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Article type: Research Article
Authors: Konstantinidis, Stavros | Nicolae, Santean | Yu, Sheng
Affiliations: Dep. of Math. and Comp. Sc. Saint Mary's University Halifax B3H 3C3, NS, Canada. E-mail: [email protected] | School of Computer Science, University of Waterloo, Waterloo N2L 3G1, ON, Canada. E-mail: [email protected] | Department of Computer Science, University of Western Ontario London N6A 5B8, ON, Canada. E-mail: [email protected]
Abstract: In this paper we present a different framework for the study of fuzzy finite machines and their fuzzy languages. Unlike the previous work on fuzzy languages, characterized by fuzzification at the level of their acceptors/generators, here we follow a top-down approach by starting our fuzzification with more abstract entities: monoids and particular families in monoids. Moreover, we replace the unit interval (in fact, a finite subset of the unit interval) as support for fuzzy values with the more general structure of a lattice. We have found that completely distributive complete lattices allow the fuzzification at the highest level, that of recognizable and rational sets. Quite surprisingly, the fuzzification process has not followed thoroughly the classical (crisp) theory. Unlike the case of rational sets, the fuzzification of recognizable sets has revealed a few remarkable exceptions from the crisp theory: for example the difficulty of proving closure properties with respect to complement, meet and inverse morphisms. Nevertheless, we succeeded to prove the McKnight and Kleene theorems for fuzzy sets by making the link between fuzzy rational/recognizable sets on the one hand and fuzzy regular languages and FT-NFA languages (languages defined by NFA with fuzzy transitions) on the other. Finally, we have drawn the attention to fuzzy rational transductions, which have not been studied extensively and which bring in a strong note of applicability.
Keywords: Fuzzy Transducer, Rational Set, Recognizable Set, Lattice
Journal: Fundamenta Informaticae, vol. 76, no. 4, pp. 413-447, 2007
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