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⊗5Summary: to accompany AM discussion⊗*

.GROUP SKIP 2
⊗2↓_The New Ideas_↓⊗*

.EVERY HEADING(⊗7Douglas B. Lenat, {DATE},page   {PAGE})

1. Theorem-proposing is interesting and quite different from Theorem-proving.

Researchers in all branches of science continually face the difficult
task of formulating problems and subproblems 
which must be soluble yet
nontrivial.
In any given field, 
it is usually easier to
tackle a specific given problem than to
propose interesting yet managable
new questions to investigate.
For example, contrast    ⊗4solving⊗* the Missionaries and Cannibals
problem,
against the more ill-defined task of 
⊗4inventing⊗* it ("create a new puzzle, which can be explained in under a minute,
which can be solved by an  intelligent person in several minutes").

For my dissertation, I am investigating
creative theory formation in ⊗4mathematics⊗*: 
how to propose interesting new concepts  and plausible
hypotheses connecting them.

2. This task is ⊗4not⊗* "mystical;" mathematical creativity can be analyzed.

There are several reasons for choosing math as a task domain, and several
domain-specific hypotheses that I have developed about doing that task.
These are discussed elsewhere [ask author for full proposal].
Suffice it to say that a detailed model of mathematical research has been
formulated. This "heuristic calculus" assures interestingness, just as
formal logic assures validity. Incidentally, this model indicates that the
order of presentation of material in a typical college math textbook is
almost the precise ⊗4opposite⊗* of the order of development.

The "secret" of the method is a corpus of local criteria to evaluate 
interestingness, utility, aesthetics, etc. of a given concept, and also
to suggest new, interesting things to investigate based on existing knowledge.
"⊗4Local⊗*" simply means that each criterion is associated only with the
subset of concepts to which it applies. Applying it to a very different
class of concepts is conciously recognized as a daring analogical experiment.

For example: how can we explain the fantastic discovery of prime numbers?
Well, given the heuristic "For operation f:A→B, if it empirically appears
that ⊗6(∀xεA)⊗* (at least P(x)), for some predicate P, then consider the subset
of A consisting of those x for which ⊗4precisely⊗* P(x). In the case of
⊗6A=natural numbers,
f=#-FACTORS-OF, P=IS-A-DOUBLETON⊗*, this simply says to consider those
numbers which have only two factors: the prime numbers. This simple 
heuristic ["examine the extrema"] has reduced the task from inventing primes
to inventing factorization and counting. A simple heuristic like
"If f:A→B is interesting, investigate the inverse operation f↑-↑1:B→A".
In case we've uncovered multiplication, this says to consider factoring.
But how to discover multiplication?  This process can be continued
until such a primitive base is reached that the needed notions are already
in AM. By reversing this "working backwards" reduction, a
scenario of rational exploration and discovery unfolds.

3. The enormous difficulties of representing the math knowledge, and controlling
the exploration of the space of growing concepts, can both be handled by Beings.

The   experimental vehicle   of my research is
a computer program called AM
(for ⊗2↓_A_↓⊗*utomated ⊗2↓_M_↓⊗*athematician),
which can actually do very simple mathematical research:
formulate new definitions out of existing ones,
propose some plausible conjectures (and, less importantly, sometimes
prove them),
and evaluate new concepts for aesthetic
interestingness. 

At any moment, AM is simply a collection of partially filled out data structures.
Each structure, called a Being, corresponds to one mathematical concept
(e.g.: bag, induction, the set of prime numbers, intersection). Each Being
has room for about 30 parts (slots) which correspond to the facets of that
particular concept (e.g.: definition, domain/range, algorithms, examples, worth).
The control structure of AM is simply to repeatedly select a blank part of a Being
(an unexplored facet of a concept; an empty slot of a Frame data structure) 
and fill in that part of that Being. In so doing, some new potential concepts
will usually be noticed, and some of them will be added to the list of Beings
that AM knows about. Some ways that these new concepts arise are:
(i) When filling in examples of B, one of those examples may be very interesting;
(ii) By generalizing or specializing B's definition; (iii) By noticing an empirical
pattern and trying to formalize that phenomenon;
(iv) As the result of executing B, where
B is any Being which is an operation (e.g., Compose).
When a new Being is first created,
most of its 30 parts will be blank; thus AM will have dozens of new 
specific tasks added to its space of possible actions (total number of blank
parts in the system).
That is,
AM's activities all serve to expand AM itself, to enlarge upon a given body
of mathematical knowledge.  Notice how the Beings formalism  inherently
unites the representation with the control structure.

An advantage of requiring that all the Beings have the same set of parts is
being able to efficiently ⊗4ask for help⊗*. Say we've filled in a few new examples
of concept X. To check them, AM simply ripples outward from X in the 
Generalization direction, asking each Being encountered for assistance.
That is, AM would ask X "How can I check examples ofX?"; AM would also ask that
question to all members of the GENL part of X; to all of ⊗4their⊗* generalizations,
etc.

Another advantage is the efficient computation of relationships related to the
chosen set of parts. Examples of X include (i) entries on the EXS part of X;
(ii) Examples of any entry on the Specializations part of X; (iii) Specializations
of any example of X. Letting "↑*" represent repeated application, we can
write this compactly as the equation
EXS(X) = SPEC↑*[ EXS(SPEC↑*(X)) ]. 

.SKIP 2

⊗2↓_Measuring AM's Success_↓⊗*

There is no specific "goal" that AM is supposed to achieve; in fact, the 
specification of any such preconceived goal would probably interfere with
creating a system general enough to explore a large space of concepts.
Rather, AM aims toward doing what can actually be called mathematical
research (although it starts at such a primitive level that it probably
won't discover anything new to Man).
Some "milestones" in its development might be the following:

.SKIP 1

.B1

(1) Define a new concept, far removed from anything it began with,
which is interesting (e.g., "prime numbers", "lattice", "gcd", "ergodicity").

(2) Find examples of some interesting concept AM has created.
(e.g., find some numbers, some prime numbers, some new compositions).

(3) Propose a non-trivial conjecture involving 
such newly-created interesting concepts.
(e.g., "The size of AxB is the same as that of BxA", "Each number has a
unique representation as a product of prime numbers", "There are 2↑A↑x↑B
different relations from a set with A elements into a set with B elements").

(4) For these new concepts and  conjectures, explain intuitively why they
are interesting, why they were
proposed, how one might go about developing the
concepts further. In fact, for each activity undertaken, justify (by a list
of meaningful reasons) why it was tried, why it was not tried before/after
this point in time, why it was worth spending as much cpu time on it as AM
spent, etc.
It is fine for AM to waste its time exploring dead-ends, as long as they are
real "traps"; it is ⊗4not⊗* permissable for AM to simply do a rapid search
in a huge space, with no justification for each point considered.

.END

.SKIP 3

↓_Some measures of success (in decreasing order) include the following:_↓

.SKIP 1

.B1

(1) Compare the given initial knowledge to the derived knowledge.
Observe the quantity, the depth of sophistication, the variety and quality.

(2) Examine the route AM took to do anything. Does it resemble blind searching?
Did AM quickly recognize truly fruitful new concepts? Quickly abandon
developing truly worthless ones? What was its reasoning at each step?

(3) Examine the character of the user-system dialogue. Must the user provide
too many hints? Does he have no control at all? Can he follow what AM does?

(4) Is AM really an experimental vehicle? Can you push it into discovering
quite different mini-theories, or (like PARRY and PUP6) does it try to get back
onto one single main line of development? Can you  excise or add a few Beings
and see interesting changes in behavior? Does changing the judgmental criteria
change the behavior in interesting ways? Can these be easily adjusted?

(5) Pragmatics:
Is AM practical on today's machines (space and time considerations)?
Has anyone shown great interest in it? Gotten a dissertation out of it?
Gotten N papers out of it?

.E

.SKIP 2

⊗2↓_Dimensions of Future Work_↓⊗*

There are several quite different directions in which AM is being
worked on:

.B1

(1) What AM creates: more new concepts, more advanced ones.
Perhaps the discovery  of concepts
hitherto unknown to Man. (?!)

(2) The quality of its reasoning: more judgmental criteria, better intuitions.
More "thoughtfulness" and less dead-ends.

(3) The user interface (more intelligible, tougher, self-explanatory, more
facilities for user interaction, model of the user's state of mind).

(4) Analysis: consider the initial basis of concepts; compute the size of
the search space involved, the power of certain concepts and heuristics.
What primitives are missing for "completeness"; what is the possible subset
of known math that AM might stumble into, and what branches are unreachable?
What implications does AM have toward 
improving math education; toward automating theory formation
in other disciplines?

(5) Experiments: empirically study the effects of removing various concepts,
various heuristic criteria. 
Vary the user's role. Vary the weights involved.

(6) Documentation: dissertation writeup, articles, talks, planning for the future.
Cleaning AM up; removing kludges, patches. Adding comments, rewriting for clarity.
Extract the heuristics used, collect them into one file, translate them into 
English, try to develop a taxonomy of heuristic criteria.
Extract the primitice concepts and the specific facts needed about each one,
translate into English, compare to Piaget's and others' studies,...


.E