Separation Logic with One Quantified Variable

International audienceWe investigate first-order separation logic with one record field restricted to a unique quantified variable (1SL1). Undecidability is known when the number of quantified variables is unbounded and the satisfiability problem is PSPACE-complete for the propositional fragment. We show that the satisfiability problem for 1SL1 is PSPACE-complete and we characterize its expressive power by showing that every formula is equivalent to a Boolean combination of atomic properties. This contributes to our understanding of fragments of first-order separation logic that can specify properties about the memory heap of programs with singly-linked lists. All the fragments we consider contain the magic wand operator and first-order quantification over a single variable

Separation Logic with One Quantified Variable

International audienceWe investigate first-order separation logic with one record field restricted to a unique quantified variable (1SL1). Undecidability is known when the number of quantified variables is unbounded and the satisfiability problem is PSPACE-complete for the propositional fragment. We show that the satisfiability problem for 1SL1 is PSPACE-complete and we characterize its expressive power by showing that every formula is equivalent to a Boolean combination of atomic properties. This contributes to our understanding of fragments of first-order separation logic that can specify properties about the memory heap of programs with singly-linked lists. When the number of program variables is fixed, the complexity drops to polynomial time. All the fragments we consider contain the magic wand operator and first-order quantification over a single variable

The Complexity of Prenex Separation Logic with One Selector

We first show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with $k\geq1$ selector fields (\seplogk{k}). Second, we show that this entails the decidability of the finite and infinite satisfiability problem for the class of prenex formulas of \seplogk{1}, by reduction to the first-order theory of one unary function symbol and unary predicate symbols. We also prove that the complexity is not elementary, by reduction from the first-order theory of one unary function symbol. Finally, we prove that the Bernays-Sch\"onfinkel-Ramsey fragment of prenex \seplogk{1} formulae with quantifier prefix in the language $\exists^*\forall^*$ is \pspace-complete. The definition of a complete (hierarchical) classification of the complexity of prenex \seplogk{1}, according to the quantifier alternation depth is left as an open problem

Completeness for a First-order Abstract Separation Logic

Existing work on theorem proving for the assertion language of separation logic (SL) either focuses on abstract semantics which are not readily available in most applications of program verification, or on concrete models for which completeness is not possible. An important element in concrete SL is the points-to predicate which denotes a singleton heap. SL with the points-to predicate has been shown to be non-recursively enumerable. In this paper, we develop a first-order SL, called FOASL, with an abstracted version of the points-to predicate. We prove that FOASL is sound and complete with respect to an abstract semantics, of which the standard SL semantics is an instance. We also show that some reasoning principles involving the points-to predicate can be approximated as FOASL theories, thus allowing our logic to be used for reasoning about concrete program verification problems. We give some example theories that are sound with respect to different variants of separation logics from the literature, including those that are incompatible with Reynolds's semantics. In the experiment we demonstrate our FOASL based theorem prover which is able to handle a large fragment of separation logic with heap semantics as well as non-standard semantics.Comment: This is an extended version of the APLAS 2016 paper with the same titl

The Effects of Adding Reachability Predicates in Propositional Separation Logic

International audienceThe list segment predicate ls used in separation logic for verifying programs with pointers is well-suited to express properties on singly-linked lists. We study the effects of adding ls to the full proposi-tional separation logic with the separating conjunction and implication, which is motivated by the recent design of new fragments in which all these ingredients are used indifferently and verification tools start to handle the magic wand connective. This is a very natural extension that has not been studied so far. We show that the restriction without the separating implication can be solved in polynomial space by using an appropriate abstraction for memory states whereas the full extension is shown undecidable by reduction from first-order separation logic. Many variants of the logic and fragments are also investigated from the computational point of view when ls is added, providing numerous results about adding reachability predicates to propositional separation logic

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.