COMMENT ⊗   VALID 00003 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.COMMENT !XGPCOMMANDS←"/PMAR=2200"
C00004 00003	We often face the difficult task of formulating new research problems
C00009 ENDMK
C⊗;
.COMMENT !XGPCOMMANDS←"/PMAR=2200";
.DEVICE XGP
.FONT 1 "BASL30"
.FONT 2 "BASB30"
.FONT 4  "BASI30"
.FONT 5  "BDR40"
.FONT 6  "NGR25"
.FONT 7  "NGR20"
.FONT 8 "SUP"
.FONT 9 "SUB"
.TURN ON "↑α↓_π[]{"
.TURN ON "⊗" FOR "%"
.TURN ON "@" FOR "%"
.PAGE FRAME 54 HIGH 83 WIDE
.AREA TEXT LINES 1 TO 53
.COUNT PAGE PRINTING "1"
.TABBREAK
.ODDLEFTBORDER←EVENLEFTBORDER←1000
.AT "ffi" ⊂ IF THISFONT ≤ 4 THEN "≠"  ELSE "fαfαi" ⊃;
.AT "ffl" ⊂ IF THISFONT ≤ 4 THEN "α∞" ELSE "fαfαl" ⊃;
.AT "ff"  ⊂ IF THISFONT ≤ 4 THEN "≥"  ELSE "fαf" ⊃;
.AT "fi"  ⊂ IF THISFONT ≤ 4 THEN "α≡" ELSE "fαi" ⊃;
.AT "fl"  ⊂ IF THISFONT ≤ 4 THEN "∨"  ELSE "fαl" ⊃;
.SELECT 1
.MACRO B ⊂ BEGIN VERBATIM GROUP ⊃
.MACRO E ⊂ APART END ⊃
.MACRO FAD ⊂ FILL ADJUST DOUBLE SPACE PREFACE 2 ⊃
.MACRO FAS ⊂ FILL ADJUST SINGLE SPACE PREFACE 1 ⊃
.FAS
.COMPACT
.SELECT 1
.PORTION THESIS
.PAGE←0
.NEXT PAGE
.INDENT 0
.TURN OFF "{∞→}"   
.BEGIN CENTER 

⊗2SEMINAR:
Friday, February 13, 1976
10:00 a.m.⊗*



⊗5↓_Automating the Discovery of Mathematical Concepts_↓⊗*


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




.END

We often face the difficult task of formulating new research problems
which  must be both soluble  and nontrivial.  It's  usually easier to
tackle a  specific  given problem  than  to propose  interesting  yet
managable  new  questions  to  investigate.   For  example,  contrast
⊗4playing⊗*  Monopoly with  the more difficult  task of ⊗4inventing⊗*
new games of the same quality.  Can such "originality" be mechanized?


My thesis -- and this  seminar -- describe one approach  to partially
automating the  development of new mathematical concepts.   First, we
shall  consider   how  to   ⊗4explain⊗*  a   discovery  D,   how   to
systematically  analyze  it   until  it  seems  obvious.     This  is
accomplished by constructing a chain of minor discoveries, stretching
from  D back  to  previously  known  concepts.    By  inverting  this
analytical procedure,  we obtain  a simple scheme  for ⊗4generating⊗*
new  discoveries,  for  synthesizing new  theories.    To  combat the
combinatorially explosive nature of this process,  heuristic rules of
thumb are used to prune away unpromising lines of investigation.


An  experimental  interactive LISP  program  called  ⊗2AM⊗* has  been
developed, which attempts to do such simple concept-growing. That is,
AM carries out some of the activities involved in simple mathematical
research:   noticing   obvious  relationships   in   empirical  data,
formulating new  definitions  out of  existing ones,  proposing  some
plausible conjectures, and estimating the potential worth of each new
concept.    It was  necessary  to  devise a  rudimentary  calculus of
"interestingness" which enabled AM  to choose which activity  to work
on  at each  moment.   AM also  relies upon  a human  in the  role of
co-researcher.

AM was initially given some scanty information about very  elementary
pre-numerical notions  (sets,  relations, composition,  set-equality,
and  a hundred  others),  plus a  few hundred  guiding  heuristics of
varying generality.   Using  these, AM  developed several  well-known
concepts (including:  numbers, arithmetic, prime numbers,  and unique
factorization).   By interacting heavily with a human, AM was able to
motivate one  original piece  of  math research  (maximally-divisible
numbers).

After explaining the workings of AM, we can discuss such issues as

.BEGIN PREFACE 0 INDENT 4,8,0

(i) Choice  of task domain: Why mathematics?   Suitability of various
other sciences.

(ii) Experiments one can perform on AM: What do we hope to learn?

(iii) The role of the human user: spectator ⊗4vs⊗* co-researcher.

(iv) How can one  judge the performance  of a concept-proposer  which
has no fixed goal?

(v) What kinds of discoveries are most difficult to mechanize?

(vi) Assimilating  new information into  data bases:  global updating
⊗4vs⊗* living with contradiction.

.END

.SKIP TO COLUMN 1