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Earl Cox understands that fuzzy set theory is a mathematical exercise, whereas fuzzy logic control is an epistemological exercise. That's enough to put this book high on my recommend list, but it's not the only reason that budding fuzzy systems designers should have a copy of The Fuzzy Systems Handbook, Second Edition.
Fuzzy logic is controversial, since the validity of its methods, except in the most trivial cases, are unprovable in terms of classic control theory. "Fahzy rogicuh," as it is known in Japan, where it is practiced earnestly, is only verifiable empirically, as if it were the chiropractic of computer science. The Fuzzy Systems Handbook demonstrates that Americans have, despite Japanese dominance of the field of practical application, a few domain experts of our own.
At 716 pages, you don't want to drop Fuzzy Systems Handbook on your foot. Cox has, in this edition, updated his original 1994 work. He has cleaned up the discussions of importance of structure of fuzzy sets, how they should be built for some kinds of applications, why bell-shaped and trapezoidal membership sets are important in business fuzzy sets while engineers tend to choose triangular sets.
Also in the second edition, the DOS 16 code on the accompanying CD-ROM has been brought up to Win32. There are tools for building fuzzy systems in Visual Basic, including a VB fuzzy set editor. There are libraries and DLLs that you can link to VB or Powerbuilder or your favorite Win32 C compiler. Source code, and much new code, is included.
The material is drawn not so much from theoretical study as from actual field application of the technology, and is imbued with the sweat of achievement rather than perfumed with punditry. These are the notebooks of a working engineer edited into an immense but eminently readable tome.
Fuzzy Systems Handbook commences with an apologia for fuzzy control itself, regaling us with humorous and humiliating experiences the author has undergone in attempting to sell projects and services. Cox has been told by prospects things like, "We're a very precise organization, we're not interested in fuzzy logic." Cox counters with case studies and consumer products and an impassioned and detailed "justification [of fuzzy logic] as a modeling tool." He attacks the "mythology" that the "One True God of Science and Fact is bedeviled by the Angel of Vagueness and Anti-Reason."
For in fact, fuzzy set theory isn't fuzzy at all. It's a very concrete method for measuring degree of membership in sets. The calculus of fuzzy set theory is in defining the sets of interest appropriately. Cox blames the tendency to misunderstanding by corporate decision makers on the brilliant/unfortunate use of the English word "fuzzy."
Fuzzicists console themselves that the principles of relativity, which constituted a century ago in physics a reassemblage of perception similar to that which fuzzy logic constitutes in control theory, were similarly misunderstood by the general public.
And then, we slip off the bank and we're deep in it, up to our chests. We're in the swamp of the technique of fuzzy decomposition of problems. The book takes us through fuzzy sets, operator, hedges, fuzzy reasoning, models, case studies, and evaluation and design methodologies. There's an appendix by William E. Combs of Boeing on his method of rapid inference designed to defeat the combinatorial explosion of rules frequently encountered in designing real-world time-critical applications based on fuzzy logic.
The text is literate and lucid, technically rich but devoid of jargon. The exposition is tightly paced, interspersed appropriately with helpful diagrams, and visually keyed by icons that help one read selectively for breadth first.
Cox himself is quite a character, described on the pages of Metus Systems group (http://www.metus.com/05_resume.html) as "Founder, Chief Executive Officer, Principal Scientist, Senior Epistemologist, and sole full-time employee of The Metus Systems Group." We're told that Earl was a U.S. Army Captain who "performed various analytical, command, and staff positions in the United States, Korea, and Vietnam. Prior to his military service, Earl was a failed poet, artist, and epistemologist living in an abandoned trailer along Highway One just south of the Little Sur River in California." (ibid.)
Earl is a five-string banjo player, spelunker, and is writing a fuzzy logic murder mystery with which he hopes to make his name as well known in fiction as it is in technical nonfiction.
In the 1970s, Cox was a TYMSHR product vendor. He worked on a zero-based budget modeler for Office of Management of the Budget, and he built a relational database into his project management system in 1975. He ran into Peter Llewellyn Jones, an early adopter of fuzzy systems, while on an extended assignment in England. Cox, who had begun enterprise modeling software featuring multiresource scheduling in the PL/I language in 1974, was impressed by Jones's work in Fortran on the first fuzzy modeling spreadsheet.
Cox teaches regular seminars on the use of fuzzy logic in data mining. The self-styled "wandering epistemologist" in Cox is never far below the surface. We can assess risk by computing the geometry of intersection of fuzzy memberships, but is that a valid way to assess risk? Coding proportional-integral-derivative loops in classic control projects, one doesn't spend a lot of time assessing the validity of PID.
Cox clearly has faith that there is order in the universe. He decides that fuzzy logic is in the doing, and he does. He makes decisions, makes alpha cuts because there's work to be done now and the heck with theory, and because attempts to prove the theory of fuzzy machine control tend to go to ground in axioms that make heirs of Judeo-Christian-Graeco thought a bit uncomfortable. It's disconcerting to the passengers in an elevator to know that the elevator feels that their office is on a floor that is "very high," and that sometimes it wonders if it is "near enough" the bottom of the shaft to start applying the brakes.
"Why do we square the membership function for the very hedge, and take the square root for the somewhat hedge?" asks the book in Chapter 5 on Fuzzy Set Hedges. "For no really good reason except that these operations seem to give a pretty good approximation of these concepts in a wide range of fuzzy spaces and operations."
I asked Earl Cox why fuzzy models work. "Elkans 'Unreasonable effectiveness of fuzzy logic' paper caused and still causes a firestorm in fuzzy community," Cox told me. "Elkans' point is that fuzzy logic doesn't obey the law of the excluded middle, nor the law of noncontradiction. The fuzzy logic community's point is that neither does reality. That's why fuzzy logic is effective."
Fuzzy Systems Handbook occasionally reaches mystic heights. The floating pyramid of variable decomposition (Fig 1.14 in the book) fairly takes the breath away. Someday, when we are certain that machines are self-aware, we may draw an all-seeing eye at the top of that pyramid.
Cox has coauthored an acclaimed work of socio-technological analysis called Beyond Humanity with Greg Paul from Johns Hopkins in Baltimore, the latter a paleontologist and artist who worked as dinosaur advisor on "Jurassic Park". More books are in the works.
Earl may have failed as a poet but he succeeds in Fuzzy Systems Handbook as a technical expositor of the first caliber.
-- Jack Woehr
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Explanation of ERCB rating scale: No stars = unacceptable, 1 Star = marginal, 2 Stars = average, 3 Stars = above average, 4 Stars = exceptional.