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The probabilistic (or quantitative) modal μ-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS’s). Two semantics have been studied for this logic, both assigning to every process state a value in the interval [0, 1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic games. The two semantics have been proved to coincide on all finite PLTS’s. A first contribution of the thesis is to extend this coincidence result to arbitrary PLTS’s. A shortcoming of the probabilistic μ-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS’s such as Probabilistic Computation Tree Logic (PCTL). To address this limitation, we extend the logic with a new pair of operators: independent product and coproduct, and we show that the resulting logic can encode the qualitative fragment of PCTL. Moreover, a further extension of the logic, with the operation of truncated sum and its dual, is expressive enough to encode full PCTL. A major contribution of the thesis is the definition of appropriate game semantics for these extended probabilistic μ-calculi. This relies on the definition of a new class of games, called tree games, which generalize standard 2-player stochastic games. In tree games, a play can be split into concurrent subplays which continue their evolution independently. Surprisingly, this simple device supports the encoding of the whole class of imperfect-information games known as Blackwell games. Moreover, interesting open problems in game theory, such as qualitative determinacy for 2-player stochastic parity games, can be reformulated as determinacy problems for suitable classes of tree games. Our main technical result about tree games is a proof of determinacy for 2-player stochastic metaparity games, which is the class of tree games that we use to give game semantics to the extended probabilistic μ-calculi. In order to cope with measure-theoretic technicalities, the proof is carried out in ZFC set theory extended with Martin’s Axiom at the first uncountable cardinal (MAℵ1). The final result of the thesis shows that the game semantics of the extended logics coincides with the denotational semantics, for arbitrary PLTS’s. However, in contrast to the earlier coincidence result, which is proved in ZFC, the proof of coincidence for the extended calculi is once again carried out in ZFC +MAℵ1.

We consider a syntactic restriction for higher-order grammars called safety that constrains occurrences of variables in the production rules according to their type-theoretic order. We transpose and generalize this restriction to the setting of the simply-typed lambda calculus, giving rise to what we call the safe lambda calculus. We analyze its expressivity and obtain a result in the same vein as Schwichtenberg's 1976 characterization of the simply-typed lambda calculus: the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a similar characterization for representable word functions. We then examine the complexity of deciding beta-eta equality of two safe simply-typed terms and show that this problem is PSPACE-hard. The safety restriction is then extended to other applied lambda calculi featuring recursion and references such as PCF and Idealized Algol (IA for short). The next contribution concerns game semantics. We introduce a new concrete presentation of this semantics using the theory of traversals. It is shown that the revealed game denotation of a term can be computed by traversing some souped-up version of the term's abstract syntax tree using adequately defined traversal rules. Based on this presentation and via syntactic reasoning we obtain a game-semantic interpretation of safety: the strategy denotations of safe lambda-terms satisfy a property called P-incremental justification which says that the player's moves are always justified by the last pending opponent's move of greater order occurring in the player's view. Next we look at models of the safe lambda calculus. We show that these are precisely captured by Incremental Closed Categories. A game model is constructed and is shown to be fully abstract for safe IA. Further, it is effectively presentable: two terms are equivalent just if they have the same set of complete O-incrementally justified plays---where O-incremental justification is defined as the dual of P-incremental justification. Finally we study safety from the point of view of algorithmic game semantics. We observe that in the third-order fragment of IA, the addition of unsafe contexts is conservative for observational equivalence. This implies that all the upper complexity bounds known for the lower-order fragments of IA also hold for the safe fragment; we show that the lower-bounds remain the same as well. At order 4, observational equivalence is known to be undecidable for IA. We conjecture that for the order-4 safe fragment of IA, the problem is reducible to the DPDA-equivalence problem and is thus decidable.

We analyse the semantics of aspect-oriented extensions to functional languages by presenting compositional translations of these primitives into languages with traditional notions of state and control. As a first step, we examine an existing semantic description of aspects which allows the labelling of program points. We show that a restriction of these semantics to aspects which do not preempt the execution of code can be fully abstractly translated into a functional calculus with higher order references, but that removing this restriction requires a notion of exception handling to be added to the target language in order to yield a sound semantics. Next, we proceed to show that abandoning the labelling technique, and consequently relaxing the so-called ``obliviousness'' property of aspectual languages, allows preemptive aspects to be included in the general references model without the need for exceptions. This means that the game model of general references is inherited by the aspect calculus. The net result is a clean semantic description of aspect-orientation, which mirrors recently published techniques for their implementation, and thereby provides theoretical justification for these systems. The practical validity of our semantics is demonstrated by implementing extensions to the basic calculus in Standard ML, and showing how a number of useful aspect-oriented features can be expressed using general references alone. Our theoretical methodology closely follows the proof structure that often appears in the game semantics literature, and therefore provides an operational perspective on notions such as ``bad variables'' and factorisation theorems.

This thesis examines the use of game semantics for the automatic equivalence checking of higher-order programs. Game semantics has proved to be a powerful method for constructing fully abstract models of logics and programming languages. Furthermore, the concrete nature of the semantics lends itself to algorithmic analysis. The game-semantic model can be used to identify fragments of languages which have a decidable observational equivalence problem. We investigate decidability results for different languages as well as the efficiency of these algorithms in practice. First we consider the call-by-value higher-order language with state, RML. This can be viewed as a canonical restriction of Standard ML to ground-type references. The O-strict fragment of RML is the largest set of type sequents for which, in the game-semantic denotation, justification pointers from O-moves are always uniquely reconstructible from the underlying move sequence. The O-strict fragment is surprisingly expressive, including higher-order types and difficult examples from the literature. By representing strategies as Visibly Pushdown Automata (VPA) we show that observational equivalence of O-strict terms is decidable (and in fact is ExpTime-complete). We then consider extensions of the O-strict fragment. Adding general recursion or using most non-O-strict types leads to undecidability. However, a limited form of recursion can be added while still preserving decidability (although the full power of DPDA is required). Next we examine languages with non-local control. This involves adding call/cc to our language and is known to correspond to dropping the game-semantic bracketing condition. In the call-by-name game-semantic model of Idealized Algol (IA), in which answers cannot justify questions, the visibility condition still implies a form of weak bracketing. By making bracketing violations explicit we show that we can still model the entire third-order fragment using VPA. We have also implemented tools based on these algorithms. Our model checkers Homer and Hector perform equivalence checking for third-order IA and O-strict RML respectively. Homer uses a naive explicit state method whereas Hector takes advantage of on-the-fly model checking. Our tools perform well on small yet challenging examples. On negative instances, the on-the-fly approach allows Hector to outperform Homer. To improve their performance, we also consider using ideas from symbolic execution. We propose a representation for finite automata using transitions labelled with formulas and guards which aims to take advantage of the symmetries of the game-semantic model so that strategies can be represented compactly. We refer to this representation as Symbolically Executed Automata (SEA). Using SEA allows much larger data types to be handled but is not as effective on larger examples with small data types.

La sémantique des jeux est une approche pour modéliser les langages de programmation dans laquelle les types sont interprétés par des jeux et les programmes par des stratégies. Ces modèles de jeux ont couvert des constructions fonctionnelles et impératives, des opérateurs de contrôle, etc. L'approche a récemment été étendue aux langages non-déterministes et concurrents, provoquant au passage un changement de perspective profond~: les parties sont maintenant organisées en une catégorie, sur laquelle les stratégies sont des préfaisceaux. La notion fondamentale d'innocence a aussi été caractérisée comme une condition de faisceau. Cette thèse s'attache à l'étude de quelques constructions apparaissant dans ces nouveaux modèles de jeux.D'abord, constatant que, dans plusieurs de ces modèles, l'étape cruciale consiste à définir une catégorie double de jeux et de parties, nous proposons une construction abstraite d'une telle catégorie double à partir de données de base, puis nous démontrons que, sous des hypothèses adéquates, le résultat obtenu permet en effet la construction des stratégies.Dans un second temps, nous établissons un lien entre deux techniques existantes pour définir les parties~: la technique standard, fondée sur les séquences justifiées, et une autre plus récente utilisant les diagrammes de cordes. Nous définissons un plongement (plein) de la première dans la seconde et prouvons qu'elles induisent essentiellement le même modèle.Enfin, nous proposons une axiomatisation des notions de jeu et de partie, de laquelle nous tirons une catégorie de jeux et stratégies. Nous raffinons ensuite les axiomes pour traiter l'innocence et nous démontrons que, sous des hypothèses adéquates, les stratégies innocentes sont stables par composition. / Game semantics is a class of models of programming languages in which types are interpreted as games and programs as strategies. Such game models have successfully covered diverse features, such as functional and imperative programming, or control operators. They have recently been extended to non-deterministic and concurrent languages, which generated an in-depth recasting of the standard approach: plays are now organised into a category, on which strategies are presheaves. The fundamental notion of innocence has also been recast as a sheaf condition. This thesis is a study of various constructions appearing in this new approach to game semantics.We first consider a pattern common to several game models of concurrent languages, in which games and plays are first organised into a double category, from which strategies are then derived. We provide an abstract construction of such a double category from more basic data, and prove that, under suitable hypotheses, the result allows the construction of strategies.Our second contribution is to relate two established techniques for defining plays: the standard one, based on justified sequences, and a more recent one, based on string diagrams. We (fully) embed the former into the latter and prove that they induce essentially the same model.Finally, we propose an axiomatisation of the notions of game and play, from which we formally derive a category of games and strategies. We also refine the axioms to deal with innocence, and prove that, under suitable hypotheses, innocent strategies are stable under composition.

In the 1980s and 1990s, Joyal and Street developed a graphical notation for various flavours of monoidal category using graphs drawn in the plane, commonly known as string diagrams. In particular, their work comprised a rigorous topological foundation of the notation. In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notions they called ⊸-schedules, ⊗-schedules and heaps. Schedules described interleavings of plays in games formed using ⊸ and ⊗, and heaps provided pointers used for backtracking. Their definitions were combinatorial in nature, but researchers often draw certain pictures when working in practice. In this thesis, we extend the framework of Joyal and Street to give a formal account of the graphical methods already informally employed by researchers in game semantics. We give a geometric formulation of ⊸-schedules and ⊗-schedules, and prove that the games they describe are isomorphic to those described in Harmer et al.’s terms, and also those given by a more general graphical representation of interleaving across games of multiple components. We further illustrate the value of the geometric methods by demonstrating that several proofs of key properties (such as that the composition of ⊸-schedules is associative) can be made straightforward, reflecting the geometry of the plane, and overstepping some of the cumbersome combinatorial detail of proofs in Harmer et al.’s terms. We further extend the framework of formal plane diagrams to account for the heaps and pointer structures used in the backtracking functors for O and P.

An interesting problem in proof theory is to find representations of proof that do not distinguish between proofs that are ‘morally’ the same. For many logics, the presentation of proofs in a traditional formalism, such as Gentzen’s sequent calculus, introduces artificial syntactic structure called ‘bureaucracy’; e.g., an arbitrary ordering of freely permutable inferences. A proof system that is free of bureaucracy is called canonical for a logic. In this dissertation two canonical proof systems are presented, for two logics: a notion of proof nets for additive linear logic with units, and ‘classical proof forests’, a graphical formalism for first-order classical logic. Additive linear logic (or sum–product logic) is the fragment of linear logic consisting of linear implication between formulae constructed only from atomic formulae and the additive connectives and units. Up to an equational theory over proofs, the logic describes categories in which finite products and coproducts occur freely. A notion of proof nets for additive linear logic is presented, providing canonical graphical representations of the categorical morphisms and constituting a tractable decision procedure for this equational theory. From existing proof nets for additive linear logic without units by Hughes and Van Glabbeek (modified to include the units naively), canonical proof nets are obtained by a simple graph rewriting algorithm called saturation. Main technical contributions are the substantial correctness proof of the saturation algorithm, and a correctness criterion for saturated nets. Classical proof forests are a canonical, graphical proof formalism for first-order classical logic. Related to Herbrand’s Theorem and backtracking games in the style of Coquand, the forests assign witnessing information to quantifiers in a structurally minimal way, reducing a first-order sentence to a decidable propositional one. A similar formalism ‘expansion tree proofs’ was presented by Miller, but not given a method of composition. The present treatment adds a notion of cut, and investigates the possibility of composing forests via cut-elimination. Cut-reduction steps take the form of a rewrite relation that arises from the structure of the forests in a natural way. Yet reductions are intricate, and initially not well-behaved: from perfectly ordinary cuts, reduction may reach unnaturally configured cuts that may not be reduced. Cutelimination is shown using a modified version of the rewrite relation, inspired by the game-theoretic interpretation of the forests, for which weak normalisation is shown, and strong normalisation is conjectured. In addition, by a more intricate argument, weak normalisation is also shown for the original reduction relation.

La sémantique des jeux permet l'étude et la modélisation abstraite des langages de programmation d'un point de vue mathématique, en gardant assez d'informations concrètes sur la structure des programmes, mais en laissant de côté les détails superflus. Durant mon doctorat, j'ai travaillé sur l'association de la sémantique des jeux avec les structures d'événements pour proposer des modèles dénotationnels vraiment concurrents de langages concurrents d'ordresupérieur. Dans un premier temps, je construis un modèle réalisant cette association, qui retient suffisamment d'informations sur le comportement des programmes pour interpréter adéquatement une grande variété de langages concurrents et non déterministes pour des notions fines de convergences. La construction de ce modèle se base surl'introduction de symétrie afin d'établir que le modèle forme une catégorie cartésienne fermée. Dans un second temps, je propose une généralisation dans ce cadre des notions d'innocence et de bon parenthésage, essentielles en sémantiquedes jeux pour comprendre les effets calculatoires, et résolvant ainsi des problèmes ouverts de la sémantique des jeux concernant l'innocence concurrente et non-déterministe. Dans un dernier temps je propose une interprétation dans ce modèle, de langages concurrents avec mémoire faible, un des premiers travaux de sémantique dénotationnelle pour ce type de langages. Bien que théoriques, ces modèles sont compositionnels et basés sur des ordres partiels, et donc pourraient permettre de faire passer la vérification de programmes concurrents à l'échelle (une problématique importante du domaine). / Game semantics is an effective tool to study and model abstractly programming languages from a mathematical point of view, by keeping enough concrete information on the structure of programs but yet leaving aside superfluous details. During my PhD thesis, I worked on merging game semantics with event structures to propose truly concurrent denotational models of higher-order concurrent languages. In the first part, I build a model based on this merge, retaining enough information about the behaviour of programs to interpret adequately a large variety of concurrent programming languages for various notions of convergence. The construction of this model is based on the introduction of symmetry to prove that the model is indeed in a cartesian-closed category. In the second part, I propose a generalization, in this setting, of innocence and well-bracketing, key notions in game semantics to understand the computational effects, and thusly closing openproblems of game semantics about concurrent and nondeterministic innocence.In the last part, I propose an interpretation in this model of concurrent languages with weak shared memory, one of the first works of denotational semantics for these kinds of languages. Althoughtheoretical, these models are compositional and based on partial orders, and thus could permit scaling verification of concurrent programs (an important problem of the domain).

Prouver l’équivalence de programmes écrits dans un langage fonctionnel avec références est un problème notoirement difficile. L’objectif de cette thèse est de proposer un système logique dans lequel de telles preuves peuvent être formalisées, et dans certains cas inférées automatiquement. Dans la première partie, une méthode générique d’extension de la théorie des types dépendants est proposée, basée sur une interprétation du forcing vu comme une traduction de préfaisceaux de la théorie des types. Cette extension dote la théorie des types de constructions récursives gardées, qui sont utilisées ensuite pour raisonner sur les références d’ordre supérieure. Dans une deuxième partie, nous définissons une sémantique des jeux nominale opérationnelle pour un langage avec références d’ordre supérieur. Elle marie la structure catégorique de la sémantique des jeux avec une représentation sous forme de traces de la dénotation des programmes, qui se calcule de manière opérationnelle et dispose donc de bonnes propriétés de modularité. Cette sémantique nous permet ensuite de prouver la complétude de relations logiques à la Kripke définit de manière directe, via l’utilisation de types récursifs gardés, sans utilisation de la biorthogonalité. Une telle définition directe nécessite l’utilisation de mondes omniscient et un contrôle fin des locations divulguées. Finalement, nous introduisons une logique temporelle qui donne un cadre pour définir ces relations logiques à la Kripke. Nous ramenons alors le problème de l’équivalence contextuelle à la satisfiabilité d’une formule de cette logique générée automatique, c’est à dire à l’existence d’un monde validant cette formule. Sous certaines conditions, cette satisfiabilité peut être décidée via l’utilisation d’un solveur SMT. La complétude de notre méthode devrait permettre d’obtenir des résultats de décidabilité pour l’équivalence contextuelle de certains fragment du langage considéré, en fournissant un algorithme pour construire de tels mondes. / Proving program equivalence for a functional language with references is a notoriously difficult problem. The goal of this thesis is to propose a logical system in which such proofs can be formalized, and in some cases inferred automatically. In the first part, a generic extension method of dependent type theory is proposed, based on a forcing interpretation seen as a presheaf translation of type theory. This extension equips type theory with guarded recursive constructions, which are subsequently used to reason on higher-order references. In the second part, we define a nominal game semantics for a language with higher-order references. It marries the categorical structure of game semantics with a trace representation of denotations of programs, which can be computed operationally and thus have good modularity properties. Using this semantics, we can prove the completeness of Kripke logical relations defined in a direct way, using guarded recursive types, without using biorthogonality. Such a direct definition requires omniscient worlds and a fine control of disclosed locations. Finally, we introduce a temporal logic which gives a framework to define these Kripke logical relations. The problem of contextual equivalence is then reduced to the satisfiability of an automatically generated formula defined in this logic, i.e. to the existence of a world validating this formula. Under some conditions, this satisfiability can be decided using a SMT solver. Completeness of our methods opens the possibility of getting decidability results of contextual equivalence for some fragments of the language, by giving an algorithm to build such worlds.