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The theory of two-player infinite games provides a framework for studying the controller synthesis problem in reactive system. This problem was solved for regular winning conditions for the first time by the fundamental Büchi-Landweber Theorem. The present work extends this result by investigating possibilities to measure the complexity of strategies in infinite games. In the first part of this work, we improve a 50 years old algorithm, the Ramsey-based Büchi automata complementation method, both on the practical as well as on the theoretical side. On the practical side, we present some heuristics for improving the Ramsey-based approach, which we also implemented in a Java program. We show that this algorithm in fact can compete with the more modern ones. On the theoretical side, we introduce a novel complementation construction, based on weak-orders, to which the improved Ramsey-based approach is tightly connected, and we prove a 2^O(n log(n)) upper bound on the size of the complement automaton. In the second part, we embed the concept of games into the domain of formal languages. By doing this, we are able to give a qualitative measure of the complexity of a winning strategy, as well as of the complexity of the corresponding winning condition. In this way, we extend and refine the fundamental Büchi-Landweber Theorem to subclasses of the class of regular languages, in particular we consider hierarchies below the starfree languages. We distinguish between weak games and strong games. Strong games rely on infinite occurrence of patterns in a word, while weak games only rely on finite occurrences of patterns. We show that for solving weak games, winning strategies lie one level above winning conditions inside the hierarchy. For strong games on level i, we show that winning strategies on level i+2 suffice. In the third part, we introduce another measure for the complexity of strategies, but this time on graph-games with parity conditions. This measure is of quantitative nature and it determines the permissiveness of a given nondeterministic strategy. The permissiveness is measured by assigning to each strategy a penalty, namely the average sum (in the limit) of the weight of edges that are to be disallowed. We reduce the problem of determining the value of such a mean-penalty parity game to the value problem of a corresponding mean-payoff parity game, and we show that both problems are in NP intersected coNP. We revisit the study of mean-payoff parity games and obtain a deterministic algorithm, which computes the value and which runs faster than all previously known algorithms. A similar algorithm for mean-penalty parity games is developed as well. |
