🤖 AI Summary
This paper investigates the computational complexity of the entailment problem for separation logic formulas equipped with user-defined inductive predicates. For the *loc-deterministic* fragment of inductive rules, we devise the first sound, complete, and polynomial-time decidable cyclic proof system, thereby establishing PTIME decidability of entailment in this fragment. We further provide a tight syntactic characterization: any nontrivial relaxation of the loc-determinism condition renders entailment undecidable. By combining semantic modeling in separation logic with fine-grained complexity analysis—distinguishing between P, NP, and undecidability—we deliver matching upper and lower bounds. This yields the first exact complexity classification for entailment in this key fragment, advancing the theoretical foundations and algorithmic guarantees for automated reasoning about recursive data structures in program verification.
📝 Abstract
We establish various complexity results for the entailment problem between formulas in Separation Logic with user-defined predicates denoting recursive data structures. The considered fragments are characterized by syntactic conditions on the inductive rules that define the semantics of the predicates. We focus on so-called P-rules, which are similar to (but simpler than) the PCE rules introduced by Iosif et al. in 2013. In particular, for a specific fragment where predicates are defined by so-called loc-deterministic inductive rules, we devise a sound and complete cyclic proof procedure running in polynomial time. Several complexity lower bounds are provided, showing that any relaxing of the provided conditions makes the problem intractable.