Separation Logic (SL) enables reasoning about programs that manipulate pointers. Its key feature is the separating conjunction ⋆, which asserts that two formulas hold on disjoint portions of memory. We consider an extension of SL, called Overlaid SL (OSL), that allows non-disjoint combinations of data structures defined over different fields, enriched with set constraints on the nodes of these structures. We prove that entailment is decidable for a broad class of data structures satisfying the so-called PCE conditions of [Iosif et al., 2013], thus extending this result to OSL. Our decision procedure is nondeterministic with doubly exponential time complexity.

Abstract The predicate definitions in Separation Logic (SL) play an important role: they capture a large spectrum of unbounded heap shapes due to their inductiveness. This expressiveness power comes with a limitation: the entailment problem is undecidable if predicates have general inductive definitions (ID). Iosif et al. [8] proposed syntactic and semantic conditions, called PCE, on the ID of predicates to ensure the decidability of the entailment problem. We provide a (possibly nonterminating) algorithm to transform arbitrary ID into equivalent PCE definitions when possible. We show that the existence of an equivalent PCE definition for a given ID is undecidable, but we identify necessary conditions that are decidable. The algorithm has been implemented, and experimental results are reported on a benchmark, including significant examples from .