Two-Variable Separation Logic and Its Inner Circle

Abstract

Separation logic is a well-known assertion language for Hoare-style proof systems. We show that first-order separation logic with a unique record field restricted to two quantified variables and no program variables is undecidable. This is among the smallest fragments of separation logic known to be undecidable, and this contrasts with the decidability of two-variable first-order logic. We also investigate its restriction by dropping the magic wand connective, known to be decidable with nonelementary complexity, and we show that the satisfiability problem with only two quantified variables is not yet elementary recursive. Furthermore, we establish insightful and concrete relationships between two-variable separation logic and propositional interval temporal logic (PITL), data logics, and modal logics, providing an inner circle of closely related logics.

Expressive Completeness of Separation Logic with Two Variables and No Separating Conjunction

“…Even though conjectured in [6,7], it is surprising that two variables suffice, and that further we are able to drop the separating conjunction, thus obtaining expressive completeness and undecidability with only two variables and the magic wand operator. In doing so, we improve previous undecidability results about separation logic [7,9,12]. Because we forbid ourselves the use of many syntactic resources, this underlines even further the power of the magic wand.…”

“…In 1SL, reachability can be expressed, and [12] gives a technique for doing so with the two-variable restriction, itself a variant of material from [7,11]. Without the separating conjunction, we can still specify reachability from f(u) to f(u), but we need a new technique: we put a fork on f(u), propagate this fork forward in the heap, and finally check whether f(u) is the endpoint of some fork.…”

“…The proof of Lemma 8 uses principles similar to what has been done in [7,12] except that all formulae belong to 1SL2(− * ) and we propose a simplified version of the lemma and proof (note also our use of f(u) ). Let anti be antiforky(u) ∧ antiknify(u) ∧ antiforky(u) ∧ antiknify(u) and let comp(u, u, k, k ) be…”

“…This is sharpened in [7] by showing that first-order separation logic with a unique record field (called herein 1SL) is also undecidable, as a consequence of the expressive equivalence between 1SL and weak second-order logic. More recently, 1SL restricted to two variables (1SL2) is shown undecidable too [12] but without touching the central question of expressive completenessthe purpose of the current paper. From the very beginning, the relationships between separation logic and second-order logic have been quite puzzling (see e.g.…”

Decidability of weak logics with deterministic transitive closure

“…The operator seems to be attractive also for applications in tasks related to verification of hardware and software, as it can express many natural properties, e.g., when added to languages containing even only two variables it allows to say that a binary relation is a linear order, a tree (with reversed edges), a forest, or a partial function. However, while there are quite a lot of related works on formalisms which assume that the admissible structures themselves are deterministic, i.e., the out-degree of their elements is at most one (let us mention here deterministic propositional dynamic logic [1] and separation logic [4,28]; regarding separation logic see in particular [7] where its two-variable variant is investigated), there are not too many papers studying the deterministic transitive closure operator in languages interpreted over structures not so constrained. We are aware of two such papers.…”