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

Indexed linear logic and higher-order model checking

In recent work, Kobayashi observed that the acceptance by an alternating tree automaton A of an infinite tree T generated by a higher-order recursion scheme G may be formulated as the typability of the recursion scheme G in an appropriate intersection type system associated to the automaton A. The purpose of this article is to establish a clean connection between this line of work and Bucciarelli and Ehrhard's indexed linear logic. This is achieved in two steps. First, we recast Kobayashi's result in an equivalent infinitary intersection type system where intersection is not idempotent anymore. Then, we show that the resulting type system is a fragment of an infinitary version of Bucciarelli and Ehrhard's indexed linear logic. While this work is very preliminary and does not integrate key ingredients of higher-order model-checking like priorities, it reveals an interesting and promising connection between higher-order model-checking and linear logic.Comment: In Proceedings ITRS 2014, arXiv:1503.0437

On the Termination Problem for Probabilistic Higher-Order Recursive Programs

In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higher-order probabilistic programming languages, not much has been done to combine those two approaches. In this paper, we initiate a study on the probabilistic higher-order model checking problem, by giving some first theoretical and experimental results. As a first step towards our goal, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higher-order programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem -- or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability. We have implemented the procedure for order-2 PHORSs, and confirmed that the procedure works well through preliminary experiments that are reported at the end of the article

Intuitionistic Non-Normal Modal Logics: A general framework

We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give a semantic characterisation of our logics in terms of neighbourhood models. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic.Comment: Preprin

Intuitionistic non-normal modal logics: A general framework

International audienceWe define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic

Probabilistic Termination by Monadic Affine Sized Typing

International audienceWe introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of probabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility, but requires a substantial adaptation of the technique

Sémantique de la logique linéaire et "model-checking" d'ordre supérieur

This thesis studies problems of higher-order model-checking from a semantic and logical perspective. Higher-order model-checking is concerned with the verification of properties expressed in monadic second-order logic, specified over infinite trees generated by a class of rewriting systems called higher-order recursion schemes. These systems are equivalent to simply-typed lambda-terms with recursion, and can therefore be studied using semantic methods.The more specific purpose of this thesis is to connect higher-order model-checking to a series of advanced ideas in contemporary semantics, such as linear logic and its relational semantics, indexed linear logic, distributive laws between comonads, parametric comonads and tensorial logic. As we will see, all these ingredients meet and combine surprisingly well with higher-order model-checking.The starting point of our approach is the study of the intersection type system of Kobayashi and Ong. This intersection type system enables one to type a higher-order recursion scheme with states of a given automaton, associated with a formula of monadic second-order logic. The recursion scheme is typable with the initial state of the automaton if and only if the infinite tree it represents satisfies the formula of interest. In spite of this soundness-and-completeness result, the original type system by Kobayashi and Ong was not designed with the connection between intersection types and models of linear logic observed by Bucciarelli, Ehrhard, de Carvalho and Terui in mind. Our work has thus been to connect these two fields.Our analysis leads us to the definition of an alternative intersection type system, which enjoys a similar soundness-and-completeness theoremwith respect to higher-order model-checking. In contrast to the original type system by Kobayashi and Ong, our modal formulation is the proof-theoretic counterpart of a finitary semantics of linear logic, obtained by composing the traditional exponential modality with a coloring comonad. We equip the semantics of linear logic with an inductive-coinductive fixpoint operator. We obtain in this way a model of the lambda-calculus with recursion in which the interpretation of a higher-order recursion scheme is the set of states from which the infinite tree it represents is accepted. The finiteness of the semantics enables us to reestablish several results of decidability for higher-order model-checking problems, among which the selection problem recently formulated and proved by Carayol and Serre.This finitary semantics are inspired from the extensional collapse theorem of Ehrhard, who shows that the relational semantics of linear logic collapses extensionally to the finitary semantics provided by Scott lattices. For that reason, we start in a preliminary approach to define the coloring comonad and the inductive-coinductive fixpoint operator in the quantitative semantics provided by an infinitary (and non-continuous) version of the relational model of linear logic.Dans cette thèse, nous envisageons des problèmes de model-checking d'ordre supérieur à l'aide d'approches issues de la sémantique et de la logique. Le model-checking d'ordre supérieur étudie la vérification de propriétés, exprimées en logique monadique du second ordre, sur des arbres infinis générés par une classe de systèmes de réécriture appelés schémas de récursion d'ordre supérieur. Ces systèmes sont équivalents au lambda-calcul simplement typé avec récursion, et peuvent donc être étudiés à l'aide d'outils sémantiques.Plus précisément, l'objet de cette thèse est de relier le model-checking d'ordre supérieur à une série de concepts de premier plan en sémantique contemporaine, tels que la logique linéaire et sa sémantique relationnelle, la logique linéaire indexée, les lois distributives entre comonades, les comonades paramétrées et la logique tensorielle. Nous verrons que ces concepts contribuent de façon particulièrement naturelle à l'étude du model-checking d'ordre supérieur.Notre approche débute par une étude du système de types intersection de Kobayashi et Ong, qui permet de typer un schéma de récursion d'ordre supérieur avec les états d'un automate donné encodant une formule de la logique monadique du second ordre. Le schéma admet pour type l'état initial de l'automate si et seulement si l'arbre infini qu'il représente satisfait la propriété encodée par l'automate. En dépit de cette adéquation, le système de types de Kobayashi et Ong a été pensé indépendamment de la connexion existant entre les types intersections et les modèles de la logique linéaire, relation observée par Bucciarelli, Ehrhard, de Carvalho et Terui. Nous avons donc cherché à relier ces deux domaines.Notre analyse nous a permis de définir un système de types intersection dérivé de celui de Kobayashi et Ong, capturant lui aussi le model-checking d'ordre supérieur de façon adéquate. Contrairement au système original, notre système est formulé de façon modale, et correspond à une sémantique finitaire de la logique linéaire obtenue en composant la modalité exponentielle usuelle avec une comonade colorant les formules. Nous équipons cette sémantique de la logique linéaire avec un opérateur de point fixe inductif-coinductif, et obtenons ainsi un modèle du lambda-calcul avec récursion dans lequel l'interprétation d'un schéma de récursion d'ordre supérieur est l'ensemble des états depuis lesquels l'arbre infini qu'il représente est accepté. La finitude de la sémantique nous permet de donner de nouvelles preuves de plusieurs résultats de décidabilité pour des problèmes de model-checking d'ordre supérieur, dont le problème de la sélection formulé récemment par Carayol et Serre.La sémantique finitaire que nous définissons est inspirée du théorème d'écrasement extensionnel d'Ehrhard, qui montre que l'écrasement extensionnel du modèle relationnel de la logique linéaire correspond à sa sémantique finitaire donnée par le modèle de Scott. Ce résultat nous permet de définir dans un premier temps la comonade de coloration et l'opérateur de point fixe inductif-coinductif dans une sémantique quantitative correspondant à une variante infinie (et non-continue) du modèle relationnel de la logique linéaire

Probabilistic Termination by Monadic Affine Sized Typing

International audienceWe introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of probabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility, but requires a substantial adaptation of the technique

Terminating Calculi and Countermodels for Constructive Modal Logics

International audienc