For untimed systems, two systems are close if every sequence of events of one system is also observable in the second system. For timed systems, the difference in timings of the two corresponding sequences is also of importance. We propose the notion of bisimulation distance which quantifies timing differences; if the bisimulation distance between two systems is epsilon, then (a) every sequence of events of one system has a corresponding matching sequence in the other, and (b) the timings of matching events in between the two corresponding traces do not differ by more than epsilon. We show that we can compute the bisimulation distance between two timed automata to within any desired degree of accuracy. We also show that the timed verification logic TCTL is robust with respect to our notion of quantitative bisimilarity, in particular, if a system satisfies a formula, then every close system satisfies a close formula.
Timed games are used for distinguishing between the actions of several agents, typically a controller and an environment. The controller must achieve its objective against all possible choices of the environment. The modeling of the passage of time leads to the presence of zeno executions, and corresponding unrealizable strategies of the controller which may achieve objectives by blocking time. We disallow such unreasonable strategies by restricting all agents to use only receptive strategies --- strategies which while not being required to ensure time divergence by any agent, are such that no agent is responsible for blocking time. Time divergence is guaranteed when all players use receptive strategies. We show that timed automaton games with receptive strategies can be solved by a reduction to finite state turn based game graphs. We define the logic timed alternating-time temporal logic for verification of timed automaton games and show that the logic can be model checked in EXPTIME. We also show that the minimum time required by an agent to reach a desired location, and the maximum time an agent can stay safe within a set of locations, against all possible actions of its adversaries are both computable.
We next study the memory requirements of winning strategies for timed automaton games. We prove that finite memory strategies suffice for safety objectives, and that winning strategies for reachability objectives may require infinite memory in general. We introduce randomized strategies in which an agent can propose a probabilistic distribution of moves and show that finite memory randomized strategies suffice for all omega-regular objectives. We also show that while randomization helps in simplifying winning strategies, and thus allows the construction of simpler controllers, it does not help a player in winning at more states, and thus does not allow the construction of more powerful controllers.
Finally we study robust winning strategies in timed games. In a physical system, a controller may propose an action together with a time delay, but the action cannot be assumed to be executed at the exact proposed time delay. We present robust strategies which incorporate such jitters and show that the set of states from which an agent can win robustly is computable.