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C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	Outline for Dissertation
C00003 00003	Chapter 1: Overview 
C00007 00004	Chapter 2: A Model for Creative Discovery in Science
C00023 00005	Chapter 3: Designing a Math Theorizer
C00025 00006	Chapter 4: Designing a Particular System
C00026 00007	Chapter 5: Some Experimental Forays
C00027 00008	Chapter 6: Discussion of Results
C00028 00009	Chapter 7: Conclusions
C00029 00010	Appendix 1:
C00030 ENDMK
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Outline for Dissertation

Orals Committee: B. Buchanan, E. Feigenbaum, C. Green, D. Knuth

Reading Committee: E. Feigenbaum, D. Knuth, A. Newell

Title: Automating the Discovery of Mathematical Concepts

Abstract
	<200 words long. Should be one of the last things written>
	A 1-sentence summary of the thesis 

Dedication and opening quotation

Table of Contents

Chapter 1: Overview 

a) 1-page summary of the project
<first pass: condense the 2-page blurb handed around earlier>

b) Motivation: why is this worthwhile?
<should this really be separate from the rest of the intro? 
  It is "meta" to the project, BUT so are most of the conclusions...>
	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.
	

c) 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>

d) Guide to reading the remainder of the thesis
	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>

Chapter 2: A Model for Creative Discovery in Science
<short, concrete, readable, many examples, relevant to the remainder of the thesis>

a) Scientific discovery as heuristic search: evolution of the model
	Each node in the space corresponds to a concept
		Concepts can be static (Sets) or Active (Composition)
		A relnship. (e.g., a theorem) is itself a concept
		An argument (e.g., a proof) is also considered a concept
	The "legal" moves (ways to expand the tree of nodes, to grow new ones)
		are too numerous to be seriously considered.
        The real operators are themselves heuristic rules of thumb
	So the "space" itself weeds out all nodes except those proposed for some
		good heuristic reason.
	As simple numerical calculations show, this space is still enormous.
	Using the big-switch idea, we can restrict our attention to 1 domain
		(e.g., math) and only use ITS nodes and heur. operators.
	Even so, space is still too big
	Refine the big-switch idea: 
		When worrying about nodes N1...,Nk, only consider heuristic
			operators known to be rele. to those concepts.
	Even so, the space is still too big
	We recurse: we use heuristics to reduce our search. These new meta-heuristics
		(strategies) guide our attention (which nodes to look at next,
		which operators are most promising to apply to each selected node).
	If these are good enuf, then we are through (else consider meta-meta-heurs.)

b) The model finally produced
	Recapping, we state explicitly how we decide what to do next at any moment.
	Indicate the data structures, the flavors of heuristics, control flow, etc.
			Note that "apply heur. operator F and add corresponding
			new node N" is considered primitive for the moment.

c) Evidence in favor of (empirical validation of) that simple model
	i) A prediction: Character of interdisciplinary research
	   For a novice in some field to do new research, he must learn the rele.
		already-known concepts, and (probably) must learn the rele. heurs.
		(e.g.: bubble-chamber physics exeriments, molec. genetics)
	   On the other hand, if he has expertise in another field, he brings with
		him many new heurs. to apply (hopefully a couple carry over and were
		never applied before in this new field), plus he brings with him the
		knowledge of a new net of concepts, from which he may draw analogies
	   Because of this, interdisciplinary research can be very productive 
		especially if you're the first such link (e.g.: Suppes)

	ii) Turn the model upside-down: Analyzing a given discovery
	   When we hear a new discovery, we try to perceive a path backwards from
		it, connecting to concepts we already know. The easier this is,
		the less mystified we are by the discovery. 
		Consequence: Discoveries in alien field seem magical and v. hard
		Consequence: Let-down after seeing how a magic trick is performed
		Consequence: Let-down after seeing how an AI program really works.
	   Reasons why going backwards is probably easier than going forwards
		Easier since you have a given starting point (the given discovery)
			The alternative is to find the "right" set of known concepts
			which will eventually lead to some interesting new concept.
		Easier since the target space is huge (get to any known concept,
			vs get to any very interesting concept)
			Since nature is unkind, and very few avenues lead to 
			interesting new discoveries, valuable new concepts.
			On the other hand, in working backwards with our heurs,
			we worry about branching also, and must be able to quickly
			tell if we are really simplifying the situation!
			(i.e., Even if the model is right, we must herein worry
			 about the branching factor when going in reverse)
		Easier if you are shown the discovery step by step
			Since then you will know the necessary intermediate concepts
			Since then each step will seem "easy" and obvious
		Consequence: Reading journal article, feel "I could have done that"
		Consequence: Work 3 years, make discovery, kick yourself for not
			having seen it earlier, since it was so obvious.
		Consequence: given a math or physics problem, you're more impressed 
			with the solution if you spend (waste?) a few hours trying 
			to solve it, than if you just read thru the soln. at once.
	iii) "Failure" is due to missing some "right" heurs/concepts, or the wisdom
		(i.e., strategies, meta-heuristics, etc.) to use them effectively.
		Consequence: Teaching by example forces students to induce the 
			(meta-)heuristics themselves, often imperfectly.
			E.g.: many GP's failed the antibiotic drug prescription test
		Consequence: One can -- and should -- teach the strategies directly
			<is this verified anywhere? counter-indicated? Polya? >
	iv) Momentous occasion in science is typically due to discovery of a brand
		new heuristic, or else due to creation of a new concept unconnected
		to the existing concepts by some plausible operator.
		Occasionally, all one finds is a daring interdisciplinary analogy.
		Occasionally, just the first to follow a perfectly plaus. path.
		Examples: 
		  Non-Euclidean geometries ("Counter-intuitive systems may
			still be consistent and interesting")
		  Relativity ("Counter-intu. sys. may even have physical
			reality; simultaneity may be a local superstition)
		  Knuth's (Conway's) Surreal Numbers (no contacts, not
			easy to see how/why their defn. was ever considered int.)
		  Schroedengier's wave equation (plucked out of thin air)
		  Methods of complex analysis (orig. based only on analogies)
		  Mendelev's periodic chart (first one to write down the
			recently-discovered data about atomic wts. systematically)
		  < ... more examples ? ... >
	v) Presence of Zeitgeist in Science: often, a discovery is made simultaneously
		Because many researchers simult. hear some fresh concepts, and apply
		the same bag of tricks. 
		Example: calculus (Newton, Leibnitz), ...
			
d) Questioning that model
	i) Misleading character of polished results
		It would seem, from reading texts and journals, that science itself
		is more a flowing, smooth development, than a backtracking search.
		This is illusory, as we scientists know. (we fend off this attack!)
	ii) Omissions:
		Serendipidy, incubation, unconscious, wealth of analogic materials
			(mass of introspections from great scientists are mystical)
		Error: accidental good fortune (Franklin using string, not wire;
			Lederberg picking one of the few bacteria that really do
			reproduce sexually; Galois forced to cram a lifetime of
			creativity into one pre-duel burst of writing).
		Inability to explain why "long-shot" investigations were undertaken.
			(eg: superconductivity discovered as a Master's project)
		Zeitgeist: popular trends/fads in science at the time;
		Cultural and political themes popular at the time.
			In CS: Heirarchy: Prussian army, bureaucracy
				Cooperating modular experts
				Tremendous dependence on existing hardware
				(also true of numerical analysis)
			Social taboos against experimenting with human subjects
		Difficulty of generating new operators and unconnected nodes
			Defense: this really IS rare, so again the model is OK
			Rebut: But it DOES happen every now and then!! How?!
		Focus of attention
			People inherently are biassed in favor of recently-considered
			nodes; they can't flit back and forth from one leaf to another.
			This can be simulated in a heuristic search (by artificially
			weighting the concepts and heuristics based on recency of
			use), but it is	not an inherent part of the model.
	iii) Getting down to earth: limitations of the model
		What kinds of creative activities is this a bad model for?
			Brainstorming (intentional lack of plausibility)
			Problem-solving (very specific goal)
			Situations where there are very few heuristics (prop. calc)
			Situations where the major difficulty is applying a heuristic
				once it's chosen (e.g., soft fields like sociology)
			Absurdly robust or delicate chains of discoveries
				A delicate chain is really like problem solving
				(there exists only 1 interesting concept in this
				part of the space)
				A very robust section of space needn't worry about
				using heuristics (if every move produces something
				very interesting)
		Exactly what situations does it honestly capture?
			Essentially undirected research in a very hard science,
			with only secondary kinds of discoveries anticipated.
			Many heuristic operators exist for any given node,
			and a few meta-heuristics exist for the domain.
			Where the percentage of int. nodes "out there" is low
			  (under 10%) but not negligible (exists only one!)

**********************************************************************
***NOTE***: The remainder of this page might be better suited to a category
like "difficulties with the model due to pragmatic considerations", and placed
somewhere in Chapter 4 (Implementing this system).
**********************************************************************
		Pragmatic limitations
			Most seriously, it treats "add new node N" as primitive.
			In real life, people spend much time "adding a node".
			They have to answer many questions about it, play with it,
			and try to relate it to other known concepts. In this way,
			the worth of the node is estimated, and new empirical data
			is gathered which may trigger some heurs. to suggest the
			next drection to take, the next concept/relnship to explore.

e) Fixing up the model
	i) The most serious pragmatic limitation is this business about how much
		work must be spent to fill in any new concept. Fix this up by
		assuming that each concept has facets, not all of which need be 
		filled in at its conception. Then some heuristics (meta-heurs) can
		be concerned with tasks at the level of filling in a facet
		(deciding which facet of which concept to work on next). The basic
		control structure can in fact be oriented around filling in facets.
		If desired, the creation of new nodes can be a side effect!
		Give diagrams illustrating all this.

Chapter 3: Designing a Math Theorizer

a) Research in various domains proceeds slightly differently
	Some examples of this...

b) By limiting our attention to one particular domain, we can add much
power to our model (although sacrificing its generality, as usual).
This brings up the choice of domain. 
Very briefly mention: Choice of task domain to test out the model
	i) Why math?
	ii) What else could it be/ not be?
The "raw materials" for adding this power will come from understanding precisely
how math researchis carried out. A first pass will be to propose some model for
this process, based on writings by Polya, Poincare, Hadamard, etc.

c) Detailed model of math research
What are the peculiarities, the details of MATH research, that enable us
to add power to our model (if we are willing to make it specific to math)?

d) Implications for an automated mathematician
What new constraints, new design features, are required/suggested by the
domain-specific features of the model of math research?

Chapter 4: Designing a Particular System

a) Gradual development of the representation and control structure

b) Final set of starting concepts; initial info. (incl. heuristics) about each.

c) Overall picture of the control structure, data structures, interaction w/user,...

Chapter 5: Some Experimental Forays

Simple examples of AM in action, presented on several levels.

Some Detailed/Advanced Examples
	Periodically, give a snapshot of the changes in the sys. itself
	(what new concepts, new slots filled in, new int. values, etc.)
Chapter 6: Discussion of Results

a) Measuring performance.

b) What was (not) done by AM?

c) Numerical data: time, space.

d) Human engineering in such a system  (and in AM in particular).

e) Experiments on AM
	Importance of various heurs.,
	Importance of kinds of heurs., 
	Vary the starting concepts, etc.

Chapter 7: Conclusions

a) What gives AM it's power? Identifying the crucial ideas.

b) Ultimately, what kinds of things could AM-like systems (never) do?

c) Uses for AM-like systems: 
	Synergy: AM most valuable as a co-researcher 

d) Implications for math education.

e) Directions for Future Research
	i) Parts of the grand plan still not realized in AM.
	ii) New ideas for future work on AM.
	iii) Extending AM to other domains in math and other fields.
	iv) Factoring out all the hack sci. discoveries.

Appendix 1:
The Theory of Maximally-Divisible Numbers

Appendix 2:
History of the "BEINGs" representation scheme

Appendix 3:
Some sample Concepts and Heuristics, as coded in LISP

Appendix 4:
Some traces of AM in action

Appendix 5:
Bibliography, Documentation, Acknowledgements