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C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.NSECP(Introductory Materials)
C00004 00003
C00005 00004	.NSECP(Overview)
C00006 00005	.SSEC(Motivation: why is this worthwhile?)
C00008 00006	.SSEC(A 10-page summary of the entire project)
C00014 00007	.SSEC(Guide to reading the remainder of the thesis)
C00018 ENDMK
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.NSECP(Introductory Materials)

.SSEC(Abstract)


<< Edit this abstract. >


A program is described  which ⊗4creatively⊗* uses a large  network of
concepts, techniques,  and data to enlarge  that same knowledge base.
The subject matter was chosen to be "elementary mathematics," and the
program manages to do some rudimentary research in number theory.

The  representation  of  concepts  as  active,  structured  knowledge
modules  provides  a definite  "space"  for the  program  to explore.
Hundreds of "local" heuristic  rules are relied upon,  to effectively
guide the investigations.

.NSECP(Overview)

.SSEC(1-page summary of the project)

<< first pass: condense the 2-page blurb handed around earlier>

   A 1-sentence summary of the thesis 
	"Creative scientific discovery (at least in elementary mathematics)
	 can be successfully represented as a heuristic search process"

.SSEC(Motivation: why is this worthwhile?)

.B

	Inherent interest of getting a handle on the task (sci. creativity)
		Personal belief that discovery can be (ought to be) demystified
		Potential for learning, from the system, more about the process 
			of sci. concept formation, thy. formation, chance discovery
			(do experiments on the implementations: eg, vary AM's heurs)
	Potential usefulness of the implementations themselves (including AM)
		Aids to research; i.e., ultimately: new discoveries.
		Potential to education: like Mycin, extract heurs. and teach them
	All the usual bad reasons:
		"Look ma, no hands" + maternal drives + ego + thesis drives +... 
	Historical: 
		Need task with no specific goal, to test BEINGs ideas.
		Disenchantment with theorem-provers that plod along, in contrast
			to the processes which my model of math demands: intu, need,
	                aesth., multiple reprs, proposing vs proving, fixed task.
	
.E

.SSEC(A 10-page summary of the entire project)

<<first pass: condense the 20-page text of the 1-hour "whirlwind tour" talk>

<potential organization: mirror the overall organization of the thesis itself>

As part of this, we will have:


.SSSEC(HISTORIAN'S SUMMARY of AM's researches)

Before diving into the  depths of AM, let's take a  moment to discuss
the  totality  of the  mathematics  which  AM carried  out.    Like a
contemporary historian summarizing the  work of Euclid, we shall  not
hesitate to use current terms, and criticize by current standards.

AM  began  its   investigations  with  scanty  knowledge   of  a  few
set-theo{muc≡{;∂↔π#M↓↓G≠↔SMbβ↔GW∞K3'SJ↓β?→π≠↔SMb↓βO↔"β?C↔⊗S'?w→%84Tk?OQε{→↓β&C∃β?↔3'?W~↓βO↔"kS#↔␈∪eβK.cπS'}sM↓#*s≥91αβ∪∃αn{K∨πr;M↓βf←M$hS←↔K*β↔[↔w#Wπ3gI↓βWv≠?[↔⊗+⊃mβ≡K;∂∃∧
5β;/3↔I↓ε3W33JβW;∪/∪OS?}!βπ∨#Kπ∂ h+π3>+K¬bβS#∃π≠SπS.k↔;Qε;⊃↓π3↔K'6K∂πSN{9β?2β↔π∂B↓β?→π##↔O*β←πMαβGW'&(4+?↔≠∂WK*q↓↓αi↓β;/3↔Iβ&+K'[.!↓β¬αβ≠?Kn1β;␈#'?9αβ?→↓εK;≠'vKSe1ε∪WQ↓εKP4+v'[↔gIβ↔O&3'≡C↔⊃β≡{;+↔∨#WK↔~↓β3'↑)↓¬π≠↔Qβ≡9β;/3↔Iβ⊗)β¬βn+7↔∩β?_4VKSO↔f1	1β∞s⊃βC⊗{∂↔∪/∪↔Mβ6{Iβ7∞[';≥ε≠#π'w→β?→εs↔]β≡+SM↓B∪';O/∪Q↓β
βO↔PhS';Sz↓β'S≡+3→	Jq↓↓↓∧s=↓↓π≠?C#O≠S'∂∂#↔⊃↓π≠↔Q↓π##↔?↔I↓↓#&Kπ∨?v3'k∂#'?9`h+?K&K;π3~Iβ←π~β↔[↔∩β∪?;*p4(4T≠S↔∩βS#'~↓β';O#'π1πβ↔K'}!β?→ε+cC3␈∪πS'}q1αεjβ∪↔∂N#↔⊃β&CπQ↓⊗+GWπfKSeλhS←πMαβ←?K&A↓↓β>+;↔K∞c'k'v91↓β∞s⊃↓β&C↔K↔↔I↓↓β&KO∂?6+K↔⊃αβS#∃α↓βK↔fS'?ph)O∞k∃7OOS∃7π~⊃9↓α≡K∪'v3'SJβ←πMαβπO.!β?9π##'Mbβπ;⊃π≠??9εk?OQπ≠'7Cf(4+π⊗KS#7/#'
β␈β↔Kπ&K?;Mπ;↔K∃ε#↔≠'v+⊃9α≡K;∂∃ε∪∪'&K?9β∂∪?O∃εMβπrβπ;πf{≤4+&yβW;N{91β∞s⊃β7.cS'CfK∂πSN{9↓β∂→βπ9ε;π3}9βS=ε≠K?O~kCK?'+∂Q1αβ'Qβ≡7∀4VMβG.KS∃↓ε	βOW↔βK'O*β←#↔rαε5βv{S'∂.!↓βSFQβSF+eβ←/∪∃βK.cπS↔"↓#;πn+3e0hR9.9k∪b9%rαO??r↓βπ≠&+Iβ∪.3';'v9↓β7.cS'CfK∂πSN{91αi↓β'w3↔OSN;πS↔"↓βS#(h+CK}≠↔OMε{→β7.cS'CgK';≥ε	β;Wn∪↔Iβ↔I↓β''≠↔3→RβOGW∂∪';≥r↓αS#*β';[/∪O∃β}04+SFKM↓β'+↓¬'rned out  to  be interesting,  and led  to the  definition of
square-root.

Although AM was very close to discovering irrationals  at this point,
it  turned aside and  was content  to work  with integer-square-root.
Raising to fourth-powers, and fourth-rooting, were discovered at this
time.

The associativity  and symmetry of  multiplication indicated  that it
could accept a  BAG (a multiset) of numbers as its argument. This led
to the  notion of  factoring a  number. Minimally-factorable  numbers
turned out to  be what we call primes.   Maximally-factorable numbers
were  also thought  to be  interesting at  the time,  and in  fact an
unusual $$ As far as  the author and his committee know, this  is the
first  such  explicit characterization  of  these  numbers, hence  is
probably "new-to-Mankind".  Since the purpose of the thesis is not to
derive  "new"  mathematics,   discussion  of  this  result   will  be
suppressed here. The  interested reader may wish to see Appendix 2. $
characterization of such numbers was discovered.

AM  conjectured  the  fundamental   theorem  of  arithmetic   (unique
factorization  into primes)  and  Goldbach's  conjecture (every  even
number >2  is the sum of two primes) in a surprisingly symmetric way.
The unary representation of numbers gave way to a representation as a
bag of  primes (based on unique factorization),  but AM never thought
of exponential  notation.   Since  the key  concepts of  modulus  and
exponentiation were never  discovered, progress in number  theory was
arrested.

.SSEC(Guide to reading the remainder of the thesis)

.B

	i) Overall organization of the thesis
	ii) Plans for what to read (and in what order), depending on your interests
		Plan for those interested in the AI ideas
		Plan for those interested in the systems ideas
		Plan for those interested in mathematics
	iii) Pre-requisites and how to satisfy them, for each chapter
		For those with little pure mathematics in their background
		For those with little computer science background
		For computer scientists with little contact to AI before
	    <either organized by "type" of reader, or by chapter/section>

.E

.SSSEC(VARIED READERSHIP of this thesis)

⊗7¬(A∨C∨M)⊗*

This thesis  -- and its  readers --  must come to  grips with a  very
interdisciplinary  problem.   For the reader  whose background  is in
Artificial  Intelligence,  most  of  the  system's  actions   --  the
"mathematics" it does  -- may seem inherently  uninteresting. For the
mathematician,  the  word "LISP"  signifies  nothing beyond  a speech
impediment  (to Artificial  Intelligence  types  it also  connotes  a
programming impediment). If I  don't describe "LISP" the first time I
mention it, a large fraction of potential readers will never  realize
that potential. If I ⊗4do⊗* stop to  describe LISP, the other readers
will  be bored.   The standard solutions  are to  sacrifice one fixed
community in favor of the other, or to be entertaining enough to keep
both of them around. In this work, a third alternative will be taken.
Sections  will be  tagged with  descriptors like  "⊗7M⊗*" (indicating
that the section will be of interest to those who enjoy mathematics),
or "⊗7¬A⊗*" (the  section will be a waste of  time for those familiar
with Artificial Intelligence). The labels will consist of the letters
⊗7A⊗* (for  ↓_A_↓I),  ⊗7M⊗* (for  ↓_M_↓ath), and  ⊗7C⊗* (for  general
↓_C_↓omputer  science), connected by standard  logical symbols: ⊗7¬⊗*
(negation), ⊗7∧⊗* (and), ⊗7∨⊗* (or).  For example, since  the current
paragraph is  labelled ⊗7¬(A∨C∨M)⊗*, the  reader can assume it  is of
little interest to anyone.

In addition, there are two glossaries of terms. Appendix 1a contains
capsule descriptions of about 100 mathemtical terms, ideas, notations,
and jokes. Appendix 1b renders the analogous service for Artificial
Intelligence jargon and computer science concepts.