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⊗5↓_Automated Math Theory Formation_↓⊗*

⊗2Douglas B. Lenat⊗*
Artificial Intelligence Lab
Stanford University
Stanford, California 94305

.END

Scientists often face the difficult
task  of formulating research problems which  must be soluble
yet nontrivial.  In any given branch of science, 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  with the  more  ill-defined
reasoning which led to ⊗4inventing⊗* it.

For my dissertation, I  am investigating creative theory formation in
mathematics: how to  propose interesting new  concepts and  plausible
hypotheses connecting them.  The experimental  vehicle of my research
is   a    computer   program   called   ⊗2AM⊗* (for   ⊗2↓_A_↓⊗*utomated
⊗2↓_M_↓⊗*athematician),   which carries out some of the activities involved
in mathematical research: noticing simple relationships in empirical data,
formulating  new  definitions  out  of  existing  ones,  proposing some
plausible conjectures (and, less importantly, sometimes  proving them),
and evaluating the aesthetic "interestingness" of new concepts.

Before discussing how to ⊗4synthesize⊗* a new theory, consider briefly
how to ⊗4analyze⊗* one, how to construct a plausible chain of
reasoning which terminates in a given discovery. One can do this
by working backwards, by reducing the creative act to simpler and simpler
creative acts.
For  example,
consider  the concept  of prime  numbers.  How might  one  be led  to
define  such a notion? Notice  the following plausible strategy:

.ONCE INDENT 9,9,9
"If f is a function which transforms elements of A into
elements of B, and B is ordered, then consider just those members of A
which are transformed into ⊗4extremal⊗* elements of B.
This  set is an interesting subset of A."

When f(x) means "factors of x", and the ordering is "by length", this
heuristic says to consider  those numbers which have a minimal⊗A1⊗* number
of factors -- that is, the primes. 
So this rule actually ⊗4reduces⊗*  our task from "proposing the concept of
prime numbers" to the more elementary problems of "inventing factorization"
and "discovering cardinality".  

But suppose we know this general rule:
"If  f  is an  interesting  function,  consider its inverse."
It reduces  the task of  discovering factorization  to the  simpler
task  of  discovering multiplication.  
Eventually, this task reduces to the discovery of very basic notions, like
substitution, set-union, and
equality.  To explain how a given researcher might have made a given
discovery, such an analysis is continued until that inductive task is
reduced to "discovering" notions which the researcher already knew.

Suppose a large collection of these  heuristics has been
assembled (e.g., by  analyzing a great many discoveries, and writing down
new heuristic rules whenever  necessary).  
Instead  of using them to ⊗4explain⊗* how a given idea might have
evolved, one can imagine
starting from a basic core of knowledge and
"running"  the
heuristics to ⊗4generate⊗* new concepts.

Such syntheses are precisely what AM does.
The program consists of
a corpus of primitive mathematical concepts and a collection of guiding
heuristics.
AM's activities  all serve  to expand AM  itself,⊗A2⊗* to  enlarge upon  a
given body  of mathematical knowledge.  
To cope with  the enormity of
the potential "search space"  involved, 
AM  uses its heuristics as judgmental  criteria to  guide
development  in  the  most  promising   direction.
It appears that  the process of inventing worthwhile new⊗A3⊗* concepts can be guided
successfully using a collection of a few hundred such heuristics.


Each  concept  is represented  as  a ⊗6BEING⊗*⊗A4⊗*, a frame-like data structure 
with  30 different
facets or slots. 
The types of slots include:
⊗6Examples, Definitions,  Generalizations, Utility, Analogies,
Interestingness, Uninterestingness,⊗* and a couple dozen others.
The  ⊗6BEINGs⊗*
representation provides a convenient scheme for organizing the
heuristics; for example, the following strategy
fits into the ⊗4Examples⊗* slot of the
⊗4Predicate⊗* concept:
"If, empirically, 
10 times as many elements ⊗4fail⊗* some predicate P, as ⊗4satisfy⊗* it, then
some ⊗4generalization⊗* (weakened version) of P might be more interesting than P".
AM considers this suggestion after trying to fill in examples of any predicate.⊗A5⊗*


AM is initially  given a large collection of core concepts, with only  a few slots
filled in for each.   Its sole  activity is to  choose some facet  of
some concept,  and fill  in that  particular slot.  In so doing,  new
notions will often  emerge.  Uninteresting ones are forgotten, mildly
interesting ones are kept  as parts of one  slot of one concept,  and
very  interesting ones  are  granted full  concept  status. Such  new
⊗6Beings⊗*  will  have dozens  of  blank  parts, hence  the  space of
possible actions (slots to fill in) grows rapidly.
The same  heuristics are used both to suggest new directions for
investigation, and to limit attention: both to grow and to prune.


The particular mathematical domains in which AM operates depend  on the choice of
initial concepts. Currently,  AM is given about a hundred concepts, all of
which are what Piaget might describe as ⊗4prenumerical⊗*:
Sets, substitution,  relations, equality, and so on.  In particular,  AM is not
told  anything about  proof,  single-valued functions,
or numbers.   Although  it  was never able to  ⊗4prove⊗*  the  unique  factorization
theorem, AM actually  did ⊗4conjecture⊗* it.⊗A6⊗*   Before this,  AM had to 
define and investigate
concepts corresponding to those we refer to as cardinality,
multiplication, factors, and primes,
based on reasoning similar to that in the example above.

The main difficulty with AM at present is getting it to accurately judge
⊗4a priori⊗* the value of each new concept, to quickly lose interest
in concepts which aren't going to develop into anything.
As with many AI programs, one aspect of working on AM is the
degree of precision with which one's ideas must be formulated.
The resultant body of detailed heuristics may be the germ of a more efficient
programme for educating math students than the current dogma.⊗A7⊗*
But perhaps the most exciting prospect opened up by AM is that
of experimentation: one could vary the concepts AM starts with, vary the
heurisitics available, etc., and study the effects on AM's behavior.
AM is a dissertation project ⊗4in progress⊗*; few conclusions have been drawn
yet.

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↑1 The other extreme, numbers with a  MAXIMAL number of factors, will
also  be  proposed  as  worth  investigating.    This leads  to  many
interesting questions; the only  "new-to-Mankind" 
mathematical result so far  is
in  fact that  such maximally-divisible  numbers must  have the  form
p⊗B1⊗*⊗Aa1⊗*  p⊗B2⊗*⊗Aa2⊗* p⊗B3⊗*⊗Aa3⊗*...   p⊗Bk⊗*⊗Aak⊗*,  where the
p⊗Bi⊗*'s are the first k consecutive primes, and the exponents a⊗Bi⊗*
decrease   with  i,  and   the  ratio  of   (a⊗Bi⊗*+1)/(a⊗Bj⊗*+1)  is
approximately   (as   closely   as    is   possibe   for    integers)
log(p⊗Bj⊗*)/log(p⊗Bi⊗*). For  example, a typical  divisor-rich number
is
n=2⊗A8⊗*3⊗A5⊗*5⊗A3⊗*7⊗A2⊗*11⊗A2⊗*13⊗A1⊗*17⊗A1⊗*19⊗A1⊗*23⊗A1⊗*29⊗A1⊗*31⊗A1⊗*37⊗A1⊗*41⊗A1⊗*43⊗A1⊗*47⊗A1⊗*53⊗A1⊗*.
The progression of its exponents+1 (9 6 4 3 3 2 2 2 2 2 2 2 2 2 2 2)
is  about as  close  as one  can  get  to satisfying the  "logarithm"
constraint.   This  number n has 3,981,312
divisors. The "AM  Conjecture" is that no  number smaller than n  has
that many divisors. By the way, this n equals 25,608,675,584.

↑2  Incidentally, these  basic concepts  include the  operators which
enlarge the space (e.g., ⊗6COMPOSE-2-RELATIONS⊗* is both a concept in
its own right and a way to generate new ones).


↑3 Of course, "new" means new to AM, not to Mankind, and "worthwhile"
can only be judged in hindsight.

↑4 Lenat, Douglas, ↓_BEINGS: Knowledge  as Interacting Experts_↓, 4th
IJCAI, 1975, pp. 126-133.

↑5  In fact, after AM  attempts to find examples  of SET-EQUALITY, so
few are  found that  AM  decides to  generalize that  predicate.  The
result is the predicate which means "Has-the-same-length-as" -- i.e.,
the rudiments of Cardinality.

↑6  Due to the firm base of  preliminary concepts which AM developed,
this relationship  was almost obvious.   AM  sought some predicate  P
which,   for  each  n,   some  member  of   FACTORS-OF(n)  satisfied.
ALL-PRIMES was such a  predicate.  AM  next constructed the  relation
which associates,  to each  number n,  all factorizations  of n  into
primes.   The full statement of the UFT  is simply that this relation
is a function; i.e., it is defined and single-valued for  all numbers
n.


↑7 Currently, the educator takes the very best work any mathematician
has  ever done,  polishes it  until its  brilliance is  blinding, and
presents it to the student  to induce upon. A few  individuals (e.g.,
Minsky and  Papert at MIT, Adams at  Stanford) are experimenting with
more realistic strategies for "teaching" creativity.