Decidable Entailments in Separation Logic with Inductive Definitions: Beyond Establishment

9 We define a class of Separation Logic [10, 16] formulæ, whose entailment problem given formulæ φ,ψ1, . . . ,ψn, 10 is every model of φ a model of some ψi? is 2-EXPTIME-complete. The formulæ in this class are existentially 11 quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures 12 that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work 13 [8, 12, 15] consider established sets of rules, meaning that every existentially quantified variable in a rule must 14 eventually be bound to an allocated location, i.e. from the domain of the heap. In particular, this guarantees 15 that each structure has treewidth bounded by the size of the largest rule in the set. In contrast, here we show 16 that establishment, although sufficient for decidability (alongside two other natural conditions), is not necessary, 17 by providing a condition, called equational restrictedness, which applies syntactically to (dis-)equalities. The 18 entailment problem is more general in this case, because equationally restricted rules define richer classes of 19 structures, of unbounded treewidth. In this paper we show that (1) every established set of rules can be converted 20 into an equationally restricted one and (2) the entailment problem is 2-EXPTIME-complete in the latter case, 21 thus matching the complexity of entailments for established sets of rules [12, 15]. 22 2012 ACM Subject Classification Theory of computation→ Logic and verification 23