Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

On Temporal and Separation Logics

International audienceThere exist many success stories about the introduction of logics designed for the formal verification of computer systems. Obviously, the introduction of temporal logics to computer science has been a major step in the development of model-checking techniques. More recently, separation logics extend Hoare logic for reasoning about programs with dynamic data structures, leading to many contributions on theory, tools and applications. In this talk, we illustrate how several features of separation logics, for instance the key concept of separation, are related to similar notions in temporal logics. We provide formal correspondences (when possible) and present an overview of related works from the literature. This is also the opportunity to present bridges between well-known temporal logics and more recent separation logics

Reasoning about reversal-bounded counter machines

International audienceIn this paper, we present a short survey on reversal-bounded counter machines. It focuses on the main techniques for model-checking such counter machines with specifications expressed with formulae from some linear-time temporal logic. All the decision procedures are designed by translation into Presburger arithmetic. We provide a proof that is alternative to Ibarra's original one for showing that reachability sets are effectively definable in Presburger arithmetic. Extensions to repeated control state reachability and to additional temporal properties are discussed in the paper. The article is written to the honor of Professor Ewa Orłowska and focuses on several topics that are developped in her works

On the Complexity of Modal Separation Logics

International audienceWe introduce a modal separation logic MSL whose models are memory states from separation logic and the logical connectives include modal operators as well as separating conjunction and implication from separation logic. With such a combination of operators, some fragments of MSL can be seen as genuine modal logics whereas some others capture standard separation logics, leading to an original language to speak about memory states. We analyse the decidability status and the computational complexity of several fragments of MSL, leading to surprising results, obtained by designing proof methods that take into account the modal and separation features of MSL. For example, the satisfiability problem for the fragment of MSL with 3, the inequality modality = and separating conjunction * is shown Tower-complete whereas the restriction either to 3 and * or to = and * is only NP-complete

Axiomatising logics with separating conjunctions and modalities

International audienceModal separation logics are formalisms that combine modal operators to reason locally, with separating connectives that allow to perform global updates on the models. In this work, we design Hilbert-style proof systems for the modal separation logics MSL(⇤, h6 =i) and MSL(⇤, 3), where ⇤ is the separating conjunction, 3 is the standard modal operator and h6 =i is the di↵erence modality. The calculi only use the logical languages at hand (no external features such as labels) and take advantage of new normal forms and of their axiomatisation

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

Petri Net Reachability Graphs: Decidability Status of FO Properties

International audienceWe investigate the decidability and complexity status of model-checking problems on unlabelled reachability graphs of Petri nets by considering first-order, modal and pattern-based languages without labels on transitions or atomic propositions on markings. We consider several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques

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

Methods for Modalities 2007

Proceedings of Methods for Modalities 200