A Simple Separation Logic

International audienceThe kinds of models that are usually considered in separation logic are structures such as words, trees, and more generally pointer structures (heaps). In this paper we introduce the separation logic of much simpler structures, viz. sets. The models of our set separation logic are nothing but valuations of classical propositional logic. Separating a valuation V consists in splitting it up into two partial valuations v 1 and v 2. Truth of a formula φ 1 * φ 2 in a valuation V can then be defined in two different ways: first, as truth of φ 1 in all total extensions of v 1 and truth of φ 2 in all total extensions of v 2; and second, as truth of φ 1 in some total extension of v 1 and truth of φ 2 in some total extension of v 2. The first is an operator of separation of resources: the update of φ 1 * φ 2 by ψ is the conjunction of the update of φ 1 by ψ and the update of φ 2 by ψ; in other words, φ 1 * φ 2 can be updated independently. The second is an operator of separation of processes: updates by ψ 1 * ψ 2 can be performed independently. We show that the satisfiability problem of our logic is decidable in polynomial space (PSPACE). We do so by embedding it into dynamic logic of propositional assignments (which is PSPACE complete). We moreover investigate its applicability to belief update and belief revision, where the separation operators allow to formulate natural requirements on independent pieces of information

Independence and concurrent separation logic

A compositional Petri net-based semantics is given to a simple language allowing pointer manipulation and parallelism. The model is then applied to give a notion of validity to the judgements made by concurrent separation logic that emphasizes the process-environment duality inherent in such rely-guarantee reasoning. Soundness of the rules of concurrent separation logic with respect to this definition of validity is shown. The independence information retained by the Petri net model is then exploited to characterize the independence of parallel processes enforced by the logic. This is shown to permit a refinement operation capable of changing the granularity of atomic actions

A Simple Separation Logic

The kinds of models that are usually considered in separation logic are structures such as words, trees, and more generally pointer structures (heaps). In this paper we introduce the separation logic of much simpler structures, viz. sets. The models of our set separation logic are nothing but valuations of classical propositional logic. Separating a valuation V consists in splitting it up into two partial valuations v 1 and v 2. Truth of a formula φ 1 * φ 2 in a valuation V can then be defined in two different ways: first, as truth of φ 1 in all total extensions of v 1 and truth of φ 2 in all total extensions of v 2; and second, as truth of φ 1 in some total extension of v 1 and truth of φ 2 in some total extension of v 2. The first is an operator of separation of resources: the update of φ 1 * φ 2 by ψ is the conjunction of the update of φ 1 by ψ and the update of φ 2 by ψ; in other words, φ 1 * φ 2 can be updated independently. The second is an operator of separation of processes: updates by ψ 1 * ψ 2 can be performed independently. We show that the satisfiability problem of our logic is decidable in polynomial space (PSPACE). We do so by embedding it into dynamic logic of propositional assignments (which is PSPACE complete). We moreover investigate its applicability to belief update and belief revision, where the separation operators allow to formulate natural requirements on independent pieces of information

Dynamic Separation Logic

This paper introduces a dynamic logic extension of separation logic. The assertion language of separation logic is extended with modalities for the five types of the basic instructions of separation logic: simple assignment, look-up, mutation, allocation, and de-allocation. The main novelty of the resulting dynamic logic is that it allows to combine different approaches to resolving these modalities. One such approach is based on the standard weakest precondition calculus of separation logic. The other approach introduced in this paper provides a novel alternative formalization in the proposed dynamic logic extension of separation logic. The soundness and completeness of this axiomatization has been formalized in the Coq theorem prover

Semantics of Separation-Logic Typing and Higher-order Frame Rules for<br> Algol-like Languages

We show how to give a coherent semantics to programs that are well-specified in a version of separation logic for a language with higher types: idealized algol extended with heaps (but with immutable stack variables). In particular, we provide simple sound rules for deriving higher-order frame rules, allowing for local reasoning

Variables as Resource for Shared-Memory Programs: Semantics and Soundness

AbstractParkinson, Bornat, and Calcagno recently introduced a logic for partial correctness in which program variables are treated as resource, generalizing earlier work based on separation logic and permissions. An advantage of their approach is that it yields a logic devoid of complex side conditions: there is no need to pepper the inference rules with “modifies” clauses. They used a simple operational semantics to prove soundness of the sequential fragment of their logic, and they showed that the inference rules of concurrent separation logic can be translated directly into their framework. Their concurrency rules are strictly more powerful than those of concurrent separation logic, since the new logic allows proofs of programs that perform concurrent reads. We provide a denotational semantics and a soundness proof for the concurrent fragment of their logic, extending our earlier work on concurrent separation logic to incorporate permissions in a natural manner

A relational model for confined separation logic

Confined separation logic is a new extension to separation logic designed to deal with problems involving dangling references within shared mutable structures. In par- ticular, it allows for reasoning about confinement in object- oriented programs. In this paper, we discuss the semantics of such an extension by defining a relational model for the overall logic, parametric on the shapes of both the store and the heap. This model provides a simple and elegant interpretation of the new confinement connectives and helps in seeking for duals. A number of properties of this logic are proved calculationally.Supported by NNSFC (No. 60573081