On the almighty wand

Abstract

We investigate decidability, complexity and expressive power issues for (first-order) separation logic with one record field (herein called SL) and its fragments. SL can specify properties about the memory heap of programs with singly-linked lists. Separation logic with two record fields is known to be undecidable by reduction of finite satisfiability for classical predicate logic with one binary relation. Surprisingly, we show that second-order logic is as expressive as SL and as a by-product we get undecidability of St.. This is refined by showing that SL without the separating conjunction is as expressive as SL, whence undecidable too. As a consequence, in SL the separating implication (also known as the magic wand) can simulate the separating conjunction. By contrast, we establish that SL without the magic wand is decidable, and we prove a non-elementary complexity by reduction from satisfiability for the first-order theory over finite words. This result is extended with a bounded use of the magic wand that appears in Hoare-style rules. As a generalization, it is shown that kSL, the separation logic over heaps with k >= 1 record fields, is equivalent to kSO, the second-order logic over heaps with k record fields. (C) 2012 Elsevier Inc. All rights reserved

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

“…Our contribution In this paper, we sharpen the main result in [7], namely we show that first-order separation logic with one record field, two quantified variables, and no separating conjunction is as expressive as weak second-order logic on heaps; in short, 1SL2(− * ) ≡ WSOL. 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.…”

“…It should be noted that the paper is structured in such a way that we provide more and more complex building blocks to establish our main results. Another contribution rests on the fact that we considerably simplify some of the technical insights borrowed from [7] and therefore the current paper proposes a self-contained proof of the equivalence between 1SL2(− * ) and weak second-order logic that in many ways is much simpler than what has been done so far, even though our results are stronger. Extensions with program variables or with heaps having k > 1 record fields are presented in Section 5.5.…”

“…So, as far as the proof for expressive completeness goes, we borrow some of the first principles from [7], but very quickly we are faced with serious problems when we need to identify in some heap at least k > 0 heap patterns (a typical example is to specify that at least k > 0 locations point to a given location). Indeed, the standard way to identify such patterns is to use an unbounded number of variables or the 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.…”

“…Indeed, it is known since [9] that first-order separation logic with two record fields (called herein 2SL) is undecidable (with a proof that does not require separating connectives and uses Trakhtenbrot's Theorem). 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.…”