Abstract
AbstractThe entailment problem$$\upvarphi \models \uppsi $$φ⊧ψin Separation Logic [12, 15], between separated conjunctions of equational ($$x \approx y$$x≈yand$$x \not \approx y$$x≉y), spatial ($$x \mapsto (y_1,\ldots ,y_\upkappa )$$x↦(y1,…,yκ)) and predicate ($$p(x_1,\ldots ,x_n)$$p(x1,…,xn)) atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, calledestablishment[10, 13, 14] andrestrictedness[8], respectively. Both classes are shown to be in$$\mathsf {2\text {EXPTIME}}$$2EXPTIMEby the independent proofs from [14] and [8], respectively, and a many-one reduction of established to restricted entailment problems has been given [8]. In this paper, we strictly generalize the restricted class, by distinguishing the conditions that apply only to the left- ($$\upvarphi $$φ) and the right- ($$\uppsi $$ψ) hand side of entailments, respectively. We provide a many-one reduction of this generalized class, calledsafe, to the established class. Together with the reduction of established to restricted entailment problems, this new reduction closes the loop and shows that the three classes of entailment problems (respectively established, restricted and safe) form a single, unified,$$\mathsf {2\text {EXPTIME}}$$2EXPTIME-complete class.
Highlights
Separation Logic [12,15] (SL) was primarily introduced for writing concise Hoare logic proofs of programs that handle pointer-linked recursive data structures
SL has evolved into a powerful logical framework, that constitutes the basis of several industrial-scale static program analyzers [3,2,5], that perform scalable compositional analyses, based on the principle of local reasoning: describing the behavior of a program statement with respect only to the small set of memory locations that are changed by that statement, with no concern for the rest of the program’s state
We describe the transformation operating on the left-hand side of the entailment problem
Summary
Separation Logic [12,15] (SL) was primarily introduced for writing concise Hoare logic proofs of programs that handle pointer-linked recursive data structures (lists, trees, etc). An important problem in program verification, arising during the construction of Hoare-style correctness proofs of programs, is the discharge of verification conditions of the form φ |= ψ, where φ and ψ are SL formulæ, asking whether every model of φ is a model of ψ These problems, called entailments, are, in general, undecidable in the presence of inductively defined predicates [11,1]. A first decidable class of entailments, described in [10], involves three restrictions on the SID rules: progress, connectivity and establishment. The rules of a progressive, connected and restricted (PCR) entailment may generate data structures with “dangling” (i.e. existentially quantified but not allocated) pointers, which was not possible with PCE entailments.
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