What Is Decidable in Separation Logic Beyond Progress, Connectivity and Establishment?

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 SL-COMP.