A Decision Procedure for Guarded Separation Logic Complete Entailment Checking for Separation Logic with Inductive Definitions

Abstract

We develop a doubly-exponential decision procedure for the satisfiability problem of guarded separation logic —a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the above connectives leads to an undecidable fragment. We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly-exponential decision procedure for entailments between (quantifier-free) symbolic heaps with inductive predicate definitions of bounded treewidth ( \({\mathbf {SL}}_{\mathsf {btw}} \) )—one of the most expressive decidable fragments of separation logic. Together with the recently shown 2ExpTime -hardness for entailments in said fragment, we conclude that the entailment problem for \({\mathbf {SL}}_{\mathsf {btw}} \) is 2ExpTime -complete—thereby closing a previously open complexity gap.

Tree-Verifiable Graph Grammars

“…One of the first fragments of SL with recursive definitions having a decidable entailment problem used an ad-hoc translation into equivalent MSO formulae, together with a syntactic guarantee of tree-width boundedness [11]. In subsequent work, a decision procedure, not based on MSO, for the entailment problem in this SL fragment was proposed and implemented [14,13] and the complexity of this entailment problem was established [9,16]. The ideas used in the definition of the SL fragment motivated the definition of tree-verifiable grammars in this paper.…”