Xref: utzoo comp.realtime:663 comp.ai:6934 Path: utzoo!utgpu!news-server.csri.toronto.edu!mailrus!umich!samsung!usc!zaphod.mps.ohio-state.edu!sunybcs!bingvaxu!cjoslyn From: cjoslyn@bingvaxu.cc.binghamton.edu (Cliff Joslyn) Newsgroups: comp.realtime,comp.ai Subject: Re: Fuzzy Logic Introduction? Message-ID: <3550@bingvaxu.cc.binghamton.edu> Date: 31 May 90 02:51:09 GMT References: <766@ssc.UUCP> <3128@se-sd.SanDiego.NCR.COM> Reply-To: cjoslyn@bingvaxu.cc.binghamton.edu (Cliff Joslyn) Organization: SUNY Binghamton, NY Lines: 101 In article wdr@wang.com (William Ricker) writes: >jim@se-sd.SanDiego.NCR.COM (Jim Ruehlin) writes: >> I've also heard that it's nothing more than probability >>theory (the rebuttal to this is that it's really more like "possibility >>theory"). >People also try to confuse it with compounded independant probabilties >or Baysian probabilities; these are potential models for fuzzy arithmetics, >but not usually the ideal ones. I like the phrase "possibility theory", >but I'm not sure it is any more intuitive than "fuzzy set". The Fuzzy world has a lot of correlates to the Classical world. Possibility Theory is a strict correlate to Probability Theory [Dubois and Prade 1988, Klir 1984]. While Zadeh [1978] identifies possibility distribution on a universe with a fuzzy set on that universe, possibility theory does not require fuzzy set theory [Shafer 1976]. The comment that fuzzy sets and possibility theory are nothing more than probability theory is rebutted directly by Klir [1989]. This view might be related to the fact that Shafer's Belief and Plausibility measures are derived from a probability measure on the power set of the power set of the universe, and that possibility measures are a class of plausibility measures. Possibility measures are also a class of non-additive fuzzy measures. Fuzzy measures do not necessarily have anything to do with fuzzy sets (alas). Possibility calculus is based on max-min algebra, not +/* algebra. And possibility distributions are normalized with a maximum of 1, not a sum to 1. This is in a sense a "local" property of one element of the distribution, not a "global" property of the whole distribution: a single element of a possibility distribution can be varied without varying any others. Thus possibility theory is useful where the universe of discourse is unknown, unbounded, or changing: introduction of a new "possibility" or changing an old one does not require rescaling of all existing distributions. Some simple semantic considerations argue for the concept of possibility distinct from that of probability. Following from Gaines and Kohout [1976], if we identify a positive possibility with a positive probability, then we are committed to a concept of possibility in which any possible event is "eventual": over a large finite time our uncertainty about the event occuring will become arbitrarily small. But we usually work with a concept of possibility which does not entail this. Certainly all probable event are possible, but the converse is not true. Also, we subjectively construct our uncertainty assesments in a local way, without rescaling all other options each time a new thought comes to mind. Possibility distributions are used to model situations where uncertainty is characterized by vagueness or non-specificity, as opposed to a decision among a set of distinct choices [Klir 1989]. A possibilistic process is a direct generalization of a non-deterministic process. A stochastic process is not. There is now a possibilistic information theory, where the U-uncertainty is a unique correlate to the Shannon entropy [Higashi and Klir 1982, Klir and Mariano 1987] and a direct generalization of the Hartley entropy. Klir has recently proposed [1990] a Principle of Uncertainty Invariance through which transformations from stochastic to possibilistic systems and vice versa can be made without loss of information. Also, since the max-min calculus is substantially more computationally efficient than +/*, possibilistic models are frequently more tractable then stochastic ones. If people have a further interest, I can email or post a copy of my dissertation prospectus on "Possibilistic State Machines" or an extensive annotated bibliography. Dubois D and Prade H: (1989) _Possibility Theory_ Gaines, Brian and Kohout: (1976) "The Logic of Automata", Int. J. Gen. Sys., v. 2:4, 191-208 Higashi, Masahiko and Klir, George: (1982) "Measures of Uncertainty and Information Based on Possibility Distributions", Int. J. Gen. Sys., v. 9 Klir, George: (1984) "Possibilistic Information Theory", Cybernetics and Systems Research, v. 2, ed. R. Trappl ------------: (1989) "Is There More to Uncertainty Than Some Probability Theorists Would Have Us Believe?", Int. J. Gen. Sys. v. 15, 347-378 ------------: (1990) "A Principle of Uncertainty Invariance", J. Approximate Reasoning, v. 17:2 Klir, George, and Mariano: (1987) "On the Uniqueness of Possibilitic Measures of Uncertainty and Information", Fuzzy Sets and Systems, v. 24, 197-219 Shafer, Glen: (1976) _A Mathematical Theory of Evidence_ Zadeh, Lofti: (1978) "Fuzzy Sets as the Basis for a Theory of Possibility", Fuzzy Sets and Systems, v. 1, 3-28 O-------------------------------------------------------------------------> | Cliff Joslyn, Cybernetician at Large, cjoslyn@bingvaxu.cc.binghamton.edu | Systems Science, SUNY Binghamton, Box 1070, Binghamton NY 13901, USA V All the world is biscuit shaped. . . -- O-------------------------------------------------------------------------> | Cliff Joslyn, Cybernetician at Large, cjoslyn@bingvaxu.cc.binghamton.edu | Systems Science, SUNY Binghamton, Box 1070, Binghamton NY 13901, USA V All the world is biscuit shaped. . .