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One of the areas of artificial intelligence (AI) that has received serious
attention in the last decade or so is symbolic math.  That is, AI programs
that duplicate what mathematicians, engineers, physicists, astronomers, etc.
do when they manipulate complex algebraic or calculus problems.  The two
most popular results of this work are MACSYMA and REDUCE, both written in
LISP.  MACSYMA is available from Symbolics for multi thousands of bucks
for some machines.  REDUCE, developed at RAND Corp., is public domain and
is available for Franz LISP systems (with full source) for $300 from
Dr. Kamal Abdali at Tektronix (I have his address and phone number if you
are interested).  To give you an idea of the scope and complexity of such
systems, REDUCE involves more than a megabyte of LISP *source* code.

We just got REDUCE running here on our P-E 3230 under Franz LISP.  If you
or anyone at your site doesn't do much math, stop reading now.  But if you
do, you may understand our excitement at it's first test.  Some engineers
had a problem that had stumped a PhD mathematician and the engineers
themselves for days.  REDUCE ran it and got the answer they had hoped for
in 17 seconds!  Incidentally, one of the test suites (reduce.tst) takes
106 seconds on the 3230 versus 90 seconds on a Vax 780.  That is, the
3230 is about 85% the speed of a Vax 780.

Note, this does not involve numerical methods, approximations or anything
of the sort.  REDUCE's input is the equations themselves (it's interactive,
you type in the equations or read them in from a file you built with vi)
and its output is the new equations you want that result from differentiation,
integration, matrix operations (i.e., inverse, determinant, trace, ....),
solution of simultaneous equations, etc.  Among its many tricks is a FORTRAN
output mode (switch selectable) in which the resulting equations are output
as legal FORTRAN code to calculate the equations, complete with continuation
cards, temp variables and all!   I could go on and on, but the reality is
that you can't believe the power of this system until you see it for yourself.

					Steve Burbeck
					Linus Pauling Institute
					440 Page Mill Road
					Palo Alto, CA 94206
					(415)327-4064