Relational semantics of linear logic and higher-order model-checking

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how his analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte

Real-time and Probabilistic Temporal Logics: An Overview

Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed

Relational Semantics of Linear Logic and Higher-order Model Checking

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how this analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes

Completeness of Flat Coalgebraic Fixpoint Logics

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The mu-calculus and its relatives are among the most expressive logics of this type. However, popular fixpoint logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL, and the logic of common knowledge. Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard mu-calculus including, e.g., flat fragments of the graded mu-calculus and the alternating-time mu-calculus (such as alternating-time temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer Science, Springer, 2010, pp. 524-53

Using models to model-check recursive schemes

We propose a model-based approach to the model checking problem for recursive schemes. Since simply typed lambda calculus with the fixpoint operator, lambda-Y-calculus, is equivalent to schemes, we propose the use of a model of lambda-Y-calculus to discriminate the terms that satisfy a given property. If a model is finite in every type, this gives a decision procedure. We provide a construction of such a model for every property expressed by automata with trivial acceptance conditions and divergence testing. Such properties pose already interesting challenges for model construction. Moreover, we argue that having models capturing some class of properties has several other virtues in addition to providing decidability of the model-checking problem. As an illustration, we show a very simple construction transforming a scheme to a scheme reflecting a property captured by a given model.Comment: Long version of a paper presented at TLCA 201