The Bernays-Schönfinkel-Ramsey Class of Separation Logic with Uninterpreted Predicates

This article investigates the satisfiability problem for Separation Logic with k record fields, with unrestricted nesting of separating conjunctions and implications. It focuses on prenex formulæ with a quantifier prefix in the language ∃*∀* that contain uninterpreted (heap-independent) predicate symbols. In analogy with first-order logic, we call this fragment Bernays-Schönfinkel-Ramsey Separation Logic [BSR(SLk)]. In contrast with existing work on Separation Logic, in which the universe of possible locations is assumed to be infinite, we consider both finite and infinite universes in the present article. We show that, unlike in first-order logic, the (in)finite satisfiability problem is undecidable for BSR(SLk). Then we define two non-trivial subsets thereof, for which the finite and infinite satisfiability problems are PSPACE-complete, respectively, assuming that the maximum arity of the uninterpreted predicate symbols does not depend on the input. These fragments are defined by controlling the polarity of the occurrences of separating implications, as well as the occurrences of universally quantified variables within their scope. These decidability results have natural applications in program verification, as they allow to automatically prove lemmas that occur in, e.g., entailment checking between inductively defined predicates and validity checking of Hoare triples expressing partial correctness conditions.