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

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 preceding 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 (SL_btw)—one of the most expressive decidable fragments of separation logic. Together with the recently shown2ExpTime-hardness for entailments in said fragment, we conclude that the entailment problem for SL_btw is 2ExpTime-complete—thereby closing a previously open complexity gap.