The window mechanism, introduced by Chatterjee et al. [17] for mean-payoff and total-payoff objectives in two-player turn-based games on graphs, refines long-term objectives with time bounds. This mechanism has proven useful in a variety of settings [14,12], and most recently in timed systems [30].In the timed setting, the so-called fixed timed window parity objectives have been studied. A fixed timed window parity objective is defined with respect to some time bound and requires that, at all times, we witness a time frame, i.e., a window, of size less than the fixed bound in which the smallest priority is even. In this work, we focus on the bounded timed window parity objective. Such an objective is satisfied if there exists some bound for which the fixed objective is satisfied. The satisfaction of bounded objectives is robust to modeling choices such as constants appearing in constraints, unlike fixed objectives, for which the choice of constants may affect the satisfaction for a given bound.We show that verification of bounded timed window objectives in timed automata can be performed in polynomial space, and that timed games with these objectives can be solved in exponential time, even for multi-objective extensions. This matches the complexity classes of the fixed case. We also provide a comparison of the different variants of window parity objectives. ACM Subject Classification Theory of computation → Formal languages and automata theoryKeywords and phrases window objectives, timed automata, timed games, parity games

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Two-player (antagonistic) games on (possibly stochastic) graphs are a prevalent model in theoretical computer science, notably as a framework for reactive synthesis. Optimal strategies may require randomisation when dealing with inherently probabilistic goals, balancing multiple objectives, or in contexts of partial information. There is no unique way to define randomised strategies. For instance, one can use so-called mixed strategies or behavioural ones. In the most general settings, these two classes do not share the same expressiveness. A seminal result in game theory -Kuhn's theorem -asserts their equivalence in games of perfect recall. This result crucially relies on the possibility for strategies to use infinite memory, i.e., unlimited knowledge of all the past of a play. However, computer systems are finite in practice. Hence it is pertinent to restrict our attention to finite-memory strategies, defined as automata with outputs. Randomisation can be implemented in these in different ways: the initialisation, outputs or transitions can be randomised or deterministic respectively. Depending on which aspects are randomised, the expressiveness of the corresponding class of finite-memory strategies differs. In this work, we study two-player turn-based stochastic games and provide a complete taxonomy of the classes of finite-memory strategies obtained by varying which of the three aforementioned components are randomised. Our taxonomy holds both in settings of perfect and imperfect information.