is a collection of programs developed at MIT which go under the name
"MACSYMA", whose purpose it is to aid mathematicians in symbolic ma-
nipulation of complex mathematical expressions. This program has in it
some sense of "where to go "-a sort of "complexity gradient" which guides
it from what we would generally consider complex expressions to simpler
ones. Part of MACSYMA's repertoire is a program called "SIN", which
does symbolic integration of functions; it is generally acknowledged to be
superior to humans in some categories. It relies upon a number of differ-
ent skills, as intelligence in general must: a vast body of knowledge, the
technique of problem reduction, a large number of heuristics, and also
some special tricks.
Another program, written by Douglas Lenat at Stanford, had as its aim
to invent concepts and discover facts in very elementary mathematics.
Beginning with the notion of sets, and a collection of notions of what is
"interesting" which had been spoon-fed into it, it "invented" the idea of
counting, then the idea of addition, then multiplication, then-among
other things-the notion of prime numbers, and it went so far as to
rediscover Goldbach's conjecture! Of course these "discoveries" were all
hundreds-even thousands--of years old. Perhaps this may be explained
in part by saying that the sense of "inteFesting" was conveyed by Lenat in a
large number of rules which may have been influenced by his twentieth-
century training; nonetheless it is impressive. The program seemed to run
out of steam after this very respectable performance. An interesting thing
about it was that it was unable to develop or improve upon its own,sense of
what is interesting. That seemed another level of difficulty up--or perhaps
several levels up.
The Crux of AI: Representation of Knowledge
Many of the examples above have been cited in order to stress that the way
a domain is represented has a huge bearing on how that domain is "under-
stood". A program which merely printed out theorems of TNT in a
preordained order would have no understanding of number theory; a
program such as Lenat's with its extra layers of knowledge could be said to
have a rudimentary sense of number theory; and one which embeds math-
ematical knowledge in a wide context of real-world experience would
probably be the most able to "understand" in the sense that we think we do.
It is thiS" representation of knowledge that is at the crux of AI.
In the early days it was assumed that knowledge came in sentence-like
"packets", and that the best way to implant knowledge into a program was
to develop a simple way of translating facts into small passive packets of
data. Then every fact would simply be a piece of data, accessible to the
programs using it. This is exemplified by chess programs, where board
positions are coded into matrices or lists of some sort and stored efficiently
in memory where they can be retrieved and acted upon by subroutines.
The fact that human beings store facts in a more complicated way wasArtificial Intelligence: Retrospects 615