Continuously valued logic

摘要

Motivated by the recognized inadequacy of conventional logic for the representationand 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 connectives 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+=1 and x=0. It is shown that every soft algebra is a bounded, distributive and symmetric lattice.A specific soft algebra, the family of all functions of n variables in the closed interval [0, 1], is analyzed in great detail. This particularlalgebra 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 two-level minimization procedure, which is a generalization of the Quine-McCluskey method, is given.