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

␈↓ ↓H␈↓␈↓ ¬∀␈↓∧␈↓&TUTORIAL  TALK␈↓)αβ␈↓


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

␈↓ ↓H␈↓␈↓ ¬Q␈↓βJanuary 5-8, 1976␈↓


␈↓ ↓H␈↓␈↓ ¬WDouglas B. Lenat




␈↓ ↓H␈↓Some␈αof␈αthe␈αmost␈αexciting␈αand␈αchallenging␈αtasks␈αthat␈αyou␈αand␈αI␈αface␈αas␈αscientists␈αare␈αthinking␈αup
␈↓ ↓H␈↓new␈α∞research␈α
problems,␈α∞picking␈α
one␈α∞to␈α
work␈α∞on,␈α
and␈α∞deciding␈α
when␈α∞to␈α
quit␈α∞and␈α
try␈α∞a␈α∞new␈α
one.
␈↓ ↓H␈↓This␈α⊗activity␈α⊗has␈α↔a␈α⊗very␈α⊗di≥erent␈α⊗∨avor␈α↔than␈α⊗simply␈α⊗solving␈α↔carefully-speci≡ed␈α⊗problems.
␈↓ ↓H␈↓Consider␈α∞the␈α∞Missionaries␈α∞and␈α∞Cannibals␈α∞problem.␈α∞ It␈α∂takes␈α∞a␈α∞few␈α∞basic␈α∞ abilities␈α∞to␈α∞be␈α∂able␈α∞to
␈↓ ↓H␈↓solve␈α
it.␈α But␈α
what␈α
did␈αit␈α
take␈α
to␈α␈↓βinvent␈↓␈α
it?␈αHow␈α
could␈α
someone␈αsit␈α
down␈α
and␈αthink␈α
up␈α
a␈αpuzzle
␈↓ ↓H␈↓like that? The abilities needed are quite di≥erent from those required just to solve it.

␈↓ ↓H␈↓How␈α∞could␈α
we␈α∞even␈α
start␈α∞to␈α∞analyze␈α
this␈α∞ill-structured␈α
problem:␈α∞how␈α
to␈α∞propose␈α∞new,␈α
interesting
␈↓ ↓H␈↓questions␈αto␈αinvestigate.␈α Let's␈αtake␈αanother␈αexample,␈αsay␈αthat␈αof␈αprime␈αnumbers.␈αIf␈αI␈αtell␈αyou␈αhow
␈↓ ↓H␈↓they're␈α
de≡ned,␈αyou␈α
can␈α
probably␈αgo␈α
o≥␈α
and␈α≡nd␈α
me␈αsome.␈α
 But␈α
how␈αwould␈α
you␈α
formulate␈αsome
␈↓ ↓H␈↓interesting␈α∂conjectures␈α∂involving␈α∂primes?␈α∂ To␈α∂be␈α∂even␈α∂more␈α∂demanding,␈α∂let's␈α∂ask␈α∂how␈α∂someone
␈↓ ↓H␈↓might␈αhave␈αbeen␈αled␈αto␈α
␈↓βpropose␈↓␈αthat␈αde≡nition␈αin␈αthe␈α≡rst␈α
place?␈αHow␈αcould␈αyou␈αmake␈αa␈α
discovery
␈↓ ↓H␈↓like␈α
that,␈αwhy␈α
would␈α
you␈αthink␈α
that␈αsuch␈α
numbers␈α
were␈αworth␈α
looking␈α
into?␈α Think␈α
about␈αthat␈α
for
␈↓ ↓H␈↓a␈α∂few␈α∂seconds.␈α∂ I'll␈α∂let␈α∂you␈α∂assume␈α∂you␈α∂already␈α∂know␈α∂about␈α∂factoring␈α∂a␈α∂number,␈α∂≡nding␈α∂all␈α∂its
␈↓ ↓H␈↓divisors.

␈↓ ↓H␈↓(pause 5 seconds)

␈↓ ↓H␈↓What␈α∞would␈α∞be␈α
some␈α∞natural␈α∞questions␈α
to␈α∞ask,␈α∞say␈α
right␈α∞after␈α∞you'd␈α
come␈α∞across␈α∞the␈α∞concept␈α
of
␈↓ ↓H␈↓≡nding the divisors of a number?

␈↓ ↓H␈↓(pause 5 seconds)

␈↓ ↓H␈↓A␈α
natural␈α∞sort␈α
of␈α
question␈α∞to␈α
ask␈α∞is␈α
"Which␈α
integers␈α∞have␈α
an␈α
unusual␈α∞number␈α
of␈α∞divisors?"␈α
 In
␈↓ ↓H␈↓particular,␈α⊃you'd␈α⊃get␈α⊃around␈α⊃to␈α⊃asking␈α⊃which␈α⊃numbers␈α⊃have␈α⊃an␈α⊃abnormally␈α⊃small␈α⊃number␈α⊃of
␈↓ ↓H␈↓divisors.␈α
 The␈α
answer,␈α
of␈αcourse,␈α
is:␈α
those␈α
numbers␈α
which␈αare␈α
prime.␈α
 This␈α
heuristic␈α
would␈αbe␈α
your
␈↓ ↓H␈↓justi≡cation for giving the primes a special name, and for spending some time investigating them.

␈↓ ↓H␈↓So␈αwe␈αhave␈αreduced␈αthe␈αproblem␈αof␈α"how␈αin␈αthe␈αworld␈αcould␈αyou␈αdiscover␈αprime␈αnumbers"␈αto␈αthe
␈↓ ↓H␈↓simpler␈α
one␈α∞of␈α
"how␈α∞in␈α
the␈α
world␈α∞could␈α
you␈α∞discover␈α
the␈α
concept␈α∞of␈α
divisors-of".␈α∞The␈α
reduction
␈↓ ↓H␈↓was␈α∂accomplished␈α∂by␈α∂citing␈α∂<heur1␈α∂slide>␈α∂the␈α∞general␈α∂heuristic␈α∂rule␈α∂that␈α∂"if␈α∂f␈α∂is␈α∂an␈α∞interesting
␈↓ ↓H␈↓function,␈α
it's␈α∞worth␈α
investigating␈α∞those␈α
elements␈α
that␈α∞f␈α
transforms␈α∞into␈α
␈↓βextremal␈↓␈α∞elements."␈α
Using
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	f␈↓εTUTORIAL      page   2␈↓


␈↓ ↓H␈↓more␈α∩mathematical␈α∩jargon,␈α⊃one␈α∩could␈α∩say␈α∩"the␈α⊃inverse␈α∩image␈α∩of␈α⊃an␈α∩interesting␈α∩set␈α∩under␈α⊃an
␈↓ ↓H␈↓interesting function is probably interesting itself".

␈↓ ↓H␈↓Now␈αwe're␈αbeginning␈αto␈αsee␈αwhat␈αit␈αmight␈α
mean␈α to␈α"explain"␈αa␈αdiscovery.␈αWe␈α just␈αkeep␈α
reducing
␈↓ ↓H␈↓it␈α⊃to␈α⊃simpler␈α⊃and␈α⊃simpler␈α⊃discoveries,␈α∩using␈α⊃some␈α⊃general␈α⊃heuristic␈α⊃rules␈α⊃like␈α⊃these,␈α∩until␈α⊃our
␈↓ ↓H␈↓problem is reduced to "discovering" a bunch of concepts which we already know.

␈↓ ↓H␈↓We'll␈α
continue␈α
with␈α
this␈α
example␈α
a␈α
little␈α
further.␈α
 How␈α
could␈α
you␈α
discover␈α
the␈α
concept␈αof␈α
"divisors-
␈↓ ↓H␈↓of";␈α
i.e.,␈α
factoring?␈α
 Suppose␈α
you␈αbelieve␈α
in␈α
the␈α
heuristic␈α
"if␈αf␈α
is␈α
an␈α
an␈α
interesting␈αfunction,␈α
consider
␈↓ ↓H␈↓its inverse".

␈↓ ↓H␈↓(pause 5 seconds)

␈↓ ↓H␈↓This␈α∞␈↓βreduces␈↓␈α∞the␈α∞problem␈α∞to␈α∞that␈α∞of␈α∞discovering␈α∞␈↓βmultiplication␈↓,␈α∞because␈α∞≡nding␈α∞the␈α∞factors␈α∞of␈α∞a
␈↓ ↓H␈↓number is the inverse of multiplying two numbers.

␈↓ ↓H␈↓Multiplication,␈α
in␈α
turn,␈α
could␈α
be␈α
discovered␈α
from␈α
more␈α
primitive␈α
concepts,␈α
like␈α
the␈αcross-product␈α
of
␈↓ ↓H␈↓two␈α
sets,␈α
and␈α
cardinality␈α
("counting").␈α The␈α
heuristic␈α
in␈α
this␈α
case␈αmight␈α
be␈α
a␈α
desire␈α
to␈α"complete␈α
the
␈↓ ↓H␈↓square" for a diagram like this one:
␈↓ ↓H␈↓        PAIRS OF SETS →→→→→→→→→→ Counting →→→→→→→→→→→ PAIRS OF NUMBERS
␈↓ ↓H␈↓                ↓                                                                                       ↓
␈↓ ↓H␈↓                ↓Cross-product                                                                  (?)
␈↓ ↓H␈↓                ↓                                                                                       ↓
␈↓ ↓H␈↓              SETS →→→→→→→→→→→→→ Counting →→→→→→→→→→→→→→→→→ NUMBERS


␈↓ ↓H␈↓This␈αsays␈αthat␈αCounting␈αtakes␈αa␈αset␈αinto␈αa␈αnumber␈α(namely,␈αthe␈αlength␈αof␈αthat␈αset).␈α Cross-product
␈↓ ↓H␈↓transforms␈αa␈αpair␈αof␈αsets␈αinto␈αa␈αsingle␈αset␈α(namely,␈αtheir␈αcross-product).␈αAnd␈αcounting␈αalso␈αtakes␈αa
␈↓ ↓H␈↓pair␈α
of␈α
sets␈α
into␈α
a␈α
pair␈α
of␈α
numbers.␈α
 It␈α
is␈α
natural␈α
to␈α
ask␈α
what␈α
this␈α
unknown␈α
function␈α
here␈α
is,␈α
which
␈↓ ↓H␈↓takes a pair of numbers into a number, and corresponds to cross-product.

␈↓ ↓H␈↓The␈α
unknown␈α
function␈α
turns␈α
out␈α
to␈α
be␈αmultiplication.␈α
 Of␈α
course,␈α
the␈α
algorithm␈α
that␈α
this␈αwould
␈↓ ↓H␈↓give␈α∂you␈α∂for␈α∞multiplication␈α∂would␈α∂be␈α∂atrocious:␈α∞given␈α∂two␈α∂numbers,␈α∞construct␈α∂2␈α∂sets␈α∂with␈α∞those
␈↓ ↓H␈↓lengths,␈α∞compute␈α∞the␈α∞cross-product␈α∞of␈α∞those␈α∞two␈α∞sets,␈α∞and␈α∞count␈α∞the␈α∞number␈α∞of␈α∞elements␈α∞in␈α∞that
␈↓ ↓H␈↓cross-product.

␈↓ ↓H␈↓The␈α
concept␈αof␈α
NUMBER␈αcan␈α
be␈αdiscovered␈α
by␈αtrying␈α
to␈α≡nd␈α
a␈αcanonical␈α
representation␈α
for␈αall
␈↓ ↓H␈↓sets␈α
with␈α
the␈α
same␈α
length.␈αHaving␈α
the␈α
same␈α
length␈α
can␈αbe␈α
noticed␈α
as␈α
a␈α
generalization␈α
of␈αhaving
␈↓ ↓H␈↓the␈α∞same␈α∞elements␈α∂as.␈α∞ By␈α∞now,␈α∞we're␈α∂talking␈α∞about␈α∞concepts␈α∞which␈α∂are␈α∞so␈α∞elementary␈α∞that␈α∂it␈α∞is
␈↓ ↓H␈↓more␈α⊂painful␈α⊂to␈α⊃try␈α⊂to␈α⊂reduce␈α⊂their␈α⊃discovery␈α⊂than␈α⊂to␈α⊂quit␈α⊃and␈α⊂assume␈α⊂that␈α⊂they␈α⊃are␈α⊂already
␈↓ ↓H␈↓known.

␈↓ ↓H␈↓By␈α∞reversing␈α∞our␈α∞chain␈α∂of␈α∞reductions,␈α∞we␈α∞have␈α∞what␈α∂I␈α∞call␈α∞a␈α∞plausible␈α∞scenario␈α∂for␈α∞discovering
␈↓ ↓H␈↓prime numbers. It looks something like this: <chain slide>
␈↓ ↓H␈↓   ␈↓εCanonical form␈↓  ...SETS\ ␈↓εcomplete-the-square␈↓␈↓ ε8␈↓εlook at the inverse␈↓␈↓ 	λ␈↓εextreme number of␈↓
␈↓ ↓H␈↓ SET------→NUMBER-----------→MULTIPLICATION-----------→DIVISORS---------→PRIMES
␈↓ ↓H␈↓ ...CROSS-PRODUCT␈↓ ε8

␈↓ ↓H␈↓Each␈α∞node␈α∞represents␈α∂a␈α∞discovery,␈α∞and␈α∞each␈α∂arc␈α∞is␈α∞labelled␈α∞with␈α∂the␈α∞heuristic␈α∞which␈α∂made␈α∞that
␈↓ ↓H␈↓discovery plausible, given the previous concepts.
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	f␈↓εTUTORIAL      page   3␈↓


␈↓ ↓H␈↓Now␈αsuppose␈αwe␈αanalyze␈αa␈αgreat␈α
many␈αmathematical␈αdiscoveries,␈αand␈αcollect␈αseveral␈αhundred␈α
such
␈↓ ↓H␈↓heuristic␈α
rules.␈α
We␈α
can␈α
use␈α
them␈α
not␈α
only␈α∞to␈α
explain␈α
a␈α
discovery,␈α
but␈α
to␈α
generate␈α
new␈α∞ones.␈α
We
␈↓ ↓H␈↓start␈αwith␈αa␈αbasic␈α
core␈αof␈αconcepts,␈αand␈αrepeatedly␈α
pick␈αa␈αheuristic␈αrule␈α
and␈αtry␈αto␈αapply␈αit.␈α
 Notice
␈↓ ↓H␈↓that␈αthis␈α
synthesis␈αprocess␈αis␈α
a␈αlot␈αlike␈α
growing␈αa␈αtree,␈α
where␈αthe␈αnodes␈α
are␈αthe␈αconcepts,␈α
and␈αthe
␈↓ ↓H␈↓operators are these rules.

␈↓ ↓H␈↓Proceeding␈αin␈α
this␈αway,␈α
from␈αprimitive␈α
concepts␈αlike␈α
Sets,␈αSubsets,␈α
and␈αUnion,␈α
we␈αcould␈α
eventually
␈↓ ↓H␈↓discover␈αthe␈αprime␈αnumbers,␈αand␈αmany␈αother␈α
interesting␈αconcepts.␈αIf␈αwe␈αactually␈αgot␈αour␈α
heuristics
␈↓ ↓H␈↓by␈α
analyzing␈α
the␈α
discoveries␈α∞of␈α
certain␈α
concepts,␈α
then␈α∞certainly␈α
the␈α
heuristics␈α
collected␈α∞should␈α
be
␈↓ ↓H␈↓su≠cient␈αto␈αguide␈αyou␈αto␈αre-discover␈αthose␈αconcepts.␈αBut␈αintuitively,␈αwe␈αhope␈αthat␈αthese␈αheuristics
␈↓ ↓H␈↓would␈α∀be␈α∪truly␈α∀general,␈α∪would␈α∀lead␈α∪to␈α∀discovering␈α∪many␈α∀di≥erent␈α∪worthwhile␈α∀concepts␈α∪and
␈↓ ↓H␈↓relationships.␈α This␈αis␈αnot␈αa␈αtrivial␈αissue,␈αand␈αwe'll␈αdiscuss␈αit␈αThursday.␈α What␈α␈↓βis␈↓␈αthe␈αjusti≡cation
␈↓ ↓H␈↓behind this whole process? behind the choice of initial core concepts?

␈↓ ↓H␈↓If␈αwe␈α
really␈αare␈α
growing␈αthe␈α
tree␈αin␈α
this␈αdirection,␈α
notice␈αhow␈α
broad␈αit␈α
probably␈αis:␈α
at␈αany␈α
moment,
␈↓ ↓H␈↓several␈α
rules␈α
might␈α∞be␈α
applicable,␈α
and␈α
each␈α∞with␈α
several␈α
possible␈α
choices␈α∞of␈α
what␈α
to␈α
apply␈α∞it␈α
to.
␈↓ ↓H␈↓For␈α
example,␈α
our␈α
"look␈αat␈α
the␈α
inverse"␈α
suggestion␈α
applies␈αto␈α
all␈α
relations.␈α
 As␈α
another␈αexample,␈α
our
␈↓ ↓H␈↓"see␈αwhat␈αmaps␈αinto␈αextremal␈αelements"␈αheuristic␈αcan␈αtell␈αus␈αto␈αseek␈αnot␈αjust␈αthose␈αintegers␈αwith␈αa
␈↓ ↓H␈↓minimum␈α
number␈α
of␈α
divisors,␈α
the␈α
primes,␈α∞but␈α
also␈α
those␈α
integers␈α
which␈α
are␈α∞␈↓βmaximally␈↓␈α
divisible.
␈↓ ↓H␈↓<draw on slide: downward arrow, to MAXIMALLY-DIVIS. Label the arc "extreme: max">

␈↓ ↓H␈↓What's␈α∞even␈α∞more␈α∞crucial,␈α∞we␈α∞don't␈α∞know␈α
back␈α∞here␈α∞ <at␈α∞divisors>␈α∞that␈α∞we're␈α∞going␈α∞to␈α
discover
␈↓ ↓H␈↓something␈αwonderful␈αby␈αtaking␈αthis␈αpath␈αrather␈αthan␈αthat␈αone,␈αso␈αhow␈αdo␈αwe␈αdecide?␈αAll␈αwe␈αcan
␈↓ ↓H␈↓do␈α∞is␈α
have␈α∞some␈α
local␈α∞evaluation␈α
criteria,␈α∞based␈α
primarily␈α∞on␈α
past␈α∞experiences,␈α
which␈α∞guide␈α
our
␈↓ ↓H␈↓path␈αat␈αeach␈αnode.␈α In␈αresearch␈αthere␈αis␈αno␈α"right"␈αproblem␈αto␈αinvestigate,␈αno␈α"correct"␈αtheorem␈αto
␈↓ ↓H␈↓propose,␈α
no␈α
speci≡c␈αgoal.␈α
This␈α
is␈α
a␈αkey␈α
di≥erence␈α
between␈α
problem-solving␈αand␈α
problem-proposing:
␈↓ ↓H␈↓the␈αpresence␈αor␈αabsence␈αof␈αa␈αgoal.␈αSuch␈αa␈αconcept-grower␈αshould␈αnever␈αstop;␈αit␈αmakes␈αno␈αsense␈αto
␈↓ ↓H␈↓talk␈αabout␈αit␈α"succeeding"␈αor␈α"failing."␈αYou␈αcan␈αrate␈αthe␈αquality␈αof␈αits␈αwork,␈αbut␈αcan't␈αdemand␈αthat
␈↓ ↓H␈↓it discover particular concepts.

␈↓ ↓H␈↓The␈α∞most␈α∂one␈α∞can␈α∞dare␈α∂to␈α∞do␈α∂is␈α∞to␈α∞compare␈α∂its␈α∞performance␈α∞against␈α∂known␈α∞mathematics,␈α∂or␈α∞to
␈↓ ↓H␈↓somehow␈α⊂measure␈α∂the␈α⊂rates␈α∂at␈α⊂which␈α∂interesting␈α⊂new␈α∂concepts␈α⊂--␈α∂and␈α⊂losers␈α∂--␈α⊂get␈α⊂created␈α∂and
␈↓ ↓H␈↓worked␈α⊃on.␈α∩ The␈α⊃automatic-discoverer␈α⊃is␈α∩doing␈α⊃well␈α⊃so␈α∩long␈α⊃as␈α⊃it␈α∩continues␈α⊃to␈α∩generate␈α⊃new,
␈↓ ↓H␈↓interesting concepts.

␈↓ ↓H␈↓Now␈α∂let's␈α∂run␈α⊂our␈α∂imaginary␈α∂system␈α⊂forward,␈α∂starting␈α∂with␈α∂the␈α⊂notions␈α∂of␈α∂prime␈α⊂numbers␈α∂and
␈↓ ↓H␈↓maximally-divisible␈αnumbers,␈αas␈αwell␈αas␈αmore␈αelementary␈αconcepts.␈α Say␈αthe␈αsystem␈αdecides␈αto␈αlook
␈↓ ↓H␈↓at examples of the function Factorings-of.  So it constructs  entries, like <Factorings slide>
␈↓ ↓H␈↓Factorings-of(18) = { (18,1),(9,2),(6,3),(3,3,2)}
␈↓ ↓H␈↓and so on.

␈↓ ↓H␈↓It␈α∂tries␈α∞to␈α∂notice␈α∞some␈α∂interesting␈α∞pattern.␈α∂One␈α∞heuristic␈α∂it␈α∞has␈α∂says␈α∞that␈α∂"An␈α∞operation␈α∂is␈α∞very
␈↓ ↓H␈↓interesting␈αif␈αeach␈αimage␈αis␈αmildly␈αinteresting␈αin␈αexactly␈αthe␈αsame␈αway".␈α So␈αhow␈αare␈αeach␈αof␈αthese
␈↓ ↓H␈↓sets␈α
interesting␈α
in␈α
the␈αsame␈α
way?␈α
 Well,␈α
how␈α
are␈αthey␈α
interesting␈α
at␈α
all?␈αWe␈α
know␈α
that␈α
"A␈α
set␈αof
␈↓ ↓H␈↓lists␈αis␈αmildly␈α
interesting␈αif,␈αfor␈α
some␈αelement␈αL␈α
of␈αthe␈α set,␈α
every␈αelement␈αof␈α
L␈αis␈αa␈α
very␈αunusual
␈↓ ↓H␈↓kind␈αof␈α
object."␈αAnother␈α
heuristic␈αtells␈αus␈α
that␈αthe␈α
most␈αe≠cient␈αway␈α
to␈αlook␈α
for␈αthese␈αproperties␈α
is
␈↓ ↓H␈↓starting␈α∞with␈α∞the␈α∞smallest␈α∞set,␈α∞namely␈α∞this␈α∞one.␈α∞ Well,␈α∞every␈α∞member␈α∞of␈α∞this␈α∞list␈α∞is␈α∞␈↓βodd␈↓␈α∞and␈α∂is␈α∞a
␈↓ ↓H␈↓␈↓βprime␈↓;␈α
is␈α
it␈α
true␈α
that␈α
each␈αof␈α
these␈α
sets␈α
contains␈α
a␈α
list␈α
of␈αall␈α
odd␈α
numbers?␈α
No.␈α
Well,␈α
there␈αgoes␈α
that
␈↓ ↓H␈↓conjecture.␈α
Well,␈α
does␈α
each␈α
of␈α
these␈α
sets␈α
contain␈α
a␈αlist␈α
of␈α
all␈α
primes?␈α
Yes,␈α
each␈α
one␈α
does.␈α
 So␈αwe␈α
can
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	f␈↓εTUTORIAL      page   4␈↓


␈↓ ↓H␈↓conjecture:␈αFactorings-of␈αis␈αvery␈α
interesting␈αbecause␈α(it␈αseems)␈α
some␈αway␈αof␈αfactoring␈α
each␈αinteger
␈↓ ↓H␈↓is␈αinto␈αall␈αprimes.␈α This␈αis␈αawkward␈αto␈αstate,␈αso␈αlet's␈αde≡ne␈αa␈αnew␈αoperation:␈αPrime-fac(n)␈αis␈αthe␈α
set
␈↓ ↓H␈↓of␈αall␈αways␈α
of␈αfactoring␈αn␈α
into␈α primes.␈α So␈α
Prime-fac(18)={(3,3,2)}.␈α<on␈αslide,␈αcircle␈α
Prime-factoring
␈↓ ↓H␈↓for␈αeach␈αn>␈αThen␈αwe␈αcan␈αphrase␈αour␈αinteresting␈αconjecture␈αas␈αfollows:␈αPrime-fac␈αis␈αde≡ned␈αfor␈αall
␈↓ ↓H␈↓integers.␈αA␈αgeneral␈αheuristic␈αwe␈α
have␈αis␈α"if␈αan␈αoperation␈αhas␈α
just␈αbeen␈αde≡ned,␈αsee␈αif␈αit␈α
is␈αsingle-
␈↓ ↓H␈↓valued".␈α
It␈α
seems␈α
that,␈αyes,␈α
the␈α
decomposition␈α
of␈αeach␈α
integer␈α
into␈α
primes␈αis␈α
unique.␈α
By␈α
the␈αway,
␈↓ ↓H␈↓this␈αis␈αthe␈αfundamental␈αtheorem␈αof␈α
arithmetic.␈αBy␈αmaking␈αgood␈αnew␈αde≡nitions,␈αeven␈α
very␈αsubtle
␈↓ ↓H␈↓relationships␈α
like␈αthis␈α
one␈αare␈α
reduced␈αto␈α
"obvious"␈αquestions␈α
to␈αask.␈α
 We've␈αjust␈α
shown␈α
that␈αthe
␈↓ ↓H␈↓unique␈αfactorization␈αconjecture␈αcan␈α
be␈αencountered␈αas␈αthe␈α
simple␈αquestion␈α"is␈αthat␈αrelation,␈α
Prime-
␈↓ ↓H␈↓fac,␈αa␈αfunction?"␈αNot␈αonly␈αdo␈αthe␈αheuristics␈αlead␈αin␈αa␈αnatural␈αway␈αto␈αnew␈αconcepts,␈αthey␈α
also␈αlead
␈↓ ↓H␈↓smoothly to interesting conjectures.

␈↓ ↓H␈↓It's␈α
not␈αtoo␈α
important␈αthat␈α
you␈αfollow␈α
every␈α
detail␈αin␈α
these␈αdevelopments;␈α
only␈αthe␈α
spirit␈α
of␈αwhat
␈↓ ↓H␈↓I'm␈α
doing␈αcounts.␈α
I'm␈αusing␈α
the␈αheuristics␈α
to␈αguide␈α
me␈αin␈α
developing␈αnew␈α
concepts,␈α
and␈αdeciding
␈↓ ↓H␈↓what relationships to look for.

␈↓ ↓H␈↓Let's␈α
look␈α
now␈α∞at␈α
our␈α
maximally-divisible␈α∞numbers,␈α
sort␈α
of␈α
the␈α∞converse␈α
of␈α
primes.␈α∞Virtually␈α
no
␈↓ ↓H␈↓attention␈α⊂has␈α⊃been␈α⊂paid␈α⊃them␈α⊂in␈α⊂history,␈α⊃but␈α⊂it␈α⊃turns␈α⊂out␈α⊂that␈α⊃the␈α⊂form␈α⊃of␈α⊂these␈α⊃numbers␈α⊂is
␈↓ ↓H␈↓necessarily␈αp␈↓
1␈↓␈↓	a1␈↓␈α p␈↓
2␈↓␈↓	a2␈↓␈α
p␈↓
3␈↓␈↓	a3␈↓...␈α  p␈↓
k␈↓␈↓	ak␈↓,␈α where␈αthe␈α
p␈↓
i␈↓'s␈αare␈αthe␈α
≡rst␈αk␈αconsecutive␈αprimes,␈α
so␈αwe␈αmay␈α
not␈αskip
␈↓ ↓H␈↓any␈α∃prime,␈α∀and␈α∃the␈α∃exponents␈α∀a␈↓
i␈↓␈α∃decrease␈α∃  with␈α∀ i,␈α∃ and␈α∃  the␈α∀ ratio␈α∃ of␈α∃  (a␈↓
i␈↓+1)/(a␈↓
j␈↓+1)␈α∀ is
␈↓ ↓H␈↓approximately␈α⊂  (as␈α⊂  closely␈α⊂  as␈α⊂   is␈α⊃  possibe␈α⊂  for␈α⊂   integers)␈α⊂log(p␈↓
j␈↓)/log(p␈↓
i␈↓).␈α⊂For␈α⊃ example,␈α⊂a
␈↓ ↓H␈↓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␈↓exponents+1␈α(9␈α6␈α
4␈α3␈α3␈α2␈α
2␈α2␈α2␈α2␈α
2␈α2␈α2␈α2␈α
2␈α2)␈αis␈α about␈α
as␈α close␈α as␈αone␈α
 can␈α get␈α to␈αsatisfying␈α
the
␈↓ ↓H␈↓"logarithm"␈α⊃constraint.␈α⊃  By␈α⊃that␈α⊂I␈α⊃mean␈α⊃that␈α⊃9/6␈α⊂is␈α⊃close␈α⊃to␈α⊃log(3)/log(2);␈α⊂that␈α⊃2/2␈α⊃is␈α⊃close␈α⊂to
␈↓ ↓H␈↓log(47)/log(43),␈α∞etc.␈α
 Changing␈α∞any␈α∞exponent␈α
by␈α∞plus␈α∞or␈α
minus␈α∞1␈α∞would␈α
make␈α∞those␈α∞ratios␈α
worse
␈↓ ↓H␈↓than␈α⊂they␈α⊂are.␈α⊂ This␈α⊂ number␈α⊂n␈α⊂<point␈α⊂to␈α⊂n␈α⊂value>␈α⊂has␈α⊂about␈α⊂4␈α⊂million␈α⊂divisors.␈α⊂ The␈α⊂"AM
␈↓ ↓H␈↓Conjecture"␈α
is␈α
that␈α
no␈α number␈α
smaller␈α
than␈α
n␈α has␈α
that␈α
many␈α
divisors.␈α This␈α
is␈α
so␈α
far␈α
the␈αonly
␈↓ ↓H␈↓piece␈αof␈αinteresting␈αmathematics,␈αpreviously␈αunknown,␈αthat␈αwas␈αdirectly␈αmotivated␈αby␈αAM.␈αAM␈αis
␈↓ ↓H␈↓the␈αexperimental␈αsystem␈αI've␈αbeen␈αworking␈α
on␈αwhich␈αactually␈αcarries␈αout␈αsimple␈αsyntheses␈α
of␈αnew
␈↓ ↓H␈↓concepts.  Maybe the time has come to begin talking about that program.

␈↓ ↓H␈↓Suppose␈αyou␈αwanted␈α
to␈αwrite␈αa␈α
program␈αwhich␈αdid␈αthis␈α
kind␈αof␈αheuristic␈α
concept-growing.␈αWhat
␈↓ ↓H␈↓exactly would you have to do?

␈↓ ↓H␈↓Well,␈αyou␈α
have␈αto␈αdecide␈α
on␈αa␈α
body␈αof␈αprimitive␈α
concepts,␈αthat␈α
is,␈αa␈αbunch␈α
of␈αnotions␈α
which␈αare
␈↓ ↓H␈↓considered␈α∞already-known␈α∞by␈α∞the␈α
system.␈α∞This␈α∞will␈α∞be␈α∞the␈α
starting␈α∞state␈α∞of␈α∞knowledge.␈α∞ Some␈α
of
␈↓ ↓H␈↓these␈α
concepts␈αwill␈α
be␈αstatic,␈α
like␈α
"Sets"␈αand␈α
"Conjectures",␈αothers␈α
will␈αbe␈α
activities,␈α
like␈α"Reverse-
␈↓ ↓H␈↓an-ordered-pair"␈αor␈α"Compose-2-relations".␈α Notice␈αthat␈αsimply␈αexcuting␈αsome␈αof␈αthe␈αactivities␈αwill
␈↓ ↓H␈↓produce␈α
new␈α
concepts.␈α
 For␈α
example,␈α
Composing␈α
the␈α
two␈α
relations␈α
Intersection␈α
and␈αComplement,
␈↓ ↓H␈↓we␈α∃get␈α∃the␈α∃relation␈α∃Intersection(x,␈α∃Complement(y))␈α∃which␈α∃is␈α∃just␈α∃Set-di≥erence(x,y).␈α∃ If␈α∀this
␈↓ ↓H␈↓operation␈αweren't␈αin␈αour␈αcore␈αof␈αconcepts,␈αit␈αcould␈αthen␈αbe␈αadded.␈α Here␈αare␈αsome␈αof␈αthe␈αconcepts
␈↓ ↓H␈↓that AM actually started with: <inital core slide>.

␈↓ ↓H␈↓In␈αaddition␈αto␈αspecifying␈αthe␈αinitial␈αconcepts,␈αwe␈αmust␈αcollect␈αthe␈αheuristics␈αthat␈αthe␈αsystem␈αwill␈αbe
␈↓ ↓H␈↓guided␈αby.␈αOne␈αway␈αto␈αdo␈αthis␈αis␈αto␈αimagine␈αthe␈αsystem␈αmaking␈αvarious␈αdiscoveries,␈αand␈αdeciding
␈↓ ↓H␈↓what␈αrules␈αit␈αwill␈α
need␈αto␈αknow.␈α Another,␈α
somewhat␈αfairer␈αway␈αis␈α
to␈αget␈αexperts␈αto␈α
introspect␈αon
␈↓ ↓H␈↓all␈αthe␈αclues␈αthey␈αuse,␈αall␈αthe␈αknowledge␈αthat␈αguides␈αtheir␈αresearch␈αe≥orts.␈α Unfortunately,␈αmany␈α
of
␈↓ ↓H␈↓these␈α
heuristics␈αare␈α
demon-like,␈αnot␈α
even␈α
occurring␈αto␈α
him␈αexcept␈α
in␈αsituations␈α
where␈α
they␈αmight
␈↓ ↓H␈↓be␈α∀used.␈α∀So␈α∪it␈α∀would␈α∀be␈α∀best␈α∪to␈α∀prepare␈α∀some␈α∀exhaustive␈α∪questionaire␈α∀for␈α∀the␈α∀experts,␈α∪to
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	f␈↓εTUTORIAL      page   5␈↓


␈↓ ↓H␈↓systematically␈αput␈αthem␈αin␈αall␈αconceivable␈αsituations,␈αwhile␈αthey␈αare␈αintrospecting.␈α We'll␈αreturn␈αto
␈↓ ↓H␈↓how to do this in a minute.

␈↓ ↓H␈↓So␈αnow␈α
we␈αhave␈αa␈α
list␈αof␈αconcepts␈α
and␈αa␈αlist␈α
of␈αheuristics.␈αHow␈α
do␈αwe␈αactually␈α
implement␈αthis␈αas␈α
a
␈↓ ↓H␈↓computer␈α
program?␈α
 What␈α
is␈α
left␈α
to␈α
do?␈α
We␈α
must␈α
make␈α
precise␈α
our␈α
representations␈αand␈α
algorithms.
␈↓ ↓H␈↓How␈α
is␈α
a␈αconcept␈α
to␈α
be␈αrepresented?␈α
Precisely␈α
what␈α
is␈αto␈α
be␈α
"given"␈αinitially␈α
about␈α
each␈α
of␈αthese
␈↓ ↓H␈↓concepts? How are the heuristics stored? How are they used? What is the control structure like?

␈↓ ↓H␈↓First,␈α∞try␈α
to␈α∞think␈α∞of␈α
what␈α∞constraints␈α∞there␈α
are␈α∞on␈α∞our␈α
answers␈α∞to␈α∞these␈α
questions.␈α∞What␈α∞␈↓βdo␈↓␈α
we
␈↓ ↓H␈↓have␈α∩to␈α⊃tell␈α∩the␈α⊃system␈α∩about␈α⊃each␈α∩concept?␈α⊃ Well,␈α∩just␈α⊃its␈α∩name␈α⊃isn't␈α∩enough.␈α∩Its␈α⊃de≡nition,
␈↓ ↓H␈↓perhaps.␈αMaybe␈αsome␈αabstract␈αway␈αof␈αviewing␈αthis␈αconcept,␈αa␈αsort␈αof␈α"intuition"␈αabout␈αit.␈α Maybe
␈↓ ↓H␈↓some␈αexamples␈αof␈α
it;␈αwell␈αno,␈α
the␈αsystem␈αshould␈αbe␈α
able␈αto␈α≡ll␈α
them␈αin.␈αIn␈α
the␈αcase␈αof␈αan␈α
operation,
␈↓ ↓H␈↓maybe␈αgiving␈α
its␈αdomain␈α
and␈αrange␈α
would␈αbe␈α
a␈αgood␈α
idea.␈αAlso,␈α
to␈αhelp␈α
the␈αsystem␈α
decide␈αwhat␈α
to
␈↓ ↓H␈↓do,␈α∂it␈α∂would␈α∞be␈α∂nice␈α∂to␈α∂supply␈α∞some␈α∂kind␈α∂of␈α∞rough␈α∂judgment␈α∂of␈α∂the␈α∞worth␈α∂of␈α∂this␈α∂concept,␈α∞its
␈↓ ↓H␈↓interest, its usefulness, its simplicity, etc.

␈↓ ↓H␈↓Let's␈αleave␈αthat␈αfor␈αthe␈αtime␈αbeing,␈αand␈α
turn␈αto␈αthe␈αheuristics.␈αWhat␈αrequirements␈αcan␈αwe␈α
put␈αon
␈↓ ↓H␈↓them?␈α⊃Well,␈α⊃it␈α⊃would␈α⊃be␈α⊃nice␈α⊃to␈α⊃have␈α∩each␈α⊃one␈α⊃come␈α⊃to␈α⊃the␈α⊃system's␈α⊃attention␈α⊃only␈α∩when␈α⊃it
␈↓ ↓H␈↓plausibly␈α∞could␈α∞be␈α∂used.␈α∞ Sometimes,␈α∞there␈α∂will␈α∞be␈α∞special␈α∞information␈α∂that␈α∞just␈α∞pertains␈α∂to␈α∞one
␈↓ ↓H␈↓kind␈α
of␈α
concept.␈α
For␈α
example,␈α
some␈α
of␈α
the␈α
heuristics␈α
tell␈α
whether␈α
a␈α
given␈α
composition␈αis␈α
interesting
␈↓ ↓H␈↓or␈α∞not.␈α
There's␈α∞no␈α∞reason␈α
to␈α∞access␈α∞them␈α
unless␈α∞we're␈α∞trying␈α
to␈α∞judge␈α∞a␈α
composition.␈α∞In␈α∞fact,␈α
by
␈↓ ↓H␈↓looking␈α∞at␈α∞the␈α∞heuristics,␈α∞it␈α∞appears␈α∞that␈α∞only␈α∞a␈α∞few␈α∞very␈α∞weak␈α∞ones␈α∞are␈α∞truly␈α∂"global".␈α∞ALmost
␈↓ ↓H␈↓every␈α∩single␈α∪one␈α∩can␈α∪be␈α∩identi≡ed␈α∪with␈α∩a␈α∩particular␈α∪concept␈α∩(and␈α∪all␈α∩specializations␈α∪of␈α∩that
␈↓ ↓H␈↓concept).␈αIt's␈αalmot␈αas␈αthough␈αthe␈αrelevant␈α
heuristics␈αwere␈αjust␈αanother␈αfacet␈αof␈αeach␈α
concept,␈αlike
␈↓ ↓H␈↓domain/range,␈αexamples,␈αand␈αde≡nition.␈α In␈αfact,␈αwe␈αcan␈αeven␈αbreak␈αthe␈αheuristics␈αdown␈αinto␈α≡ner
␈↓ ↓H␈↓categories,␈α
for␈αeach␈α
concept␈α
C,␈αlike␈α
heuristics␈αthat␈α
deal␈α
with␈αchecking␈α
a␈α
C,␈αheuristics␈α
that␈αdeal␈α
with
␈↓ ↓H␈↓creating␈αa␈αnew␈αC,␈αthose␈αthat␈αdeal␈αwith␈αjudging␈αhow␈αinteresting␈αa␈αparticular␈αinstance␈αof␈αC␈αis,␈αand
␈↓ ↓H␈↓so on.

␈↓ ↓H␈↓Hmmm.␈αThat␈αwould␈αbe␈αa␈αnice␈α
way␈αto␈αthink␈αof␈αthings,␈αsince␈α
it␈αprovides␈αus␈αwith␈αthe␈αframework␈α
we
␈↓ ↓H␈↓talked␈αabout␈αto␈αpick␈αthe␈αexperts'␈αbrains,␈αa␈α
kind␈αof␈α"questionaire"␈αstructuring.␈α We␈αjust␈αget␈αthem␈α
to
␈↓ ↓H␈↓think␈αof␈αheuristics␈αfor␈αeach␈αparticular␈αfacet␈αof␈αeach␈αconcept␈αin␈αturn.␈α There's␈αno␈αreason␈αto␈αput␈αthe
␈↓ ↓H␈↓resulting␈α⊃little␈α⊃bundles␈α⊂of␈α⊃heuristics␈α⊃together;␈α⊂just␈α⊃leave␈α⊃them␈α⊂attached␈α⊃to␈α⊃the␈α⊃concepts'␈α⊂facets.
␈↓ ↓H␈↓Whenever␈α
some␈α
concept␈α
C␈α
is␈αthe␈α
center␈α
of␈α
attention,␈α
the␈αonly␈α
heuristics␈α
that␈α
the␈α
system␈αneed␈α
worry
␈↓ ↓H␈↓about␈αare␈α those␈αattached␈αto␈αC,␈αor␈αsome␈αgeneralization␈αof␈αC,␈αand␈αprobably␈αonly␈αthose␈αattached␈αto
␈↓ ↓H␈↓the relevant facets that the system is working on.

␈↓ ↓H␈↓But␈α
what␈α∞if␈α
we're␈α
thinking␈α∞about␈α
performing␈α∞an␈α
operation,␈α
like␈α∞Composing␈α
two␈α∞relations.␈α
Then
␈↓ ↓H␈↓the␈α⊂heuristics␈α⊃from␈α⊂all␈α⊃3␈α⊂concepts␈α⊃have␈α⊂to␈α⊂work␈α⊃together.␈α⊂How␈α⊃can␈α⊂we␈α⊃arrange␈α⊂that?␈α⊃Well,␈α⊂a
␈↓ ↓H␈↓rational␈α⊃AI␈α⊃answer␈α⊃is:␈α⊂give␈α⊃them␈α⊃all␈α⊃the␈α⊃same␈α⊂representation,␈α⊃let␈α⊃them␈α⊃be␈α⊃simple␈α⊂cooperating
␈↓ ↓H␈↓knowledge␈α∞sources.␈α∞For␈α∞example,␈α∞they␈α∞could␈α∞all␈α∞be␈α∞predicate␈α∞calculus␈α∞statements,␈α∞and␈α∞combining
␈↓ ↓H␈↓them␈α⊂would␈α⊂mean␈α⊂conjoining␈α∂them.␈α⊂But␈α⊂that␈α⊂would␈α∂involve␈α⊂logical␈α⊂manipulations␈α⊂to␈α∂conclude
␈↓ ↓H␈↓anything.␈α∞ Or␈α
they␈α∞could␈α∞all␈α
be␈α∞productions,␈α
and␈α∞combining␈α∞them␈α
would␈α∞mean␈α∞simply␈α
dumping
␈↓ ↓H␈↓them␈α
together␈α
into␈α
one␈α
large␈α
production␈α
system.␈α
I␈αlike␈α
that␈α
image␈α
better.␈α
We␈α
still␈α
have␈α
to␈αdecide
␈↓ ↓H␈↓what the left and right hand sides of each production rule will look like.  But let that go for now.

␈↓ ↓H␈↓Oh,␈αanother␈αproblem!␈αSome␈α
of␈αthe␈αheuristics␈αapply␈α
not␈αto␈αconcepts␈αbut␈α
to␈αfacets;␈αlike␈αheuristics␈α
for
␈↓ ↓H␈↓how␈α∂and␈α∂when␈α∂ to␈α∂≡ll␈α⊂in␈α∂examples␈α∂of␈α∂anything.␈α∂They␈α⊂apply␈α∂to␈α∂the␈α∂whole␈α∂notion␈α⊂of␈α∂examples.
␈↓ ↓H␈↓Well,␈α
maybe␈α
it␈α
won't␈α
hurt␈α
to␈α
consider␈α
giving␈α
the␈α
system␈α
a␈α
full-∨edged␈α
concept␈α
for␈α
each␈α
facet.␈α
So
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	f␈↓εTUTORIAL      page   6␈↓


␈↓ ↓H␈↓there␈α∂would␈α⊂be␈α∂some␈α⊂concept␈α∂named␈α⊂Examples,␈α∂some␈α∂concept␈α⊂named␈α∂De≡nition,␈α⊂and␈α∂so␈α⊂on,␈α∂in
␈↓ ↓H␈↓addition␈α
to␈α
the␈α
kinds␈α
of␈α
static␈α
and␈α
ooperational␈α
concepts␈α
we␈α
already␈α
decided␈α
to␈α
include.␈α
Then␈αit
␈↓ ↓H␈↓makes␈α
sense␈α
to␈α
talk␈α∞about␈α
some␈α
heuristics␈α
for␈α
dealing␈α∞with␈α
Examples;␈α
we␈α
simply␈α
tack␈α∞them␈α
onto
␈↓ ↓H␈↓the␈α
concpet␈αnamed␈α
Examples.␈α Well,␈α
that's␈αnot␈α
so␈αbad.␈α
Oh,␈α
but␈αthen␈α
the␈αset␈α
of␈αfacets␈α
a␈αconcept␈α
can
␈↓ ↓H␈↓have␈α∞must␈α∞be␈α∞≡xed␈α∞once␈α∞and␈α∞for␈α∞all.␈α
 Otherwise␈α∞we'd␈α∞have␈α∞to␈α∞keep␈α∞adding␈α∞new␈α∞concepts␈α
every
␈↓ ↓H␈↓time␈αa␈αnew␈α
kind␈αof␈αfacet␈αwas␈α
wanted.␈αSo␈αthe␈αset␈α
of␈αpossible␈αfacets␈αis␈α
decided␈αand␈α≡xed.␈α Well,␈α
that
␈↓ ↓H␈↓may␈αbe␈αactually␈αbe␈αbetter␈αthan␈αletting␈αeach␈αconcept␈αhave␈αwhatever␈αfacets␈αit␈αwants.␈αSince␈αwe␈αknow
␈↓ ↓H␈↓what␈αthe␈αfacets␈αare␈αgoing␈αto␈αbe,␈αwe␈αcan␈α collect␈αknowledge␈αabout␈αeach␈αof␈αthose␈αfacets.␈α So␈αlet's␈α
add
␈↓ ↓H␈↓these␈α
facets,␈αas␈α
individual␈α
concepts,␈αto␈α
the␈α
initial␈αcore␈α
which␈αthe␈α
system␈α
starts␈αwith.␈α
Some␈α
of␈αthis
␈↓ ↓H␈↓might␈αbe␈αtricky,␈αlike␈αasking␈αfor␈αthe␈αde≡nition␈αof␈αa␈αde≡nition.␈αMaybe␈αthe␈αanswer␈αis␈αthat␈αif␈αit's␈αthat
␈↓ ↓H␈↓tricky␈α⊃we␈α⊃have␈α⊃no␈α∩business␈α⊃asking␈α⊃about␈α⊃it.␈α∩If␈α⊃you␈α⊃know␈α⊃enough␈α∩to␈α⊃ask␈α⊃for␈α⊃it,␈α∩you␈α⊃already
␈↓ ↓H␈↓understand it.

␈↓ ↓H␈↓Well,␈αI've␈αstopped␈αfollowing␈αwhat␈αI'm␈αsaying,␈αso␈αlet␈αme␈αgo␈αback␈αto␈αhow␈αto␈αimplement␈αthe␈αconcepts.
␈↓ ↓H␈↓We␈α
may␈α
as␈αwell␈α
make␈α
their␈αfacets␈α
explicit,␈α
say␈α
as␈αproperties␈α
on␈α
a␈αproperty␈α
list.␈α
Hmmm.␈α
Some␈αof
␈↓ ↓H␈↓them␈αare␈αpretty␈αtrivial;␈αdomain/range␈αshould␈αbe␈αjust␈αa␈αproperly-formatted,␈αstatic␈αbit␈αof␈αdata.␈α But
␈↓ ↓H␈↓some␈α∞are␈α
very␈α∞sophisticated;␈α
the␈α∞algorithms␈α
part␈α∞for␈α
instance.␈α∞ They␈α
will␈α∞have␈α
to␈α∞be␈α
procedural.
␈↓ ↓H␈↓Well that's all right.

␈↓ ↓H␈↓What␈αdoes␈αthe␈αsystem␈α
actually␈αdo,␈αnow?␈αIt␈αexpands␈α
its␈αknowledge,␈αboth␈αby␈αcreating␈α
new␈αconcepts
␈↓ ↓H␈↓and␈α
by␈α
≡nding␈α
out␈αmore␈α
about␈α
the␈α
already-known␈α
ones.␈α That␈α
means␈α
it␈α
creates␈α
new␈αdata␈α
structures
␈↓ ↓H␈↓like␈αthese,␈αand␈αit␈α
≡lls␈αout␈αthe␈αproperties␈αof␈α
existing␈αones.␈αIn␈αgeneral,␈αthe␈α
creation␈αof␈αa␈αnew␈α
one␈αis
␈↓ ↓H␈↓directly␈αsuggested␈αwhile␈α≡lling␈α
in␈αsome␈αdetail␈αsomewhere.␈α So␈α
the␈αonly␈αreal␈αtop-level␈α
activity␈αneed
␈↓ ↓H␈↓be:␈α∂pick␈α⊂some␈α∂facet␈α⊂of␈α∂some␈α⊂concept,␈α∂and␈α⊂try␈α∂to␈α∂≡nd␈α⊂some␈α∂new␈α⊂knowledge␈α∂to␈α⊂put␈α∂in␈α⊂that␈α∂slot.
␈↓ ↓H␈↓During this process, some new concepts may be created.

␈↓ ↓H␈↓What␈αdoes␈αit␈αmean␈αfor␈αa␈αnew␈αone␈αto␈αbe␈αcreated?␈α Well,␈αwhoever␈αcreated␈αit␈αshould␈αprobably␈αknow
␈↓ ↓H␈↓its␈α⊂de≡nition,␈α⊃and␈α⊂something␈α⊂about␈α⊃its␈α⊂worth.␈α⊂Probably␈α⊃also␈α⊂how␈α⊂it␈α⊃connects␈α⊂to␈α⊃some␈α⊂existing
␈↓ ↓H␈↓concepts.␈α
Once␈α
it's␈α
creatd,␈α
most␈α
of␈α
its␈α
facets␈α
will␈αbe␈α
empty,␈α
so␈α
there␈α
will␈α
be␈α
many␈α
new␈α
things␈αfor␈α
the
␈↓ ↓H␈↓system to do. The space of possible actions will grow very rapidly.

␈↓ ↓H␈↓For␈αthose␈αfacets␈αwhich␈αare␈αprocedures,␈α≡lling␈αthem␈αin␈αwill␈αmean␈αwriting␈αnew␈αlittle␈αprograms.␈αWell,
␈↓ ↓H␈↓we␈α⊂know␈α⊃enough␈α⊂about␈α⊃automatic␈α⊂programming␈α⊃to␈α⊂do␈α⊃that.␈α⊂The␈α⊃knowledge␈α⊂stored␈α⊃under␈α⊂the
␈↓ ↓H␈↓Algorithms␈α
concept,␈αfor␈α
example,␈αwill␈α
be␈αautomatic␈α
programming␈αtechniques,␈α
which␈αcan␈α
use,␈αsay,
␈↓ ↓H␈↓the de≡nition of any concept to produce an executable algorithm for it.

␈↓ ↓H␈↓Sometimes␈αwe'll␈α≡nd␈αout␈αthat␈αtwo␈αconcepts␈αare␈αreally␈αthe␈αsame;␈αin␈αthat␈αcase,␈αI␈αsuppose␈αwe␈αcan␈αjust
␈↓ ↓H␈↓merge the corresponding facets of each concept.  No problem there.

␈↓ ↓H␈↓Well,␈αin␈αany␈αevent,␈αthat␈αwas␈αhow␈αAM␈αwas␈αdesigned␈αand␈αcreated.␈αYou␈αnow␈αknow␈αall␈αthe␈αessentials
␈↓ ↓H␈↓of its representations and algorithms.

␈↓ ↓H␈↓The␈αonly␈αpiece␈αmissing␈αfrom␈αthe␈αpicture␈αis␈αa␈αconcrete␈αlook␈αat␈αAM␈αitself.␈α Tomorrow,␈αif␈αyou␈αwant,
␈↓ ↓H␈↓you␈α∩can␈α∩work␈α∪with␈α∩AM␈α∩during␈α∪the␈α∩morning␈α∩demo␈α∩period.␈α∪ I␈α∩have␈α∩some␈α∪detailed␈α∩examples
␈↓ ↓H␈↓prepared,␈α
and␈α
will␈α
discuss␈α
them␈α
tomorrow␈α∞morning,␈α
around␈α
10.␈α
 If␈α
youire␈α
interested,␈α∞they'll␈α
help
␈↓ ↓H␈↓you get a feeling for how the current version of AM actually does its stu≥.

␈↓ ↓H␈↓Step␈αback␈αfor␈αa␈αmoment␈αand␈αlook␈αover␈αwhat's␈αbeen␈αdone.␈α Notice␈αthat␈αthis␈αwhole␈αdiscussion␈αis␈α
not
␈↓ ↓H␈↓really␈α∀speci≡c␈α∪to␈α∀doing␈α∪math␈α∀research.␈α∀ The␈α∪design␈α∀for␈α∪the␈α∀concept-grower␈α∪can␈α∀be␈α∀used␈α∪to
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	f␈↓εTUTORIAL      page   7␈↓


␈↓ ↓H␈↓implement␈α
almost␈α∞any␈α
kind␈α∞of␈α
theory␈α∞formation␈α
system.␈α
 The␈α∞only␈α
change␈α∞from␈α
one␈α∞program␈α
to
␈↓ ↓H␈↓another␈α
would␈α
be␈α
the␈α
particular␈α
starting␈α
concepts,␈α
the␈α
particular␈α
facets␈α
each␈α
concept␈α∞could␈α
have,
␈↓ ↓H␈↓and␈α
of␈α∞course␈α
the␈α
individual␈α∞heuristics␈α
would␈α∞change.␈α
So␈α
the␈α∞design␈α
of␈α
the␈α∞system␈α
could␈α∞be␈α
the
␈↓ ↓H␈↓same. Perhaps if several such systems were created, they could be run together, and cooperate.

␈↓ ↓H␈↓But␈α⊃how␈α⊂should␈α⊃we␈α⊂choose␈α⊃the␈α⊂particular␈α⊃scienti≡c␈α⊂≡eld␈α⊃for␈α⊂the␈α⊃theory␈α⊂formation?␈α⊃ Why␈α⊂was
␈↓ ↓H␈↓mathematics␈αchosen?␈α Well,␈αfor␈αone␈αthing,␈α
mathematicians␈αdon't␈αhave␈αto␈αworry␈αabout␈α
uncertainties
␈↓ ↓H␈↓in␈α∞their␈α∞data,␈α∞as␈α∞one␈α∞would␈α∞from,␈α∞eg.,␈α∞a␈α∞mass␈α∞spectrometer.␈α∞A␈α∞personal␈α∞reason␈α∞was␈α∞that␈α∂I␈α∞know
␈↓ ↓H␈↓something␈αabout␈αmathematics,␈αso␈α
I␈αwouldn't␈αhave␈αto␈αelicit␈α
the␈αheuristics␈αfrom␈αexperts␈α
outside␈αthe
␈↓ ↓H␈↓AI␈α≡eld;␈αI␈αcould␈αuse␈αintrospection.␈αSo␈αthat␈α
problem␈αis␈αeliminated.␈αBut␈αthe␈αbiggest␈αdistinction␈αis␈α
the
␈↓ ↓H␈↓idea␈αthat␈αeach␈α␈↓βnatural␈↓␈α
science␈αtries␈αto␈αexplain␈αobserved␈α
data;␈αmathematics␈αis␈αthe␈αonly␈α
science␈αnot
␈↓ ↓H␈↓limited␈α∂to␈α∞explaining␈α∂nature.␈α∞ Ed␈α∂and␈α∞Josh␈α∂came␈α∞up␈α∂with␈α∞a␈α∂similar␈α∞list␈α∂of␈α∞distinctions␈α∂about␈α∞a
␈↓ ↓H␈↓decade␈α∞ago,␈α∞but␈α∞they␈α∞viewed␈α
them␈α∞as␈α∞reasons␈α∞for␈α∞doing␈α
theory␈α∞formation␈α∞in␈α∞chemistiry␈α∞and␈α
␈↓βnot␈↓
␈↓ ↓H␈↓graph theory.  "Choosing a good domain" is one of the topics we'll discuss on Thursday.

␈↓ ↓H␈↓To␈α∪prepare␈α∩for␈α∪that,␈α∪let␈α∩me␈α∪stress␈α∩today␈α∪that␈α∪mathematics␈α∩is␈α∪an␈α∩empirical␈α∪science.␈α∪ This␈α∩is
␈↓ ↓H␈↓something␈αwhich␈αis␈αhard␈αto␈α
believe␈αunless␈αyou've␈αever␈αdone␈α
some␈αmath␈αresearch.␈αYou␈αquickly␈α
≡nd
␈↓ ↓H␈↓that␈α∞it␈α∞has␈α∞very␈α∞little␈α∞to␈α∞do␈α∞with␈α∞the␈α∞smooth␈α∞∨owing␈α∞proofs␈α∞in␈α∞textbooks.␈α∞It's␈α∞more␈α∞like␈α∞physics,
␈↓ ↓H␈↓where␈α
you␈α∞gather␈α
some␈α
data,␈α∞look␈α
for␈α
regularities,␈α∞and␈α
induce␈α
some␈α∞conjectures.␈α
 But␈α∞often␈α
they
␈↓ ↓H␈↓can't␈αbe␈αphrased␈αconcisely␈αwith␈αthe␈αexisting␈αterminology.␈αThis␈αis␈αa␈αclue␈αthat␈αsome␈αnew␈αde≡nitions
␈↓ ↓H␈↓should␈αbe␈αmade.␈α
 Or␈αperhaps␈αsome␈αconcept␈α
is␈αtoo␈αrigid,␈αtoo␈α
narrow␈αto␈αpermit␈αsome␈α
conjecture␈αto
␈↓ ↓H␈↓hold.␈α
Then␈α
we␈αcan␈α
modify␈α
it,␈αgeneralize␈α
or␈α
relax␈α
it␈αin␈α
some␈α
way.␈α So␈α
if␈α
anything,␈α
math␈αresearch
␈↓ ↓H␈↓proceeds in almost the opposite order from the way it is ≡nally reported in books and journals.

␈↓ ↓H␈↓Now␈αlet's␈αsee␈αwhat␈αlessons␈αwe␈αcan␈αlearn␈αfrom␈αAM,␈αthat␈αapply␈αto␈αprograms␈αyou␈αmight␈αbe␈αworking
␈↓ ↓H␈↓on.␈α⊂ If␈α⊂you␈α⊂try␈α⊂to␈α⊂imagine␈α⊂the␈α⊂number␈α∂of␈α⊂possible␈α⊂avenues␈α⊂to␈α⊂explore,␈α⊂for␈α⊂any␈α⊂given␈α⊂state␈α∂of
␈↓ ↓H␈↓mathematical␈αknowledge,␈αthe␈α
mind␈αboggles.␈α It␈α
is␈αsheer␈αfolly␈α
to␈αeven␈αthink␈α
of␈αwalking␈αaround␈αin␈α
it
␈↓ ↓H␈↓unguided.␈α Heuristic␈αrules␈α--␈αlike␈αthe␈αkind␈αthat␈αAM␈αcontains␈α--␈αcan␈αtransform␈αthis␈αso␈αthat␈αthe␈α
only
␈↓ ↓H␈↓legal␈α∞moves␈α∞are␈α∞those␈α∞which␈α∞are␈α∞locally␈α∞plausible,␈α∞those␈α∞which␈α∞can␈α∞be␈α∞justi≡ed␈α∞by␈α∞some␈α
general
␈↓ ↓H␈↓heuristic.␈α This␈αis␈αlike␈αconstraining␈αthe␈αgenerator␈αin␈αany␈αheuristic␈αsearch␈αprogram.␈α But␈αeven␈αthis
␈↓ ↓H␈↓isn't␈α
enough;␈α
some␈α
meta-heuristics␈α
are␈α
needed␈α
to␈αdecide␈α
which␈α
of␈α
all␈α
the␈α
legal␈α
heuristics␈α
to␈αapply␈α
in
␈↓ ↓H␈↓a␈α∂given␈α⊂situation,␈α∂and␈α∂how.␈α⊂ Experimental␈α∂results␈α∂from␈α⊂AM␈α∂indicate␈α∂that␈α⊂if␈α∂the␈α⊂heuristics␈α∂are
␈↓ ↓H␈↓good,␈αthe␈αmeta-heuristics␈αcan␈αbe␈α
very␈αsimple,␈αlike␈αnumerical␈αweights,␈αand␈α
no␈αmeta-meta-heuristics
␈↓ ↓H␈↓at all are needed.  It would be interesting to see if that is true in some sense for other tasks as well.

␈↓ ↓H␈↓One␈αe≥ective␈αtechnique␈αfor␈αlearning␈αis␈αto␈αbe␈αexposed␈αto␈αothers'␈αmistakes.␈α So␈αlet␈αme␈αmention␈αsome
␈↓ ↓H␈↓of␈αthe␈αproblems␈αI␈αhad␈αprogramming␈αAM.␈α One␈α problem␈αwe␈αall␈αface␈αis␈αhow␈αto␈αcope␈αwith␈αchanges
␈↓ ↓H␈↓in␈α∂the␈α∂system's␈α∂knowledge␈α∂base.␈α∂The␈α∂standard␈α∂solution␈α∂is␈α∂to␈α∂track␈α∂down␈α∂all␈α∂the␈α∂e≥ects␈α⊂of␈α∂each
␈↓ ↓H␈↓change,␈α
and␈αupdate␈α
the␈α
whole␈αdata␈α
base.␈α
The␈αsolution␈α
AM␈α
uses,␈αand␈α
perhaps␈α
people␈αtoo,␈α
is␈αto␈α
just
␈↓ ↓H␈↓record␈α∂the␈α∞changed␈α∂information,␈α∞and␈α∂␈↓βnot␈↓␈α∞to␈α∂update␈α∞everything.␈α∂ When␈α∞a␈α∂contradiction␈α∂of␈α∞some
␈↓ ↓H␈↓kind␈αoccurs,␈αthen␈αthe␈αcon∨icting␈αknowledge␈αis␈αchecked,␈αthe␈αknowledge␈α␈↓βit␈↓␈αwas␈αbased␈αon␈αis␈αchecked,
␈↓ ↓H␈↓and␈α∞so␈α∞on,␈α∞until␈α∞we␈α∞encounter␈α∂some␈α∞changed␈α∞opinion␈α∞which␈α∞explains␈α∞the␈α∞disagreement.␈α∂ So␈α∞the
␈↓ ↓H␈↓system␈α
is␈αallowed␈α
to␈αhold␈α
contradictory␈α
viewpoints,␈αso␈α
long␈αas␈α
it␈α
could␈αresolve␈α
any␈αdispute␈α
if␈αit␈α
had
␈↓ ↓H␈↓to.  This is one solution you might keep in mind for your knowledge-based systems.

␈↓ ↓H␈↓Of␈α
course␈α␈↓βthe␈↓␈α
standard␈αproblem␈α
in␈α
our␈αbusiness␈α
is␈αthe␈α
tremendous␈α
gap␈αbetween␈α
English␈αprose␈α
that
␈↓ ↓H␈↓sounds␈α
plausible,␈α
that␈α
will␈αconvince␈α
a␈α
human,␈α
and␈αLISP␈α
code␈α
that␈α
will␈αrun␈α
on␈α
a␈α
PDP10.␈α
I␈αcan't
␈↓ ↓H␈↓just␈αsay␈αthat␈αa␈αComposition␈αis␈αinteresting␈αif␈αits␈αdomain␈αand␈αrange␈αsets␈αcoincide;␈αI␈αhave␈αto␈αspecify
␈↓ ↓H␈↓precisely␈αhow␈αinterest␈αis␈αcalculated␈αand␈αused,␈αand␈αin␈αparticular␈αhow␈αto␈αcompute␈αthe␈αinterest␈αvalue
␈↓ ↓H␈↓␈↓εDouglas B. Lenat␈↓␈↓ ¬!␈↓β␈↓&Automated Mathematician␈↓)αβ␈↓␈↓ 	f␈↓εTUTORIAL      page   8␈↓


␈↓ ↓H␈↓for␈α∞such␈α∞a␈α∞composition.␈α∞ The␈α∞precision␈α∞I␈α∞need␈α∞to␈α∞instruct␈α∞the␈α∞machine␈α∞still␈α∞catches␈α∞me␈α∞unaware
␈↓ ↓H␈↓sometimes,␈αand␈α
I␈α≡nd␈αmyself␈α
underestimating␈αthe␈α
time␈αI'll␈αneed␈α
to␈αprogram␈αsome␈α
idea␈αI␈α
was␈αsure
␈↓ ↓H␈↓would␈α∩be␈α∪a␈α∩cinch.␈α∩ This␈α∪problem␈α∩can␈α∪be␈α∩summed␈α∩up␈α∪as␈α∩"Just␈α∩because␈α∪you␈α∩can␈α∪talk␈α∩about
␈↓ ↓H␈↓something␈αdoesn't␈αmean␈αyou␈αunderstand␈αit".␈α There␈αis␈αno␈αsolution␈αto␈αthis.␈αIt␈αis␈αone␈αreason␈αthat␈αwe
␈↓ ↓H␈↓actually␈αwrite␈αthese␈αprograms.␈α It␈αimplies␈αthat␈αwe␈αcan␈αtalk␈αabout␈αmore␈αthan␈αwe␈αreally␈αunderstand.
␈↓ ↓H␈↓Which suggests to me that it's about time for me to stop talking.

␈↓ ↓H␈↓There␈αare␈αseveral␈αother␈α
questions␈αwhich␈αshould␈αbe␈αdiscussed␈α
sometime,␈αlike␈αhow␈αto␈αmeasure␈α
AM's
␈↓ ↓H␈↓performance,␈αwhat␈αthe␈αjusti≡cation␈αis␈αfor␈αits␈αparticular␈αinitial␈αcore␈αof␈αknowledge,␈αwhat␈αthe␈αrole␈αof
␈↓ ↓H␈↓the␈α
user␈α
is,␈α
what␈α∞it␈α
would␈α
take␈α
to␈α∞be␈α
able␈α
to␈α
do␈α
research␈α∞in␈α
various␈α
areas␈α
of␈α∞mathematics,␈α
what
␈↓ ↓H␈↓characteristics␈α∂make␈α∂a␈α∂task␈α∂ripe␈α∂for␈α∂attack␈α∂by␈α∂AI␈α∂methods,␈α∂what␈α∂considerations␈α∂come␈α∂into␈α∂play
␈↓ ↓H␈↓when␈αchoosing␈αa␈αname␈αfor␈αthe␈αsystem,␈αand␈αmany␈αmore.␈α On␈αThursday,␈αI'll␈αtalk␈αabout␈α
some␈αmore
␈↓ ↓H␈↓of␈αthese␈αproblems,␈αand␈αthose␈αwe␈αmiss␈αcan␈αbe␈αcovered␈αat␈αa␈αSIGLUNCH␈αsometime.␈αI'll␈αalso␈αgo␈αinto
␈↓ ↓H␈↓more␈α
detail␈αabout␈α
the␈α
state␈αthat␈α
AM␈α
is␈αactually␈α
in,␈αand␈α
we␈α
can␈αdiscuss␈α
plans␈α
for␈αits␈α
future.␈α Let␈α
me
␈↓ ↓H␈↓leave the rest of ␈↓βthis␈↓ hour open for questions or discussion.