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␈↓ ↓H␈↓∧␈↓ ∧D*** Introduction ***

␈↓ ↓H␈↓This␈α→talk␈α_is␈α→about␈α→one␈α_particular␈α→heuristic␈α_program,␈α→called␈α→AM,␈α_which
␈↓ ↓H␈↓emulates␈α∃some␈α⊗of␈α∃the␈α⊗activities␈α∃involved␈α⊗in␈α∃doing␈α⊗mathematics␈α∃research:
␈↓ ↓H␈↓namely,␈α∃AM␈α∃investigates␈α∃a␈α∃given␈α∀collection␈α∃of␈α∃objects␈α∃and␈α∃operators,␈α∀and
␈↓ ↓H␈↓gradually␈α≠enlarges␈α~it␈α≠by␈α~defining␈α≠new␈α~concepts.␈α≠ This␈α≠is␈α~accomplished
␈↓ ↓H␈↓under the guidance of a large body of heuristic rules.

␈↓ ↓H␈↓But␈α∪before␈α∪discussing␈α∪AM,␈α∪let's␈α∩spend␈α∪a␈α∪few␈α∪minutes␈α∪thinking␈α∪about␈α∩how
␈↓ ↓H␈↓scientific␈α∀discoveries␈α∀are␈α∃made;␈α∀at␈α∀least,␈α∀about␈α∃how␈α∀they're␈α∀made␈α∃in␈α∀very
␈↓ ↓H␈↓elementary mathematics.

␈↓ ↓H␈↓We'll␈α∃gather␈α∀a␈α∃few␈α∀hints␈α∃from␈α∀that␈α∃analysis,␈α∀and␈α∃use␈α∀them␈α∃to␈α∃propose␈α∀a
␈↓ ↓H␈↓simple ␈↓βmodel␈↓ of how mathematical theories are developed.

␈↓ ↓H␈↓At␈α⊗that␈α⊗time␈α⊗we␈α⊗can␈α⊗turn␈α↔to␈α⊗AM,␈α⊗and␈α⊗view␈α⊗it␈α⊗as␈α⊗a␈α↔computer␈α⊗program
␈↓ ↓H␈↓embodying that model.

␈↓ ↓H␈↓The␈α⊃first␈α⊂thing␈α⊃I␈α⊂want␈α⊃to␈α⊂stress␈α⊃is␈α⊂the␈α⊃difference␈α⊂between␈α⊃solving␈α⊃a␈α⊂given
␈↓ ↓H␈↓problem␈α#and␈α#thinking␈α#it␈α#up␈α#in␈α#the␈α#first␈α#place.␈α# Contrast␈α#playing
␈↓ ↓H␈↓monopoly,␈α≠to␈α~inventing␈α≠new␈α~games␈α≠of␈α~the␈α≠same␈α~quality.␈α≠ Or␈α~contrast
␈↓ ↓H␈↓solving␈α≠the␈α≠missionaries␈α≠and␈α≠cannibals␈α≠problem,␈α≠against␈α≠the␈α~ill-defined
␈↓ ↓H␈↓reasoning which led to formulating it.

␈↓ ↓H␈↓Turning␈α∩to␈α∩mathematics,␈α∩If␈α∩I␈α∪show␈α∩you␈α∩the␈α∩definition␈α∩of␈α∪prime␈α∩numbers,
␈↓ ↓H␈↓<SLIDE␈α_primes>␈α_you␈α_can␈α_find␈α↔some␈α_for␈α_me␈α_without␈α_any␈α_trouble.␈α↔ But
␈↓ ↓H␈↓whoever first formulated that concept performed a much harder task.
␈↓ ↓H␈↓∧␈↓ β(*** Discovering the Primes ***

␈↓ ↓H␈↓How␈α∀did␈α∃he␈α∀do␈α∀it?␈α∃ How␈α∀can␈α∃we␈α∀rationally␈α∀account␈α∃for␈α∀the␈α∃discovery␈α∀of
␈↓ ↓H␈↓prime␈α≥numbers?␈α≤ We'd␈α≥like␈α≥to␈α≤propose␈α≥a␈α≤plausible␈α≥scenario␈α≥in␈α≤which
␈↓ ↓H␈↓someone␈α≥decides␈α≥to␈α≡look␈α≥at␈α≥these␈α≡numbers,␈α≥and␈α≥then␈α≡decides␈α≥they're
␈↓ ↓H␈↓interesting.

␈↓ ↓H␈↓Imagine␈α_that␈α↔a␈α_bright␈α_undergraduate␈α↔math␈α_major␈α↔has␈α_been␈α_given␈α↔some
␈↓ ↓H␈↓formal␈α∨system␈α∨of␈α∨objects␈α∨and␈α∨operators,␈α∨which␈α∨he␈α∨doesn't␈α∨realize␈α∨is
␈↓ ↓H␈↓isomorphic␈α∩to␈α⊃set,␈α∩set-operations,␈α∩numbers,␈α⊃counting,␈α∩and␈α∩arithmetic.␈α⊃ He's
␈↓ ↓H␈↓told␈α⊃to␈α⊂EXPLORE␈α⊃that␈α⊃formal␈α⊂system,␈α⊃finding␈α⊂out␈α⊃some␈α⊃new␈α⊂relationships
␈↓ ↓H␈↓between␈α≠those␈α~concepts␈α≠which␈α~were␈α≠given␈α~to␈α≠him,␈α~and␈α≠encouraged␈α~to
␈↓ ↓H␈↓define␈α↔some␈α↔new␈α_concepts␈α↔if␈α↔they␈α↔appear␈α_useful␈α↔or␈α↔interesting␈α_to␈α↔him.
␈↓ ↓H␈↓This␈α⊃student␈α∩will␈α⊃also␈α∩be␈α⊃called␈α∩the␈α⊃RESEARCHER,␈α∩since␈α⊃he's␈α∩doing␈α⊃what
␈↓ ↓H␈↓I mean by math research.

␈↓ ↓H␈↓In␈α~that␈α~context,␈α≠let␈α~me␈α~suggest␈α~to␈α≠you␈α~one␈α~plausible␈α~scenario␈α≠of␈α~the
␈↓ ↓H␈↓discovery of prime numbers.

␈↓ ↓H␈↓Suppose␈α⊃the␈α⊃researcher␈α⊂has␈α⊃just␈α⊃been␈α⊃playing␈α⊂with␈α⊃the␈α⊃notion␈α⊃of␈α⊂"divisors-
␈↓ ↓H␈↓of"␈α⊃of␈α⊃a␈α⊃number.␈α⊃ One␈α⊃rule␈α⊂of␈α⊃thumb␈α⊃he␈α⊃sometimes␈α⊃follows␈α⊃when␈α⊂devising
␈↓ ↓H␈↓new␈α∃theories␈α⊗is␈α∃to␈α⊗␈↓βconsider␈α∃extreme␈α∃cases␈↓.␈α⊗<SLIDE␈α∃goto␈α⊗extremes>␈α∃In
␈↓ ↓H␈↓more␈α⊗precise␈α⊗terms,␈α⊗this␈α⊗rule␈α⊗says␈α⊗that␈α⊗if␈α⊗you've␈α⊗just␈α⊗been␈α⊗studying␈α⊗an
␈↓ ↓H␈↓interesting␈α∩function␈α∩f␈α∩which␈α∪takes␈α∩elements␈α∩of␈α∩set␈α∪A␈α∩into␈α∩those␈α∩of␈α∪set␈α∩B,
␈↓ ↓H␈↓and␈α↔there␈α↔is␈α↔some␈α↔extremal␈α↔subset␈α↔of␈α↔B,␈α↔it's␈α↔worth␈α↔taking␈α↔the␈α↔time␈α⊗to
␈↓ ↓H␈↓consider which elements from A map into that distinguished subset.

␈↓ ↓H␈↓In␈α_the␈α_current␈α_situation,␈α→the␈α_function␈α_f␈α_is␈α_"Divisors-of".␈α→ For␈α_example,
␈↓ ↓H␈↓<SLIDE:␈α∀Heur␈α∀overlay>␈α∀it␈α∀takes␈α∀the␈α∀number␈α∀12␈α∀into␈α∀the␈α∀set␈α∃of␈α∀numbers
␈↓ ↓H␈↓which␈α_divide␈α_evenly␈α↔into␈α_12,␈α_namely␈α_{1,2,3,4,6,12}.␈α↔So␈α_f␈α_maps␈α_a␈α↔number
␈↓ ↓H␈↓into␈α⊃a␈α⊃set␈α⊃of␈α⊂numbers.␈α⊃ An␈α⊃interesting␈α⊃or␈α⊂extreme␈α⊃subset␈α⊃of␈α⊃B␈α⊃would␈α⊂thus
␈↓ ↓H␈↓be␈α∃some␈α∀extreme␈α∃kind␈α∀of␈α∃set.␈α∃ We␈α∀have␈α∃names␈α∀for␈α∃extremely␈α∃small␈α∀sets,
␈↓ ↓H␈↓for␈α⊗example:␈α↔empty␈α⊗sets,␈α↔singletons,␈α⊗doubletons,␈α↔tripletons.␈α⊗ The␈α↔rule␈α⊗of
␈↓ ↓H␈↓thumb␈α→thus␈α→says␈α→it's␈α→worth␈α_considering␈α→the␈α→set␈α→of␈α→numbers␈α→over␈α_here
␈↓ ↓H␈↓which␈α⊂map␈α⊃into␈α⊂empty␈α⊂sets;␈α⊃it's␈α⊂worth␈α⊂isolating␈α⊃those␈α⊂numbers␈α⊃which␈α⊂map
␈↓ ↓H␈↓into␈α∀singletons;␈α∀the␈α∃set␈α∀of␈α∀numbers␈α∀whose␈α∃set␈α∀of␈α∀divisors␈α∀is␈α∃a␈α∀doubleton;
␈↓ ↓H␈↓numbers whose image under the function divisors-of is a tripleton.
␈↓ ↓H␈↓In␈α other␈α words,␈α this␈α rule␈α has␈α suggested␈α defining␈α and␈α studying␈α∨the
␈↓ ↓H␈↓following␈α∃classes␈α∃of␈α∃numbers:␈α∃the␈α∀set␈α∃of␈α∃numbers␈α∃with␈α∃no␈α∃divisors,␈α∀with
␈↓ ↓H␈↓only␈α≥1␈α≡divisor,␈α≥with␈α≥2␈α≡divisors␈α≥<SLIDE␈α≥n␈α≡divisors>,␈α≥and␈α≡with␈α≥three
␈↓ ↓H␈↓divisors.␈α∃ A␈α∃priori,␈α∀any␈α∃or␈α∃all␈α∀of␈α∃these␈α∃sets␈α∀could␈α∃be␈α∃interesting.␈α∃ It␈α∀will
␈↓ ↓H␈↓turn␈α∩out␈α∩that␈α∩␈↓βthis␈↓␈α∩set␈α∩(2␈α∪divisors)␈α∩is␈α∩the␈α∩most␈α∩interesting␈α∩and␈α∪useful␈α∩one.
␈↓ ↓H␈↓Why?

␈↓ ↓H␈↓Just␈α⊃the␈α⊂knowing␈α⊃their␈α⊃definitions␈α⊂is␈α⊃no␈α⊃help.␈α⊂ The␈α⊃researcher␈α⊃might␈α⊂take
␈↓ ↓H␈↓the␈α⊗trouble␈α⊗to␈α⊗examine␈α⊗all␈α↔the␈α⊗numbers␈α⊗less␈α⊗than␈α⊗a␈α⊗thousand,␈α↔let's␈α⊗say,
␈↓ ↓H␈↓and␈α∃place␈α∃them␈α∀in␈α∃the␈α∃appropriate␈α∃set,␈α∀depending␈α∃on␈α∃how␈α∃many␈α∀divisors
␈↓ ↓H␈↓they␈α∪had.␈α∀Based␈α∪on␈α∀this␈α∪empirical␈α∪evidence,␈α∀he␈α∪might␈α∀be␈α∪able␈α∀to␈α∪perceive
␈↓ ↓H␈↓some regularity in one of these kinds of numbers.

␈↓ ↓H␈↓Well,␈α∃these␈α∃two␈α∃sets␈α∃of␈α∃numbers␈α∀are␈α∃very␈α∃small␈α∃and␈α∃there's␈α∃nothing␈α∀too
␈↓ ↓H␈↓interesting␈α∩about␈α∩them.␈α∩ Now␈α⊃we␈α∩turn␈α∩our␈α∩attention␈α⊃to␈α∩these␈α∩two␈α∩sets␈α⊃of
␈↓ ↓H␈↓numbers...␈α≠ Nothing␈α≤obviously␈α≠interesting␈α≠about␈α≤these␈α≠(2␈α≤divisors),␈α≠but
␈↓ ↓H␈↓these␈α∃(3␈α⊗divisors)␈α∃seem␈α⊗to␈α∃all␈α∃be␈α⊗perfect␈α∃squares.␈α⊗ A␈α∃natural␈α⊗question␈α∃to
␈↓ ↓H␈↓ask␈α∪is:␈α∪what␈α∩are␈α∪square-roots␈α∪of␈α∩these␈α∪numbers?␈α∪ We␈α∩then␈α∪notice␈α∪that␈α∩all
␈↓ ↓H␈↓the␈α∩members␈α∩here␈α∪just␈α∩seem␈α∩to␈α∪be␈α∩the␈α∩squares␈α∪of␈α∩the␈α∩members␈α∪from␈α∩this
␈↓ ↓H␈↓set.␈α↔ This␈α↔may␈α⊗cause␈α↔us␈α↔to␈α⊗spend␈α↔some␈α↔more␈α⊗time␈α↔studying␈α↔both␈α⊗these
␈↓ ↓H␈↓sets.

␈↓ ↓H␈↓After␈α→examining␈α~some␈α→more␈α~empirical␈α→data,␈α~the␈α→researcher␈α~may␈α→notice
␈↓ ↓H␈↓that␈α∃all␈α∃numbers␈α∃seem␈α∃to␈α∀factor␈α∃uniquly␈α∃into␈α∃numbers␈α∃drawn␈α∃from␈α∀this
␈↓ ↓H␈↓set.␈α⊃ This␈α∩is␈α⊃far␈α∩from␈α⊃trivial,␈α∩and␈α⊃we'll␈α∩go␈α⊃into␈α∩how␈α⊃he␈α∩does␈α⊃this␈α∩later␈α⊃on.
␈↓ ↓H␈↓He'll␈α≤conjecture␈α≤the␈α≤unique␈α≤factorization␈α≤theorem␈α≤<SLIDE␈α≤UFT>␈α≠and
␈↓ ↓H␈↓notice␈α⊂that␈α⊃it␈α⊂can␈α⊂be␈α⊃made␈α⊂much␈α⊂less␈α⊃awkward␈α⊂if␈α⊂we␈α⊃give␈α⊂this␈α⊂set␈α⊃a␈α⊂name,
␈↓ ↓H␈↓<SLIDE: Primes overlay> say PRIME NUMBERS.

␈↓ ↓H␈↓Once␈α⊃he's␈α⊃got␈α∩the␈α⊃definition␈α⊃of␈α∩primes,␈α⊃and␈α⊃he's␈α∩got␈α⊃a␈α⊃couple␈α∩good␈α⊃reasons
␈↓ ↓H␈↓for␈α⊂giving␈α⊂them␈α⊂a␈α⊂name,␈α⊂theorems␈α⊂involving␈α⊂primes,␈α⊂we␈α⊂shall␈α⊂say␈α⊂that␈α⊂he's
␈↓ ↓H␈↓actually ␈↓βdiscovered␈↓ them.
␈↓ ↓H␈↓∧␈↓ αS*** We Did a Problem-Reduction ***

␈↓ ↓H␈↓We␈α→haven't␈α→completely␈α→accounted␈α_for␈α→the␈α→discovery␈α→of␈α→prime␈α_numbers,
␈↓ ↓H␈↓we've␈α∀just␈α∀reduced␈α∀our␈α∀problem␈α∀from␈α∀"how␈α∀in␈α∀the␈α∀world␈α∀could␈α∀somebody
␈↓ ↓H␈↓invent␈α↔the␈α↔notion␈α⊗of␈α↔prime␈α↔numbers"␈α↔to␈α⊗the␈α↔somewhat␈α↔simpler␈α↔task␈α⊗of
␈↓ ↓H␈↓"how␈α≤in␈α≤the␈α≤world␈α≤could␈α≠somebody␈α≤invent␈α≤the␈α≤notion␈α≤of␈α≠divisors-of".
␈↓ ↓H␈↓<SLIDE␈α∀Reduc␈α∀1>␈α∀The␈α∃reduction␈α∀was␈α∀accomplished␈α∀by␈α∀citing␈α∃a␈α∀heuristic,
␈↓ ↓H␈↓which said:

␈↓ ↓H␈↓␈↓β"Investigate␈α⊂the␈α⊂inverse␈α⊂image␈α⊂of␈α⊂unusual␈α⊂or␈α⊂extreme␈α⊂subsets␈α⊂of␈α⊂the
␈↓ ↓H␈↓βrange of some operation."␈↓

␈↓ ↓H␈↓In␈α∀this␈α∀case,␈α∀the␈α∀extreme␈α∀subset␈α∀was␈α∀"doubletons",␈α∀and␈α∀the␈α∀operation␈α∪was
␈↓ ↓H␈↓divisors-of.␈α∩ The␈α∩inverse␈α∩image␈α∩of␈α∩doubletons␈α∩turned␈α∩out␈α∩to␈α∩be␈α∩the␈α∩set␈α⊃of
␈↓ ↓H␈↓numbers we know as the primes.

␈↓ ↓H␈↓"Looking for extreme cases" is a pretty general heuristic.

␈↓ ↓H␈↓It's␈α⊃not␈α∩a␈α⊃rule␈α⊃of␈α∩inference,␈α⊃it's␈α⊃just␈α∩a␈α⊃rule␈α⊃of␈α∩thumb;␈α⊃it␈α∩doesn't␈α⊃guarantee
␈↓ ↓H␈↓anything,␈α⊃in␈α⊃the␈α∩sense␈α⊃that␈α⊃modus␈α∩ponens␈α⊃guarantees␈α⊃to␈α∩preserve␈α⊃validity,
␈↓ ↓H␈↓or␈α∩in␈α∩the␈α∩sense␈α∩that␈α∩Waltz's␈α∩constraints␈α∩on␈α∩shadowing␈α∩line␈α∩drawings␈α⊃were
␈↓ ↓H␈↓absolute.

␈↓ ↓H␈↓This␈α∪heuristic␈α∪rule␈α∪certainly␈α∪isn't␈α∀applicable␈α∪or␈α∪useful␈α∪all␈α∪the␈α∀time.␈α∪ And
␈↓ ↓H␈↓yet␈α≥it's␈α≥worth␈α≥remembering␈α≥and␈α≥using␈α≥because,␈α≥in␈α≥my␈α≥experience,␈α≤it
␈↓ ↓H␈↓frequently␈α∀leads␈α∀to␈α∀interesting,␈α∀valuable␈α∀new␈α∀concepts␈α∀to␈α∀investigate.␈α∀ In
␈↓ ↓H␈↓the long run, using this heuristic is cost-effective.

␈↓ ↓H␈↓A␈α≠natural␈α≤and␈α≠fair␈α≤question,␈α≠now,␈α≤is␈α≠"OK,␈α≤so␈α≠you've␈α≤rationalized␈α≠the
␈↓ ↓H␈↓discovery␈α_of␈α_prime␈α_numbers,␈α_but␈α_why␈α_was␈α_the␈α_researcher␈α_investigating
␈↓ ↓H␈↓the␈α%concept␈α%`divisors-of'␈α%to␈α%begin␈α%with?".␈α% Have␈α%we␈α&really␈α%gotten
␈↓ ↓H␈↓anywhere?␈α⊃ If␈α⊃"divisors-of"␈α⊃was␈α⊃one␈α⊃of␈α⊃the␈α⊃primitive␈α⊃conepts␈α⊃we␈α⊃gave␈α⊂that
␈↓ ↓H␈↓researcher␈α≥to␈α≥begin␈α≥with,␈α≥then␈α≥the␈α≤answer␈α≥is␈α≥Yes,␈α≥we␈α≥have␈α≥in␈α≤fact
␈↓ ↓H␈↓explained␈α_how␈α_he␈α→might␈α_have␈α_discovered␈α→primes.␈α_ But␈α_assume␈α→that␈α_we
␈↓ ↓H␈↓didn't give him divisors-of, just simpler arithmetic operations.

␈↓ ↓H␈↓Maybe␈α⊂we␈α⊃use␈α⊂the␈α⊃same␈α⊂technique␈α⊃to␈α⊂explain␈α⊃why␈α⊂the␈α⊃researcher␈α⊂proposed
␈↓ ↓H␈↓this␈α⊂concept,␈α⊂divisors-of.␈α⊂ We'd␈α⊂like␈α⊂to␈α⊂grab␈α⊂some␈α⊂rule␈α⊂of␈α⊂thumb␈α⊂and␈α⊃use␈α⊂it
␈↓ ↓H␈↓to rationalize that discovery.

␈↓ ↓H␈↓But␈α∪look␈α∩at␈α∪the␈α∪definition␈α∩of␈α∪divisors-of(n)␈α∩<SLIDE:␈α∪Defn␈α∪of␈α∩Divisors-of>.
␈↓ ↓H␈↓It's␈α∨very␈α≡closely␈α∨related␈α≡to␈α∨the␈α≡␈↓βinverse␈↓␈α∨of␈α≡the␈α∨function␈α∨Times.␈α≡ A
␈↓ ↓H␈↓plausible␈α∃heuristic␈α∃tactic␈α∃<SLIDE:␈α∃Look␈α∃at␈α∃inverse>␈α∃is␈α∃to␈α⊗investigate␈α∃the
␈↓ ↓H␈↓inverse␈α⊂of␈α⊂each␈α⊂interesting␈α⊂operation.␈α⊂ So␈α⊂that␈α⊂rule␈α⊂would␈α⊂account␈α⊃for␈α⊂this
␈↓ ↓H␈↓discovery,␈α⊗reducing␈α∃it␈α⊗to␈α⊗the␈α∃discovery␈α⊗of␈α∃the␈α⊗concept␈α⊗of␈α∃multiplication,
␈↓ ↓H␈↓because␈α⊗looking␈α↔at␈α⊗the␈α↔inverse␈α⊗of␈α⊗multiplication␈α↔would␈α⊗lead␈α↔to␈α⊗defining
␈↓ ↓H␈↓and␈α↔studying␈α↔this␈α↔concept,␈α↔divisors-of.␈α↔ Actually,␈α↔one␈α↔also␈α↔has␈α_to␈α↔notice
␈↓ ↓H␈↓that␈α≡multiplication␈α∨is␈α≡␈↓βassociative␈α∨and␈α≡commutative␈↓,␈α∨i.e,␈α≡it␈α∨can␈α≡be
␈↓ ↓H␈↓viewed as taking a ␈↓βbag␈↓ or multiset of numbers as its argument.

␈↓ ↓H␈↓If␈α≥you␈α≥can␈α≤stand␈α≥it,␈α≥we␈α≤might␈α≥try␈α≥to␈α≤continue␈α≥this␈α≥reduction␈α≤even
␈↓ ↓H␈↓further.␈α∃ How␈α∀could␈α∃we␈α∀rationally␈α∃explain␈α∀how␈α∃␈↓βthis␈↓␈α∀concept␈α∃might␈α∀have
␈↓ ↓H␈↓been␈α≡discovered.␈α≡ You␈α≡might␈α≥consider␈α≡discovering␈α≡multiplication␈α≡as␈α≥a
␈↓ ↓H␈↓numeric␈α≠analog␈α≠to␈α≤the␈α≠operation␈α≠CARTESIAN-product,␈α≠or␈α≤as␈α≠repeated
␈↓ ↓H␈↓addition.␈α_ Addition␈α_could␈α→have␈α_been␈α_found␈α→as␈α_the␈α_numeric␈α→analogue␈α_of
␈↓ ↓H␈↓union,␈α∀or␈α∪as␈α∀repeated␈α∪successoring.␈α∀ Successor␈α∪is␈α∀the␈α∪numeric␈α∀analogue␈α∪of
␈↓ ↓H␈↓CONS or insertion.


␈↓ ↓H␈↓∧␈↓ ↓[*** This General Sequence of Reductions ***

␈↓ ↓H␈↓<chain␈α≡SLIDE>␈α≥We've␈α≡performed␈α≥a␈α≡sequence␈α≥of␈α≡reductions,␈α≥<Overlay:
␈↓ ↓H␈↓arrows␈α∪down>␈α∪from␈α∪primes␈α∪to␈α∪divisors-of␈α∪to␈α∪multiplication␈α∪to␈α∪addition␈α∪to
␈↓ ↓H␈↓counting␈α⊃to␈α⊂set-insertion.␈α⊃ Each␈α⊃step␈α⊂was␈α⊃justified␈α⊃by␈α⊂some␈α⊃rule␈α⊃of␈α⊂thumb:
␈↓ ↓H␈↓Here␈α~it␈α→is␈α~"consider␈α→extremals",␈α~here␈α→it␈α~is␈α→"consider␈α~the␈α→inverse␈α~of␈α→an
␈↓ ↓H␈↓interesting relation", here and here it is "repeat an interesting operation".

␈↓ ↓H␈↓We␈α⊂could␈α⊂continue␈α⊂this␈α⊂analysis,␈α⊂but␈α⊂at␈α⊂some␈α⊂point␈α⊂the␈α⊂concepts␈α⊃we'd␈α⊂have
␈↓ ↓H␈↓to␈α∀deal␈α∀with␈α∀get␈α∀so␈α∀elementary␈α∀that␈α∀we'd␈α∀feel␈α∀it␈α∀makes␈α∀no␈α∀sense␈α∃to␈α∀talk
␈↓ ↓H␈↓about␈α⊃"discovering"␈α∩them.␈α⊃ At␈α∩some␈α⊃point,␈α⊃they␈α∩are␈α⊃the␈α∩primitive␈α⊃concepts
␈↓ ↓H␈↓we initially provided the student researcher with.

␈↓ ↓H␈↓I␈α⊃BELIEVE␈α⊃THAT␈α⊃when␈α⊃we␈α⊃hear␈α⊃of␈α⊃a␈α⊃new␈α⊃discovery,␈α⊃we␈α⊃mentally␈α⊃try␈α⊂to
␈↓ ↓H␈↓analyze␈α≠it␈α≤in␈α≠this␈α≠fashion␈α≤ourselves.␈α≠ We␈α≠try␈α≤to␈α≠perceive␈α≠a␈α≤chain␈α≠of
␈↓ ↓H␈↓concepts,␈α⊂stretching␈α⊂from␈α⊂that␈α⊃␈↓βlatest␈↓␈α⊂discovery␈α⊂backwards,␈α⊂not␈α⊂all␈α⊃the␈α⊂way
␈↓ ↓H␈↓to␈α→our␈α_conceptual␈α→primitives,␈α_but␈α→at␈α→least␈α_all␈α→the␈α_way␈α→to␈α→concepts␈α_we
␈↓ ↓H␈↓already␈α∃are␈α∀familiar␈α∃with.␈α∀ If␈α∃we␈α∀succeed␈α∃right␈α∀away,␈α∃we␈α∃probably␈α∀think
␈↓ ↓H␈↓that the discovery wasn't so hard, that we could have done it ourselves.

␈↓ ↓H␈↓<Remove␈α∪down-arrow␈α∪overlay>␈α∪Very␈α∪often,␈α∩a␈α∪new␈α∪discovery␈α∪is␈α∪really␈α∩just
␈↓ ↓H␈↓a␈α∃couple␈α∃steps␈α∃in␈α∃this␈α⊗kind␈α∃of␈α∃process,␈α∃one␈α∃or␈α∃two␈α⊗heuristic␈α∃applications
␈↓ ↓H␈↓away␈α∩from␈α∩what␈α⊃is␈α∩already␈α∩known.␈α∩ That␈α⊃is␈α∩why␈α∩it's␈α⊃often␈α∩easy␈α∩for␈α∩us␈α⊃to
␈↓ ↓H␈↓believe␈α≤that␈α≠there␈α≤wasn't␈α≠really␈α≤so␈α≠much␈α≤effort␈α≠needed␈α≤to␈α≤make␈α≠the
␈↓ ↓H␈↓discovery.␈α∪ If␈α∪we␈α∀already␈α∪know␈α∪this␈α∪concept,␈α∀and␈α∪this␈α∪heuristic,␈α∀it␈α∪almost
␈↓ ↓H␈↓seems obvious once we've heard that someone has taken this step.

␈↓ ↓H␈↓Why␈α∀then␈α∀don't␈α∀we␈α∀make␈α∀new␈α∀discoveries␈α∀every␈α∀few␈α∀minutes?␈α∀ Why␈α∪are
␈↓ ↓H␈↓even␈α∀some␈α∀1-step␈α∀advances␈α∀worth␈α∀publishing?␈α∀ The␈α∀answer␈α∀can␈α∀be␈α∀found
␈↓ ↓H␈↓by␈α∀seeing␈α∃how␈α∀(according␈α∃to␈α∀this␈α∃model)␈α∀very␈α∃simple␈α∀new␈α∃discoveries␈α∀are
␈↓ ↓H␈↓made.

␈↓ ↓H␈↓We're␈α_talking␈α→now␈α_about␈α_reversing␈α→the␈α_flow␈α_of␈α→time␈α_here.␈α→ Instead␈α_of
␈↓ ↓H␈↓analyzing␈α∪a␈α∩discovery␈α∪by␈α∩breaking␈α∪it␈α∩into␈α∪simpler␈α∩and␈α∪simpler␈α∪ones,␈α∩we're
␈↓ ↓H␈↓talking␈α~about␈α~the␈α~synthesis␈α~of␈α~new␈α~concepts,␈α~␈↓βthe␈α~activity␈α≠which␈α~the
␈↓ ↓H␈↓βresearcher␈α∀was␈α∀carrying␈α∀out␈α∀all␈α∪along␈↓.␈α∀ He␈α∀was␈α∀continually␈α∀trying␈α∪to
␈↓ ↓H␈↓grow␈α∪new␈α∪nodes␈α∪on␈α∪this␈α∪tree,␈α∪using␈α∪whatever␈α∪heuristic␈α∪rules␈α∪he's␈α∩learned
␈↓ ↓H␈↓over the years.

␈↓ ↓H␈↓The␈α∀researcher␈α∪has␈α∀a␈α∪bunch␈α∀of␈α∪concepts␈α∀that␈α∪he␈α∀knows␈α∪about,␈α∀say␈α∀a␈α∪few
␈↓ ↓H␈↓thousand.␈α∃ I␈α∃won't␈α∃even␈α∃consider␈α∀his␈α∃␈↓βlegal␈↓␈α∃choices,␈α∃since␈α∃they␈α∃are␈α∀quite
␈↓ ↓H␈↓astronomical␈α∪in␈α∀number.␈α∪ Say␈α∪he␈α∀knows␈α∪a␈α∪few␈α∀hundred␈α∪heuristics␈α∀he␈α∪can
␈↓ ↓H␈↓apply.␈α⊃ In␈α⊂any␈α⊃situation,␈α⊃then,␈α⊂he␈α⊃may␈α⊃literally␈α⊂have␈α⊃a␈α⊃million␈α⊂alternatives
␈↓ ↓H␈↓to␈α∀consider,␈α∀<OVERLAY␈α∀of␈α∀red␈α∀lines>.␈α∀ These␈α∀are␈α∀not␈α∀his␈α∀␈↓βlegal␈↓␈α∀choices,
␈↓ ↓H␈↓but␈α∩his␈α⊃␈↓βplausible␈↓␈α∩ones;␈α⊃each␈α∩and␈α⊃every␈α∩one␈α⊃is␈α∩justified␈α⊃by␈α∩some␈α⊃heuristic,
␈↓ ↓H␈↓like this link is.

␈↓ ↓H␈↓But␈α∀nature␈α∪is␈α∀not␈α∪kind␈α∀to␈α∪us,␈α∀and␈α∪only␈α∀a␈α∪very␈α∀small␈α∪percentage␈α∀of␈α∪those
␈↓ ↓H␈↓new␈α↔concepts␈α⊗will␈α↔turn␈α⊗out␈α↔to␈α⊗be␈α↔interesting.␈α⊗ What␈α↔is␈α⊗even␈α↔worse,␈α⊗he
␈↓ ↓H␈↓can't␈α∨whiz␈α∨through␈α∨this␈α≡space,␈α∨stopping␈α∨at␈α∨the␈α∨interesting␈α≡concepts,
␈↓ ↓H␈↓because␈α∃it␈α∃may␈α∃require␈α∃hours␈α⊗--␈α∃sometimes␈α∃even␈α∃years␈α∃--␈α∃of␈α⊗research␈α∃to
␈↓ ↓H␈↓decide whether the concept he's studying is worthwhile or a dead-end.
␈↓ ↓H␈↓Even␈α⊗here,␈α⊗where␈α⊗we␈α⊗have␈α⊗just␈α⊗a␈α⊗few␈α⊗concepts␈α⊗and␈α⊗heuristics,␈α⊗the␈α⊗tree
␈↓ ↓H␈↓gets␈α∩bushy␈α∪when␈α∩we␈α∩try␈α∪to␈α∩go␈α∪in␈α∩this␈α∩upward␈α∪direction.␈α∩ We␈α∪could␈α∩apply
␈↓ ↓H␈↓"looking␈α∃at␈α∃inverse"␈α⊗here,␈α∃(CARTESIAN-product)␈α∃to␈α∃decide␈α⊗to␈α∃investigate
␈↓ ↓H␈↓the␈α∃kind␈α⊗of␈α∃operation␈α⊗we␈α∃normally␈α∃call␈α⊗projection.␈α∃ Or␈α⊗here␈α∃(divisors-of)
␈↓ ↓H␈↓even␈α∪if␈α∩we␈α∪decide␈α∩to␈α∪look␈α∪for␈α∩extremals,␈α∪which␈α∩way␈α∪do␈α∩we␈α∪go?␈α∪ Why␈α∩not
␈↓ ↓H␈↓examine␈α∩numbers␈α∩which␈α∪have␈α∩very␈α∩many,␈α∪rather␈α∩than␈α∩very␈α∪few,␈α∩divisors?
␈↓ ↓H␈↓Why␈α∩not␈α∩keep␈α∩creating␈α∩new␈α∩operations,␈α∩by␈α∩repeating␈α∩the␈α∩last␈α∩one␈α⊃created,
␈↓ ↓H␈↓thereby discovering exponentiation, hyper-exponentiation, and so on?

␈↓ ↓H␈↓The␈α⊂analysis␈α⊂procedure␈α⊂was␈α⊂a␈α⊂search␈α⊂for␈α⊂a␈α⊂path␈α⊂from␈α⊂a␈α⊂given␈α⊂node␈α⊃back␈α⊂to
␈↓ ↓H␈↓primitives,␈α≠but␈α≠the␈α≠synthesis␈α≠procedure␈α≠is␈α≠blind,␈α≠doesn't␈α≠have␈α≤a␈α≠well-
␈↓ ↓H␈↓specified goal, so it is combinatorially more explosive.

␈↓ ↓H␈↓So␈α⊃it's␈α⊃harder␈α⊃to␈α⊃MAKE␈α⊃a␈α⊃valuable␈α⊃new␈α⊃discovery␈α⊃than␈α⊃to␈α∩rationalize␈α⊃how
␈↓ ↓H␈↓someone␈α→might␈α~plausibly␈α→have␈α→discovered␈α~something␈α→he's␈α→just␈α~told␈α→you
␈↓ ↓H␈↓about.␈α↔ This␈α↔explains␈α↔why␈α↔discoveries␈α↔sometimes␈α↔seem␈α↔so␈α↔obvious␈α⊗␈↓βafter
␈↓ ↓H␈↓βthe␈α→fact␈↓,␈α_even␈α→to␈α_their␈α→inventor␈α→who␈α_may␈α→have␈α_labored␈α→years␈α→on␈α_the
␈↓ ↓H␈↓problem.

␈↓ ↓H␈↓The␈α∀same␈α∀let-down␈α∃occurs␈α∀when␈α∀a␈α∀magician␈α∃shows␈α∀you␈α∀how␈α∃he␈α∀manages
␈↓ ↓H␈↓some␈α~magic␈α≠trick,␈α~or␈α≠equivalently␈α~when␈α~an␈α≠AI␈α~researcher␈α≠like␈α~myself
␈↓ ↓H␈↓shows␈α→you␈α→how␈α→his␈α→program␈α→was␈α→really␈α→able␈α→to␈α→manage␈α→some␈α→amazing
␈↓ ↓H␈↓behavior.


␈↓ ↓H␈↓∧␈↓ αN*** Introduce Heuristic Guidance ***

␈↓ ↓H␈↓Now␈α↔it's␈α_clear␈α↔that␈α↔this␈α_synthesis␈α↔task,␈α↔exploring␈α_outward␈α↔to␈α_find␈α↔new
␈↓ ↓H␈↓valuable concepts, is a very explosive search process.

␈↓ ↓H␈↓The␈α∪standard␈α∪Artificial␈α∪Intelligence␈α∀method␈α∪for␈α∪muffling␈α∪an␈α∀explosion␈α∪of
␈↓ ↓H␈↓red␈α→arrows␈α→is␈α→to␈α→introduce␈α→purple␈α→and␈α→green␈α→heuristic␈α→guidance.␈α→ <2nd
␈↓ ↓H␈↓OVERLAY:␈α∩green/purple>:␈α∩purple␈α∩strategies␈α∩highlight␈α∩the␈α∩most␈α∩promising
␈↓ ↓H␈↓nodes␈α∩to␈α∩work␈α∪on␈α∩at␈α∩a␈α∩given␈α∪moment,␈α∩and␈α∩␈↓βwiggly␈α∩green␈α∪ideas␈α∩suggest
␈↓ ↓H␈↓βfuriously␈↓ one or two operators to try first on a given node.

␈↓ ↓H␈↓Recall␈α∩that␈α⊃each␈α∩red␈α⊃operator␈α∩is␈α⊃a␈α∩rule␈α⊃of␈α∩thumb.␈α⊃ So␈α∩the␈α∩green␈α⊃strategies
␈↓ ↓H␈↓tell us, in a given situation what the best heuristics are to apply.
␈↓ ↓H␈↓From␈α↔now␈α↔on,␈α↔when␈α⊗I␈α↔use␈α↔the␈α↔term␈α⊗"heuristic␈α↔rule",␈α↔I'll␈α↔really␈α↔mean␈α⊗a
␈↓ ↓H␈↓two-sided␈α∩creature␈α∩like␈α∩this␈α∩one,␈α⊃<SLIDE:␈α∩heurs.;␈α∩a␈α∩situation/action␈α∩rule,␈α⊃a
␈↓ ↓H␈↓production.␈α≥ It␈α≥says␈α≥that␈α≥in␈α≤some␈α≥empirical␈α≥situation,␈α≥here␈α≥are␈α≤some
␈↓ ↓H␈↓plausible things to spend your time doing.

␈↓ ↓H␈↓Here,␈α∨the␈α rule's␈α∨IF-part,␈α or␈α∨left-hand-side,␈α was␈α∨asking␈α whether␈α∨two
␈↓ ↓H␈↓concepts␈α~have␈α~some␈α→nontrivial␈α~yet␈α~unexpected␈α→overlap.␈α~If␈α~so,␈α~the␈α→rule
␈↓ ↓H␈↓concludes␈α∀that␈α∀it's␈α∪worth␈α∀the␈α∀trouble␈α∀to␈α∪isolate␈α∀and␈α∀explicitly␈α∀study␈α∪that
␈↓ ↓H␈↓overlap.


␈↓ ↓H␈↓∧␈↓ ↓k*** We Write a Theory Formation Pgm ***

␈↓ ↓H␈↓The␈α_basic␈α↔idea␈α_to␈α↔remember␈α_from␈α↔all␈α_this␈α↔is␈α_that␈α_simple␈α↔mathematical
␈↓ ↓H␈↓discovery␈α'might␈α&be␈α'successfully␈α&represented␈α'as␈α&a␈α'heuristic␈α&search
␈↓ ↓H␈↓procedure:␈α∃<SLIDE:␈α∀3␈α∃tasks>␈α∀a␈α∃continual␈α∀expansion␈α∃of␈α∀a␈α∃tree␈α∃of␈α∀concepts
␈↓ ↓H␈↓and␈α⊃relationships,␈α⊃guided␈α⊂by␈α⊃some␈α⊃judgmental␈α⊂criteria,␈α⊃some␈α⊃informal␈α⊂rules
␈↓ ↓H␈↓of thumb.

␈↓ ↓H␈↓To␈α∀test␈α∀this␈α∪hypothesis,␈α∀I␈α∀wrote␈α∀a␈α∪computer␈α∀program,␈α∀called␈α∀A.M.,␈α∪which
␈↓ ↓H␈↓proposes␈α→new␈α_concepts␈α→in␈α_simple␈α→mathematical␈α_domains.␈α→ This␈α_program
␈↓ ↓H␈↓goes␈α⊃through␈α⊂the␈α⊃same␈α⊃kinds␈α⊂of␈α⊃concept␈α⊂exploration␈α⊃and␈α⊃development␈α⊂that
␈↓ ↓H␈↓the␈α∪hypothetical␈α∩undergraduate␈α∪researcher␈α∪did.␈α∩ AM␈α∪builds␈α∩up␈α∪a␈α∪model␈α∩of
␈↓ ↓H␈↓some␈α→simple␈α→parts␈α→of␈α→mathematics.␈α~ Let␈α→me␈α→just␈α→pause␈α→a␈α~moment␈α→and
␈↓ ↓H␈↓emphasize␈α⊂that␈α⊃AM␈α⊂is␈α⊂writtn␈α⊃and␈α⊂is␈α⊂running,␈α⊃and␈α⊂has␈α⊂managed␈α⊃to␈α⊂discover
␈↓ ↓H␈↓all␈α⊗the␈α↔concepts␈α⊗and␈α↔theorems␈α⊗which␈α⊗I␈α↔have␈α⊗mentioned␈α↔so␈α⊗far,␈α↔and␈α⊗the
␈↓ ↓H␈↓ones I will mention.

␈↓ ↓H␈↓AM␈α⊃is␈α∩a␈α⊃heuristic␈α∩search␈α⊃program,␈α∩and␈α⊃in␈α∩the␈α⊃tradition␈α∩of␈α⊃AI␈α∩programs␈α⊃of
␈↓ ↓H␈↓this␈α↔form,␈α↔there␈α↔are␈α⊗three␈α↔separate␈α↔things␈α↔I␈α⊗should␈α↔tell␈α↔you␈α↔about:␈α⊗the
␈↓ ↓H␈↓same things you always have to do to implement a heuristic search:
␈↓ ↓H␈↓(1) specify the initial core of knowledge that it will start with,
␈↓ ↓H␈↓(2) specify the legal ways of adding to that knowledge
␈↓ ↓H␈↓(3)␈α∃collect␈α∃the␈α∃strategies␈α∀and␈α∃tactics␈α∃that␈α∃experts␈α∀in␈α∃that␈α∃field␈α∃use␈α∀when
␈↓ ↓H␈↓␈↓ ↓xtrying to form theories.

␈↓ ↓H␈↓At␈α∃a␈α∃higher␈α∃level,␈α∃the␈α∃"goal"␈α⊗of␈α∃AM␈α∃is␈α∃to␈α∃find␈α∃interesting␈α⊗new␈α∃concepts
␈↓ ↓H␈↓and conjectures, by exploring a space of partially-investigated concepts.
␈↓ ↓H␈↓∧␈↓ α *** Mathematical Theory Formation ***

␈↓ ↓H␈↓If␈α⊂anyone␈α⊂asks␈α⊂at␈α⊃the␈α⊂end,␈α⊂I'll␈α⊂discuss␈α⊂why␈α⊃mathematics␈α⊂is␈α⊂an␈α⊂ideal␈α⊃field␈α⊂in
␈↓ ↓H␈↓which␈α⊂to␈α⊂try␈α⊃out␈α⊂these␈α⊂ideas.␈α⊃ For␈α⊂now,␈α⊂assume␈α⊃that␈α⊂we've␈α⊂settled␈α⊃on␈α⊂very
␈↓ ↓H␈↓elementary␈α_math.␈α_ Before␈α_carrying␈α_out␈α_these␈α_3␈α_steps,␈α_I␈α_should␈α_at␈α↔least
␈↓ ↓H␈↓mention what I mean by a ␈↓βmathematical concept␈↓ and a ␈↓βtheory␈↓.

␈↓ ↓H␈↓A␈α≡mathematical␈α≡theory␈α≥<SLIDE:␈α≡thy>␈α≡is␈α≥built␈α≡on␈α≡a␈α≡␈↓&␈↓↓FOUNDATION␈↓␈↓)αβ␈α≥of
␈↓ ↓H␈↓primitive,␈α≡undefined␈α≡␈↓&objects␈↓)αβ,␈α≡and␈α≡␈↓&operators␈↓)αβ,␈α≡plus␈α≡some␈α∨postulates␈α≡and
␈↓ ↓H␈↓axioms␈α_which␈α_interrelate␈α_them.␈α_ Each␈α↔of␈α_these␈α_qualifies␈α_to␈α_be␈α_called␈α↔a
␈↓ ↓H␈↓mathematical ␈↓&concept␈↓)αβ.

␈↓ ↓H␈↓Onto␈α⊗this␈α⊗foundation␈α⊗is␈α⊗built␈α⊗a␈α⊗tower␈α⊗of␈α⊗theorems␈α⊗--␈α⊗statements␈α⊗which
␈↓ ↓H␈↓are␈α⊃implied␈α⊃by␈α⊃the␈α⊂foundation␈α⊃--␈α⊃plus␈α⊃some␈α⊂definitions␈α⊃of␈α⊃new␈α⊃concepts␈α⊂in
␈↓ ↓H␈↓terms of preexisting ones.

␈↓ ↓H␈↓This␈α!is␈α!the␈α final,␈α!breath-taking␈α!architecture␈α that␈α!we␈α!see␈α!in␈α math
␈↓ ↓H␈↓textbooks.␈α⊗ ␈↓βThe␈α⊗foundation␈α⊗is␈α⊗laid␈α⊗carefully␈↓,␈α⊗and␈α⊗the␈α⊗edifice␈α⊗is␈α⊗built
␈↓ ↓H␈↓smoothly, brick by brick, with no mistakes, no backtracking.

␈↓ ↓H␈↓What␈α∃they␈α∀␈↓βdon't␈↓␈α∃show␈α∀you␈α∃in␈α∀textbooks␈α∃and␈α∀journals␈α∃is␈α∃the␈α∀unaesthetic
␈↓ ↓H␈↓scaffolding␈α≠that␈α≠was␈α≠required␈α≠to␈α~erect␈α≠that␈α≠tower:␈α≠that␈α≠scaffolding␈α~is
␈↓ ↓H␈↓empirical␈α∩induction␈α∩and␈α∩good␈α⊃old-fasioned␈α∩search,␈α∩with␈α∩guessing,␈α⊃intuition,
␈↓ ↓H␈↓back-tracking and everything.

␈↓ ↓H␈↓At␈α≤the␈α≤very␈α≤elementary␈α≥levels,␈α≤at␈α≤least,␈α≤mathematics␈α≤is␈α≥an␈α≤empirical
␈↓ ↓H␈↓science.

␈↓ ↓H␈↓Most␈α⊂of␈α⊃the␈α⊂mathematician's␈α⊂empirical␈α⊃forays␈α⊂are␈α⊂fruitless,␈α⊃but␈α⊂there␈α⊃is␈α⊂no
␈↓ ↓H␈↓way␈α∃to␈α∃know␈α∃this␈α∃before␈α∃carrying␈α∃them␈α∃out.␈α∃ His␈α∃only␈α∃guidance␈α∃are␈α∃his
␈↓ ↓H␈↓intuitions␈α∪about␈α∪what␈α∪to␈α∪investigate␈α∪next,␈α∪his␈α∪purple␈α∪and␈α∀green␈α∪heuristic
␈↓ ↓H␈↓strategies.

␈↓ ↓H␈↓This same character of searching a large space also carries over to AM.

␈↓ ↓H␈↓<3tasks,␈α∀revisited␈α∀SLIDE>␈α∀Recall␈α∀that␈α∃there␈α∀is␈α∀a␈α∀standard␈α∀set␈α∀of␈α∃tasks␈α∀to
␈↓ ↓H␈↓perform, to create such a system.
␈↓ ↓H␈↓∧␈↓ ∧F*** Writing AM ***

␈↓ ↓H␈↓The␈α∪first␈α∪step␈α∩was␈α∪to␈α∪list␈α∩all␈α∪the␈α∪concepts␈α∩the␈α∪system␈α∪would␈α∪know␈α∩about
␈↓ ↓H␈↓initially.␈α↔That␈α_is,␈α↔a␈α_bunch␈α↔of␈α_primitive␈α↔notions␈α_which␈α↔are␈α_the␈α↔starting
␈↓ ↓H␈↓state of the system's knowledge.

␈↓ ↓H␈↓<SLIDE␈α⊗concept>␈α∃Here␈α⊗are␈α∃most␈α⊗of␈α∃the␈α⊗concepts␈α∃that␈α⊗AM␈α⊗started␈α∃with.
␈↓ ↓H␈↓For␈α∩reasons␈α∪I'll␈α∩go␈α∩into␈α∪if␈α∩anyone␈α∩asks,␈α∪I␈α∩chose␈α∩a␈α∪small␈α∩set␈α∪of␈α∩elementary,
␈↓ ↓H␈↓pre-numerical concepts.

␈↓ ↓H␈↓Included␈α_here␈α_are␈α↔static␈α_structural␈α_concepts␈α↔like␈α_sets␈α_and␈α↔ordered-pairs,
␈↓ ↓H␈↓and active ones like union, reverse, compose-2-relations, repeat, and so on.

␈↓ ↓H␈↓Note that there is no notion of numbers or proof.

␈↓ ↓H␈↓At␈α⊗a␈α∃slightly␈α⊗deeper␈α⊗level,␈α∃we␈α⊗had␈α⊗to␈α∃decide␈α⊗precisely␈α⊗what␈α∃information
␈↓ ↓H␈↓the system would be told about each of these concepts.

␈↓ ↓H␈↓Certainly we had to provide a ␈↓&definition␈↓)αβ for each concept, and a name.

␈↓ ↓H␈↓For␈α∩each␈α∩operation,␈α∩we␈α⊃also␈α∩mentioned␈α∩its␈α∩␈↓&domain␈α⊃and␈α∩range␈↓)αβ,␈α∩and␈α∩gave␈α⊃an
␈↓ ↓H␈↓algorithm for computing it.

␈↓ ↓H␈↓What else should be supplied initially about each concept?

␈↓ ↓H␈↓To␈α∪help␈α∪AM␈α∪decide␈α∪what␈α∩to␈α∪do,␈α∪it␈α∪would␈α∪be␈α∩nice␈α∪to␈α∪supply␈α∪some␈α∪kind␈α∩of
␈↓ ↓H␈↓rough␈α"judgment␈α"of␈α#the␈α"worth␈α"of␈α#each␈α"concept:␈α"its␈α#usefulness,␈α"its
␈↓ ↓H␈↓simplicity,␈α∃symmetry,␈α∀etc.␈α∃ Instead␈α∀of␈α∃grappling␈α∀with␈α∃the␈α∃metaphysics␈α∀or
␈↓ ↓H␈↓this␈α∪problem,␈α∩I␈α∪simply␈α∩attached,␈α∪ad-hoc,␈α∩a␈α∪number␈α∩to␈α∪each␈α∪concept,␈α∩which
␈↓ ↓H␈↓was supposed to represent its overall worth.

␈↓ ↓H␈↓If,␈α∃by␈α∃introspecting,␈α∃we␈α∃knew␈α⊗of␈α∃any␈α∃heuristic␈α∃rules␈α∃relevant␈α⊗to␈α∃dealing
␈↓ ↓H␈↓with that type of concept, they were tacked onto it.

␈↓ ↓H␈↓So␈α∃there␈α∃are␈α∃many␈α∃facets␈α∃to␈α∀each␈α∃concept,␈α∃<SLIDE␈α∃facets>.␈α∃Some␈α∃of␈α∀the
␈↓ ↓H␈↓ones␈α5we␈α4didn't␈α5mention␈α4yet␈α5were:␈α5Examples,␈α4Generalizations,
␈↓ ↓H␈↓Specializations,␈α≡and␈α≡Analogies.␈α≡Thus␈α≡by␈α≡a␈α≡"concept"␈α≡I'll␈α≡now␈α≡mean␈α≥a
␈↓ ↓H␈↓concrete␈α≤kind␈α≤of␈α≠data␈α≤structure,␈α≤similar␈α≠to␈α≤Frames,␈α≤Beings,␈α≤and␈α≠KRL
␈↓ ↓H␈↓Units.

␈↓ ↓H␈↓The␈α∩set␈α∩of␈α∩facets␈α⊃is␈α∩fixed␈α∩once␈α∩and␈α∩for␈α⊃all.␈α∩ Each␈α∩concept␈α∩can␈α∩possess␈α⊃any
␈↓ ↓H␈↓or␈α⊂all␈α⊂of␈α⊃these␈α⊂facets.␈α⊂But␈α⊂that's␈α⊃all␈α⊂we␈α⊂can␈α⊂ever␈α⊃know␈α⊂about␈α⊂a␈α⊃concept.␈α⊂ A
␈↓ ↓H␈↓concept,␈α_as␈α_represented␈α_inside␈α_AM,␈α_␈↓βis␈↓␈α_simply␈α_a␈α_bunch␈α_of␈α_facets,␈α_drawn
␈↓ ↓H␈↓from this fixed set.

␈↓ ↓H␈↓In␈α∪the␈α∪LISP␈α∪program␈α∪implementing␈α∪these␈α∪ideas,␈α∪each␈α∪concept␈α∪is␈α∀stored␈α∪as
␈↓ ↓H␈↓an␈α∩atom,␈α∩and␈α∩its␈α⊃facets␈α∩are␈α∩implemented␈α∩as␈α⊃the␈α∩properties␈α∩on␈α∩its␈α⊃property
␈↓ ↓H␈↓list.

␈↓ ↓H␈↓Let's␈α∀consider␈α∀one␈α∀particular␈α∃concept,␈α∀and␈α∀see␈α∀what␈α∀kinds␈α∃of␈α∀information
␈↓ ↓H␈↓could␈α↔go␈α⊗into␈α↔each␈α↔facet.␈α⊗ We'll␈α↔consider␈α⊗the␈α↔concept␈α↔COMPOSE,␈α⊗which
␈↓ ↓H␈↓refers␈α∃to␈α∀the␈α∃composition␈α∀of␈α∃two␈α∀relations.␈α∃ We␈α∀define␈α∃AoB(x)␈α∃as␈α∀A(B(x)).
␈↓ ↓H␈↓That␈α∪is,␈α∪apply␈α∩B␈α∪and␈α∪then␈α∩apply␈α∪A␈α∪to␈α∩the␈α∪result.␈α∪ <COMPOSE␈α∪slide>␈α∩<go
␈↓ ↓H␈↓over each facet>.  Here is an abbreviated description of that concept.

␈↓ ↓H␈↓Several␈α≤equivalent␈α≥definitions␈α≤are␈α≥provided.␈α≤Some␈α≥are␈α≤computationally
␈↓ ↓H␈↓efficient,␈α_and␈α↔others␈α_are␈α↔slow␈α_but␈α↔especially␈α_easy␈α↔to␈α_for␈α↔the␈α_system␈α↔to
␈↓ ↓H␈↓analyze.␈α∃ Each␈α∃definition␈α∃is␈α∃an␈α∃executable␈α∃LISP␈α∃predicate,␈α∃which␈α∃takes␈α∃3
␈↓ ↓H␈↓arguments␈α∃and␈α∃tries␈α∃to␈α∃see␈α∃whether␈α∃the␈α∃composition␈α∃of␈α∃the␈α∃first␈α∃two␈α∃is
␈↓ ↓H␈↓equal to the third.

␈↓ ↓H␈↓The␈α∀Domain/range␈α∃facet␈α∀notes␈α∃that␈α∀Compose␈α∀accepts␈α∃a␈α∀pair␈α∃of␈α∀activities,
␈↓ ↓H␈↓and creates a new one.

␈↓ ↓H␈↓The␈α↔specializations␈α↔facet␈α_points␈α↔to␈α↔another␈α_concept␈α↔which␈α↔is␈α_similar␈α↔to
␈↓ ↓H␈↓Compose.  This specialization only composes an activity with itself.

␈↓ ↓H␈↓An ␈↓βexample␈↓ of Composition is provided here.

␈↓ ↓H␈↓One␈α_conjecture␈α→that␈α_AM␈α_found␈α→and␈α_stored␈α_here␈α→is␈α_that␈α→composition␈α_is
␈↓ ↓H␈↓associative.

␈↓ ↓H␈↓Down␈α⊗here␈α⊗we␈α∃have␈α⊗some␈α⊗heuristics␈α∃for␈α⊗dealing␈α⊗with␈α⊗compositions.␈α∃ An
␈↓ ↓H␈↓overall␈α⊃rating␈α⊃of␈α⊃this␈α⊃concept,␈α∩how␈α⊃to␈α⊃spot␈α⊃an␈α⊃interesting␈α∩compostion␈α⊃(e.g.,
␈↓ ↓H␈↓if␈α⊃the␈α∩d-r␈α⊃of␈α⊃the␈α∩composition␈α⊃are␈α∩equal),␈α⊃hints␈α⊃for␈α∩filling␈α⊃in␈α∩certain␈α⊃facets
␈↓ ↓H␈↓of␈α"any␈α"new␈α!composition␈α"which␈α"gets␈α!created,␈α"and␈α"something␈α!worth
␈↓ ↓H␈↓checking about each composition.

␈↓ ↓H␈↓The␈α⊗format␈α⊗of␈α⊗each␈α⊗facet␈α⊗is␈α⊗fixed,␈α⊗but␈α⊗the␈α⊗formats␈α⊗vary␈α⊗from␈α⊗facet␈α⊗to
␈↓ ↓H␈↓facet,␈α≤from␈α≤productions␈α≤<heurs>␈α≤to␈α≤little␈α≤programs␈α≤<algs>,␈α≤to␈α≥bits␈α≤of
␈↓ ↓H␈↓declarative data <d-r>.

␈↓ ↓H␈↓Let␈α me␈α put␈α back␈α the␈α list␈α of␈α concepts␈α again,␈α and␈α also␈α our␈α list␈α of
␈↓ ↓H␈↓facets.<SLIDE>

␈↓ ↓H␈↓I␈α∩provided␈α∩the␈α⊃contents␈α∩of␈α∩a␈α⊃few␈α∩of␈α∩these␈α⊃facets␈α∩by␈α∩hand,␈α⊃for␈α∩each␈α∩of␈α⊃the
␈↓ ↓H␈↓initial␈α⊃100␈α⊃concepts.␈α∩ AM␈α⊃later␈α⊃filled␈α⊃out␈α∩the␈α⊃other␈α⊃facets.␈α∩ Whenever␈α⊃AM
␈↓ ↓H␈↓created␈α⊂a␈α⊂new␈α⊂concept,␈α⊂it␈α⊂had␈α⊂of␈α⊂course␈α⊂no␈α⊂"hand-supplied"␈α⊂facets␈α⊂at␈α⊂all␈α⊂for
␈↓ ↓H␈↓it.


␈↓ ↓H␈↓∧␈↓ ∧~*** What AM Does ***

␈↓ ↓H␈↓What␈α!does␈α the␈α!system␈α!actually␈α do,␈α!now?␈α  From␈α!a␈α!heuristic␈α search
␈↓ ↓H␈↓viewpoint,␈α⊃the␈α⊃primary␈α⊃activity␈α⊃of␈α⊃the␈α⊃system␈α⊃is␈α⊃to␈α⊃␈↓βcreate␈α⊃new␈α⊂concepts␈↓.
␈↓ ↓H␈↓These␈α⊗are␈α↔the␈α⊗highly-visible␈α↔"legal␈α⊗moves":␈α↔taking␈α⊗existing␈α↔concepts␈α⊗and
␈↓ ↓H␈↓combining␈α≤them,␈α≠conjoning␈α≤them,␈α≤disjoining␈α≠them,␈α≤running␈α≤them,␈α≠and
␈↓ ↓H␈↓modifying␈α⊗their␈α⊗definitions.␈α↔ This␈α⊗plethora␈α⊗of␈α↔alternatives␈α⊗is␈α⊗too␈α↔big␈α⊗to
␈↓ ↓H␈↓consider,␈α_so␈α_AM␈α_only␈α_creates␈α→a␈α_new␈α_concept␈α_when␈α_some␈α→heuristic␈α_rule
␈↓ ↓H␈↓indicates␈α⊗that␈α⊗it␈α⊗might␈α⊗be␈α⊗worth␈α⊗the␈α⊗gamble.␈α⊗So␈α⊗the␈α⊗heuristics␈α⊗act␈α∃like
␈↓ ↓H␈↓little plausible move generators.

␈↓ ↓H␈↓When␈α∩AM␈α∪does␈α∩create␈α∩a␈α∪new␈α∩concept,␈α∩most␈α∪of␈α∩its␈α∩facets␈α∪will␈α∩be␈α∪blank,␈α∩so
␈↓ ↓H␈↓that␈α∪in␈α∀practice␈α∪most␈α∪of␈α∀the␈α∪system's␈α∀time␈α∪is␈α∪spent␈α∀in␈α∪fleshing␈α∀out␈α∪some
␈↓ ↓H␈↓recently-discovered concept, filling in some of its facets.

␈↓ ↓H␈↓In␈α∪other␈α∪words,␈α∪there␈α∪are␈α∪␈↓βTWO␈α∪WAYS␈↓␈α∪that␈α∪AM␈α∪expands␈α∪its␈α∪knowledge.
␈↓ ↓H␈↓One␈α∩way␈α∩is␈α∩to␈α∩pick␈α∩an␈α∩already-known␈α∩concept␈α∩<SLIDE:␈α∩Compose>␈α∩and␈α∩try
␈↓ ↓H␈↓to␈α≡find␈α≥out␈α≡more␈α≡about␈α≥it,␈α≡by␈α≥filling␈α≡in␈α≡some␈α≥facet␈α≡of␈α≡it␈α≥<SLIDE:
␈↓ ↓H␈↓Compose.exs␈α⊗overlay>.␈α∃The␈α⊗second␈α∃way␈α⊗AM␈α∃expands␈α⊗its␈α∃knowledge␈α⊗is␈α∃by
␈↓ ↓H␈↓defining new concepts out of old ones <SLIDE: ∪o' overlay>.
␈↓ ↓H␈↓In␈α!the␈α!LISP␈α!implementation,␈α!that␈α!translates␈α!to␈α!(1)␈α!filling␈α!out␈α!the
␈↓ ↓H␈↓properties␈α∩of␈α∩existing␈α⊃data␈α∩stuctures␈α∩(here␈α⊃some␈α∩examples␈α∩of␈α⊃Compositions
␈↓ ↓H␈↓have␈α⊃been␈α⊃found),␈α⊃and␈α∩sometimes␈α⊃(2)␈α⊃creating␈α⊃entirely␈α⊃new␈α∩data␈α⊃structures
␈↓ ↓H␈↓(Here␈α∪the␈α∀system␈α∪takes␈α∀one␈α∪particular␈α∪example␈α∀of␈α∪Compose␈α∀and␈α∪promotes
␈↓ ↓H␈↓it␈α∃from␈α⊗being␈α∃just␈α⊗one␈α∃entry␈α⊗on␈α∃one␈α∃facet␈α⊗of␈α∃one␈α⊗concept,␈α∃into␈α⊗being␈α∃a
␈↓ ↓H␈↓full-fledged concept module with facets of its own).

␈↓ ↓H␈↓Since␈α∃the␈α∃"fleshing␈α∃out"␈α∃activity␈α∃is␈α∃quite␈α∃expensive,␈α∃there␈α∃must␈α∃be␈α∀some
␈↓ ↓H␈↓"fine␈α∃structure"␈α∃to␈α∃the␈α⊗process␈α∃of␈α∃growing␈α∃a␈α⊗new␈α∃node␈α∃onto␈α∃the␈α⊗tree␈α∃of
␈↓ ↓H␈↓concepts.␈α∀ This␈α∀is␈α∀a␈α∀non-standard␈α∪problem,␈α∀not␈α∀faced␈α∀during␈α∀your␈α∪typical
␈↓ ↓H␈↓heuristic␈α∪search.␈α∩ We␈α∪can't␈α∪permit␈α∩the␈α∪system␈α∪to␈α∩completely␈α∪fill␈α∪out␈α∩each
␈↓ ↓H␈↓new node (concept) it wants to look at.

␈↓ ↓H␈↓The␈α∩solution␈α∩I␈α∩use␈α∩is␈α∩to␈α∪have␈α∩AM␈α∩maintain␈α∩a␈α∩list␈α∩of␈α∩very␈α∪plausible␈α∩tasks,
␈↓ ↓H␈↓and␈α∩repeatedly␈α∩choose␈α⊃the␈α∩most␈α∩promising␈α∩of␈α⊃them␈α∩to␈α∩execute.␈α∩ Each␈α⊃task
␈↓ ↓H␈↓is␈α≠at␈α≠the␈α~level␈α≠of␈α≠fillng␈α~out␈α≠a␈α≠particular␈α~blank␈α≠facet␈α≠of␈α≠a␈α~particular
␈↓ ↓H␈↓concept.␈α→So␈α→a␈α→few␈α_facets␈α→of␈α→some␈α→concept␈α_will␈α→get␈α→filled␈α→out,␈α→and␈α_the
␈↓ ↓H␈↓situation␈α⊂at␈α⊂that␈α⊂moment␈α⊂might␈α⊂then␈α⊂cause␈α⊂AM␈α⊂to␈α⊂shift␈α⊂to␈α⊂filling␈α⊂in␈α⊂some
␈↓ ↓H␈↓facets␈α⊂of␈α⊃an␈α⊂entirely␈α⊃different␈α⊂concept.␈α⊃ When␈α⊂a␈α⊃new␈α⊂concept␈α⊃gets␈α⊂created,
␈↓ ↓H␈↓most of its facets may remain blank for a long period of time.

␈↓ ↓H␈↓The␈α⊃LISP␈α⊃implementation␈α⊃of␈α⊃this␈α⊃idea␈α⊃is␈α⊃to␈α⊃have␈α⊃a␈α⊃global␈α⊃job-list,␈α⊂<SLIDE
␈↓ ↓H␈↓joblist>␈α∨an␈α∨agenda␈α of␈α∨suggested␈α∨activities␈α∨to␈α do,␈α∨an␈α∨ordered␈α list␈α∨of
␈↓ ↓H␈↓candidates␈α⊗for␈α↔the␈α⊗system's␈α↔attention.␈α⊗ Each␈α⊗entry␈α↔consists␈α⊗of␈α↔a␈α⊗specific
␈↓ ↓H␈↓task␈α⊗for␈α⊗the␈α⊗system␈α⊗to␈α⊗work␈α⊗on,␈α⊗a␈α⊗priority␈α⊗value,␈α⊗and␈α⊗a␈α⊗list␈α↔of␈α⊗reasons
␈↓ ↓H␈↓why␈α→this␈α_is␈α→a␈α→reasonable␈α_task␈α→to␈α_do.␈α→ Except␈α→for␈α_this␈α→list␈α→of␈α_symbolic
␈↓ ↓H␈↓reasons,␈α∃this␈α∀is␈α∃the␈α∃same␈α∀as␈α∃the␈α∀agenda␈α∃mechanism␈α∃used␈α∀in␈α∃KRL,␈α∃in␈α∀the
␈↓ ↓H␈↓Dendral Predictor, and in LT.

␈↓ ↓H␈↓At␈α∩some␈α∩moment,␈α⊃there␈α∩might␈α∩be␈α⊃hundreds␈α∩of␈α∩entries␈α⊃on␈α∩the␈α∩agenda;␈α⊃this
␈↓ ↓H␈↓one␈α∪indicates␈α∀that␈α∪the␈α∀system␈α∪spend␈α∪some␈α∀time␈α∪filling␈α∀in␈α∪example␈α∀of␈α∪the
␈↓ ↓H␈↓concept Primes.

␈↓ ↓H␈↓This␈α⊃agenda␈α⊃governs␈α⊃the␈α⊃top-level␈α⊂control␈α⊃structure␈α⊃of␈α⊃the␈α⊃program.␈α⊂ The
␈↓ ↓H␈↓system picks the highest priority job off this list, and executes it.

␈↓ ↓H␈↓The␈α⊃priority␈α⊃rating␈α⊃is␈α⊃a␈α⊃function␈α⊂of␈α⊃the␈α⊃values␈α⊃of␈α⊃these␈α⊃reasons.␈α⊃ They␈α⊂in
␈↓ ↓H␈↓turn␈α∀are␈α∀proposed␈α∀locally,␈α∀by␈α∀the␈α∀heuristic␈α∀rule␈α∀which␈α∀suggested␈α∀this␈α∀job
␈↓ ↓H␈↓in the first place. We'll see this in more detail in a minute.
␈↓ ↓H␈↓In the process of performing a job, 3 kinds of things can happen:
␈↓ ↓H␈↓␈↓ ↓x(1)␈α≥blank␈α≤facets␈α≥may␈α≤get␈α≥filled␈α≤in.␈α≥ In␈α≤the␈α≥case␈α≤of␈α≥this␈α≥task,␈α≤the
␈↓ ↓H␈↓Examples facet of the Primes concept would hopefully get filled in.
␈↓ ↓H␈↓␈↓ ↓x(2)␈α⊃new␈α⊂concepts␈α⊃may␈α⊂get␈α⊃created.␈α⊃ Up␈α⊂here,␈α⊃when␈α⊂this␈α⊃task␈α⊃is␈α⊂executed,
␈↓ ↓H␈↓the resultant gnneralizations will be full-fledged concepts themselves.
␈↓ ↓H␈↓␈↓ ↓x(3)␈α⊃new␈α⊃tasks␈α⊂may␈α⊃get␈α⊃added␈α⊂to␈α⊃the␈α⊃agenda.␈α⊂After␈α⊃filling␈α⊃in␈α⊃examples␈α⊂of
␈↓ ↓H␈↓primes,␈α∃they␈α∀might␈α∃be␈α∃so␈α∀rare␈α∃that␈α∃the␈α∀system␈α∃would␈α∃add␈α∀a␈α∃task␈α∃of␈α∀the
␈↓ ↓H␈↓form "Generalize the concept of Prime numbers".

␈↓ ↓H␈↓<SLIDE:␈α∪3␈α∪actions>␈α∪Each␈α∪of␈α∪these␈α∪activities␈α∪is␈α∪guided␈α∪by␈α∪a␈α∪heuristic␈α∩rule.
␈↓ ↓H␈↓I'd␈α∪like␈α∪to␈α∪take␈α∪a␈α∪minute␈α∪and␈α∪illustrate␈α∪how␈α∪a␈α∪heuristic␈α∪rule␈α∀can␈α∪manage
␈↓ ↓H␈↓each␈α↔of␈α⊗these␈α↔kinds␈α⊗of␈α↔actions.␈α⊗ The␈α↔basic␈α⊗idea␈α↔is␈α⊗that␈α↔a␈α↔heuristic␈α⊗rule
␈↓ ↓H␈↓triggers,␈α⊂because␈α⊂its␈α⊂left␈α⊃side␈α⊂is␈α⊂relevant␈α⊂to␈α⊃the␈α⊂current␈α⊂state␈α⊂of␈α⊃the␈α⊂world.
␈↓ ↓H␈↓The␈α⊃right␈α⊃side␈α⊂of␈α⊃the␈α⊃rule␈α⊃contains␈α⊂a␈α⊃list␈α⊃of␈α⊂actions␈α⊃to␈α⊃perform,␈α⊃and␈α⊂those
␈↓ ↓H␈↓actions are of these three types.


␈↓ ↓H␈↓∧␈↓ α~*** Action 1: Fill in entries of facets ***

␈↓ ↓H␈↓The␈α∪first␈α∪kind␈α∪of␈α∪action␈α∪that␈α∪a␈α∪heuristic␈α∪rule␈α∪can␈α∪direct␈α∪is␈α∪the␈α∪filling␈α∩in
␈↓ ↓H␈↓of specific pieces of information about a particular concept.

␈↓ ↓H␈↓For␈α∩example,␈α∩here's␈α∩a␈α∩rule␈α∩<SLIDE:␈α∩fill␈α∩exs>␈α∩which␈α∩gives␈α∩one␈α∩strategy␈α⊃for
␈↓ ↓H␈↓filling␈α∩in␈α∩examples␈α∪of␈α∩any␈α∩concept␈α∪C.␈α∩ It␈α∩says␈α∪to␈α∩find␈α∩an␈α∪operation␈α∩whose
␈↓ ↓H␈↓range␈α⊂is␈α⊂C,␈α⊂then␈α⊂find␈α⊂examples␈α⊃of␈α⊂the␈α⊂domain␈α⊂of␈α⊂the␈α⊂operation,␈α⊃and␈α⊂simply
␈↓ ↓H␈↓run␈α∩that␈α⊃operation␈α∩and␈α⊃gather␈α∩up␈α⊃all␈α∩the␈α⊃values␈α∩it␈α⊃returns.␈α∩ Those␈α⊃values
␈↓ ↓H␈↓should then be examples of C.

␈↓ ↓H␈↓Here␈α∩is␈α⊃how␈α∩it␈α∩might␈α⊃be␈α∩used␈α∩to␈α⊃fill␈α∩in␈α∩some␈α⊃examples␈α∩for␈α∩Sets,␈α⊃assuming
␈↓ ↓H␈↓a␈α⊂few␈α⊃were␈α⊂already␈α⊃known.␈α⊂It␈α⊃would␈α⊂say␈α⊃to␈α⊂find␈α⊃an␈α⊂operation␈α⊃whose␈α⊂range
␈↓ ↓H␈↓is␈α∃Sets.␈α∃ One␈α∃such␈α⊗known␈α∃operation␈α∃is␈α∃Union.␈α⊗Then␈α∃it␈α∃says␈α∃to␈α⊗run␈α∃that
␈↓ ↓H␈↓operation,␈α⊃and␈α⊃the␈α⊃results␈α⊃will␈α⊃be␈α⊃examples␈α⊃of␈α⊃Sets.␈α⊃Sure␈α⊃enough,␈α⊃if␈α∩we␈α⊃do
␈↓ ↓H␈↓that,␈α≠we␈α≠do␈α≠produce␈α≠lots␈α≠of␈α≠sets.␈α≠ Some␈α≠of␈α≠them␈α≠will␈α≠be␈α≠sets␈α≠which
␈↓ ↓H␈↓weren't known before.

␈↓ ↓H␈↓A␈α⊃similar␈α⊃kind␈α⊃of␈α⊃heuristic␈α⊃rule␈α⊃is␈α⊂used␈α⊃to␈α⊃find␈α⊃conjectures:␈α⊃Here␈α⊃is␈α⊃␈↓&a␈α⊂rule␈↓'%
␈↓ ↓H␈↓␈↓&<SLIDE:␈α_find␈α_conjec>␈α_which␈α_has␈α→AM␈↓)αβ␈α_ask␈α_if␈α_a␈α_known␈α→generalization␈α_of
␈↓ ↓H␈↓concept␈α∩X␈α∩is␈α∩really␈α∪equivalent␈α∩to␈α∩X.␈α∩If␈α∩so,␈α∪this␈α∩gets␈α∩recorded,␈α∩as␈α∪an␈α∩entry
␈↓ ↓H␈↓on␈α⊃the␈α⊃Conjecs␈α⊃facet␈α∩of␈α⊃the␈α⊃concepts␈α⊃involved.␈α∩ This␈α⊃is␈α⊃the␈α⊃only␈α∩way␈α⊃that
␈↓ ↓H␈↓AM␈α∀noticed␈α∀any␈α∀conjectures:␈α∀when␈α∀specifically␈α∀directed␈α∀to␈α∀do␈α∀so␈α∀by␈α∪some
␈↓ ↓H␈↓heuristic rule or other.


␈↓ ↓H␈↓∧␈↓ α%*** Action 2: Creating new concepts ***

␈↓ ↓H␈↓<SLIDE:␈α∃remove␈α∃overlay␈α∃from␈α∃3␈α∃actions>␈α∃We've␈α∃seen␈α∃how␈α∃heuristics␈α∀can
␈↓ ↓H␈↓add entries.  How can they cause new concepts to be created?

␈↓ ↓H␈↓Consider␈α≤this␈α≤heuristic:␈α≠<SLIDE:␈α≤Create-con>.␈α≤It␈α≠says␈α≤to␈α≤look␈α≤for␈α≠the
␈↓ ↓H␈↓inverse image of extreme elements under an interesting operation.

␈↓ ↓H␈↓We␈α⊃already␈α⊃saw␈α⊃how␈α⊃this␈α⊃heuristic␈α⊃can␈α⊃lead␈α⊃to␈α⊃proposing␈α⊃the␈α⊃definition␈α⊂of
␈↓ ↓H␈↓prime numbers, when the operation F is divisors-of.

␈↓ ↓H␈↓The␈α_heuristic␈α_contains␈α_enough␈α_information␈α_to␈α_define␈α_the␈α→new␈α_concept,
␈↓ ↓H␈↓and␈α∃to␈α∀fill␈α∃in␈α∃any␈α∀other␈α∃information␈α∀which␈α∃is␈α∃trivial␈α∀now␈α∃but␈α∃might␈α∀be
␈↓ ↓H␈↓very hard to reconstruct later on (like why the concept was created).


␈↓ ↓H␈↓∧␈↓ αG*** Action 3: Proposing a new job ***

␈↓ ↓H␈↓<SLIDE:␈α∩remove␈α∩overlay␈α∩from␈α∩3␈α∩actions>␈α∩The␈α∩third␈α∩kind␈α∩of␈α∩action␈α∪that␈α∩a
␈↓ ↓H␈↓heuristic␈α⊃rule␈α⊃can␈α⊃initiate␈α⊃is␈α⊃to␈α⊃suggest␈α⊃a␈α⊃new␈α⊃task␈α⊃which␈α⊃then␈α⊃gets␈α⊃added
␈↓ ↓H␈↓to the agenda.  Let's see how this happens.

␈↓ ↓H␈↓Consider␈α↔a␈α_heuristic␈α↔rule,␈α_like␈α↔this␈α_one␈α↔<SLIDE:␈α_New-job>.␈α↔ Somewhere
␈↓ ↓H␈↓among␈α_the␈α_actions␈α_to␈α_take,␈α_listed␈α→here␈α_on␈α_its␈α_right␈α_side,␈α_is␈α→one␈α_which
␈↓ ↓H␈↓explicitly␈α∪calls␈α∪for␈α∪a␈α∪new␈α∪task␈α∪to␈α∪be␈α∪added␈α∪to␈α∪the␈α∪agenda.␈α∪ The␈α∀rule␈α∪says
␈↓ ↓H␈↓that␈α∩if␈α∩a␈α⊃predicate␈α∩appears␈α∩to␈α⊃be␈α∩rarely␈α∩satisfied,␈α⊃Then␈α∩the␈α∩following␈α⊃task
␈↓ ↓H␈↓is␈α∩plausible:␈α∪fill␈α∩in␈α∩some␈α∪generalized␈α∩forms␈α∩of␈α∪that␈α∩predicate.␈α∪ Notice␈α∩that
␈↓ ↓H␈↓the␈α⊗rule␈α⊗also␈α↔specifies␈α⊗an␈α⊗English␈α⊗phrase␈α↔which␈α⊗is␈α⊗the␈α⊗␈↓βreason␈↓␈α↔why␈α⊗the
␈↓ ↓H␈↓task␈α⊃is␈α⊂now␈α⊃plausible,␈α⊂and␈α⊃also␈α⊂supplies␈α⊃a␈α⊂way␈α⊃to␈α⊂rate␈α⊃the␈α⊂overall␈α⊃worth␈α⊂of
␈↓ ↓H␈↓that reason.
␈↓ ↓H␈↓The precise numbers turn out not to be important.

␈↓ ↓H␈↓Let's␈α⊂see␈α⊂how␈α⊂this␈α⊂particular␈α⊂rule␈α⊂might␈α⊂be␈α⊂used␈α⊂to␈α⊂suggest␈α⊂a␈α⊂new␈α⊃task␈α⊂for
␈↓ ↓H␈↓the␈α≠agenda.␈α≤ Say␈α≠AM␈α≤recently␈α≠worked␈α≠on␈α≤finding␈α≠things␈α≤which␈α≠were
␈↓ ↓H␈↓Equal␈α⊗to␈α⊗each␈α⊗other,␈α∃but␈α⊗that␈α⊗very␈α⊗few␈α∃random␈α⊗pairs␈α⊗of␈α⊗objects␈α∃turned
␈↓ ↓H␈↓out␈α∃to␈α⊗be␈α∃equal.␈α⊗ Then␈α∃this␈α∃rule␈α⊗might␈α∃trigger,␈α⊗and␈α∃would␈α⊗propose␈α∃this
␈↓ ↓H␈↓task:␈α∩<SLIDE:␈α∩task1>.␈α⊃ If␈α∩this␈α∩task␈α⊃wasn't␈α∩already␈α∩on␈α⊃the␈α∩agenda,␈α∩it␈α⊃would
␈↓ ↓H␈↓get inserted there.

␈↓ ↓H␈↓But␈α∪suppose␈α∩␈↓βthis␈↓␈α∪<SLIDE:␈α∩partial␈α∪agenda>␈α∩was␈α∪the␈α∩state␈α∪of␈α∩the␈α∪agenda␈α∩at
␈↓ ↓H␈↓the␈α∪time.␈α∪ We␈α∩see␈α∪right␈α∪here␈α∪that␈α∩the␈α∪task␈α∪is␈α∩already␈α∪on␈α∪the␈α∪agenda,␈α∩but
␈↓ ↓H␈↓for␈α⊃a␈α⊃different␈α⊃reason.␈α∩ In␈α⊃such␈α⊃a␈α⊃case,␈α∩the␈α⊃new␈α⊃reason␈α⊃would␈α∩be␈α⊃recorded,
␈↓ ↓H␈↓and␈α⊂the␈α⊂priority␈α⊃value␈α⊂of␈α⊂the␈α⊂task␈α⊃would␈α⊂get␈α⊂boosted.␈α⊂ <Full␈α⊃agenda>␈α⊂Here
␈↓ ↓H␈↓is␈α⊃the␈α∩way␈α⊃that␈α∩task␈α⊃would␈α∩look.␈α⊃The␈α∩new␈α⊃reason␈α∩has␈α⊃raised␈α∩it␈α⊃to␈α∩the␈α⊃top
␈↓ ↓H␈↓of the agenda.

␈↓ ↓H␈↓This␈α~turned␈α~out␈α~to␈α≠be␈α~a␈α~surprisingly␈α~important␈α~mechanism.␈α≠ When␈α~a
␈↓ ↓H␈↓newly-proposed␈α~job␈α≠turns␈α~out␈α≠to␈α~be␈α≠␈↓βalready␈↓␈α~on␈α≠the␈α~agenda,␈α≠then␈α~its
␈↓ ↓H␈↓reasons␈α∀are␈α∀examined.␈α∀ If␈α∀the␈α∀job␈α∀is␈α∀being␈α∀proposed␈α∀for␈α∀pretty␈α∃much␈α∀the
␈↓ ↓H␈↓same␈α≥reasons␈α≥as␈α≥before,␈α≥then␈α≥its␈α≥priority␈α≥rating␈α≥is␈α≥incremented␈α≤only
␈↓ ↓H␈↓slightly.␈α↔ But␈α⊗if␈α↔it␈α⊗is␈α↔being␈α⊗proposed␈α↔for␈α⊗a␈α↔new␈α⊗reason,␈α↔as␈α⊗in␈α↔this␈α⊗case,
␈↓ ↓H␈↓then␈α∨its␈α priority␈α∨is␈α increased␈α∨tremendously.␈α  ␈↓βThis␈α∨is␈α the␈α∨practical
␈↓ ↓H␈↓βimportance of maintaining reasons for each job.␈↓

␈↓ ↓H␈↓AM␈α⊂can␈α⊃compare␈α⊂two␈α⊃reasons␈α⊂to␈α⊂see␈α⊃if␈α⊂they␈α⊃are␈α⊂equal,␈α⊂and␈α⊃that's␈α⊂all␈α⊃it␈α⊂has
␈↓ ↓H␈↓to␈α∩do␈α⊃manage␈α∩this␈α⊃scheme.␈α∩ Other␈α⊃than␈α∩that,␈α⊃there␈α∩is␈α⊃very␈α∩little␈α∩that␈α⊃AM
␈↓ ↓H␈↓can␈α⊂do␈α⊂to␈α⊂a␈α⊂reason,␈α⊂except␈α⊂type␈α⊂it␈α⊂out␈α⊂to␈α⊂the␈α⊂user.␈α⊂ Each␈α⊂reason␈α⊂is␈α⊂actually
␈↓ ↓H␈↓just␈α∩a␈α⊃token,␈α∩but␈α∩as␈α⊃you␈α∩can␈α⊃imagine␈α∩they're␈α∩quite␈α⊃useful␈α∩for␈α∩keeping␈α⊃the
␈↓ ↓H␈↓human user aware of what the system is doing and why.

␈↓ ↓H␈↓In␈α↔general,␈α↔the␈α↔priority␈α↔value␈α↔of␈α⊗a␈α↔task␈α↔on␈α↔the␈α↔agenda␈α↔depends␈α↔on␈α⊗the
␈↓ ↓H␈↓ratings␈α⊗of␈α∃all␈α⊗the␈α⊗supporting␈α∃reasons.␈α⊗ The␈α⊗reasons␈α∃are␈α⊗rated␈α⊗locally,␈α∃by
␈↓ ↓H␈↓the␈α∪same␈α∪heuristic␈α∪rules␈α∪which␈α∪proposed␈α∪them.␈α∪ A␈α∪single␈α∪␈↓βglobal␈↓␈α∪formula
␈↓ ↓H␈↓then␈α_looks␈α_at␈α_the␈α↔job␈α_and␈α_the␈α_reasons,␈α↔and␈α_assigns␈α_it␈α_a␈α_single␈α↔priority
␈↓ ↓H␈↓rating␈α⊃from␈α⊃0␈α⊃to␈α⊃1000.␈α⊃ <SLIDE:␈α⊃Formula>␈α⊃That␈α⊃formula␈α⊃was␈α∩just␈α⊃guessed
␈↓ ↓H␈↓at␈α∩as␈α∩a␈α∪first␈α∩pass,␈α∩and␈α∩it's␈α∪worked␈α∩well␈α∩enough␈α∩not␈α∪to␈α∩tamper␈α∩with␈α∪it.␈α∩ It
␈↓ ↓H␈↓involves␈α↔squareroots␈α_and␈α↔dot-products␈α_of␈α↔vectors␈α_and␈α↔isn't␈α_really␈α↔worth
␈↓ ↓H␈↓delving into.
␈↓ ↓H␈↓Each␈α∪concept,␈α∩each␈α∪facet,␈α∩each␈α∪reason␈α∪supporting␈α∩a␈α∪job␈α∩on␈α∪the␈α∪agenda␈α∩has
␈↓ ↓H␈↓a␈α∪number␈α∪attached␈α∪to␈α∪it.␈α∪ This␈α∪formula␈α∪uses␈α∪those␈α∪numbers␈α∪to␈α∀attach␈α∪an
␈↓ ↓H␈↓interestingness␈α_factor,␈α→a␈α_priority␈α→rating,␈α_a␈α→number,␈α_to␈α→each␈α_job␈α→on␈α_the
␈↓ ↓H␈↓agenda.

␈↓ ↓H␈↓<SLIDE:␈α∀Put␈α∀back␈α∀full␈α∀agenda>␈α∀AM␈α∪looks␈α∀over␈α∀the␈α∀job-list,␈α∀picks␈α∀the␈α∪job
␈↓ ↓H␈↓with␈α∩the␈α⊃highest␈α∩priority,␈α∩and␈α⊃tries␈α∩to␈α⊃execute␈α∩it.␈α∩ The␈α⊃flow␈α∩of␈α∩control␈α⊃is
␈↓ ↓H␈↓so distributed that it's almost unpredictable.

␈↓ ↓H␈↓That␈α∩priority␈α∩rating␈α∩of␈α∩the␈α∩chosen␈α∩task␈α∩then␈α∩dictates␈α∩how␈α∩to␈α∩manage␈α⊃the
␈↓ ↓H␈↓system's␈α⊃resources:␈α⊃it␈α⊃indicates␈α⊃the␈α⊃number␈α⊃of␈α⊃cpu␈α⊃seconds␈α⊃to␈α⊃spend␈α⊃before
␈↓ ↓H␈↓giving␈α⊗up␈α⊗and␈α⊗trying␈α⊗the␈α⊗next␈α⊗task␈α⊗on␈α⊗the␈α⊗agenda,␈α⊗and␈α⊗it␈α⊗specifies␈α∃the
␈↓ ↓H␈↓number␈α∀of␈α∪list␈α∀cells␈α∪that␈α∀the␈α∪task␈α∀is␈α∪permitted␈α∀to␈α∪use␈α∀up␈α∪before␈α∀it␈α∪must
␈↓ ↓H␈↓quit,␈α⊗and␈α⊗so␈α⊗on.␈α⊗ (This␈α↔is␈α⊗like␈α⊗Danny␈α⊗Bobrow's␈α⊗idea␈α↔of␈α⊗Resource-limited
␈↓ ↓H␈↓computation.)

␈↓ ↓H␈↓We␈α⊃are␈α⊃assuming␈α⊂that␈α⊃there␈α⊃is␈α⊃a␈α⊂self-consistent␈α⊃set␈α⊃of␈α⊃computational␈α⊂rules
␈↓ ↓H␈↓for␈α∪manipulating␈α∪the␈α∩estimated␈α∪worth␈α∪of␈α∪all␈α∩the␈α∪concepts␈α∪and␈α∪jobs.␈α∩ This
␈↓ ↓H␈↓is␈α∀at␈α∀best␈α∀a␈α∀first-order␈α∀theory␈α∀of␈α∀interestingness.␈α∀ Comparing␈α∀the␈α∪worths
␈↓ ↓H␈↓of "equality" and "primes" is really comparing apples to oranges.

␈↓ ↓H␈↓I␈α∩think␈α∪that␈α∩this␈α∩scheme␈α∪works␈α∩adequately␈α∩here␈α∪only␈α∩because␈α∪neither␈α∩the
␈↓ ↓H␈↓heuristics␈α_nor␈α_the␈α→jobs␈α_are␈α_strongly␈α_coupled;␈α→they␈α_are␈α_all␈α→pretty␈α_much
␈↓ ↓H␈↓independent of each other.

␈↓ ↓H␈↓For␈α⊗example,␈α⊗it␈α⊗doesn't␈α⊗really␈α⊗matter␈α⊗which␈α⊗order␈α⊗the␈α⊗top␈α⊗few␈α⊗jobs␈α∃get
␈↓ ↓H␈↓executed␈α≥in;␈α≥no␈α≥one␈α≥cares␈α≥whether␈α≥we␈α≥look␈α≥for␈α≥primes␈α≥right␈α≤before
␈↓ ↓H␈↓generalizing␈α⊂equality,␈α⊂or␈α⊂right␈α⊂afterwards.␈α⊂ What␈α⊂␈↓βdoes␈↓␈α⊂matter␈α⊂is␈α⊂that␈α⊂they
␈↓ ↓H␈↓are␈α⊂near␈α⊂the␈α⊂top,␈α⊂not␈α⊂near␈α⊂the␈α⊂bottom,␈α⊂of␈α⊂the␈α⊂agenda.␈α⊂ That's␈α⊂why␈α⊃a␈α⊂crude
␈↓ ↓H␈↓kind of evaluation function is all you need.
␈↓ ↓H␈↓∧␈↓ αX*** Locating Relevant Heuristics ***

␈↓ ↓H␈↓<SLIDE:␈α⊂where␈α⊃we␈α⊂are>␈α⊂We've␈α⊃seen␈α⊂3␈α⊃kinds␈α⊂of␈α⊂actions␈α⊃a␈α⊂heuristic␈α⊃rule␈α⊂can
␈↓ ↓H␈↓take.␈α∀ Furthermore,␈α∀we've␈α∀seen␈α∀how␈α∀the␈α∀agenda␈α∀mechanism␈α∀always␈α∀keeps
␈↓ ↓H␈↓AM␈α∩working␈α⊃on␈α∩a␈α∩plausible␈α⊃task.␈α∩ But␈α⊃we're␈α∩not␈α∩really␈α⊃in␈α∩business␈α∩until␈α⊃I
␈↓ ↓H␈↓explain␈α≡one␈α≡more␈α≡thing.␈α≡Once␈α≡a␈α≡task␈α≡is␈α≡chosen,␈α≡how␈α≡are␈α≡the␈α≥right
␈↓ ↓H␈↓heuristic␈α∃rules␈α∃located,␈α∃the␈α∀ones␈α∃which␈α∃will␈α∃be␈α∀relevant␈α∃to␈α∃the␈α∃task␈α∀just
␈↓ ↓H␈↓chosen?

␈↓ ↓H␈↓There␈α∩are␈α∩about␈α∩250␈α∩heuristic␈α∩rules␈α⊃altogether␈α∩in␈α∩AM;␈α∩that's␈α∩too␈α∩many␈α⊃to
␈↓ ↓H␈↓simply evaluate the left sides of ␈↓βall␈↓ of them.

␈↓ ↓H␈↓After explaining this, the overall behavior of AM will be described.

␈↓ ↓H␈↓Like␈α∪a␈α∀human␈α∪mathematician,␈α∀AM␈α∪should␈α∀only␈α∪access␈α∀a␈α∪strategy␈α∀in␈α∪those
␈↓ ↓H␈↓situations␈α∩where␈α∪it␈α∩could␈α∪plausibly␈α∩be␈α∪used.␈α∩ By␈α∪looking␈α∩at␈α∪the␈α∩heuristics,
␈↓ ↓H␈↓it␈α≠seemed␈α≠that␈α~each␈α≠one␈α≠has␈α~a␈α≠very␈α≠clear␈α~domain␈α≠of␈α≠applicability,␈α~of
␈↓ ↓H␈↓relevance.

␈↓ ↓H␈↓For␈α(example,␈α(<SLIDE:␈α'compose␈α(heur>␈α(one␈α'heuristic␈α(says␈α(that␈α'a
␈↓ ↓H␈↓composition␈α≡FoG␈α≡is␈α≡interesting␈α≡if␈α≡it␈α≡preserves␈α≡some␈α≡of␈α∨the␈α≡unusual
␈↓ ↓H␈↓properties␈α⊗of␈α∃F␈α⊗and␈α∃G.␈α⊗This␈α⊗is␈α∃a␈α⊗useful␈α∃thing␈α⊗to␈α∃know␈α⊗when␈α⊗trying␈α∃to
␈↓ ↓H␈↓judge␈α∃the␈α∀interestingness␈α∃of␈α∀a␈α∃composition,␈α∀but␈α∃it␈α∀won't␈α∃help␈α∀you␈α∃if␈α∀the
␈↓ ↓H␈↓current␈α⊃task␈α⊃is␈α⊃to␈α⊃check␈α⊃examples␈α∩of␈α⊃Lists.␈α⊃ We␈α⊃may␈α⊃as␈α⊃well␈α∩associate␈α⊃this
␈↓ ↓H␈↓heuristic␈α≠with␈α≤the␈α≠concept␈α≤Compose.␈α≠ We'll␈α≤record␈α≠this␈α≤heuristic␈α≠rule
␈↓ ↓H␈↓inside␈α≥the␈α≥Interest␈α≤facet␈α≥of␈α≥the␈α≤Compose␈α≥concpet.␈α≥ That's␈α≥what␈α≤this
␈↓ ↓H␈↓orange line means.

␈↓ ↓H␈↓Other␈α⊂heuristics␈α⊂are␈α⊃a␈α⊂little␈α⊂bit␈α⊂more␈α⊃general,␈α⊂they␈α⊂relate␈α⊂to␈α⊃any␈α⊂operation
␈↓ ↓H␈↓<SLIDE:␈α~Operation␈α~heurs␈α~overlay>,␈α~so␈α~they␈α~would␈α~be␈α~tacked␈α≠onto␈α~the
␈↓ ↓H␈↓concept␈α∩Operation.␈α∪ This␈α∩one␈α∩is␈α∪an␈α∩entry␈α∩on␈α∪the␈α∩Fillin␈α∩facet␈α∪there.␈α∩ This
␈↓ ↓H␈↓one␈α∀is␈α∃on␈α∀the␈α∀Interest␈α∃facet␈α∀of␈α∃the␈α∀concept␈α∀Operation.␈α∃ It␈α∀says␈α∃that␈α∀any
␈↓ ↓H␈↓operation is interesting if its domain and range are identical.

␈↓ ↓H␈↓As␈α_AM␈α_was␈α↔written,␈α_I␈α_gathered␈α↔heuristics␈α_by␈α_introspection,␈α_and␈α↔tacked
␈↓ ↓H␈↓them␈α∃onto␈α∀the␈α∃concept␈α∃I␈α∀thought␈α∃they␈α∃were␈α∀most␈α∃relevant␈α∃to.␈α∀ <SLIDE:
␈↓ ↓H␈↓Genl␈α~branch>␈α≠The␈α~more␈α≠general␈α~the␈α≠heuristic␈α~rule,␈α≠the␈α~weaker␈α≠it␈α~is,
␈↓ ↓H␈↓typically.␈α≥ So␈α≤each␈α≥individual␈α≥heuristic␈α≤rule␈α≥is␈α≤tacked␈α≥onto␈α≥the␈α≤most
␈↓ ↓H␈↓general␈α_kind␈α_of␈α_entity␈α_to␈α→which␈α_it␈α_applies.␈α_ Whenever␈α_I␈α_thought␈α→of␈α_a
␈↓ ↓H␈↓heuristic,␈α⊃I␈α⊃pushed␈α⊃it␈α⊃as␈α⊃high␈α∩as␈α⊃possible␈α⊃in␈α⊃this␈α⊃tree␈α⊃of␈α⊃concepts.␈α∩ In␈α⊃fact,
␈↓ ↓H␈↓half of AM's heuristics are tacked onto these two topmost concepts.

␈↓ ↓H␈↓When␈α~dealing␈α→with␈α~any␈α→concept␈α~X,␈α→all␈α~the␈α→heuristics␈α~attached␈α~to␈α→any
␈↓ ↓H␈↓generalization␈α↔of␈α⊗X,␈α↔that␈α⊗is,␈α↔any␈α↔higher␈α⊗node,␈α↔are␈α⊗also␈α↔considered␈α↔to␈α⊗be
␈↓ ↓H␈↓relevant to X.

␈↓ ↓H␈↓Say␈α⊗AM␈α⊗wants␈α⊗to␈α⊗see␈α⊗how␈α⊗interesting␈α⊗this␈α⊗particular␈α⊗concept␈α⊗is.␈α∃ Then
␈↓ ↓H␈↓AM␈α∪could␈α∪use␈α∀evaluation␈α∪heuristics␈α∪gathered␈α∀from␈α∪any␈α∪of␈α∀these␈α∪concepts.
␈↓ ↓H␈↓Any reason why an operation might be interesting will also be valid here.

␈↓ ↓H␈↓The␈α&system␈α&would␈α&ripple␈α%upward,␈α&in␈α&the␈α&direction␈α&of␈α%increasing
␈↓ ↓H␈↓generality,␈α⊃collecting␈α⊃heuristics␈α⊃as␈α⊃it␈α⊃went.␈α⊃ They␈α⊃would␈α⊃be␈α⊃evaluated,␈α⊃and
␈↓ ↓H␈↓their␈α(weighted␈α(average␈α(result␈α'would␈α(be␈α(taken␈α(as␈α(the␈α'estimated
␈↓ ↓H␈↓interestingness of this concept.

␈↓ ↓H␈↓In␈α_fact,␈α_<OVERLAY>␈α↔␈↓βthese␈↓␈α_features␈α_are␈α↔satisfied␈α_by␈α_this␈α_operation,␈α↔so
␈↓ ↓H␈↓they␈α⊗combine␈α⊗to␈α⊗indicate␈α⊗it␈α⊗is␈α⊗a␈α⊗mildly␈α⊗promising␈α⊗concept.␈α⊗ Notice␈α⊗that
␈↓ ↓H␈↓the␈α⊗further␈α⊗upward␈α↔we␈α⊗go,␈α⊗the␈α⊗less␈α↔specific␈α⊗the␈α⊗heuristics␈α⊗get,␈α↔and␈α⊗the
␈↓ ↓H␈↓less useful they are.

␈↓ ↓H␈↓If␈α∃AM␈α∃wanted␈α⊗to␈α∃fill␈α∃in␈α∃generalizations␈α⊗of␈α∃this␈α∃concept,␈α∃it␈α⊗would␈α∃ripple
␈↓ ↓H␈↓upwards␈α→again,␈α→gathering␈α→heuristics␈α→for␈α→filling␈α→in␈α~generalizations.␈α→ Any
␈↓ ↓H␈↓technique for generalizing an operation would certainly work here too.

␈↓ ↓H␈↓The␈α⊂same␈α⊂kind␈α⊃of␈α⊂rippling␈α⊂would␈α⊂collect␈α⊃relevant␈α⊂heuristics␈α⊂if␈α⊃AM␈α⊂wanted
␈↓ ↓H␈↓to␈α~fill␈α~in␈α~examples␈α~of␈α~a␈α→certain␈α~concept,␈α~or␈α~check␈α~the␈α~entries␈α~of␈α→the
␈↓ ↓H␈↓domain/range facet of this concept.

␈↓ ↓H␈↓Since␈α≠the␈α≠heuristics␈α≠attached␈α≠to␈α≠these␈α≠(specific)␈α≠concepts␈α≤are␈α≠typically
␈↓ ↓H␈↓more␈α∩powerful␈α∪than␈α∩the␈α∪heuristics␈α∩way␈α∪up␈α∩here,␈α∪AM␈α∩always␈α∪executes␈α∩the
␈↓ ↓H␈↓heuristics␈α∃in␈α∃the␈α∃order␈α∃they␈α∃are␈α∃collected.␈α∃ No␈α∃sophisticated␈α∃reasoning␈α∃is
␈↓ ↓H␈↓done to order them in some way prior to obeying them.

␈↓ ↓H␈↓This␈α→works␈α→[in␈α→practice]␈α~because␈α→the␈α→heuristics␈α→rarely␈α→couple;␈α~at␈α→most,
␈↓ ↓H␈↓they␈α⊂superimpose␈α⊂their␈α⊂effects;␈α⊂in␈α⊂either␈α⊂case␈α⊂the␈α⊂␈↓βorder␈↓␈α⊂in␈α⊂which␈α⊂they␈α⊂are
␈↓ ↓H␈↓executed␈α is␈α not␈α too␈α∨important.␈α  They␈α don't␈α interact␈α much.␈α∨ Simple
␈↓ ↓H␈↓schemes␈α⊃like␈α⊃this␈α⊂only␈α⊃work␈α⊃when␈α⊂the␈α⊃components␈α⊃of␈α⊂the␈α⊃system␈α⊃are␈α⊂very
␈↓ ↓H␈↓independent␈α∪of␈α∪each␈α∪other,␈α∪when␈α∩the␈α∪problem␈α∪was␈α∪easily␈α∪decomposable␈α∩or
␈↓ ↓H␈↓factorable␈α↔to␈α↔begin␈α↔with.␈α⊗ Tight␈α↔interdependence␈α↔would␈α↔require␈α↔a␈α⊗more
␈↓ ↓H␈↓sophisticated approach.


␈↓ ↓H␈↓∧␈↓ βT*** Cardinality Example ***

␈↓ ↓H␈↓Let's␈α∃finally␈α∃take␈α∃a␈α∃look␈α∃at␈α∃the␈α∃program␈α∃in␈α∃action.␈α∃ <SLIDE:␈α∃Cardinality
␈↓ ↓H␈↓slide>

␈↓ ↓H␈↓This␈α∩is␈α∩a␈α∩condensed␈α∪excerpt␈α∩from␈α∩a␈α∩session␈α∪with␈α∩AM,␈α∩as␈α∩it␈α∪first␈α∩discovers
␈↓ ↓H␈↓concept␈α↔of␈α↔numbers.␈α↔ AM␈α↔already␈α↔knew␈α↔about␈α↔sets,␈α↔bags,␈α↔lists,␈α_etc.,␈α↔and
␈↓ ↓H␈↓equality␈α→of␈α→2␈α→sets.␈α→ A␈α~bag␈α→is␈α→a␈α→multiset,␈α→an␈α→unordered␈α~structure␈α→with
␈↓ ↓H␈↓repeated elements allowed.

␈↓ ↓H␈↓I'll␈α↔read␈α↔through␈α_it␈α↔once␈α↔to␈α↔show␈α_you␈α↔that␈α↔AM␈α↔really␈α_performs␈α↔magic,
␈↓ ↓H␈↓then␈α↔we'll␈α↔peek␈α↔behind␈α↔the␈α↔curtains␈α↔to␈α↔see␈α↔that␈α↔AM␈α↔is␈α↔really␈α↔no␈α↔more
␈↓ ↓H␈↓than␈α(the␈α(straight-forward␈α)data-structure␈α(expander␈α(that␈α)I've␈α(been
␈↓ ↓H␈↓describing.

␈↓ ↓H␈↓::␈α∩After␈α∩testing␈α∩random␈α∩structures␈α∪to␈α∩see␈α∩if␈α∩they␈α∩are␈α∩equal,␈α∪AM␈α∩concludes
␈↓ ↓H␈↓that␈α→it␈α→would␈α_be␈α→worthwhile␈α→to␈α_generalize␈α→the␈α→predicate␈α→EQUAL.␈α_The
␈↓ ↓H␈↓user␈α→asks␈α~"why"␈α→and␈α~AM␈α→mentions␈α→that␈α~few␈α→randomly␈α~selected␈α→objects
␈↓ ↓H␈↓turned out to be equal to each other.

␈↓ ↓H␈↓::␈α⊃Two␈α⊃generalizations␈α∩of␈α⊃EQUAL␈α⊃are␈α⊃constructed.␈α∩ The␈α⊃first␈α⊃turns␈α∩out␈α⊃to
␈↓ ↓H␈↓be␈α∃the␈α∃relationship␈α∃"has␈α∃the␈α∃same␈α∀length␈α∃as",␈α∃and␈α∃the␈α∃second␈α∃is␈α∃"has␈α∀the
␈↓ ↓H␈↓same first element as".

␈↓ ↓H␈↓::␈α≡After␈α∨testing␈α≡random␈α∨structures␈α≡to␈α∨see␈α≡if␈α∨they␈α≡have␈α∨the␈α≡SAME-
␈↓ ↓H␈↓LENGTH,␈α∪AM␈α∪decides␈α∪to␈α∪try␈α∀to␈α∪find␈α∪a␈α∪canonical␈α∪form␈α∪for␈α∀all␈α∪structures.
␈↓ ↓H␈↓This␈α↔would␈α↔mean␈α↔that␈α↔two␈α↔structures␈α↔were␈α↔of␈α↔the␈α↔same␈α↔length␈α↔if␈α↔and
␈↓ ↓H␈↓only if their canonical forms were identically equal.

␈↓ ↓H␈↓::␈α∀AM␈α∪comes␈α∀up␈α∀with␈α∪the␈α∀awkward␈α∪idea␈α∀of␈α∀making␈α∪this␈α∀conversion␈α∀by␈α∪(i)
␈↓ ↓H␈↓converting␈α⊗each␈α⊗structure␈α⊗to␈α⊗a␈α⊗BAG,␈α⊗and␈α⊗(ii)␈α⊗replacing␈α⊗each␈α↔element␈α⊗by
␈↓ ↓H␈↓the␈α⊃constant␈α⊃T.␈α⊃ Thus␈α⊂the␈α⊃canonical␈α⊃form␈α⊃of␈α⊂our␈α⊃alphabet␈α⊃would␈α⊃be␈α⊃a␈α⊂bag
␈↓ ↓H␈↓26␈α!T's.␈α! AM␈α!decides␈α!to␈α!restrict␈α!various␈α!operations␈α!like␈α"Union␈α!and
␈↓ ↓H␈↓Intersection to such canonical structures, and see what happens.

␈↓ ↓H␈↓The␈α∨user␈α∨tells␈α AM␈α∨that␈α∨such␈α canonical␈α∨structures␈α∨should␈α be␈α∨called
␈↓ ↓H␈↓NUMBERS.␈α⊗Those␈α⊗restricted␈α⊗bag-operations␈α∃will␈α⊗turn␈α⊗into␈α⊗what␈α⊗we␈α∃call
␈↓ ↓H␈↓arithmetic.

␈↓ ↓H␈↓Well, now that you've seen what AM says, let's see how it does it.


␈↓ ↓H␈↓∧␈↓ αH*** Finding Examples of Equality ***

␈↓ ↓H␈↓Recall␈α∃that␈α∀Initially,␈α∃I␈α∀supplied␈α∃by␈α∀hand␈α∃a␈α∀definition␈α∃or␈α∀two␈α∃for␈α∃each␈α∀of
␈↓ ↓H␈↓the␈α≠115␈α≠concepts␈α≠AM␈α≠started␈α≠with.␈α≠Each␈α≠such␈α≠definition␈α≠was␈α≤a␈α≠LISP
␈↓ ↓H␈↓predicate.␈α∀I␈α∀also␈α∀gave␈α∀each␈α∃concept␈α∀a␈α∀rough␈α∀numeric␈α∀Worth␈α∃value.␈α∀ Each
␈↓ ↓H␈↓concept␈α∩which␈α∪was␈α∩an␈α∩operation␈α∪also␈α∩received␈α∩an␈α∪algorithm␈α∩or␈α∩two,␈α∪and␈α∩a
␈↓ ↓H␈↓statement␈α⊗of␈α∃domain/range.␈α⊗ Finally,␈α∃I␈α⊗introspected␈α∃and␈α⊗collected␈α⊗all␈α∃the
␈↓ ↓H␈↓general␈α⊃heuristics␈α⊃I␈α⊃could,␈α∩and␈α⊃tacked␈α⊃them␈α⊃onto␈α⊃the␈α∩appropriate␈α⊃concepts.
␈↓ ↓H␈↓Most␈α∃of␈α∃these␈α∃heuristics␈α⊗applied␈α∃to␈α∃the␈α∃general␈α∃concepts,␈α⊗like␈α∃Operation.
␈↓ ↓H␈↓Very few applied to the specific concepts, like Equality or Reverse.

␈↓ ↓H␈↓Equality␈α⊃was␈α⊃one␈α⊃of␈α⊃the␈α⊃initial␈α⊂base␈α⊃concepts.␈α⊃It␈α⊃originally␈α⊃had␈α⊃a␈α⊃very␈α⊂low
␈↓ ↓H␈↓Worth␈α∪rating.␈α∪ AM␈α∪began␈α∪to␈α∀notice,␈α∪e.g.,␈α∪sets␈α∪all␈α∪of␈α∪whose␈α∀members␈α∪were
␈↓ ↓H␈↓equal,␈α≠and␈α~gradually␈α≠boosted␈α≠Equality's␈α~Worth␈α≠rating.␈α≠ When␈α~Equality
␈↓ ↓H␈↓became␈α⊂valuable␈α⊂enough,␈α⊂a␈α⊂general␈α⊂heuristic␈α⊂rule␈α⊂said␈α⊂to␈α⊂consider␈α⊃filling␈α⊂in
␈↓ ↓H␈↓some␈α∀examples␈α∀of␈α∪it.␈α∀ This␈α∀was␈α∪added␈α∀to␈α∀the␈α∪agenda,␈α∀and␈α∀eventually␈α∪AM
␈↓ ↓H␈↓got around to that task.

␈↓ ↓H␈↓How␈α∀in␈α∃fact␈α∀did␈α∃AM␈α∀find␈α∀examples␈α∃of␈α∀objects␈α∃which␈α∀were␈α∃EQUAL,␈α∀once
␈↓ ↓H␈↓that␈α⊃job␈α⊃was␈α⊃chosen?␈α⊃ The␈α⊃job␈α⊃said␈α⊃to␈α⊃fill␈α⊃in␈α⊃the␈α⊃EXAMPLES␈α⊃facet␈α∩of␈α⊃the
␈↓ ↓H␈↓EQUAL␈α⊃concept.␈α⊂ To␈α⊃do␈α⊂that,␈α⊃AM␈α⊂gathered␈α⊃heuristics␈α⊂by␈α⊃rippling␈α⊂outward
␈↓ ↓H␈↓<SLIDE: Genls(Equality)> from Equal.

␈↓ ↓H␈↓There␈α∀were␈α∀no␈α∃heuristics␈α∀located␈α∀here,␈α∀but␈α∃some␈α∀were␈α∀found␈α∀on␈α∃each␈α∀of
␈↓ ↓H␈↓the higher, more general concepts.
␈↓ ↓H␈↓One␈α of␈α these␈α∨(Any-concept)␈α rules␈α of␈α∨thumb␈α said␈α to␈α instantiate␈α∨the
␈↓ ↓H␈↓definition␈α~of␈α≠the␈α~predicate␈α≠EQUAL.␈α~ Three␈α≠definitions␈α~of␈α≠Equal␈α~were
␈↓ ↓H␈↓supplied␈α→by␈α→me␈α~initially.␈α→ Here␈α→is␈α~one␈α→recursive␈α→definition␈α~of␈α→EQUAL:
␈↓ ↓H␈↓<defn␈α#of␈α#EQUAL␈α"slide>.␈α# AM␈α#possesses␈α"several␈α#syntactic␈α#rules␈α"for
␈↓ ↓H␈↓instantiating LISP predicates, which is all that this definition really is.

␈↓ ↓H␈↓An␈α⊂easy␈α⊃way␈α⊂to␈α⊃instantiate␈α⊂this,␈α⊃to␈α⊂satisfy␈α⊃this,␈α⊂would␈α⊃be␈α⊂simply␈α⊃to␈α⊂satisfy
␈↓ ↓H␈↓this␈α∀base␈α∀step.␈α∀ To␈α∀do␈α∀that,␈α∀the␈α∀two␈α∀arguments␈α∀must␈α∀be␈α∀identical␈α∀atoms.
␈↓ ↓H␈↓So␈α_some␈α_examples␈α↔like␈α_NIL=NIL,␈α_T=T,␈α↔are␈α_produced.␈α_ Also,␈α_AM␈α↔inverts
␈↓ ↓H␈↓this␈α⊂recursive␈α⊂step,␈α⊂by␈α⊂plugging␈α⊂in␈α⊂here␈α⊂objects␈α⊂already␈α⊂known␈α⊂to␈α⊂be␈α⊂equal.
␈↓ ↓H␈↓But␈α⊗AM␈α⊗just␈α∃found␈α⊗that␈α⊗NIL=NIL.␈α∃ So␈α⊗if␈α⊗the␈α∃CARs␈α⊗of␈α⊗2␈α⊗structures␈α∃are
␈↓ ↓H␈↓both␈α∩T,␈α∩and␈α∩their␈α∩CDRs␈α∩are␈α∩both␈α∩NIL,␈α∩then␈α∩the␈α∩structures␈α∩will␈α∩be␈α∩Equal.
␈↓ ↓H␈↓So an example like (List T) = (List T) is generated.

␈↓ ↓H␈↓An␈α⊂entirely␈α⊃different␈α⊂approach␈α⊃is␈α⊂suggested␈α⊃by␈α⊂a␈α⊃heuristic␈α⊂tacked␈α⊃onto␈α⊂the
␈↓ ↓H␈↓Activity␈α∀concept.␈α∀ It␈α∪says␈α∀that,␈α∀when␈α∀filling␈α∪in␈α∀examples␈α∀of␈α∀any␈α∪activity,
␈↓ ↓H␈↓including␈α_a␈α_predicate,␈α_we␈α_can␈α_randomly␈α_instantiate␈α_the␈α_arguments,␈α_run
␈↓ ↓H␈↓the␈α⊗concept,␈α⊗and␈α∃observe␈α⊗the␈α⊗results.␈α∃ So␈α⊗AM␈α⊗accesses␈α⊗the␈α∃domain/range
␈↓ ↓H␈↓facet␈α∩of␈α∪EQUAL,␈α∩sees␈α∪there␈α∩the␈α∪entry:␈α∩a␈α∪pair␈α∩of␈α∪objects␈α∩gets␈α∪mapped␈α∩into
␈↓ ↓H␈↓True/false.␈α∃So␈α∃AM␈α∃accesses␈α∃the␈α∀Examplls␈α∃facet␈α∃of␈α∃Object,␈α∃and␈α∃acquires␈α∀a
␈↓ ↓H␈↓huge␈α↔collection␈α↔of␈α↔objects.␈α_ Then␈α↔AM␈α↔repeatedly␈α↔picks␈α↔random␈α_pairs␈α↔of
␈↓ ↓H␈↓these␈α≤objects␈α≤and␈α≤sees␈α≤if␈α≤they␈α≤satisfy␈α≤EQUAL,␈α≤by␈α≤running␈α≤the␈α≤little
␈↓ ↓H␈↓program␈α~stored␈α≠in␈α~the␈α≠Algorithm␈α~facet␈α≠of␈α~Equal.␈α≠ This␈α~is␈α≠where␈α~the
␈↓ ↓H␈↓empirical data comes from that tells AM how rarely EQUAL is satisfied.

␈↓ ↓H␈↓We␈α≥already␈α≥saw␈α≡the␈α≥heuristic␈α≥that␈α≡said␈α≥to␈α≥consider␈α≡generalizing␈α≥the
␈↓ ↓H␈↓definition␈α∀of␈α∀a␈α∀predicate,␈α∃if␈α∀very␈α∀few␈α∀examples␈α∃can␈α∀be␈α∀found.␈α∀ So␈α∃a␈α∀new
␈↓ ↓H␈↓job␈α_is␈α→added␈α_to␈α_the␈α→agenda,␈α_which␈α_says␈α→to␈α_generalize␈α_the␈α→definition␈α_of
␈↓ ↓H␈↓EQUAL.␈α∪ Since␈α∪it␈α∩has␈α∪a␈α∪good␈α∪reason,␈α∩it␈α∪gets␈α∪a␈α∩high␈α∪priority␈α∪value,␈α∪and␈α∩it
␈↓ ↓H␈↓soon rises to the top of the job-list.

␈↓ ↓H␈↓␈↓βNote␈↓␈α↔that␈α↔the␈α↔techniques␈α↔for␈α_filling␈α↔in␈α↔all␈α↔those␈α↔examples␈α_of␈α↔Equality
␈↓ ↓H␈↓were␈α⊃drawn␈α⊃from␈α⊃very␈α⊃general␈α⊃concepts:␈α⊃ways␈α⊃to␈α⊃fill␈α⊃in␈α⊃any␈α⊃concept,␈α⊂ways
␈↓ ↓H␈↓to␈α⊗fill␈α⊗in␈α⊗any␈α⊗active␈α⊗concept.␈α⊗ There␈α⊗were␈α⊗no␈α⊗special-purpose␈α↔tricks,␈α⊗no
␈↓ ↓H␈↓specialized heuristics for filling in examples of Equality.

␈↓ ↓H␈↓Similarly,␈α!AM␈α"used␈α!very␈α!general␈α"techniques␈α!for␈α!noticing␈α"that␈α!this
␈↓ ↓H␈↓predicate␈α⊗should␈α∃be␈α⊗generalized,␈α∃and␈α⊗for␈α∃generalizing␈α⊗its␈α⊗definition.␈α∃ AM
␈↓ ↓H␈↓in␈α~fact␈α→was␈α~given␈α→only␈α~a␈α→few␈α~syntactic␈α→rules␈α~for␈α→generalizing␈α~a␈α→LISP
␈↓ ↓H␈↓predicate,␈α~and␈α~therefore␈α~some␈α~semantically-uninteresting␈α~generalizations
␈↓ ↓H␈↓of Equal were created.


␈↓ ↓H␈↓∧␈↓ β?*** Generalizing Equality ***

␈↓ ↓H␈↓Assume␈α⊃that␈α⊃AM␈α⊃has␈α∩plucked␈α⊃that␈α⊃job␈α⊃off␈α∩the␈α⊃agenda,␈α⊃the␈α⊃one␈α∩which␈α⊃says
␈↓ ↓H␈↓to␈α→generalize␈α~the␈α→definition␈α→of␈α~EQUAL.␈α→ AM␈α→magically␈α~turns␈α→Equality
␈↓ ↓H␈↓into Equal-length.

␈↓ ↓H␈↓It␈α⊗does␈α⊗this␈α⊗using␈α⊗the␈α⊗syntactic␈α⊗rules␈α⊗for␈α⊗generalizing␈α⊗the␈α↔definition␈α⊗of
␈↓ ↓H␈↓any␈α≤concept.␈α≠ One␈α≤rule␈α≠says␈α≤to␈α≤remove␈α≠a␈α≤conjunct,␈α≠so␈α≤that␈α≤the␈α≠new
␈↓ ↓H␈↓predicate would recur in only one direction.

␈↓ ↓H␈↓In␈α⊗particular,␈α⊗by␈α⊗doing␈α⊗away␈α⊗with␈α∃the␈α⊗test␈α⊗in␈α⊗the␈α⊗CAR␈α⊗direction␈α∃<defn
␈↓ ↓H␈↓OVERLAY␈α⊗slide>,␈α∃we␈α⊗get␈α⊗a␈α∃predicate␈α⊗which␈α⊗counts␈α∃down␈α⊗two␈α⊗lists,␈α∃and
␈↓ ↓H␈↓returns␈α→T␈α→if␈α→and␈α→only␈α→if␈α→they␈α→both␈α→become␈α→empty␈α→at␈α→the␈α→same␈α_time.
␈↓ ↓H␈↓That is, iff they have the same length.

␈↓ ↓H␈↓A␈α∃few␈α∃other␈α∃concepts␈α∃were␈α∃generated␈α∃which␈α∃were␈α∃boring␈α∀generalizations
␈↓ ↓H␈↓of␈α⊗Equality.␈α⊗There␈α↔are␈α⊗two␈α⊗reasons␈α⊗why␈α↔syntactic␈α⊗rules␈α⊗suffice:␈α↔(1)␈α⊗AM
␈↓ ↓H␈↓never␈α⊗generalizes␈α⊗X␈α⊗unless␈α⊗it␈α⊗has␈α⊗a␈α∃very␈α⊗good␈α⊗reason,␈α⊗so␈α⊗this␈α⊗is␈α⊗a␈α∃very
␈↓ ↓H␈↓rare␈α∩occurrence;␈α∪(2)␈α∩very␈α∪quickly,␈α∩AM␈α∪will␈α∩notice␈α∪interesting␈α∩relationships
␈↓ ↓H␈↓involving some of these, and lose interest in the other, boring ones.

␈↓ ↓H␈↓<SLIDE:␈α_User␈α_sees␈α→it␈α_again>␈α_The␈α_second␈α→magic␈α_trick␈α_was␈α→deriving␈α_the
␈↓ ↓H␈↓canonical␈α∩form␈α∩of␈α∩a␈α∪structure.␈α∩ AM␈α∩did␈α∩this␈α∩by␈α∪selective␈α∩experimentation,
␈↓ ↓H␈↓which I'll describe later if anyone asks.

␈↓ ↓H␈↓This␈α∪whole␈α∪development␈α∪was␈α∪done␈α∪to␈α∩satisfy␈α∪just␈α∪3␈α∪jobs␈α∪from␈α∪the␈α∩agenda
␈↓ ↓H␈↓list:
␈↓ ↓H␈↓␈↓ αλ(Fillin Exs. of Equality)
␈↓ ↓H␈↓␈↓ αλ(Generalize Defn. of Equality)
␈↓ ↓H␈↓␈↓ αλ(Canonicalize Same-length wrt Equality)

␈↓ ↓H␈↓This␈α→example␈α→is␈α→really␈α→compressed␈α→by␈α→a␈α→factor␈α→of␈α→10;␈α→it␈α~distorts␈α→AM's
␈↓ ↓H␈↓abilites␈α→and␈α→limitations.␈α→ Let's␈α→move␈α→at␈α→a␈α→faster␈α→pace␈α→through␈α→another,
␈↓ ↓H␈↓longer, but more faithful excerpt of AM in action.


␈↓ ↓H␈↓∧␈↓ ∧b*** Example 2 ***

␈↓ ↓H␈↓After␈α≤going␈α≤through␈α≤this␈α≤final␈α≠example,␈α≤I'll␈α≤just␈α≤describe␈α≤the␈α≠overall
␈↓ ↓H␈↓results of this project.

␈↓ ↓H␈↓We'll␈α⊃start␈α⊂in␈α⊃the␈α⊂middle␈α⊃of␈α⊂a␈α⊃sessions␈α⊂with␈α⊃AM,␈α⊂after␈α⊃64␈α⊂other␈α⊃tasks␈α⊂have
␈↓ ↓H␈↓been␈α(chosen␈α(and␈α(executed.␈α( AM␈α(has␈α(already␈α(discovered␈α(numbers,
␈↓ ↓H␈↓arithmetic,␈α!squaring,␈α!square-rooting,␈α!odd␈α!numbers,␈α!and␈α!several␈α other
␈↓ ↓H␈↓elementary␈α∃objects␈α⊗and␈α∃operators.␈α∃ The␈α⊗concept␈α∃of␈α∃Divisors-of␈α⊗a␈α∃number
␈↓ ↓H␈↓has just been defined.

␈↓ ↓H␈↓<SLIDE: Ex2>

␈↓ ↓H␈↓Task␈α→65:␈α→AM␈α~was␈α→interested␈α→in␈α→the␈α~concept␈α→of␈α→Divisors-of,␈α~and␈α→found
␈↓ ↓H␈↓several examples of it.

␈↓ ↓H␈↓66:␈α∩The␈α⊃inverse␈α∩images␈α∩of␈α⊃various␈α∩extremal␈α∩kinds␈α⊃of␈α∩sets␈α∩were␈α⊃considered.
␈↓ ↓H␈↓AM␈α⊂noticed␈α⊂that␈α⊂numbers-with-3-divisors␈α⊃all␈α⊂seemed␈α⊂to␈α⊂be␈α⊃perfect␈α⊂squares.
␈↓ ↓H␈↓How␈α≥did␈α≥it␈α≥notice␈α≥that␈α≥relationship?␈α≥ It␈α≥took␈α≥one␈α≥example␈α≡of␈α≥these
␈↓ ↓H␈↓numbers,␈α∨49.␈α∨ AM␈α∨then␈α∨asked␈α∨what␈α∨49␈α∨was␈α∨an␈α∨example␈α of,␈α∨besides
␈↓ ↓H␈↓Numbers-with-3-divisors.␈α≠ The␈α~answer␈α≠was:␈α≠odd-number,␈α~perfect-square.
␈↓ ↓H␈↓So␈α⊗one␈α⊗hypothesis␈α⊗was␈α⊗that␈α∃all␈α⊗such␈α⊗numbers␈α⊗might␈α⊗be␈α⊗perfect␈α∃squares.
␈↓ ↓H␈↓This␈α∩was␈α∪tested␈α∩on␈α∪the␈α∩other␈α∩10␈α∪examples␈α∩of␈α∪of␈α∩Numbers-with-3-divisors,
␈↓ ↓H␈↓and␈α_sure␈α↔enough␈α_they␈α_all␈α↔were␈α_squares.␈α_ So␈α↔AM␈α_conjectured␈α_that␈α↔Nos-
␈↓ ↓H␈↓with-3-divisors is a specialization of the concept Perfect-squares.

␈↓ ↓H␈↓␈↓α␈↓&                                                                                  ␈↓ h ␈↓)αβ␈↓ 

␈↓ ↓H␈↓67:␈α∩The␈α∩square-roots␈α∩of␈α∩those␈α∩numbers␈α∩turned␈α∩out␈α∩to␈α∩be␈α∩numbers-with-2-
␈↓ ↓H␈↓divisors.␈α∀ Which␈α∃the␈α∀user␈α∀then␈α∃renamed␈α∀as␈α∀"Primes".␈α∃ The␈α∀value␈α∃of␈α∀both
␈↓ ↓H␈↓concepts was raised.

␈↓ ↓H␈↓␈↓α␈↓&                                                                                  ␈↓ h ␈↓)αβ␈↓ 
␈↓ ↓H␈↓79:␈α∪Later,␈α∀AM␈α∪examined␈α∀the␈α∪inverse␈α∪of␈α∀Times.␈α∪ For␈α∀example,␈α∪Times␈↓#
-␈↓#␈↓#
1␈↓#(12)
␈↓ ↓H␈↓is␈α⊃the␈α⊃set␈α⊃of␈α⊃all␈α⊃possible␈α⊃ways␈α⊃of␈α⊃factoring␈α⊃12.␈α⊃ It␈α⊃contains␈α⊃the␈α⊃bag␈α⊃(2␈α∩2␈α⊃3),
␈↓ ↓H␈↓since␈α∃2x2x3␈α∃is␈α∃12.␈α∀ AM␈α∃noticed␈α∃that␈α∃Times␈↓#
-␈↓#␈↓#
1␈↓#␈α∀of␈α∃each␈α∃number␈α∃seemed␈α∀to
␈↓ ↓H␈↓contain␈α∃a␈α∃bag␈α∃of␈α∀primes.␈α∃ So␈α∃a␈α∃new␈α∀concept␈α∃was␈α∃created,␈α∃a␈α∃new␈α∀relation
␈↓ ↓H␈↓which␈α⊃maps␈α∩a␈α⊃number␈α∩into␈α⊃all␈α∩possible␈α⊃factorizations␈α∩of␈α⊃that␈α∩number␈α⊃into
␈↓ ↓H␈↓primes.

␈↓ ↓H␈↓␈↓α␈↓&                                                                                  ␈↓ h ␈↓)αβ␈↓ 

␈↓ ↓H␈↓80:␈α∪Lo␈α∪and␈α∩behold,␈α∪that␈α∪relation␈α∪turned␈α∩out␈α∪to␈α∪be␈α∩a␈α∪function.␈α∪This␈α∪is␈α∩the
␈↓ ↓H␈↓fundamental theorem of arithmetic.

␈↓ ↓H␈↓␈↓α␈↓&                                                                                  ␈↓ h ␈↓)αβ␈↓ 

␈↓ ↓H␈↓84:␈α∀Lest␈α∀you␈α∀think␈α∀that␈α∀AM␈α∀was␈α∀always␈α∀this␈α∀brilliant,␈α∀let's␈α∀continue␈α∀and
␈↓ ↓H␈↓look␈α⊂at␈α⊂one␈α⊂of␈α⊂the␈α⊂next␈α⊃things␈α⊂it␈α⊂did.␈α⊂It␈α⊂examined␈α⊂the␈α⊂inverse␈α⊃of␈α⊂Addition.
␈↓ ↓H␈↓For␈α∃example,␈α∃Add␈↓#
-␈↓#␈↓#
1␈↓#(6)␈α∃contains␈α∃the␈α∃bag␈α⊗(1␈α∃1␈α∃2␈α∃2),␈α∃since␈α∃1+1+2+2␈α⊗equals␈α∃6.
␈↓ ↓H␈↓This␈α∀is␈α∀related␈α∃to␈α∀the␈α∀Partition␈α∀function.␈α∃ One␈α∀of␈α∀many␈α∃conjectures␈α∀AM
␈↓ ↓H␈↓notices␈α⊃then␈α⊂is␈α⊃that␈α⊂Add␈↓#
-␈↓#␈↓#
1␈↓#␈α⊃of␈α⊂a␈α⊃number␈α⊂often␈α⊃contains␈α⊂a␈α⊃pair␈α⊂of␈α⊃primes.␈α⊂A
␈↓ ↓H␈↓new␈α∃relation␈α∃is␈α⊗created,␈α∃which␈α∃takes␈α⊗a␈α∃number␈α∃and␈α⊗returns␈α∃a␈α∃list␈α⊗of␈α∃all
␈↓ ↓H␈↓possible ways of representing it as the sum of two primes.

␈↓ ↓H␈↓␈↓α␈↓&                                                                                  ␈↓ h ␈↓)αβ␈↓ 

␈↓ ↓H␈↓106:␈α∩Eventually,␈α∩AM␈α∩studies␈α∩this,␈α∩and␈α∩notices␈α∩that␈α∩all␈α∩even␈α∩numbers␈α∩seem
␈↓ ↓H␈↓to␈α∃have␈α∃at␈α∃least␈α∃one␈α∃such␈α∃representation.␈α∃ This␈α∃is␈α∃Goldbach's␈α∃conjecture.
␈↓ ↓H␈↓AM␈α doesn't␈α rate␈α!this␈α conjecture␈α very␈α!highly,␈α since␈α primes␈α!are␈α not
␈↓ ↓H␈↓intimately connected with addition.

␈↓ ↓H␈↓107:␈α⊂Here,␈α⊂AM␈α⊃isolates␈α⊂those␈α⊂number␈α⊂which␈α⊃can␈α⊂be␈α⊂represented␈α⊂as␈α⊃the␈α⊂sum
␈↓ ↓H␈↓of␈α⊗2␈α↔primes␈α⊗in␈α⊗only␈α↔one␈α⊗way.␈α⊗While␈α↔this␈α⊗is␈α⊗a␈α↔novel␈α⊗set␈α⊗of␈α↔numbers␈α⊗to
␈↓ ↓H␈↓investigate, AM wasn't able to find anything interesting out about them.

␈↓ ↓H␈↓␈↓α␈↓&                                                                                  ␈↓ h ␈↓)αβ␈↓ 
␈↓ ↓H␈↓∧␈↓ ¬∩*** Results ***

␈↓ ↓H␈↓Now␈α∩that␈α⊃we've␈α∩seen␈α⊃a␈α∩few␈α⊃examples␈α∩of␈α⊃AM␈α∩in␈α⊃action,␈α∩let's␈α⊃step␈α∩back␈α⊃and
␈↓ ↓H␈↓look at the whole run, of which these were just excerpts.

␈↓ ↓H␈↓This␈α!run␈α!occurred␈α"a␈α!few␈α!months␈α!ago.␈α" AM␈α!started␈α!with␈α"the␈α!115
␈↓ ↓H␈↓prenumerical␈α∪concepts␈α∪I␈α∪showed␈α∪you␈α∪earlier,␈α∪with␈α∪about␈α∪8␈α∪facets␈α∪filled␈α∪in
␈↓ ↓H␈↓for␈α∃each:␈α∀typically,␈α∃the␈α∀name,␈α∃definition,␈α∀worth,␈α∃domain/range,␈α∀algorithm,
␈↓ ↓H␈↓fillin␈α*heuristics,␈α*checking␈α*heurisics,␈α*and␈α*a␈α*pointer␈α*to␈α*a␈α*known
␈↓ ↓H␈↓generalization.

␈↓ ↓H␈↓AM␈α⊃worked␈α⊃completely␈α⊃by␈α⊃itself␈α⊃for␈α⊃1␈α⊃cpu␈α⊃hour,␈α⊃when␈α⊃it␈α⊃ran␈α⊃out␈α⊃of␈α⊃space
␈↓ ↓H␈↓and␈α⊂died.␈α⊂ AM␈α⊂had␈α⊃filled␈α⊂in␈α⊂a␈α⊂total␈α⊃of␈α⊂200␈α⊂previously-blank␈α⊂facets␈α⊃on␈α⊂these
␈↓ ↓H␈↓primitive␈α∀concepts,␈α∪and␈α∀it␈α∀had␈α∪created␈α∀about␈α∀135␈α∪new␈α∀concepts.␈α∀The␈α∪new
␈↓ ↓H␈↓concepts␈α∀ended␈α∀up␈α∀with␈α∀an␈α∀average␈α∀of␈α∀10␈α∀facets␈α∀filled␈α∀in.␈α∀ Half␈α∃of␈α∀those
␈↓ ↓H␈↓new␈α~ones␈α→were␈α~not␈α~really␈α→worth␈α~defining,␈α→and␈α~were␈α~technically␈α→called
␈↓ ↓H␈↓losers,␈α∪but␈α∪the␈α∀rest␈α∪were␈α∪interesting.␈α∀ <SLIDE:␈α∪Winners>.␈α∪Note␈α∀AM␈α∪went
␈↓ ↓H␈↓through␈α⊃the␈α⊂whole␈α⊃chain␈α⊂of␈α⊃discoveries␈α⊃I␈α⊂talked␈α⊃about␈α⊂during␈α⊃this␈α⊃talk,␈α⊂all
␈↓ ↓H␈↓the␈α⊗way␈α↔from␈α⊗set-theoretic␈α↔notions␈α⊗to␈α↔numbers␈α⊗to␈α↔unique␈α⊗factorization.
␈↓ ↓H␈↓Several new concepts of questionable value were derived.

␈↓ ↓H␈↓AM␈α∀did␈α∃this␈α∀all␈α∀on␈α∃its␈α∀own;␈α∃if␈α∀I␈α∀sit␈α∃at␈α∀the␈α∀keyboard␈α∃and␈α∀guide␈α∃it␈α∀along,
␈↓ ↓H␈↓the␈α⊂same␈α⊂discoveries␈α⊃can␈α⊂be␈α⊂made␈α⊃in␈α⊂about␈α⊂a␈α⊃third␈α⊂the␈α⊂cpu␈α⊃time.␈α⊂ <Results
␈↓ ↓H␈↓again: out of order!>

␈↓ ↓H␈↓200␈α∃jobs␈α∃from␈α∃the␈α∃agenda␈α∃were␈α∃executed␈α∃during␈α∃the␈α∃cpu␈α∃hour␈α∃AM␈α∀used.
␈↓ ↓H␈↓Each␈α≠job␈α≤was␈α≠allocated␈α≤from␈α≠1␈α≠to␈α≤100␈α≠cpu␈α≤seconds,␈α≠depending␈α≤on␈α≠its
␈↓ ↓H␈↓priority␈α∪value.␈α∪ Most␈α∩jobs␈α∪were␈α∪granted␈α∪about␈α∩30␈α∪seconds,␈α∪but␈α∪used␈α∩under
␈↓ ↓H␈↓20.

␈↓ ↓H␈↓The␈α∨1-hour␈α∨run␈α∨ended␈α∨because␈α∨of␈α∨space␈α∨problems,␈α∨and␈α∨because␈α≡AM
␈↓ ↓H␈↓couldn't␈α∀fill␈α∪in␈α∀very␈α∪good␈α∀heuristics␈α∪for␈α∀the␈α∪new␈α∀concepts␈α∪it␈α∀created.␈α∪ As
␈↓ ↓H␈↓it␈α∀got␈α∀further␈α∀and␈α∪further␈α∀from␈α∀its␈α∀initial␈α∪starting␈α∀basis,␈α∀the␈α∀methods␈α∪it
␈↓ ↓H␈↓was␈α∀able␈α∪to␈α∀bring␈α∀to␈α∪bear␈α∀on␈α∪its␈α∀new␈α∀concepts␈α∪got␈α∀relatively␈α∀weaker␈α∪and
␈↓ ↓H␈↓weaker.␈α≠ For␈α≤example,␈α≠it␈α≤was␈α≠dealing␈α≤with␈α≠primes␈α≤using␈α≠set-theoretic
␈↓ ↓H␈↓heuristics.
␈↓ ↓H␈↓To␈α∀correct␈α∀this,␈α∀AM␈α∀would␈α∃need␈α∀a␈α∀body␈α∀of␈α∀good␈α∀meta-heuristics;␈α∃that␈α∀is,
␈↓ ↓H␈↓rules␈α↔for␈α↔finding␈α⊗and␈α↔fiilling␈α↔in␈α⊗new␈α↔rules.␈α↔ At␈α⊗the␈α↔moment,␈α↔it␈α⊗doesn't
␈↓ ↓H␈↓have␈α↔any.␈α↔ Since␈α↔most␈α↔heuristics␈α⊗are␈α↔based␈α↔on␈α↔hindsight,␈α↔I␈α↔would␈α⊗guess
␈↓ ↓H␈↓that␈α⊃most␈α⊃of␈α⊃the␈α⊃meta-heuristics␈α⊂couldn't␈α⊃be␈α⊃applied␈α⊃until␈α⊃AM␈α⊃has␈α⊂already
␈↓ ↓H␈↓made␈α∃most␈α⊗of␈α∃the␈α∃possible␈α⊗mistakes␈α∃it␈α⊗could␈α∃make.␈α∃ I'm␈α⊗not␈α∃sure,␈α⊗and␈α∃I
␈↓ ↓H␈↓leave this as an open research problem <SLIDE: open problem>.

␈↓ ↓H␈↓All␈α≤was␈α≤not␈α≤perfect,␈α≤of␈α≤course.␈α≤Several␈α≤tasks␈α≤were␈α≤chosen␈α≤at␈α≤bizarre
␈↓ ↓H␈↓moments,␈α∃and␈α∀many␈α∃poor␈α∃tasks␈α∀were␈α∃chosen.␈α∀<SLIDE:␈α∃Losers␈α∃again>␈α∀One
␈↓ ↓H␈↓danger␈α∪that␈α∪AM␈α∪is␈α∪prone␈α∪to␈α∀is␈α∪that␈α∪of␈α∪regular␈α∪or␈α∪looping␈α∀behavior.␈α∪ Any
␈↓ ↓H␈↓kind␈α⊗of␈α⊗repetition␈α⊗should␈α⊗--␈α⊗but␈α↔doesn't␈α⊗--␈α⊗warn␈α⊗AM␈α⊗to␈α⊗be␈α↔careful␈α⊗and
␈↓ ↓H␈↓look␈α⊗for␈α⊗a␈α⊗unifying␈α⊗generalization␈α↔of␈α⊗what␈α⊗it's␈α⊗doing,␈α⊗to␈α⊗rise␈α↔above␈α⊗the
␈↓ ↓H␈↓infinite␈α.loop␈α.it's␈α.fallen␈α.into.␈α. That's␈α.how␈α.this␈α.concept␈α.arose
␈↓ ↓H␈↓(ComposeoCompose...).␈α. Similarly,␈α/AM␈α.repeated␈α/multiplication,␈α.then
␈↓ ↓H␈↓repeated exponentiation, then repeated hyper-exponentiation,...

␈↓ ↓H␈↓Although␈α⊂any␈α⊃specific␈α⊂example␈α⊃I␈α⊂give␈α⊂you␈α⊃might␈α⊂have␈α⊃fixed␈α⊂by␈α⊃adding␈α⊂one
␈↓ ↓H␈↓simple␈α∪rule,␈α∩the␈α∪general␈α∪problem␈α∩of␈α∪detecting␈α∩regularity,␈α∪infinite␈α∪loops,␈α∩is
␈↓ ↓H␈↓quite difficult and I leave that as another open problem.

␈↓ ↓H␈↓IF ASKED: A HANDLE ON THE SOLUTION IS:

␈↓ ↓H␈↓␈↓ α_By␈α'investigating␈α'each␈α(concept␈α'it␈α'defined,␈α'and␈α(basing␈α'its
␈↓ ↓H␈↓␈↓ α_interestingness␈α⊃judgments␈α⊃on␈α⊃those␈α⊂results,␈α⊃AM␈α⊃was␈α⊃able␈α⊃to␈α⊂avoid
␈↓ ↓H␈↓␈↓ α_most such traps.


␈↓ ↓H␈↓∧␈↓ β.*** Experiments With AM ***

␈↓ ↓H␈↓One␈α%important␈α%reason␈α%for␈α%actually␈α%constructing␈α%AM␈α%was␈α%that␈α%of
␈↓ ↓H␈↓␈↓βexperimentation␈↓:␈α⊗one␈α⊗can␈α↔vary␈α⊗the␈α⊗concepts␈α↔AM␈α⊗starts␈α⊗with,␈α↔vary␈α⊗the
␈↓ ↓H␈↓heuristics␈α⊂available,␈α⊂etc.,␈α⊂and␈α⊂study␈α⊂the␈α⊂effects␈α⊂on␈α⊂AM's␈α⊃behavior.␈α⊂ <SLIDE:
␈↓ ↓H␈↓expts> Several such experiments were performed.

␈↓ ↓H␈↓One␈α∀involved␈α∀adding␈α∀a␈α∀couple␈α∀dozen␈α∀new␈α∀concepts␈α∀from␈α∀an␈α∃entirely␈α∀new
␈↓ ↓H␈↓domain:␈α⊂plane␈α⊂geometry.␈α⊃ AM␈α⊂busied␈α⊂itself␈α⊂exploring␈α⊃elementary␈α⊂geometric
␈↓ ↓H␈↓concepts,␈α∩but␈α∩was␈α∩not␈α∩quite␈α∩as␈α∩productive␈α∩there␈α∩as␈α∩in␈α∩its␈α∪original␈α∩domain.
␈↓ ↓H␈↓New␈α#concepts␈α$were␈α#defined,␈α#and␈α$new␈α#conjectures␈α$formulated.␈α# For
␈↓ ↓H␈↓example,␈α⊂AM␈α⊂had␈α⊂the␈α⊂notion␈α⊂of␈α⊂identical␈α⊂equality␈α⊂of␈α⊂lines␈α⊂and␈α⊂of␈α⊂triangles,
␈↓ ↓H␈↓and␈α↔generalized␈α↔these␈α⊗into␈α↔similarity,␈α↔parallel,␈α⊗congruence␈α↔etc.␈α↔ It␈α⊗noted
␈↓ ↓H␈↓many␈α∀new␈α∪concepts,␈α∀like␈α∪triangles␈α∀sharing␈α∪a␈α∀common␈α∪side␈α∀and␈α∀a␈α∪common
␈↓ ↓H␈↓angle.␈α∨AM␈α∨also␈α∨found␈α≡a␈α∨cute␈α∨geometric␈α∨interpretation␈α∨of␈α≡Goldbach's
␈↓ ↓H␈↓conjecture,␈α≥which␈α≥it␈α≥had␈α≥proposed␈α≥earlier:<SLIDE:␈α≥prime␈α≡angles>␈α≥Any
␈↓ ↓H␈↓angle,␈α∪from␈α∀0␈α∪to␈α∪180␈↓#
o␈↓#,␈α∀can␈α∪be␈α∪approximated␈α∀(to␈α∪within␈α∪1␈↓#
o␈↓#)␈α∀as␈α∪the␈α∀sum␈α∪of
␈↓ ↓H␈↓two␈α"angles␈α"of␈α"prime␈α"degree.␈α! If␈α"our␈α"culture␈α"and␈α"technology␈α!were
␈↓ ↓H␈↓different,␈α-this␈α.might␈α-have␈α-been␈α.an␈α-important,␈α.familiar␈α-result.
␈↓ ↓H␈↓Incidentally,␈α∩much␈α∩sharper␈α∩results␈α∩and␈α∩much␈α∩more␈α∩general␈α∩ones␈α∩were␈α∩also
␈↓ ↓H␈↓obtained.

␈↓ ↓H␈↓<SLIDE:␈α→expts␈α_again>␈α→Several␈α_of␈α→the␈α_experiments␈α→on␈α_AM␈α→involved␈α_de-
␈↓ ↓H␈↓tuning␈α~the␈α~system,␈α~e.g.␈α~ radically␈α~altering␈α~the␈α~formula␈α~which␈α~assigned
␈↓ ↓H␈↓each␈α≥task␈α≥a␈α≤priority␈α≥rating.␈α≥ Or:␈α≤radically␈α≥altering␈α≥the␈α≥initial␈α≤Worth
␈↓ ↓H␈↓numbers␈α→of␈α→the␈α→concepts.␈α→ Generally,␈α→AM's␈α→behavior␈α→was␈α→degraded,␈α→but
␈↓ ↓H␈↓overall it was much more robust than anticipated.

␈↓ ↓H␈↓An␈α≠interesting␈α≠result␈α≠was␈α≠that␈α≠AM␈α≠uses␈α≠precise␈α≠numeric␈α≠values␈α~very
␈↓ ↓H␈↓rarely,␈α≥only␈α≥in␈α≥very␈α≥blind␈α≥situations,␈α≥where␈α≥no␈α≥decent␈α≥reasons␈α≥exist.
␈↓ ↓H␈↓When␈α≥AM␈α≥enters␈α≤a␈α≥chain␈α≥of␈α≥discoveries,␈α≤it's␈α≥sustained␈α≥by␈α≥masses␈α≤of
␈↓ ↓H␈↓reasons, not by nuances in their numeric ratings.

␈↓ ↓H␈↓Another␈α∪unexpected␈α∪result␈α∀of␈α∪this␈α∪experiment␈α∪was␈α∀that␈α∪the␈α∪top␈α∀few␈α∪jobs
␈↓ ↓H␈↓on␈α∩the␈α⊃agenda␈α∩almost␈α⊃always␈α∩suggest␈α∩new␈α⊃high-priority␈α∩jobs,␈α⊃so␈α∩that␈α∩a␈α⊃job
␈↓ ↓H␈↓sitting␈α_in␈α↔tenth␈α_place␈α_on␈α↔the␈α_agenda␈α↔at␈α_some␈α_moment␈α↔has␈α_a␈α_very␈α↔low
␈↓ ↓H␈↓chance␈α↔of␈α⊗␈↓βever␈↓␈α↔being␈α⊗chosen,␈α↔unless␈α↔of␈α⊗course␈α↔some␈α⊗new␈α↔reasons␈α↔for␈α⊗it
␈↓ ↓H␈↓emerge.␈α∃ So␈α∃even␈α∃if␈α∃you␈α∃had␈α∃a␈α∃kilo-processor␈α∃machine,␈α∃executing␈α⊗all␈α∃the
␈↓ ↓H␈↓tasks␈α∀on␈α∀the␈α∀agenda␈α∀simultaneously,␈α∀it␈α∀wouldn't␈α∀really␈α∀speed␈α∀up␈α∀the␈α∪rate
␈↓ ↓H␈↓at␈α⊃which␈α⊃AM␈α⊂makes␈α⊃great␈α⊃discoveries␈α⊂more␈α⊃than␈α⊃by␈α⊂a␈α⊃single␈α⊃factor␈α⊃of␈α⊂ten.
␈↓ ↓H␈↓Not␈α↔only␈α↔wouldn't␈α_it␈α↔help,␈α↔it␈α↔would␈α_reduce␈α↔the␈α↔user's␈α↔opinion␈α_of␈α↔AM's
␈↓ ↓H␈↓rationality:␈α∩a␈α∩lot␈α⊃of␈α∩that␈α∩stems␈α∩from␈α⊃AM's␈α∩choice␈α∩of␈α⊃which␈α∩task␈α∩is␈α∩best␈α⊃at
␈↓ ↓H␈↓each moment.

␈↓ ↓H␈↓Let␈α∀me␈α∀move␈α∀on␈α∪to␈α∀another␈α∀kind␈α∀of␈α∪experiment␈α∀that␈α∀was␈α∀performed.␈α∪ It
␈↓ ↓H␈↓was␈α∪expected␈α∀that␈α∪a␈α∀few␈α∪key␈α∀concepts␈α∪(e.g.␈α∀ Equality)␈α∪and␈α∀heuristics␈α∪(e.g.,
␈↓ ↓H␈↓define␈α∀"f"␈α∀as␈α∀repeated␈α∀"g"-ing,␈α∀where␈α∀g␈α∀is␈α∀an␈α∀interesting␈α∀operation)␈α∪would
␈↓ ↓H␈↓have␈α∪a␈α∀tremendous␈α∪impact␈α∀on␈α∪AM's␈α∪behavior:␈α∀that␈α∪removing␈α∀them␈α∪would
␈↓ ↓H␈↓destroy␈α∪most␈α∪of␈α∩AM's␈α∪discoveries.␈α∪ This␈α∪was␈α∩partly␈α∪true,␈α∪but␈α∪a␈α∩surprising
␈↓ ↓H␈↓effect␈α∪was␈α∩noticed␈α∪when␈α∪these␈α∩experiments␈α∪were␈α∩actually␈α∪done␈α∪last␈α∩week.
␈↓ ↓H␈↓Many␈α$of␈α%the␈α$very␈α%common␈α$and␈α$valuable␈α%concepts␈α$manage␈α%to␈α$be
␈↓ ↓H␈↓rediscovered␈α_in␈α_many␈α_separate␈α_ways,␈α↔so␈α_that␈α_the␈α_absence␈α_of␈α_any␈α↔single
␈↓ ↓H␈↓concept␈α⊗or␈α⊗heuristic␈α∃isn't␈α⊗critical.␈α⊗ Multiplication␈α∃arises␈α⊗in␈α⊗four␈α∃separate
␈↓ ↓H␈↓ways,␈α∃so␈α∃one␈α∀would␈α∃have␈α∃to␈α∀remove␈α∃many␈α∃concepts␈α∀to␈α∃prevent␈α∃its␈α∀being
␈↓ ↓H␈↓noticed.

␈↓ ↓H␈↓Several␈α⊗more␈α⊗experiments␈α⊗have␈α⊗been␈α⊗planned␈α⊗for␈α⊗the␈α⊗near␈α↔future.␈α⊗This
␈↓ ↓H␈↓Fall,␈α∪I␈α∪intend␈α∀to␈α∪add␈α∪about␈α∪twenty␈α∀new␈α∪concepts,␈α∪dealing␈α∪with␈α∀proof␈α∪and
␈↓ ↓H␈↓proof␈α≠techniques,␈α≠to␈α≠see␈α≤how␈α≠the␈α≠reasons␈α≠for␈α≠conjecturing␈α≤a␈α≠theorem
␈↓ ↓H␈↓might aid in its subsequent proof.

␈↓ ↓H␈↓To␈α⊃me␈α⊃personally,␈α⊃this␈α⊃whole␈α⊃businss␈α⊃of␈α⊃experimenting␈α⊃with␈α⊃AM␈α⊃has␈α⊂been
␈↓ ↓H␈↓the␈α∀most␈α∪exciting␈α∀aspect␈α∀of␈α∪the␈α∀project.␈α∀ It's␈α∪like␈α∀using␈α∀AM␈α∪as␈α∀a␈α∀tool,␈α∪to
␈↓ ↓H␈↓explore␈α∀the␈α∪process␈α∀of␈α∀discovery␈α∪--␈α∀at␈α∀least␈α∪discovery␈α∀in␈α∀very␈α∪elementary
␈↓ ↓H␈↓mathematics.


␈↓ ↓H␈↓∧␈↓ αQ*** Maximally-Divisible Numbers ***

␈↓ ↓H␈↓While␈α⊂working␈α⊂on␈α⊂its␈α⊂own,␈α⊂AM␈α⊂has␈α⊂not␈α⊂discovered␈α⊂any␈α⊂mathematics␈α⊂which
␈↓ ↓H␈↓is␈α∃new␈α∃and␈α∃publishable,␈α⊗although␈α∃it␈α∃has␈α∃done␈α⊗so␈α∃when␈α∃working␈α∃as␈α⊗a␈α∃co-
␈↓ ↓H␈↓researcher␈α⊗with␈α⊗a␈α⊗human.␈α⊗AM␈α⊗noticed␈α⊗simple␈α⊗new␈α⊗concepts␈α⊗which␈α∃most
␈↓ ↓H␈↓humans␈α≤had␈α≤overlooked,␈α≥but␈α≤AM␈α≤by␈α≤itself␈α≥was␈α≤not␈α≤able␈α≥to␈α≤precisely
␈↓ ↓H␈↓formulate interesting statements about those concepts.

␈↓ ↓H␈↓Besides␈α⊂the␈α⊃cute␈α⊂Goldbach␈α⊂interpretation␈α⊃just␈α⊂described,␈α⊂AM␈α⊃motivated␈α⊂the
␈↓ ↓H␈↓development␈α∩of␈α∩a␈α∩new␈α∩result␈α∪in␈α∩number␈α∩theory.␈α∩ I'll␈α∩close␈α∩this␈α∪slide␈α∩show
␈↓ ↓H␈↓with one slide about it. <SLIDE: AM conjec>

␈↓ ↓H␈↓Just␈α∃as␈α∃AM␈α∃decided␈α∃to␈α∃look␈α∃at␈α∃numbers␈α∃with␈α∃extremely␈α∃␈↓βfew␈↓␈α∃divisors,␈α∀it
␈↓ ↓H␈↓thought␈α≡to␈α≥look␈α≡at␈α≡those␈α≥with␈α≡abnormally␈α≥␈↓βmany␈↓␈α≡divisors.␈α≡ For␈α≥any
␈↓ ↓H␈↓number␈α⊂d,␈α⊂the␈α⊂smallest␈α⊂number␈α⊂with␈α⊂at␈α⊂least␈α⊂d␈α⊂divisors␈α⊂is␈α⊂at␈α⊂least␈α⊂this␈α⊂big.
␈↓ ↓H␈↓If␈α⊗its's␈α↔factored␈α⊗into␈α↔primes,␈α⊗then␈α⊗their␈α↔exponents␈α⊗decrease␈α↔inversely␈α⊗as
␈↓ ↓H␈↓the logs of the primes.
␈↓ ↓H␈↓For␈α_example,␈α_here's␈α_the␈α_smallest␈α↔number␈α_with␈α_6␈α_million␈α_divisors.␈α↔ The
␈↓ ↓H␈↓AM␈α∩conjecture␈α∩predicted␈α∪that␈α∩this␈α∩number␈α∪n␈α∩would␈α∩be␈α∪this␈α∩big,␈α∩so␈α∪it␈α∩was
␈↓ ↓H␈↓off␈α~by␈α~less␈α~than␈α~a␈α~factor␈α≠of␈α~two,␈α~so␈α~it's␈α~a␈α~very␈α~sharp␈α≠bound.␈α~ Erdos
␈↓ ↓H␈↓recently␈α→showed␈α~me␈α→the␈α→best␈α~previously-known␈α→formula␈α~bounding␈α→n(d),
␈↓ ↓H␈↓and␈α⊂it␈α⊃could␈α⊂only␈α⊂gurantee␈α⊃that␈α⊂n␈α⊂would␈α⊃be␈α⊂bigger␈α⊂than␈α⊃this,␈α⊂so␈α⊂it␈α⊃was␈α⊂off
␈↓ ↓H␈↓by more than a whole order of magnitude.

␈↓ ↓H␈↓␈↓βAM␈α≠made␈α≠this␈α≠definition␈↓,␈α≠found␈α≠examples␈α≠of␈α≤such␈α≠highly-composite
␈↓ ↓H␈↓numbers,␈α⊗and␈α⊗then␈α⊗suggested␈α↔something␈α⊗like␈α⊗this␈α⊗relationship.␈α↔Later,␈α⊗by
␈↓ ↓H␈↓hand, I was able to derive these formulae, and with Erdos' help this one.

␈↓ ↓H␈↓A␈α↔couple␈α↔months␈α↔ago,␈α↔I␈α↔found␈α↔out␈α↔that␈α↔Ramanujan␈α↔defined␈α_these␈α↔same
␈↓ ↓H␈↓numbers␈α∪and␈α∩found␈α∪a␈α∪similar␈α∩regularity␈α∪to␈α∩this␈α∪one.␈α∪ As␈α∩far␈α∪as␈α∪is␈α∩known,
␈↓ ↓H␈↓this particular formula is new.

␈↓ ↓H␈↓It␈α∪was␈α∀AM␈α∪which␈α∀expected␈α∪that␈α∪this␈α∀kind␈α∪of␈α∀maximally-divisible␈α∪number
␈↓ ↓H␈↓would␈α⊃be␈α∩very␈α⊃interesting.␈α∩ I␈α⊃thought␈α∩I␈α⊃knew␈α∩better,␈α⊃but␈α∩in␈α⊃this␈α∩case␈α⊃AM
␈↓ ↓H␈↓really was right.

␈↓ ↓H␈↓In␈α∩dozens␈α∩of␈α∩other␈α∪instances,␈α∩AM␈α∩seemed␈α∩to␈α∪me␈α∩to␈α∩be␈α∩going␈α∪nowhere␈α∩and
␈↓ ↓H␈↓in␈α∃reality␈α∃it␈α∃was␈α∀actually␈α∃going␈α∃nowhere.␈α∃It␈α∀outsmarted␈α∃me␈α∃in␈α∃this␈α∀cute
␈↓ ↓H␈↓way only twice.


␈↓ ↓H␈↓∧␈↓ ∧R*** Conclusions ***

␈↓ ↓H␈↓What is the final conclusion of all this?

␈↓ ↓H␈↓AM␈α∀is␈α∀forced␈α∀to␈α∀judge␈α∀␈↓βa␈α∀priori␈↓␈α∪the␈α∀value␈α∀of␈α∀each␈α∀new␈α∀concept;␈α∀it␈α∪␈↓βmust␈↓
␈↓ ↓H␈↓quickly␈α≥lose␈α≤interest␈α≥in␈α≥concepts␈α≤which␈α≥aren't␈α≤going␈α≥to␈α≥develop␈α≤into
␈↓ ↓H␈↓anything.  Often, such judgments can only be based on hindsight.

␈↓ ↓H␈↓For␈α∀similar␈α∀reasons,␈α∀AM␈α∪has␈α∀difficulty␈α∀formulating␈α∀new␈α∀heuristics␈α∪which
␈↓ ↓H␈↓are␈α≡relevant␈α≥to␈α≡the␈α≥new␈α≡concepts␈α≥it␈α≡creates.␈α≥ It␈α≡needed␈α≡good␈α≥meta-
␈↓ ↓H␈↓heuristics, and I don't even know whether any exist for this domain.

␈↓ ↓H␈↓While␈α∃AM's␈α∃"approach"␈α∃to␈α∃empirical␈α∃research␈α∃may␈α∃be␈α∃used␈α∃in␈α∃other␈α∃very
␈↓ ↓H␈↓hard␈α⊗scientific␈α↔domains,␈α⊗the␈α⊗main␈α↔limitation␈α⊗(reliance␈α⊗on␈α↔hindsight)␈α⊗will
␈↓ ↓H␈↓probably␈α∩recur.␈α∩ This␈α∩prevents␈α∩AM␈α∩--␈α∩and␈α∩future␈α∩similar␈α∩systems␈α∪--␈α∩from
␈↓ ↓H␈↓progressing indefinitely far on their own.

␈↓ ↓H␈↓Before␈α∀this␈α∃ultimate␈α∀limitation␈α∀was␈α∃reached,␈α∀AM␈α∀had␈α∃in␈α∀fact␈α∃carried␈α∀out
␈↓ ↓H␈↓some␈α∃math␈α⊗research.␈α∃ The␈α⊗research␈α∃was␈α∃of␈α⊗a␈α∃very␈α⊗elementary␈α∃character,
␈↓ ↓H␈↓and␈α the␈α∨same␈α program␈α∨may␈α not␈α be␈α∨able␈α to␈α∨carry␈α out␈α much␈α∨more
␈↓ ↓H␈↓sophisticated␈α~theory␈α~formation,␈α~but␈α~nevertheless␈α~we␈α~can␈α≠conclude␈α~that
␈↓ ↓H␈↓certain␈α↔kinds␈α⊗of␈α↔hack␈α⊗discoveries,␈α↔certain␈α⊗routine␈α↔research␈α⊗tasks,␈α↔can␈α⊗be
␈↓ ↓H␈↓represented adequately as a rule-guided heuristic search.
␈↓ ↓H␈↓*********************************************************************

␈↓ ↓H␈↓Ideally,␈α∀one␈α∀could␈α∀run␈α∀AM␈α∀overnight,␈α∀and␈α∀wake␈α∀up␈α∀the␈α∀next␈α∀morning␈α∪to
␈↓ ↓H␈↓find␈α_five␈α_or␈α_ten␈α_more␈α_or␈α↔less␈α_interesting␈α_new␈α_concepts␈α_to␈α_look␈α_at.␈α↔ It
␈↓ ↓H␈↓would␈α_still␈α_be␈α_the␈α→researcher␈α_who␈α_made␈α_the␈α_highest-level␈α→selections␈α_of
␈↓ ↓H␈↓what␈α⊃to␈α⊃do␈α∩next,␈α⊃much␈α⊃like␈α⊃a␈α∩professor␈α⊃directing␈α⊃graduate␈α∩students.␈α⊃ This
␈↓ ↓H␈↓is␈α⊃the␈α⊂ultimate␈α⊃use␈α⊂that␈α⊃I␈α⊂hope␈α⊃such␈α⊂systems␈α⊃will␈α⊂be␈α⊃put␈α⊂to␈α⊃--␈α⊃probably␈α⊂in
␈↓ ↓H␈↓the␈α⊂next␈α⊂centry:␈α⊂to␈α⊂work␈α⊂synergetically␈α⊂alongside␈α⊂a␈α⊂researcher,␈α⊂to␈α⊃free␈α⊂him
␈↓ ↓H␈↓from doing all the routine detective work.


␈↓ ↓H␈↓∧␈↓ ∧P*** Supplement ***

␈↓ ↓H␈↓␈↓& Finding canonizing function for numbers␈↓)αβ

␈↓ ↓H␈↓Experiment␈α⊗number␈α⊗1␈α⊗was␈α⊗to␈α↔convert␈α⊗the␈α⊗arguments␈α⊗from␈α⊗one␈α↔type␈α⊗of
␈↓ ↓H␈↓structure␈α≡to␈α≡another,␈α≥and␈α≡see␈α≡if␈α≡that␈α≥changed␈α≡the␈α≡value␈α≡of␈α≥SAME-
␈↓ ↓H␈↓LENGTH␈α_when␈α_it␈α_was␈α_applied␈α_to␈α_them.␈α_ It␈α_turned␈α_out␈α_not␈α→to,␈α_which
␈↓ ↓H␈↓meant that one single canonical type of structure could be chosen.

␈↓ ↓H␈↓But␈α∀there␈α∀are␈α∀4␈α∀kinds␈α∀it␈α∀could␈α∀be:␈α∀a␈α∀set,␈α∀a␈α∀list,␈α∀a␈α∀bag,␈α∀or␈α∀an␈α∃ordered␈α∀set.
␈↓ ↓H␈↓The␈α⊗particular␈α⊗kind␈α∃of␈α⊗structure␈α⊗depends␈α∃on␈α⊗2␈α⊗things:␈α⊗whether␈α∃altering
␈↓ ↓H␈↓the␈α~order␈α→of␈α~the␈α~elements␈α→in␈α~the␈α~arguments␈α→makes␈α~any␈α~difference␈α→to
␈↓ ↓H␈↓SAME-LENGTH,␈α↔and␈α⊗whether␈α↔adding␈α↔multiple␈α⊗occurrences␈α↔of␈α↔the␈α⊗same
␈↓ ↓H␈↓element makes any difference to SAME-LENGTH.

␈↓ ↓H␈↓But␈α⊗changing␈α⊗the␈α⊗order␈α⊗makes␈α∃no␈α⊗difference,␈α⊗so␈α⊗the␈α⊗canonical␈α∃structure
␈↓ ↓H␈↓type␈α≡must␈α≡be␈α≡unordered,␈α≡either␈α≥BAG␈α≡or␈α≡SET.␈α≡ But␈α≡adding␈α≥multiple
␈↓ ↓H␈↓elements␈α_␈↓βdoes␈↓␈α_make␈α→a␈α_difference,␈α_so␈α_the␈α→type␈α_must␈α_be␈α_BAG␈α→or␈α_LIST.
␈↓ ↓H␈↓The only possible canonical type is therefore BAG.

␈↓ ↓H␈↓Next,␈α∃AM␈α∃notices␈α∃a␈α∃message␈α∃in␈α∃the␈α∃Ties␈α∃facet␈α∃of␈α⊗SAME-LENGTH,␈α∃that
␈↓ ↓H␈↓explicitly␈α⊂says␈α⊂that␈α⊂it's␈α⊃the␈α⊂same␈α⊂as␈α⊂Equality,␈α⊂except␈α⊃that␈α⊂SAME-LENGTH
␈↓ ↓H␈↓does␈α∪not␈α∪recurse␈α∩in␈α∪the␈α∪CAR␈α∪direction.␈α∩The␈α∪CAR␈α∪of␈α∩an␈α∪atom␈α∪is␈α∪its␈α∩value
␈↓ ↓H␈↓cell,␈α⊗so␈α⊗that␈α⊗message␈α⊗is␈α⊗enough␈α⊗evidence␈α⊗to␈α⊗warrant␈α⊗testing␈α⊗whether␈α∃or
␈↓ ↓H␈↓not␈α$the␈α$value␈α$cells␈α$of␈α$any␈α$of␈α$these␈α$elements␈α$are␈α%ever␈α$accessed.
␈↓ ↓H␈↓Empirically,␈α∪they␈α∪never␈α∪are.␈α∩So␈α∪AM␈α∪assumes␈α∪it␈α∩can␈α∪replace␈α∪them␈α∪all␈α∪by␈α∩a
␈↓ ↓H␈↓single, fixed value: say "T".
␈↓ ↓H␈↓Putting␈α this␈α all␈α together,␈α we␈α see␈α the␈α canonizing␈α transformation␈α∨as:
␈↓ ↓H␈↓convert the structure to a BAG, and substitute T for each element.

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

␈↓ ↓H␈↓Another␈α∀thought␈α∀I␈α∀hope␈α∪you'll␈α∀leave␈α∀with␈α∀today␈α∪is␈α∀that␈α∀(for␈α∀this␈α∀task␈α∪at
␈↓ ↓H␈↓least)␈α⊗if␈α⊗the␈α↔heuristic␈α⊗operators␈α⊗are␈α↔sophisticated␈α⊗enough,␈α⊗then␈α↔a␈α⊗simple
␈↓ ↓H␈↓numeric␈α↔calculus␈α⊗is␈α↔all␈α↔the␈α⊗meta-level␈α↔heuristics␈α⊗you␈α↔need␈α↔for␈α⊗adequate
␈↓ ↓H␈↓performance.␈α⊂ And␈α⊂you␈α⊂don't␈α⊂need␈α⊂any␈α⊂explicit␈α⊂meta-meta-heuristics␈α⊃at␈α⊂all.
␈↓ ↓H␈↓It␈α⊂would␈α⊂be␈α⊂interesting␈α⊂to␈α⊂see␈α⊂if␈α⊃this␈α⊂is␈α⊂really␈α⊂true␈α⊂for␈α⊂other␈α⊂tasks␈α⊃as␈α⊂well.
␈↓ ↓H␈↓I␈α∪suspect␈α∪tht␈α∪this␈α∪is␈α∀true␈α∪whenever␈α∪the␈α∪heuristics␈α∪are␈α∀fairly␈α∪independent
␈↓ ↓H␈↓of each other.

␈↓ ↓H␈↓One␈α~problem␈α~we␈α≠all␈α~face␈α~is␈α≠how␈α~to␈α~cope␈α≠with␈α~changes␈α~in␈α≠a␈α~system's
␈↓ ↓H␈↓knowledge␈α∩base.␈α∩The␈α∩standard␈α∩solution␈α∩is␈α∩to␈α∩track␈α∩down␈α∩all␈α∩the␈α∩effects␈α⊃of
␈↓ ↓H␈↓each␈α⊃change,␈α∩and␈α⊃update␈α⊃the␈α∩whole␈α⊃data␈α⊃base.␈α∩ The␈α⊃solution␈α⊃AM␈α∩uses,␈α⊃and
␈↓ ↓H␈↓perhaps␈α∩people␈α∪too,␈α∩is␈α∪to␈α∩just␈α∩record␈α∪the␈α∩changed␈α∪information,␈α∩and␈α∪␈↓βnot␈↓␈α∩to
␈↓ ↓H␈↓update␈α∀everything.␈α∀ When␈α∀a␈α∀contradiction␈α∀of␈α∀some␈α∀kind␈α∀occurs,␈α∀then␈α∪the
␈↓ ↓H␈↓conflicting␈α≡knowledge␈α≡is␈α≡checked,␈α≡the␈α≥knowledge␈α≡␈↓βit␈↓␈α≡was␈α≡based␈α≡on␈α≥is
␈↓ ↓H␈↓checked,␈α≠and␈α≤so␈α≠on,␈α≤until␈α≠we␈α≠encounter␈α≤some␈α≠changed␈α≤opinion␈α≠which
␈↓ ↓H␈↓explains␈α⊃the␈α⊃disagreement.␈α⊂ So␈α⊃the␈α⊃system␈α⊂is␈α⊃allowed␈α⊃to␈α⊃hold␈α⊂contradictory
␈↓ ↓H␈↓viewpoints,␈α∀so␈α∀long␈α∀as␈α∀it␈α∀could␈α∀resolve␈α∀any␈α∀dispute␈α∀if␈α∀it␈α∀had␈α∀to.␈α∀ This␈α∀is
␈↓ ↓H␈↓one␈α⊃solution␈α⊃you␈α⊂might␈α⊃keep␈α⊃in␈α⊃mind␈α⊂for␈α⊃your␈α⊃knowledge-based␈α⊃systems.␈α⊂I
␈↓ ↓H␈↓think␈α↔that␈α↔␈↓βthis␈↓␈α↔works␈α↔because␈α_there␈α↔are␈α↔so␈α↔few␈α↔contradictions;␈α_that␈α↔in
␈↓ ↓H␈↓turn is due to the independence of the resoning involved.