On the Complexity of Validity Degrees in Łukasiewicz Logic

Abstract

Łukasiewicz logic is an established formal system of many-valued logic. Decision problems in both propositional and first-order case have been classified as to their computational complexity or degrees of undecidability; for the propositional fragment, theoremhood and provability from finite theories are complete. This paper extends the range of results by looking at validity degree in propositional Łukasiewicz logic, a natural optimization problem to find the minimal value of a term under a finite theory in a fixed complete semantics interpreting the logic. A classification for this problem is provided using the oracle class, where it is shown complete under metric reductions.