The Complexity of Lukasiewicz Logic

Lukasiewicz logic is a [0, 1]-valued fuzzy logic with two disjunctions, corresponding to the maximum operation and (truncated) addition. The set of valid formulas for this logic is a complete Π₂⁰ set. We give a proof of this result and discuss two variants of this logic: a syntactic variant in which only the classical connectives are allowed, and a semantic variant where we restrict the models to structures with decidable relations.