A THEORY OF CONTINUOUSLY VALUED LOGIC.
TEXAS UNIV AUSTIN ELECTRONICS RESEARCH CENTER
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Motivated by the recognized inadequacy of conventional logic for the representation and manipulation of variables in areas related to artificial intelligence, this paper addresses itself to the investigation of the formal systems obtained by extending well-known operators to continuous arguments. The studied systems, called soft algebras, are generalizations of boolean algebras in that they satisfy all the axioms of the latter ones except the laws of complementarity, i.e., x x bar 1 and xx bar 0. It is shown that every soft algebra is a bounded, distributive and symmetric lattice. A specific soft algebra, namely, the family of all expressions of variables valued over the closed interval 0,1, is analyzed in great detail. This particular algebra is a formal unification of many recent results concerning fuzzy logic. It is shown that every soft function can be canonically represented by a pair of normal expressions, i.e., each soft function is representable by a double-array of tables a generalization of the truth-table representation of boolean functions. Also, a synthesis and a two-level minimization procedure, which is a generalization of the Quine-McCluskey method, are given. Author
- Theoretical Mathematics