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В подходе Artificial General Intelligence (Universal AI) интеллект рассматривается как информационный процессор, потребляющий и выдающий информацию, которой и определяется поведение системы. В рамках этого подхода M. Hutter получил способ выбора оптимальной траектории агента в абстрактной среде. Однако этот способ требовал численных оценок вознаграждения при том или ином движении, способ получения которых оставался открытым. Также этот метод не подходит для оценок движения группы агентов. В настоящей работе оценки вознаграждений предоставляются самой средой и предложен способ выбора траектории, который применим для группы агентов. Параллельное выполнение группой интеллектуальных агентов ряда задач, представляется тензорным произведением соответствующих процессов в категории игр Конвея, которая сопоставлена среде и движениям агентов. Оптимальный маршрут группы определяется как игра с наибольшим суммарным выигрышем в этой категории. Выигрыш представлен степенью определенности (видимости) цели агента, т.е. некоторым множеством, а не числом. Доказано, что такое определение выигрыша может быть использовано в категорной конструкции для игр Конвея. В этой категории тензорное произведение является операцией линейной логики. Также линейная логика применяется для выбора целей, которые система может достичь, из всего множества видимых целей. Для этого все множество целей представляется в виде решетки, на которой задана структура линейной логики. Решетка целей в этом случае становится множеством истинностных значений логики. Целям, которые достигаются параллельно, также соответствует тензорное произведение (как и параллельным процессам в среде), но только теперь это произведение элементов решетки целей. Цели выбираются по наибольшей степени истинности того элемента решетки, который соответствует их параллельному достижению. В результате получена формула для оценки наиболее выигрышного маршрута в абстрактной среде для группы агентов., Intelligence is represented in the Artificial General Intelligence (or Universal AI) approach as an information processor which consumes and gives out information. This information completely determines the system behavior. M. Hutter has obtained an optimal agent trajectory selecting in an abstract environment within this approach. However, the method demands number assessments of the reward in such a movement, and the way to obtain these assessments was not defined. The method does not also fit to estimate an agent group movement. In this paper, reward assessments are represented by the environment. The way to choose a trajectory for an agent group is also suggested. A group of intelligent agents which fulfill a set of tasks in parallel is represented by the tensor multiplication of corresponding processes in the Conway game category. The category corresponds to the environment and agents' moves. An optimal itinerary of the group is defined as a play with maximal total reward in the category. The reward is represented as a degree of certainty (visibility) of an agent goal, i.e., by some set, not a number. It is proved that such a reward determination may be used in the categorical construction of Conway games. The tensor product in the category is the operation of linear logic. Such logic is also used in selecting goals which the system can achieve of all visible ones. In the purpose, the whole goal set is represented as a lattice with the linear logic structure. The goal lattice is the truth value set of the logic in this case. The tensor product of the lattice elements corresponds to goals which are achieved in parallel. The system goals are chosen by the highest truth value of the lattice element which corresponds to they parallel achieving processes. Such a construction results in the formula of the optimal itinerary estimation in an abstract environment for an agent group., №78 (2019)

Attack trees provide a structure to an attack scenario, where disjunctions represent choices decomposing attacker’s goals into smaller subgoals. This paper investigates the nature of choices in attack trees. For some choices, the attacker has the initiative, but for other choices either the environment or an active defender decides. A semantics for attack trees combining both types of choice is expressed in linear logic and connections with extensive-form games are highlighted. The linear logic semantics defines a specialisation preorder enabling trees, not necessarily equal, to be compared in such a way that all strategies are preserved.

International audience; Game semantics is the art of interpreting types as games and programs as strategies interacting in space and time with their environment. In order to reflect the interactive behavior of programs, strategies are required to follow specific scheduling policies. Typically, in the case of a purely sequential programming language, the program (Player) and its environment (Opponent) will play one after the other, in a strictly alternating way. On the other hand, in the case of a concurrent language, Player and Opponent will be allowed to play several moves in a row, in a non-alternating way. In both cases, the scheduling policy is designed very carefully in order to ensure that the strategies synchronize properly and compose well when plugged together. A longstanding conceptual problem has been to understand when and why a given scheduling policy works and is compositional in that sense. In this paper, we exhibit a number of simple and fundamental combinatorial structures which ensure that a given scheduling policy encoded as synchronization template defines a symmetric monoidal closed (and in fact *-autonomous) bicategory of games, strategies and simulations. To that purpose, we choose to work at a very general level, and illustrate our method by constructing two template game models of linear logic with different flavors (alternating and non-alternating) using the same categorical combinatorics, performed in the category of small categories. As a whole, the paper may be seen as a hymn in praise of synchronization, building on the notion of synchronization algebra in process calculi and adapting it smoothly to programming language semantics, using a combination of ideas at the converging point of game semantics and of categorical algebra.

This thesis presents a general framework for enriching causal concurrent games model with annotations. These annotations can be viewed as meta-data on strategies: they are modified throughout interactions but do not affect their general flow of control. These data can be of various nature, in particular our enrichment is parametrised over any multi-sorted equational theory and can also reflect structure upon these data such as a partial order. From a semantics point of view, this construction is motivated by problems from both logic and programming languages: On the logic side, the annotated games model specialised to first-order terms enables us to give a novel interpretation of first-order classical proofs as concurrent strategies carrying first-order witnesses. In particular this answer the question of giving a compositional version to Herbrand’s theorem while avoiding the usual proof sequentialization of other denotational approaches. On the programming language side, annotations on games offer intrinsic quantitative models. We show that those can be used to provide denotational semantics for resource consumption analysis of concurrent higher order programming language with shared memory.These enrichments, strongly connected to the causal structure of concurrent games, give an argument in favor of a causal meaning of computations., La sémantique des jeux est une sémantique dénotationnelle centrée sur l’interaction : preuves et programmes y sont représentés par des stratégies modélisant, par le flot d’exécution, leur manière de réagir à leur environnement. Malgré cette présentation intensionnelle, les sémantiques de jeux ne suffisent pas à capturer certaines informations calculatoires annexes au flot d’exécution telles que, par exemple, la production de témoins en logique du premier ordre ou la consommation de ressources dans les langages de programmation. Dans cette thèse nous proposons un enrichissement du modèle des jeux concurrent à base de structures d’événements permettant de garder trace de ces informations.Nous construisons d’abord un modèle de jeux concurrent dans lequel les coups joueurs d’une stratégie sont annotés par les termes d’une théorie (in)équationnelle. Cette théorie est un paramètre de notre modèle et les annotations permettent de refléter de manière compacte des informations d’exécution n’ayant pas d’influence sur le flot d’exécution. Nous montrons que le modèle ainsi construit préserve la structure catégorique compacte fermée du modèle sans annotation.Nous explorons ensuite l’expressivité de notre modèle et présentons deux interprétations nouvelles en sémantique des preuves et des programmes : l’une interprétant les preuves de la logique classique du premier ordre par des stratégies concurrentes avec échange de témoins, donnant une version compositionnelle au théorème de Herbrand ; l’autre permettant de refléter les aspects quantitatifs liés à la consommation de ressources telles que le temps, dans l’exécution de programmes concurrents d’ordre supérieur avec mémoire partagée.

Commuting conversions of Linear Logic induce a notion of dependency between rules inside a proof derivation: a rule depends on a previous rule when they cannot be permuted using the conversions. We propose a new interpretation of proofs of Linear Logic as causal invariants which captures exactly this dependency. We represent causal invariants using game semantics based on general event structures, carving out, inside the model of [6], a submodel of causal invariants. This submodel supports an interpretation of unit-free Multiplicative Additive Linear Logic with MIX (MALL\(^-\)) which is (1) fully complete: every element of the model is the denotation of a proof and (2) injective: equality in the model characterises exactly commuting conversions of MALL\(^-\). This improves over the standard fully complete game semantics model of MALL\(^-\).

International audience; Template games [P.-A. Melliès, 2019] unify various approaches to game semantics, by exhibiting them as instances of a double-categorical variant of the slice construction. However, in the particular case of simple games [R. Harmer et al., 2007; C. Jacq and P.-A. Melliès, 2018], template games do not quite yield the standard (bi)category. We refine the construction using factorisation systems, obtaining as an instance a slight generalisation of simple games and strategies. This proves that template games have the descriptive power to capture combinatorial constraints defining well-known classes of games. Another instance is Day's convolution monoidal structure on the category of presheaves over a strict monoidal category [B. Day, 1970], which answers a question raised in [C. Eberhart, 2018].

In natural language (NL), quantifiers are often used to make statements about states of affairs, like ���Many people like football���, and ���About half the people are female���. In particular, many and few express that some set of objects is relatively big, or small respectively. The semantics of those two quantifiers is not fixed once and for all, but rather depends on contextual information. Likewise, quantifiers like about half and almost all show a comparable behavior, as the tolerance margins that make corresponding statements acceptable can change from one situation to another. Fuzzy logic is often used to model such NL constructs, in particular contemporary t-norm based mathematical fuzzy logics (MFLs). Hintikka expressed Classical Logic (CL) game semantically, and Giles expressed ��ukasiewicz logic (��), a MFL, game semantically. The shared underpinning is a two player zero sum game of perfect information, where the two players act strategically. Ferm��ller and Roschger have augmented Giless game by a third non-strategic player, thereby introducing what we call the random witness selection principle into the framework. The latter principle allows us to also express two other MFLs, G��del logic and Product logic, game semantically. We achieve this by allowing for propositional quantification, which enables us to model the Delta operator, which is basically a projection operator, evoking discontinuous truth functions. This is needed to express G��del implication in Giless framework. Moreover, the propositional quantifier based on the random witness selection principle, together with the Delta operator and the existential propositional quantifier, allows us to model multiplication and division of truth functions, which we need to define the connectives of Product logic. Building on this result, we show how to define all MFLs that are finitely representable in our framework. Furthermore, the gained expressibility is used to model a variety of NL quantifiers within the framework. This pursuit is conducted in a step-by-step manner, that guarantees neat interpretability of statements. First, we model semi-fuzzy quantifiers, i.e. quantifiers that can only take classical arguments, i.e. predicates that evaluate to either (definitely) true or false. Then we lift those to fully-fuzzy quantifiers in a systematic and principle guided way, by means of quantifier fuzzification mechanisms (QFMs). As a final contribution of this thesis, we define and test, by means of an implementation, a full-fledged query language, featuring quantifiers based on the random witness selection principle. The results show that probabilistic evaluations not only are suitable to model vagueness in NL, but also increase efficiency in presence of large amounts of data.

In Osorio et al. [Revisiting Z. Notre Dame Journal of Formal Logic, 55(1):129–155, 2014] introduced a paraconsistent three-valued logic, the logic CG 3 ′ which was named after the logic G 3 ′ due to the close relation between them. Authors defined CG 3 ′ via the three-valued matrix that defines G 3 ′ but changing the set of designated truth values. In this article we present a brief study of the Kripke-type semantics for some logics related with CG 3 ′ before constructing a Kripke-type semantics for it.

We define a new logic programming semantics in terms of abducible atoms. We use it to characterize the stage extensions of an argumentation framework AF by means of an associated normal program PAF. We also define the stage semantics for a special type of normal programs and present a similar characterization.

Topological fixpoint logics are a family of logics that admits topological models and where the fixpoint operators are defined with respect to the topological interpretations. Here we consider a topological fixpoint logic for relational structures based on Stone spaces, where the fixpoint operators are interpreted via clopen sets. We develop a game-theoretic semantics for this logic. First we introduce games characterising clopen fixpoints of monotone operators on Stone spaces. These fixpoint games allow us to characterise the semantics for our topological fixpoint logic using a two-player graph game. Adequacy of this game is the main result of our paper. Finally, we define bisimulations for the topological structures under consideration and use our game semantics to prove that the truth of a formula of our topological fixpoint logic is bisimulation-invariant., Comment: In Proceedings GandALF 2016, arXiv:1609.03648

© 2016 ACM. In the past, compelling arguments in favour of the well-founded semantics for autoepistemic logic have been presented. In this article, we show that for certain classes of theories, this semantics fails to identify the unique intended model. We solve this problem by refining the well-founded semantics. We develop our work in approximation fixpoint theory, an abstract algebraical study of semantics of nonmonotonic logics. As such, our results also apply to logic programming, default logic, Dung's argumentation frameworks, and abstract dialectical frameworks. ispartof: ACM Transactions on Computational Logic vol:17 issue:4 pages:27- status: published

One of the many research projects of Jaakko Hintikka was entitled “Logical tools for human thinking and their history”. This is in fact an apt summary of the lifetime work of this master logician who developed several new methods and systems in mathematical and philosophical logic, among them distributive normal forms, model sets, possible-worlds semantics, epistemic logic, doxastic logic, inductive logic, semantic information, game-theoretical semantics, interrogative approach to inquiry, and independence-friendly logic. He applied them to study problems in philosophy of language, formal epistemology, and philosophy of science. He combined systematic work with novel interpretations of important historical figures like Aristotle, Leibniz, Kant, Peirce, and Wittgenstein. Hintikka was one of the most cited analytic philosophers, and he influenced logic and philosophy also as a successful teacher and the long-time editor of the journal Synthese.

Game semantics is a flexible semantic theory that has led in recent years to an unprecedented number of full abstraction results for various programming paradigms. We present a gentle introduction to the subject, focussing on high-level ideas and examples with a view to providing a bridge to more technical literature.

We define and study a slight variation on the semantics of Strategy Logic: while in the classical semantics, all strategies are shifted during the evaluation of temporal modalities, we propose to only shift the strategies that have been assigned to a player, thus matching the intuition that we can assign the very same strategy to the players at different points in time. We prove that surprisingly, this renders the model-checking problem undecidable. Strategy logic is a powerful formalism for expressing properties of multi-player games;We propose a (natural) variant of the semantics of Strategy Logic;While the change may look minor, we prove that the model-checking problem becomes undecidable.

Stochastic model checking is a recent extension and generalization of the classical model checking, which focuses on quantitatively checking the temporal property of a system model. PCTL* is one of the important quantitative property specification languages, which is strictly more expressive than either PCTL (probabilistic computation tree logic) or LTL (linear temporal logic) with probability bounds. At present, PCTL* stochastic model checking algorithm is very complicated, and cannot provide any relevant explanation of why a formula does or does not hold in a given model. For dealing with this problem, an intuitive and succinct approach for PCTL* stochastic model checking with evidence is put forward in this paper, which includes: presenting the game semantics for PCTL* in release-PNF (release-positive normal form), defining the PCTL* stochastic model checking game, using strategy solving in game to achieve the PCTL* stochastic model checking, and refining winning strategy as the evidence to certify stochastic model checking result. The soundness and the completeness of game-based PCTL* stochastic model checking are proved, and its complexity matches the known lower and upper bounds. The game-based PCTL* stochastic model checking algorithm is implemented in a visual prototype tool, and its feasibility is demonstrated by an illustrative example.

The present paper introduces a novel notion of `(effective) computability', called viability, of strategies in game semantics in an intrinsic (i.e., without recourse to the standard Church-Turing computability), non-inductive and non-axiomatic manner, and shows, as a main technical achievement, that viable strategies are Turing complete. Consequently, we have given a mathematical foundation of computation in the same sense as Turing machines but beyond computation on natural numbers, e.g., higher-order computation, in a more abstract fashion. As immediate corollaries, some of the well-known theorems in computability theory such as the smn-theorem and the first recursion theorem are generalized. Notably, our game-semantic framework distinguishes `high-level' computational processes that operate directly on mathematical objects such as natural numbers (not on their symbolic representations) and their `symbolic implementations' that define their `computability', which sheds new light on the very concept of computation. This work is intended to be a stepping stone towards a new mathematical foundation of computation, intuitionistic logic and constructive mathematics.

International audience; We compare three models of the probabilistic λ-calculus: the probabilistic Böhm trees of Leventis, the probabilistic concurrent games of Winskel et al., and the weighted relational model of Ehrhard et al. Probabilistic Böhm trees and probabilistic strategies are shown to be related by a precise correspondence theorem, in the spirit of existing work for the pure λ-calculus. Using Leventis' theorem (probabilistic Böhm trees characterise observational equivalence), we derive a full abstraction result for the games model. Then, we relate probabilistic strategies to the weighted relational model, using an interpretation-preserving functor from the former to the latter. We obtain that the relational model is also fully abstract.

International audience; Herbrand's theorem, widely regarded as a cornerstone of proof theory, exposes some of the constructive content of classical logic. In its simplest form, it reduces the validity of a first-order purely existential formula to that of a finite disjunction. In the general case, it reduces first-order validity to propositional validity, by understanding the structure of the assignment of first-order terms to existential quantifiers, and the causal dependency between quantifiers. In this paper, we show that Herbrand's theorem in its general form can be elegantly stated and proved as a theorem in the framework of concurrent games, a denotational semantics designed to faithfully represent causality and independence in concurrent systems, thereby exposing the concurrency underlying the computational content of classical proofs. The causal structure of concurrent strategies, paired with annotations by first-order terms, is used to specify the dependency between quantifiers implicit in proofs. Furthermore concurrent strategies can be composed, yielding a compositional proof of Herbrand's theorem, simply by interpreting classical sequent proofs in a well-chosen denotational model.

This paper surveys a new perspective on tree automata and Monadic second-order logic (MSO) on infinite trees. We show that the operations on tree automata used in the translations of MSO-formulae to automata underlying Rabin’s Tree Theorem (the decidability of MSO) correspond to the connectives of Intuitionistic Multiplicative Exponential Linear Logic (IMELL). Namely, we equip a variant of usual alternating tree automata (that we call uniform tree automata) with a fibered monoidal-closed structure which in particular handles a linear complementation of alternating automata. Moreover, this monoidal structure is actually Cartesian on non-deterministic automata, and an adaptation of a usual construction for the simulation of alternating automata by non-deterministic ones satisfies the deduction rules of the !(–) exponential modality of IMELL. (But this operation is unfortunately not a functor because it does not preserve composition.) Our model of IMLL consists in categories of games which are based on usual categories of two-player linear sequential games called simple games, and which generalize usual acceptance games of tree automata. This model provides a realizability semantics, along the lines of Curry–Howard proofs-as-programs correspondence, of a linear constructive deduction system for tree automata. This realizability semantics, which can be summarized with the slogan “automata as objects, strategies as morphisms,” satisfies an expected property of witness extraction from proofs of existential statements. Moreover, it makes it possible to combine realizers produced as interpretations of proofs with strategies witnessing (non-)emptiness of tree automata.

We define a new games model of Probabilistic PCF (PPCF) by enriching thin concurrent games with symmetry, recently introduced by Castellan et al, with probability. This model supports two interpretations of PPCF, one sequential and one parallel. We make the case for this model by exploiting the causal structure of probabilistic concurrent strategies. First, we show that the strategies obtained from PPCF programs have a deadlock-free interaction, and therefore deduce that there is an interpretation-preserving functor from our games to the probabilistic relational model recently proved fully abstract by Ehrhard et al. It follows that our model is intensionally fully abstract. Finally, we propose a definition of probabilistic innocence and prove a finite definability result, leading to a second (independent) proof of full abstraction.

Concurrent separation logic (CSL) is a specification logic for concurrent imperative programs with shared memory and locks. In this paper, we develop a concurrent and interactive account of the logic inspired by asynchronous game semantics. To every program $C$, we associate a pair of asynchronous transition systems $[C]_S$ and $[C]_L$ which describe the operational behavior of the Code when confronted to its Environment or Frame --- both at the level of machine states ($S$) and of machine instructions and locks ($L$). We then establish that every derivation tree $\pi$ of a judgment $\Gamma\vdash\{P\}C\{Q\}$ defines a winning and asynchronous strategy $[\pi]_{Sep}$ with respect to both asynchronous semantics $[C]_S$ and $[C]_L$. From this, we deduce an asynchronous soundness theorem for CSL, which states that the canonical map $\mathcal{L}:[C]_S\to[C]_L$ from the stateful semantics $[C]_S$ to the stateless semantics $[C]_L$ satisfies a basic fibrational property. We advocate that this provides a clean and conceptual explanation for the usual soundness theorem of CSL, including the absence of data races., Comment: Full version of an extended abstract published at LICS 2018

International audience; We propose LMSO, a proof system inspired from Linear Logic, as a proof-theoretical framework to extract finite-state stream transducers from linear-constructive proofs of omega-regular specifications. We advocate LMSO as a stepping stone toward semi-automatic approaches to Church's synthesis combining computer assisted proofs with automatic decisions procedures. LMSO is correct in the sense that it comes with an automata-based realizability model in which proofs are interpreted as finite-state stream transducers. It is moreover complete, in the sense that every solvable instance of Church's synthesis problem leads to a linear-constructive proof of the formula specifying the synthesis problem.

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.; 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.

International audience; The hiding operation, crucial in the compositional aspect of game semantics, removes computation paths not leading to observable results. Accordingly, games models are usually biased towards angelic non-determinism: diverging branches are forgotten.We present here new categories of games, not suffering from this bias. In our first category, we achieve this by avoiding hiding altogether; instead morphisms are uncovered strategies (with neutral events) up to weak bisimulation. Then, we show that by hiding only certain events dubbed inessential we can consider strategies up to isomorphism, and still get a category – this partial hiding remains sound up to weak bisimulation, so we get a concrete representations of programs (as in standard concurrent games) while avoiding the angelic bias. These techniques are illustrated with an interpretation of affine nondeterministic PCF which is adequate for weak bisimulation; and may, must and fair convergences.

We present a trace model for Strachey parametric polymorphism. The model is built using operational nominal game semantics and captures parametricity by using names. It is used here to prove an operational version of a conjecture of Abadi, Cardelli, Curien and Plotkin which states that Strachey equivalence implies Reynolds equivalence in System F.

We present a model of dependent type theory ( DTT ) with Π-, 1-, Σ- and intensional Id -types, which is based on a slight variation of the (call-by-name) category of AJM-games and history-free winning well-bracketed strategies. The model satisfies Streicher's criteria of intensionality and refutes function extensionality. The principle of uniqueness of identity proofs is satisfied. We show it contains a submodel as a full subcategory which gives a faithful interpretation of DTT with Π-, 1-, Σ- and intensional Id -types and, additionally, finite inductive type families. This smaller model is fully (and faithfully) complete with respect to the syntax at the type hierarchy built without Id -types, as well as at the more general class of types where we allow for one strictly positive occurrence of an Id -type. Definability for the full type hierarchy with Id -types remains to be investigated.

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., 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.

We present a model of concurrent games in which strategies are probabilistic and support both discrete and continuous distributions. This is a generalisation of the probabilistic concurrent strategies of Winskel, based on event structures. We first introduce measurable event structures, discrete fibrations of event structures in which each fibre is turned into a measurable space. We then construct a bicategory of measurable games and measurable strategies based on measurable event structures, and add probability to measurable strategies using standard techniques of measure theory. We illustrate the model by giving semantics to an affine, higher-order, probabilistic language with a type of real numbers and continuous distributions.

Game-theoretical interpretation of logical semantics has been first suggested in 1950s. Although it is well know among professionals, it is often excluded from introductory courses in logic. In this paper we examine whether and to what extent would including game semantics in the syllabus of an introductory course of predicate and propositional logic prove to be beneficial to students’ understanding of the subject. To that purpose, we have exposed two groups of undergraduate students of computer science to game semantics as part of recitations for the course Propositional and Predicate Logic. We focus mainly on how this affects students’ mastery of two particular topics – alternating quantification and restricted quantification. Observations were made during the recitations to evaluate any changes in students’ approach to solving relevant problems in classroom. Moreover, before the final test of the recitations one of the groups was given a brief review of game semantics and the other was not. We have measured the difference in performance of these two groups and the results support the hypothesis that game semantics helps students master certain topics of predicate logic. We reference some of the later research in psychology and pedagogy to help us explain this effect.

This essays examines Hintikka’s trajectory through Semantic Representationalism from the classical, i.e. Wittgensteinian, Picture Theory of Meaning to Game-Theoretical Semantics. It starts by asking what makes a sentence a representation of a fact and what conditions enable a sentence to represent something. It is argued that at the end of his journey Hintikka conceives of the representational function of sentences as arising from the norms that regulate their place in verification practices. As a consequence of it, the analysis of semantic representation in terms of an isomorphic relation between sentences and facts is replaced by a theory based on the concept of winning strategy in a semantical game. This maneuver allows Hintikka to emphasize that semantical representation is shaped by normative constraints that rule the specific activities involved in those games.

We present an extension of propositional dynamic logic called HOT-PDL for specifying temporal properties of higher-order functional programs. The semantics of HOT-PDL is defined over Higher-Order Traces (HOTs) that model execution traces of higher-order programs. A HOT is a sequence of events such as function calls and returns, equipped with two kinds of pointers inspired by the notion of justification pointers from game semantics: one for capturing the correspondence between call and return events, and the other for capturing higher-order control flow involving a function that is passed to or returned by a higher-order function. To allow traversal of the new kinds of pointers, HOT-PDL extends PDL with new path expressions. The extension enables HOT-PDL to specify interesting properties of higher-order programs, including stack-based access control properties and those definable using dependent refinement types. We show that HOT-PDL model checking of higher-order functional programs over bounded integers is decidable via a reduction to modal \(\mu \)-calculus model checking of higher-order recursion schemes.

Game semantics is a rich and successful class of denotational models for programming languages. Most game models feature a rather intuitive setup, yet surprisingly difficult proofs of such basic results as associativity of composition of strategies. We set out to unify these models into a basic abstract framework for game semantics, game settings. Our main contribution is the generic construction, for any game setting, of a category of games and strategies. Furthermore, we extend the framework to deal with innocence, and prove that innocent strategies form a subcategory. We finally show that our constructions cover many concrete cases, mainly among the early models and the very recent sheaf-based ones., 17 pages