Timed Parity Games: Complexity and Robustness

Abstract

We consider two-player games played in real time on game structures withclocks where the objectives of players are described using parity conditions.The games are \emph{concurrent} in that at each turn, both playersindependently propose a time delay and an action, and the action with theshorter delay is chosen. To prevent a player from winning by blocking time, werestrict each player to play strategies that ensure that the player cannot beresponsible for causing a zeno run. First, we present an efficient reduction ofthese games to \emph{turn-based} (i.e., not concurrent) \emph{finite-state}(i.e., untimed) parity games. Our reduction improves the best known complexityfor solving timed parity games. Moreover, the rich class of algorithms forclassical parity games can now be applied to timed parity games. The states ofthe resulting game are based on clock regions of the original game, and thestate space of the finite game is linear in the size of the region graph. Second, we consider two restricted classes of strategies for the player thatrepresents the controller in a real-time synthesis problem, namely,\emph{limit-robust} and \emph{bounded-robust} winning strategies. Using alimit-robust winning strategy, the controller cannot choose an exactreal-valued time delay but must allow for some nonzero jitter in each of itsactions. If there is a given lower bound on the jitter, then the strategy isbounded-robust winning. We show that exact strategies are more powerful thanlimit-robust strategies, which are more powerful than bounded-robust winningstrategies for any bound. For both kinds of robust strategies, we presentefficient reductions to standard timed automaton games. These reductionsprovide algorithms for the synthesis of robust real-time controllers