Constant-only multiplicative linear logic is NP-complete

Abstract

Linear logic is a resource-aware logic that is based on an analysis of the classical proof rules of contraction (copying) and weakening (throwing away). In this paper we study the decision problem for the multiplicative fragment of linear logic without quantifiers or propositions: the constant-only case. We show that this fragment is NP-complete. Earlier work by Kanovich showed that propositional multiplicative linear logic is NP-complete. With Natarajan Shankar, the first author developed a simplified proof for the propositional case. The structure of this simplified proof is utilized here with a new encoding which uses only constants. The end product is the somewhat surprising result that simply evaluating expressions in true, false, and, and or in multiplicative linear logic (1, ⊥ ⊗ ▪) is NP-complete. By conversativity results not proven here, the NP-hardness of larger fragments of linear logic follows.