Propositional fuzzy logics: Decidable for some (algebraic) operators; undecidable for more complicated ones

Abstract

If we view fuzzy logic as a logic, i.e., as a particular case of a multi-valued logic, then one of the most natural questions to ask is whether the corresponding propositional logic is decidable, i.e., does there exist an algorithm that, given two propositional formulas F and G, decides whether these two formulas always have the same truth value. It is known that the simplest fuzzy logic, in which &=min and ∨=max, is decidable. In this paper, we prove a more general result: that all propositional fuzzy logics with algebraic operations are decidable. We also show that this result cannot be generalized further, e.g., no deciding algorithm is possible for logics in which operations are algebraic with constructive (nonalgebraic) coefficients. ©1999 John Wiley & Sons, Inc.14: 935–947, 1999