Reasoning in the Bernays-Schönfinkel-Ramsey Fragment of Separation Logic

Abstract

Separation Logic (\(\mathsf {SL}\)) is a well-known assertion language used in Hoare-style modular proof systems for programs with dynamically allocated data structures. In this paper we investigate the fragment of first-order \(\mathsf {SL}\) restricted to the Bernays-Schonfinkel-Ramsey quantifier prefix \(\exists ^*\forall ^*\), where the quantified variables range over the set of memory locations. When this set is uninterpreted (has no associated theory) the fragment is PSPACE-complete, which matches the complexity of the quantifier-free fragment [7]. However, \(\mathsf {SL}\) becomes undecidable when the quantifier prefix belongs to \(\exists ^*\forall ^*\exists ^*\) instead, or when the memory locations are interpreted as integers with linear arithmetic constraints, thus setting a sharp boundary for decidability within \(\mathsf {SL}\). We have implemented a decision procedure for the decidable fragment of \(\exists ^*\forall ^*\mathsf {SL}\) as a specialized solver inside a DPLL(T) architecture, within the CVC4 SMT solver. The evaluation of our implementation was carried out using two sets of verification conditions, produced by (i) unfolding inductive predicates, and (ii) a weakest precondition-based verification condition generator. Experimental data shows that automated quantifier instantiation has little overhead, compared to manual model-based instantiation.