Cost automata are traditional finite state automata enriched with a finite set of counters that can be manipulated on each transition. Based on the evolution of counter values, a cost automaton defines a function from the set of structures under consideration to the natural numbers extended with infinity, modulo a boundedness relation that ignores exact values but preserves boundedness properties. Historically, variants of cost automata have been used to solve problems in language theory such as the star height problem. They also have a rich theory in their own right as part of the theory of regular cost functions, which was introduced by Colcombet as an extension to the theory of regular languages. It subsumes the classical theory since a language can be associated with the function that maps every structure in the language to 0 and everything else to infinity; it is a strict extension since cost functions can count some behaviour within the input. Regular cost functions have been previously studied over finite words and trees. This thesis extends the theory to infinite trees, where classical parity automata are enriched with a finite set of counters. Weak cost automata, which have priorities {0,1} or {1,2} and an additional restriction on the structure of the transition function, are shown to be equivalent to a weak cost monadic logic. A new notion of quasi-weak cost automata is also studied and shown to arise naturally in this cost setting. Moreover, a decision procedure is given to determine whether or not functions definable using weak or quasi-weak cost automata are equivalent up to the boundedness relation, which also proves the decidability of the weak cost monadic logic over infinite trees. The semantics of these cost automata over infinite trees are defined in terms of cost-parity games which are two-player infinite games where one player seeks to minimize the counter values and satisfy the parity condition, and the other player seeks to maximize the counter values or sabotage the parity condition. The main contributions and key technical results involve proving that certain cost-parity games admit positional or finite-memory strategies. These results also help settle the decidability of some special cases of long-standing open problems in the classical theory. In particular, it is shown that it is decidable whether a regular language of infinite trees is recognizable using a nondeterministic co-Büchi automaton. Likewise, given a Büchi or co-Büchi automaton as input, it is decidable whether or not there is a weak automaton recognizing the same language.

Cette thèse examine certaines questions quantitatives dans le cadre de deux modèles stochastiques différents. Il est divisé en deux parties : la première partie examine une nouvelle classe de jeux stochastiques avec une fonction de paiement particulière que nous appelons « de priorité ». Cette classe de jeux contient comme sous-classes propre les jeux de parité, largement étudiés en informatique, et les jeux de limsup et liminf, étudiés dans la théorie des jeux. La deuxième partie de la thèse examine certaines questions naturelles mais complexes sur les distributions, étudiées dans le cadre plus simple des chaînes de Markov à espace d'états fini. Dans la première partie, nous examinons les jeux à somme nulle à deux joueurs en se centrant sur la fonction de paiement de priorité. Cette fonction de paiement génère le gain utilisé dans les jeux de parité. Nous considérons à la fois les jeux de priorité stochastiques à tour de rôle et les jeux de priorité simultanés. Notre approche des jeux de priorité est basée sur le concept du point fixe le plus proche (« nearest fixed point ») des applications monotones non expansives et étend l'approche mu-calcul aux jeux de priorité.La deuxième partie de la thèse concerne les questions de population. De manière simplifiée, nous examinons comment une distribution de probabilité sur les états évolue dans le temps. Plus précisément, nous sommes intéressés par des questions comme la suivante : à partir d'une distribution initiale, la population peut-elle atteindre à un moment donné une distribution avec une probabilité dépassant un seuil donné dans l'état visé? Il s'avère que ce type de questions est beaucoup plus difficile à gérer que les questions concernant les trajectoires individuelles : on ne connaît pas, pour le modèle des chaînes de Markov, si les questions de population soient décidables. Nous étudions les restrictions des chaînes de Markov assurant la décision des questions de population. / This thesis examines some quantitative questions in the framework of two different stochastic models. It is divided into two parts: the first part examines a new class of stochastic games with priority payoff. This class of games contains as proper subclasses the parity games extensively studied in computer science, and limsup and liminf games studied in game theory. The second part of the thesis examines some natural but involved questions about distributions, studied in the simple framework of finite state Markov chain.In the first part, we examine two-player zero-sum games focusing on a particular payoff function that we call the priority payoff. This payoff function generalizes the payoff used in parity games. We consider both turn-based stochastic priority games and concurrent priority games. Our approach to priority games is based on the concept of the nearest fixed point of monotone nonexpansive mappings and extends the mu-calculus approach to priority games.The second part of the thesis deals with population questions. Roughly speaking, we examine how a probability distribution over states evolves in time. More specifically, we are interested in questions like the following one: from an initial distribution, can the population reach at some moment a distribution with a probability mass exceeding a given threshold in state Goal? It turns out that this type of questions is much more difficult to handle than the questions concerning individual trajectories: it is not known for the simple model of Markov chains whether population questions are decidable. We study restrictions of Markov chains ensuring decidability of population questions.

<p>Computer programs keep finding their ways into new safety-critical applications, while at the same time growing more complex. This calls for new and better methods to verify the correctness of software. We focus on one approach to verifying systems, namely that of <i>model checking</i>. At first, we investigate two categories of problems related to model checking: <i>games</i> and <i>stochastic infinite-state systems</i>. In the end, we join these two lines of research, by studying <i>stochastic infinite-state games</i>.</p><p>Game theory has been used in verification for a long time. We focus on finite-state 2-player parity and limit-average (mean payoff) games. These problems have applications in model checking for the <i>μ</i>-calculus, one of the most expressive logics for programs. We give a simplified proof of memoryless determinacy. The proof applies <i>both</i> to parity and limit-average games. Moreover, we suggest a strategy improvement algorithm for limit-average games. The algorithm is discrete and strongly subexponential.</p><p>We also consider probabilistic infinite-state systems (Markov chains) induced by three types of models. <i>Lossy channel systems (LCS)</i> have been used to model processes that communicate over an unreliable medium. <i>Petri nets</i> model systems with unboundedly many parallel processes. <i>Noisy Turing machines</i> can model computers where the memory may be corrupted in a stochastic manner. We introduce the notion of <i>eagerness</i> and prove that all these systems are eager. We give a scheme to approximate the value of a reward function defined on paths. Eagerness allows us to prove that the scheme terminates. For probabilistic LCS, we also give an algorithm that approximates the limit-average reward. This quantity describes the long-run behavior of the system.</p><p>Finally, we investigate Büchi games on probabilistic LCS. Such games can be used to model a malicious cracker trying to break a network protocol. We give an algorithm to solve these games.</p>

Computer programs keep finding their ways into new safety-critical applications, while at the same time growing more complex. This calls for new and better methods to verify the correctness of software. We focus on one approach to verifying systems, namely that of model checking. At first, we investigate two categories of problems related to model checking: games and stochastic infinite-state systems. In the end, we join these two lines of research, by studying stochastic infinite-state games. Game theory has been used in verification for a long time. We focus on finite-state 2-player parity and limit-average (mean payoff) games. These problems have applications in model checking for the μ-calculus, one of the most expressive logics for programs. We give a simplified proof of memoryless determinacy. The proof applies both to parity and limit-average games. Moreover, we suggest a strategy improvement algorithm for limit-average games. The algorithm is discrete and strongly subexponential. We also consider probabilistic infinite-state systems (Markov chains) induced by three types of models. Lossy channel systems (LCS) have been used to model processes that communicate over an unreliable medium. Petri nets model systems with unboundedly many parallel processes. Noisy Turing machines can model computers where the memory may be corrupted in a stochastic manner. We introduce the notion of eagerness and prove that all these systems are eager. We give a scheme to approximate the value of a reward function defined on paths. Eagerness allows us to prove that the scheme terminates. For probabilistic LCS, we also give an algorithm that approximates the limit-average reward. This quantity describes the long-run behavior of the system. Finally, we investigate Büchi games on probabilistic LCS. Such games can be used to model a malicious cracker trying to break a network protocol. We give an algorithm to solve these games.