈↓Modular␈αRepresentations␈α
of␈αKnowledge␈α in␈α
AI␈αSystems:␈α(1)␈α
 De≡nition:␈αKnowledge␈α
is␈αpartitioned
␈↓ ↓H␈↓into␈α⊗packets␈α⊗(called␈α⊗modules,␈α⊗frames,␈α↔units,␈α⊗experts,␈α⊗  actors)␈α⊗  along␈α⊗ lines␈α↔  of:␈α⊗  di≥erent
␈↓ ↓H␈↓applicabilities,␈α∀expertise,␈α∀purpose,␈α∪ importance,␈α∀ generality,␈α∀etc.␈α∪   Each␈α∀packet␈α∀ is␈α∪structurally
␈↓ ↓H␈↓similar␈α⊃to␈α⊂all␈α⊃the␈α⊂ rest.␈α⊃ (2)␈α⊃Advantages:␈α⊂By␈α⊃having␈α⊂the␈α⊃knowledge␈α⊂discretized,␈α⊃pieces␈α⊃ can␈α⊂be
␈↓ ↓H␈↓added␈α
and/or␈α
 removed␈α∞with␈α
 no␈α
trouble.␈α∞  The␈α
 knowledge␈α
 of␈α∞ the␈α
 system␈α
is␈α∞ easily␈α
 inspected
␈↓ ↓H␈↓and␈α⊂analyzed.␈α⊂ The␈α⊃ structural␈α⊂similarity␈α⊂ yields␈α⊂ several␈α⊃ advantages:␈α⊂a␈α⊂simple␈α⊃control␈α⊂ system
␈↓ ↓H␈↓su≠ces␈α
 to␈α
 "run"␈α∞all␈α
 the␈α
 knowledge,␈α
 the␈α∞modules␈α
 can␈α
intercommunicate␈α
 easily,␈α∞ new␈α
modules
␈↓ ↓H␈↓can␈α∞be␈α∞ inserted␈α∞without␈α∞ knowing␈α∞precisely␈α∞"who␈α∞else"␈α∞is␈α∞already␈α∞ in␈α∞the␈α∞system.␈α∞ (3)␈α
Examples:
␈↓ ↓H␈↓Some␈α
modular␈α
schemes␈α
(and␈α
their␈α
 program␈α
incarnations)␈α
are:␈α
Actors␈α
 (Plasma),␈α
 Frames,␈α
Beings
␈↓ ↓H␈↓(PUP6),␈α∀ Production␈α∀Systems␈α∀ (PSG,␈α∀Dendral,␈α∀Mycin),␈α∀Predicate␈α∀ Calculus.␈α∀ (4)␈α∃ Relation␈α∀to
␈↓ ↓H␈↓"Cooperating␈α⊗Knowledge␈α⊗Sources"␈α⊗Although␈α↔ modular␈α⊗representation␈α⊗is␈α⊗a␈α⊗ natural␈α↔way␈α⊗to
␈↓ ↓H␈↓implement␈α cooperating␈αknowledge␈α sources,␈αthe␈α two␈α concepts␈αare␈αdistinct.␈αFor␈αexample,␈αHearsay
␈↓ ↓H␈↓uses␈α⊃opaque␈α⊃modules,␈α⊃which␈α⊃ do␈α⊃␈↓βnot␈↓␈α⊂have␈α⊃similar␈α⊃structures,␈α⊃who␈α⊃communicate␈α⊃ via␈α⊃a␈α⊂global
␈↓ ↓H␈↓blackboard.␈α∪In␈α∪general,␈α∪if␈α∪the␈α∪modules␈α∩are␈α∪to␈α∪have␈α∪non-standard␈α∪structures,␈α∪then␈α∪the␈α∩inter-
␈↓ ↓H␈↓communication␈α∀media␈α∀ must␈α∀be␈α∀ a␈α∀ simple␈α∃scheme␈α∀ (like␈α∀a␈α∀ global␈α∀assertional␈α∀data␈α∃base,␈α∀a
␈↓ ↓H␈↓blackboard).

␈↓ ↓H␈↓␈↓∧␈↓&5. Documentation␈↓)αβ␈↓
␈↓ ↓H␈↓1. Thesis Proposal: SU-AI ≡le SYS4[TLK,DBL]
␈↓ ↓H␈↓2. This supplementary ≡le: SU-AI ≡le SUP[HPP,DBL]
␈↓ ↓H␈↓3. The text of the tutorial: SU-AI ≡le TUT[HPP,DBL]
␈↓ ↓H␈↓4. The text of the planning talk: SU-AI ≡le DET[HPP,DBL]
␈↓ ↓H␈↓5.␈α∂The␈α∞system␈α∂itself:␈α∞SUMEX␈α∂≡les␈α∞<LENAT>␈α∂TOP6,␈α∞CON6,␈α∂and␈α∞UTIL6.␈α∂    To␈α∞run:␈α∂get␈α∞into
␈↓ ↓H␈↓␈↓ ↓xLISP, load <LENAT>L, follow instructions.
␈↓ ↓H␈↓6.␈αThe␈αuse␈αof␈αBEINGS␈αrepresentation␈αin␈αAM␈αis␈αdescribed␈αin␈αthe␈αpaper:␈α   ␈↓βDuplication␈αof␈αHuman
␈↓ ↓H␈↓β␈↓ ↓xActions␈α∩by␈α∩an␈α∩Interacting␈α∩Community␈α∩of␈α∩Knowledge␈α∩Modules␈↓,␈α∩   Proceedings␈α∩of␈α∩the␈α⊃Third
␈↓ ↓H␈↓␈↓ ↓xInternational Congress of Cybernetics and Systems,    Bucharest, Roumania, August, 1975.
␈↓ ↓H␈↓7.␈αAn␈αEnglish-like␈αdescription␈α
of␈αthe␈αheuristics␈αfor␈α
each␈αfacet␈αof␈αeach␈α
   concept␈αcan␈αbe␈αperused␈α
as
␈↓ ↓H␈↓␈↓ ↓xSU-AI ≡le GIVEN[TLK,DBL].