The Importance of Being Fuzzy: And Other Insights from the Border between Math and Computers (Hardcover)

Most people know that the mathematics behind computers has something to do with binary arithmetic and things called bits and bytes and logic gates. What they may not know is that there is a well-established science of computability; it even has its own Einstein in the person of the tragic English genius Alan Turing.

In this book, Canadian academic Arturo Sangalli describes Turing's work, and examines three mathematical techniques which modern research is using in attempts to make computers more efficient and possibly more intelligent. All three approaches started life as esoteric mathematical ideas and lay dormant for some time before computer researchers began to use them.

The most interesting and most easily understood is what is known as fuzzy logic. This may seem like a contradiction in terms, but as Sangalli points out, it is a logic of fuzziness, not a logic which is itself fuzzy. In its broadest sense, it is synonymous with the mathematical theory of fuzzy sets.

The concept of a Set was one of the core elements of the New Maths of the 1970s which many parents will gratefully recall as the excuse they were able to give for not being able to help their children with school mathematics. I won't try to change that situation at this late stage, beyond saying that a set has to be precisely determined - you cannot have vaguely defined sets like "tall people" or "funny television programs" or "interesting books."

The mathematical theory of fuzzy sets was taken down from the shelf and dusted off some years ago by computer engineers, particularly in Japan where the government persuaded a consortium of local heavies - Matsushita, Canon, Hitachi, and Mitsubishi among others - to invest $50 million in the Laboratory for International Fuzzy Engineering Research (LIFE). The result has been that most electronic goods now produced in Japan contain what an advertisement currently running on television calls "fuzzy thingys."

To give an idea of how fuzziness is used in computer programs, consider the way a child balances a stick upright on the palm of one hand. The feat requires a complex combination of movements and adjustments and feedback. How can a computer be taught to do the trick?

What computer engineeres have done is to construct a mechanical model controlled by a computer which has been programmed with the kinds of rules which a child might give to explain how he/she was performing the trick. So, there is rule which says "If the stick is balanced, do not move your hand," and another "If the stick is slightly tilted away from you and falling slowly, move your hand forward, but not too quickly."

Phrases like "slightly tilted" or "falling slowly" or "not too quickly" would be anathema to conventional mathematics, but fuzzy mathematics is able to put a range of possible numbers on them, and the computer can be taught to understand and respond.

Surprisingly, to balance the rod, only five other rules along with the two above are needed. The system works perfectly every time and appears to be extraordinarily robust, working equally well with rods of different lengths and weights. Moreover, if one rule is omitted or minor changes are made to a rule, the system still works most of the time - a far cry from the annoyance which is the constant companion of computer programmers who see their program crash as a result of a missing comma or semi-colon somewhere in the middle of metres of code.

Of course, the solution is as much a triumph of smart electronics as of clever computer programming. But the point is that it demonstrates a program which works well even though it has been given imprecise instructions. Just as with the child, the logic may be fuzzy, but the result is not.

By 1993, there were more than 600 consumer products in Japan which used fuzzy chips. And the Japanese hit a marketing jackpot when they found that their country was happy to accept the English word fuzzy rather than its local equivalent. Everything from washing machines to wheelchairs to photocopiers and cameras had fuzzy logic chips - not to mention the television set which the chap in the advertisement so enthusiastically demonstrates to his friend by opening and closing his curtains while their kids wonder at the childlike joy of their elders.

"Computers can do only what we tell them to do," is a paraphrase of the famous statement by Ada Lovelace, daughter of Lord Byron and one of the pioneers of computer programming. With fuzzy logic, Lady Lovelace's dictum may not be as close to Holy Writ as we thought.

Well done, Dr Sangalli.