# Time-optimal Winning Strategies in Infinite Games

Martin Zimmermann

Infinite Games are an important tool in the synthesis of finite-state controllers for reactive systems. The interaction between the environment and the system is modeled by a finite graph. The specification that has to be satisfied by the controlled system is translated into a winning condition on the infinite paths of the graph. Then, a winning strategy is a controller that is correct with respect to the given specification. Winning strategies are often finitely described by automata with output.

While classical optimization of synthesized controllers focuses on the size of the automaton we consider a different quality measure. Many winning conditions allow a natural definition of waiting times that reflect periods of waiting in the original system. We investigate time-optimal strategies for Request-Response Games, Poset Games – a novel type of infinite games that extends Request-Response Games – and games with winning conditions in Parametric Linear Temporal Logic. Here, the temporal operators of classical Temporal Logics can be subscribed with free variables that represent bounds on the satisfaction. Then, one is interested in winning strategies with respect to optimal valuations of the free variables. The optimization objective, maximization respectively minimization of the variable values, depends on the formula.

For Request-Response Games and Poset Games, we prove the existence of time- optimal finite-state winning strategies. For games with winning conditions in Parametric Linear Temporal Logic, we prove that optimal winning strategies are computable for solitary games.

Diploma Thesis.