The Complexity of Fuzzy Logic

We show that the set of valid formulas in Lukasiewicz predicate logic is a complete Π 2 set. We also show that the classically valid formulas are exactly those formulas in the classical language whose fuzzy value is 1/2. Lukasiewicz’ infinite valued logic can be seen as a particular “implementation” of fuzzy logic. The set of possible “truth values” (or, in another interpretation, degrees of certainty) is the real interval [0, 1]. Minimum, maximum, and truncated addition are the basic operations. It is well known that the propositional fragment version of Lukasiewicz logic is decidable. The exact complexity of Lukasiewicz predicate logic was a more diffcult problem. For an upper bound, it is known that the set of valid formulas in this logic is a Π2. (An explicit Π2 representation can be found through the axiomatisation of Novak and Pavelka. See [1] for references.) For a lower bound, Scarpellini [4] showed that the set of valid formulas is not r.e., and in fact Π1-hard. He also remarks in a footnote that this set is not Σ 0 2, either. In his unpublished thesis [3], Mathias Ragaz showed that the set of valid formulas in Lukasiewicz predicate logic is actually Π2-complete. The proof of this theorem that we give here was found independently. Furthermore, we show that if we restrict our attention to classical formulas, the classically valid formulas are exactly those formulas which have value ≥ 1/2 in every fuzzy model.