The game of Hex is of interest to the mathematics, algorithms, and artificial intelligence communities. It is a classical PSPACE-complete problem, and its invention is intrinsically tied to the Four Colour Theorem and the well-known strategy-stealing argument. Nash, Shannon, Tarjan, and Berge are among the mathematicians who have researched and published about this game. In this thesis we expand on previous research, further developing the mathematical theory and algorithmic techniques relating to Hex. In particular, we identify new classes of moves that can be pruned from consideration, and devise new algorithms to identify connection strategies efficiently. As a result of these theoretical improvements, we produce an automated solver capable of solving all 8 x 8 Hex openings and most 9 x 9 Hex openings; this marks the first time that computers have solved all Hex openings solved by humans. We also produce the two strongest automated Hex players in the world --- Wolve and MoHex --- and obtain both the gold and silver medals in the 2008 and 2009 International Computer Olympiads.

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Cette thèse s’inscrit dans le cadre de la programmation par contraintes quantifiées, un formalisme étendantla programmation par contraintes classique en ajoutant aux variables des quantificateurs existentiels ouuniversels, ce qui apporte en théorie une expressivité suffisante pour modéliser des problèmes avec adversaireou incertitude sur certains paramètres sous forme de problèmes appelés QCSP (Quantified Constraintsatisfaction Problem).Nous commençons par apporter une réponse aux difficultés de modélisation de problèmes réels dont estfrappée la programmation par contraintes quantifiées en introduisant une extension aux QCSP permettantd’expliciter les actions possibles de l’agent principal et de son adversaire. Puis, nous décrivons différentproblèmes grâce à ce formalisme, et discutons de la place de cette extension parmi les formalismes voisins créésen réponse à cette même difficulté de modélisation. Enfin, nous nous intéressons à la notion d’optimisationdans le cas des contraintes quantifiées, et apportons un formalisme d’optimisation de contraintes quantifiéespermettant d’exprimer des problèmes multi-niveaux non linéaires. / This thesis presents works in the research area of quantified constraint programming, which extends theconstraint programming framework by setting (existential and universal) quantifiers to the problem’s variables.This framework is theoretically expressive enough to model problems where an opponent or uncertainparameters are involved, under the form of Quantified Constraint Safisfaction Problems (QCSP).QCSPs suffer from a modeling difficulty that we solve by presenting an extension to this framework, in whichpossible moves for the principal agent and its opponent may be explicitely declared. Then, we describe realproblems using this extention, and discuss of its pros and cons against neighbour framework thar were createdto solve the same difficulty. Finally, we focus on quantifies optimization problems, and present a quantifiedoptimization framework thet allows the modeling of nonlinear multi-level problems.

Unification modulo the theory of Boolean algebras has been investigated by several autors. Nevertheless, the exact complexity of the decision problem for unification with constants and general unification was not known. In this research note, we show that the decision problem is complete for unification with constants and PSPACE-complete for general unification. In contrast, the decision problem for elementary unification (where the terms to be unified contain only symbols of the signature of Boolean algebras) is 'only' NP-complete.