The entailment between separation logic formulæ with inductive predicates, also known as sym- bolic heaps, has been shown to be decidable for a large class of inductive definitions [7]. Recently, a 2-EXPTIME algorithm was proposed [10, 14] and an EXPTIME-hard bound was established in [8]; however no precise lower bound is known. In this paper, we show that deciding entailment between predicate atoms is 2-EXPTIME-hard. The proof is based on a reduction from the membership problem for exponential-space bounded alternating Turing machines [5].

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This paper investigates the satisfiability problem for Separation Logic with k record fields, with unrestricted nesting of separating conjunctions and implications, for prenex formulae with quantifier prefix ∃ * ∀ * . In analogy with first-order logic, we call this fragment Bernays-Schönfinkel-Ramsey Separation Logic [BSR(SL k )]. In contrast to existing work in Separation Logic, in which the universe of possible locations is assumed to be infinite, both finite and infinite universes are considered. We show that, unlike in first-order logic, the (in)finite satisfiability problem is undecidable for BSR(SL k ). Then we define two non-trivial subsets thereof, that are decidable for finite and infinite satisfiability respectively, by controlling the occurrences of universally quantified variables within the scope of separating implications, as well as the polarity of the occurrences of the latter. Beside the theoretical interest, our work has natural applications in program verification, for checking that constraints on the shape of a data-structure are preserved by a sequence of transformations.

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International audienceState-of-the-art theory solvers generally rely on an instantiation of the axioms of the theory, and depending on the solvers, this instantiation is more or less explicit. This paper introduces a generic instantiation scheme for solving SMT problems, along with syntactic criteria to identify the classes of clauses for which it is complete. The instantiation scheme itself is simple to implement, and we have produced an implementation of the syntactic criteria that guarantee a given set of clauses can be safely instantiated. We used our implementation to test the completeness of our scheme for several theories of interest in the SMT community, some of which are listed in the last section of this paper