Memory and delay in regular infinite games

Infinite two-player games constitute a powerful and flexible framework for the design and verification of reactive systems. It is well-known that, for example, the construction of a controller acting indefinitely within its environment can be reduced to the computation of a winning strategy in an infinite game (Pnueli and Rosner, 1989). For the class of regular games, Büchi and Landweber (1969) showed that one of the players has a winning strategy which can be realized by a finite-state automaton. Based on this fundamental result, many efforts have been made by the research community to improve the solution methods for infinite games. This is meant with respect to both the time needed to compute winning strategies and the amount of space required to implement them. In the present work we are mainly concerned with the second aspect. We study two problems related to the construction of small controller programs. In the first part of the thesis, we address the classical problem of computing finite-state winning strategies which can be realized by automata with as little memory, i.e., number of states, as possible. Even though it follows from a result of Dziembowski et al. (1997) that there exist exotic regular games for which the size of automata implementing winning strategies is at least exponential in the size of the game arena, most practical cases require far less space. We propose an efficient algorithm which reduces the memory used for the implementation of winning strategies, for several classes of regular conditions, and we show that our technique can effect an exponential gain in the size of the memory. In the second part of this thesis, we introduce a generalized notion of a strategy. One of the players is allowed to delay each of his moves for a finite number of steps, or, in other words, exploit in his strategy some lookahead on the moves of the opponent. This setting captures situations in distributed systems, for example, when buffers are present in communication between remote components. Our concept of strategies corresponds to the class of continuous operators, thereby extending the work of Hosch and Landweber (1972) and, in particular, that of Büchi and Landweber (1969). We show that the problem whether a given regular specification is solvable by a continuous operator is decidable and that each continuous solution can be reduced to one of bounded lookahead. From our results, we derive a generalized determinacy of regular conditions.