Games given by transition graphs of pushdown processes are considered. It is shown that if there is a winning strategy in such a game then there is a winning strategy that is realized by a pushdown process. This fact turns out to be connected with the model checking problem for the pushdown automata and the propositional µ-calculus. It is show that this model checking problem is DEXPTIME-complete.

The synthesis of controllers for discrete event systems, as introduced by Ramadge and Wonham, amounts to computing winning strategies in parity games. We show that in this framework it is possible to extend the specifications of the supervised systems as well as the constraints on the controllers by expressing them in the modal µ-calculus. In order to express unobservability constraints, we propose an extension of the modal µ-calculus in which one can specify whether an edge of a graph is a loop. This extended µ-calculus still has the interesting properties of the classical one. In particular it is equivalent to automata with loop testing. The problems such as emptiness testing and elimination of alternation are solvable for such automata. The method proposed in this paper to solve a control problem consists in transforming this problem into a problem of satisfiability of a µ-calculus formula so that the set of models of this formula is exactly the set of controllers that solve the problem. This transformation relies on a simple construction of the quotient of automata with loop testing by a deterministic transition system. This is enough to deal with centralized control problems. The solution of decentralized control problems uses a more involved construction of the quotient of two automata. This work extends the framework of Ramadge and Wonham in two directions. We consider infinite behaviours and arbitrary regular specifications, while the standard framework deals only with specifications on the set of finite paths of processes. We also allow dynamic changes of the set of observable and controllable events.

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Guarded fixed point logics are obtained by adding least and greatest fixed points to the guarded fragments of firstorder logic that were recently introduced by Andréka, van Benthem and Németi. Guarded fixed point logics can also be viewed as the natural common extensions of the modal -calculus and the guarded fragments. We prove that the satisfiability problems for guarded fixed point logics are decidable and complete for deterministic double exponential time. For guarded fixed point sentences of bounded width, the most important case for applications, the satisfiability problem is EXPTIME-complete.

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It is proposed to compare strategies in a parity game by comparing the sets of behaviours they allow. For such a game, there may be no winning strategy that encompasses all the behaviours of all winning strategies. It is shown, however, that there always exists a permissive strategy that encompasses all the behaviours of all memoryless strategies. An algorithm for finding such a permissive strategy is presented. Its complexity matches currently known upper bounds for the simpler problem of finding the set of winning positions in a parity game. The algorithm can be seen as a reduction of a parity to a safety game and computation of the set of winning positions in the resulting game.

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