On the Expressive Completeness of Bernays-Sch\"onfinkel-Ramsey Separation Logic

TL;DR

This paper explores the satisfiability of a specific fragment of Separation Logic with complex quantifier structures, revealing undecidability in general but identifying decidable subclasses through controlled variable occurrences.

Contribution

It introduces the Bernays-Sch"onfinkel-Ramsey Separation Logic fragment, proves its general undecidability, and defines decidable subclasses via variable occurrence restrictions.

Findings

Undecidability of satisfiability for the general fragment.

Decidability of finite and infinite satisfiability in specific subclasses.

Effective translation methods to reduce Separation Logic formulas to first-order logic.

Abstract

This paper investigates the satisfiability problem for Separation Logic, with unrestricted nesting of separating conjunctions and implications, for prenex formulae with quantifier prefix in the language $\exists^*\forall^*$, in the cases where the universe of possible locations is either countably infinite or finite. In analogy with first-order logic with uninterpreted predicates and equality, we call this fragment Bernays-Sch\"onfinkel-Ramsey Separation Logic [BSR(SLk)]. We show that, unlike in first-order logic, the (in)finite satisfiability problem is undecidable for BSR(SLk) and 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. The decidability results are obtained by a controlled elimination of separating connectives, described as (i) an effective translation of a prenex form Separation Logic formula into a combination of a small number of \emph{test formulae}, using only first-order connectives, followed by (ii) a translation of the latter into an equisatisfiable first-order formula.