As a mathematician, operations researcher, and computer programmer, I feel I have to respond to Raphael Needleman's March editorial ("Mutant Chips") that warns against depending on heuristic methods, because they may work, but "we don't know why."
Neural networks are used for absolutely everything that humans do, and deterministic algorithms cannot duplicate human performance in many cases. This isn't a temporary situation. Chaos theory and Godel's incompleteness theorem both guarantee that we won't be able to solve every problem in a deterministic way.
Artificial intelligence--like real intelligence--depends on heuristic methods, and computers won't be doing anything really interesting for us until heuristics are built into chips.
I also have to make a comment about the "How To Bruise an Integer" text box in Tom R. Halfhill's article "The Trut
h Behind the Pentium Bug" (March). A number like 4.1 or 1.1 or 0.1 cannot be exactly represented in binary floating-point values. The binary equivalent of 0.1 (decimal) is 0.0001100110011 (binary), where the "0011" sequence repeats infinitely. That is, the fraction 1/10 has a repeating binary representation, in the same way that 1/3 has a repeating decimal representation.
When we use Calculator to do arithmetic, we forget that we are doing things approximately, through binary floating-point notation. That leads to disconcerting results. This is not because we have "bruised an integer," but because we have truncated a floating-point number without realizing it. We are disconcerted because we forget the approximations, not because we use them.
It isn't particularly difficult to do such computations accurately. If I wrote Calculator in Smalltalk, with its Fraction class used to represent every number entered in the display, there would be no such errors in ordinary arithmetic. The problem isn't bru
ised integers; the problem is a poor substitute for arithmetic.
Dr. Bobby R. Treat
Arlington, VA
Bobby.Treat@dp.hq.af.mil
Comments from Treat
