Decidable Entailments in Separation Logic with Inductive Definitions: Beyond Established Systems

TL;DR

This paper expands the class of decidable entailment problems in separation logic with inductive definitions by introducing equational restrictedness, allowing for richer structures with unbounded treewidth while maintaining 2EXPTIME-completeness.

Contribution

It introduces equational restrictedness as a new condition that generalizes established rules, enabling decidability for broader classes of separation logic formulas.

Findings

Entailment problem is 2EXPTIME-complete for the new class.

Established rules can be transformed into equationally restricted rules.

The new class includes structures with unbounded treewidth.

Abstract

We define a class of Separation Logic formulae, whose entailment problem: given formulae $\phi, \psi_1, \ldots, \psi_n$, is every model of $\phi$ a model of some $\psi_i$? is 2EXPTIME-complete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work consider established sets of rules, meaning that every existentially quantified variable in a rule must eventually be bound to an allocated location, i.e. from the domain of the heap. In particular, this guarantees that each structure has treewidth bounded by the size of the largest rule in the set. In contrast, here we show that establishment, although sufficient for decidability (alongside two other natural conditions), is not necessary, by providing a condition, called equational restrictedness, which applies syntactically to (dis-)equalities. The entailment problem is more general in this case, because equationally restricted rules define richer classes of structures, of unbounded treewidth. In this paper we show that (1) every established set of rules can be converted into an equationally restricted one and (2) the entailment problem is 2EXPTIME-complete in the latter case, thus matching the complexity of entailments for established sets of rules.