Detecting Loops During Proof Search in Propositional Affine Logic

Abstract

It is well-known that proof search, in general, does not terminate. In some decidable logics (e.g. intuitionistic propositional logic) it is possible to give a terminating sequent calculus, i.e. one in which a naive backward proof search will always terminate. However, such calculi are not always available, even for decidable logics. In this paper we investigate the incorporation of a loop detection mechanism into an inference system for propositional affine logic (i.e. propositional linear logic with arbitrary weakening). This logic is decidable, but no terminating sequent calculus for it is known. We adapt the history techniques used for intuitionistic and modal logics, but in this case we cannot assume that the context will always be non-decreasing. We show how to overcome this problem, and hence to provide a loop detection mechanism for propositional affine logic.