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International audienceSeparation Logic is a widely used formalism for describing dynamically allocated linked data structures, such as lists, trees, etc. The decidability status of various fragments of the logic constitutes a long standing open problem. Current results report on techniques to decide satisfiability and validity of entail-ments for Separation Logic(s) over lists (possibly with data). In this paper we establish a more general decidability result. We prove that any Separation Logic formula using rather general recursively defined predicates is decidable for satis-fiability, and moreover, entailments between such formulae are decidable for validity. These predicates are general enough to define (doubly-) linked lists, trees, and structures more general than trees, such as trees whose leaves are chained in a list. The decidability proofs are by reduction to decidability of Monadic Second Order Logic on graphs with bounded tree width

We address the problem of verifying programs manipulating one-selector linked data structures. We propose and study in detail an application of counter automata as an accurate abstract model for this problem. We let control states of the counter automata correspond to abstract heap graphs where list segments without sharing are collapsed, and use counters to keep track of the number of elements in these segments. As a significant theoretical result, we show that the obtained counter automata are bisimilar to the original programs. Moreover, from a practical point of view, our translation allows one to apply efficient automatic analysis techniques and tools developed for counter automata (integer programs) in order to verify both safety as well as termination of list-manipulating programs. As another This work is a full and revised version of the extended abstract [10] published in Proceedings of CAV'06. Form Methods Syst Des (2011) 38: 158-192 159 theoretical contribution, we prove that if the control of the generated counter automata does not contain nested loops (i.e., these automata are flat), both safety and termination are decidable for the original programs. Subsequently, we generalise our counter-automata-based model to keep track of ordering properties over lists storing ordered data. Finally, we show effectiveness of our approach by verifying automatically safety as well as termination of several sorting programs.

Abstract. Computing transitive closures of integer relations is the key to finding precise invariants of integer programs. In this paper, we describe an efficient algorithm for computing the transitive closures of difference bounds, octagonal and finite monoid affine relations. On the theoretical side, this framework provides a common solution to the acceleration problem, for all these three classes of relations. In practice, according to our experiments, the new method performs up to four orders of magnitude better than the previous ones, making it a promising approach for the verification of integer programs.

International audienceIn this paper we study the reachability problem for parametric flat counter automata, in relation with the satisfiability problem of three fragments of integer arithmetic. The equivalence between non-parametric flat counter au-tomata and Presburger arithmetic has been established previously by Comon and Jurski [5]. We simplify their proof by introducing finite state automata defined over alphabets of a special kind of graphs (zigzags). This framework allows one to express also the reachability problem for parametric automata with one control loop as the existence of solutions of a 1-parametric linear Diophantine systems. The latter problem is shown to be decidable, using a number-theoretic argument. Finally, the general reachability problem for parametric flat counter automata with more than one loops is shown to be undecidable, by reduction from Hilbert's Tenth Problem [9]

International audienceSeparation Logic (SL) with inductive definitions is a natural formalism for specifying complex recursive data structures, used in compositional verification of programs manipulating such structures. The key ingredient of any automated verification procedure based on SL is the decidability of the entailment problem. In this work, we reduce the entailment problem for a non-trivial subset of SL describing trees (and beyond) to the language inclusion of tree automata (TA). Our reduction provides tight complexity bounds for the problem and shows that entailment in our fragment is EXPTIME-complete. For practical purposes, we leverage from recent advances in automata theory, such as inclusion checking for non-deterministic TA avoiding explicit determinization. We implemented our method and present promising preliminary experimental results

Abstract. In this paper we prove that the transitive closure of a nondeterministic octagonal relation using integer counters can be expressed in Presburger arithmetic. The direct consequence of this fact is that the reachability problem is decidable for flat counter automata with octagonal transition relations. This result improves the previous results of Comon and Jurski [7] and Bozga, Iosif and Lakhnech [6] concerning the computation of transitive closures for difference bound relations. The importance of this result is justified by the wide use of octagons to computing sound abstractions of real-life systems [15]. We have implemented the octagonal transitive closure algorithm in a prototype system for the analysis of counter automata, called FLATA, and we have experimented with a number of test cases.

Abstract. This paper presents a publicly available toolkit and a benchmark suite for rigorous verification of Integer Numerical Transition Systems (INTS), which can be viewed as control-flow graphs whose edges are annotated by Presburger arithmetic formulas. We present FLATA and ELDARICA, two verification tools for INTS. The FLATA system is based on precise acceleration of the transition relation, while the ELDARICA system is based on predicate abstraction with interpolation-based counterexample-driven refinement. The ELDARICA verifier uses the PRINCESS theorem prover as a sound and complete interpolating prover for Presburger arithmetic. Both systems can solve several examples for which previous approaches failed, and present a useful baseline for verifying integer programs. The infrastructure is a starting point for rigorous benchmarking, competitions, and standardized communication between tools.

International audienceWe introduce a new decidable logic for reasoning about infinite arrays of integers. The logic is in the ∃ * ∀ * first-order fragment and allows (1) Presburger constraints on existentially quantified variables, (2) difference constraints as well as periodicity constraints on universally quantified indices, and (3) difference constraints on values. In particular, using our logic, one can express constraints on consecutive elements of arrays (e.g. ∀i. 0 ≤ i < n → a[i + 1] = a[i] − 1) as well as periodic facts (e.g. ∀i. i ≡ 2 0 → a[i] = 0). The decision procedure follows the automata-theoretic approach: we translate formulae into a special class of Büchi counter automata such that any model of a formula corresponds to an accepting run of the automaton, and vice versa. The emptiness problem for this class of counter automata is shown to be decidable, as a consequence of earlier results on counter automata with a flat control structure and transitions based on difference constraints. We show interesting program properties expressible in our logic, and give an example of invariant verification for programs that handle integer arrays

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