The complexity of Horn fragments of Linear Logic

Abstract

The question at issue is to develop a computational interpretation of Girard's Linear Logic [Girard, 1987] and to obtain efficient decision algorithms for this logic, based on the bottom-up approach. It involves starting with the simplest natural fragment of linear logic and then expanding it step-by-step. We give a complete computational interpretation for the Horn fragment of Linear Logic and some natural generalizations of it enriched by the two additive connectives: ⊛ and &. Within the framework of this interpretation, it becomes possible to explicitly formalize and clarify the computational aspects of the fragments of Linear Logic in question and establish exactly the complexity level of these fragments. In particular, the simplest natural Horn fragment of Linear Logic is proved to be NP-complete. As a corollary, we obtain the affirmative solution for the problem (formulated by Lincoln et al. (1992)): whether the multiplicative fragment of linear logic is NP-complete.