Questions of Computational Complexity and Undecidability

Abstract

The present chapter is devoted to an analysis of computational complexity of fuzzy propositional calculi and to the determination of the degree of undecidability of fuzzy predicate calculi. The results are important but will be only rarely used in other chapters, so the reader unfamiliar with the theory of (polynomial) complexity and/or arithmetical hierarchy may skip the chapter as a whole. On the other hand, in Section 1 we summarize all necessary material so that the non-expert in complexity or arithmetical hierarchy may also read the chapter and understand the results. In Section 2 we study the computational complexity of Ł, G, Π (propositional Łukasiewicz, Gödel and product logic) and show, among other things, that their sets of 1-tautologies T AUT ^Ł₁ , T AUT ^G₁ , T AUT ^Π₁ are all co-NP-complete. In Section 3 we study the corresponding predicate calculi Ł∀, G∀, Π∀. We show that the set T AUT ^Π₁ of Gödel predicate logic is Σ₂; it remains open if it is in Π₂ or is still more complex). Hence all these predicate calculi are undecidable.