We consider two-player games played in real time on game structures with clocks where the objectives of players are described using parity conditions. The games are \emph{concurrent} in that at each turn, both players independently propose a time delay and an action, and the action with the shorter delay is chosen. To prevent a player from winning by blocking time, we restrict each player to play strategies that ensure that the player cannot be responsible for causing a zeno run. First, we present an efficient reduction of these games to \emph{turn-based} (i.e., not concurrent) \emph{finite-state} (i.e., untimed) parity games. Our reduction improves the best known complexity for solving timed parity games. Moreover, the rich class of algorithms for classical parity games can now be applied to timed parity games. The states of the resulting game are based on clock regions of the original game, and the state 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 that represents the controller in a real-time synthesis problem, namely, \emph{limit-robust} and \emph{bounded-robust} winning strategies. Using a limit-robust winning strategy, the controller cannot choose an exact real-valued time delay but must allow for some nonzero jitter in each of its actions. If there is a given lower bound on the jitter, then the strategy is bounded-robust winning. We show that exact strategies are more powerful than limit-robust strategies, which are more powerful than bounded-robust winning strategies for any bound. For both kinds of robust strategies, we present efficient reductions to standard timed automaton games. These reductions provide algorithms for the synthesis of robust real-time controllers.
Timed automata [AD94] are models of real-time systems in which states consist of discrete locations and values for real-time clocks. The transitions between locations are dependent on the clock values. Timed automaton games, introduced in [MPS95], and explored further in [dAFH + 03, AdAF05, CDF + 05, FTM02b, FTM02a] (amongst others), are played by two players on timed automata, e.g., a "controller" and a "plant" for modeling real-time
Figure 1: A timed automaton game T.
controller synthesis problems. We consider timed automaton games with ω-regular objectives specified as parity conditions. The class of ω-regular objectives can express all safety and liveness specifications that arise in the synthesis and verification of reactive systems, and parity conditions are a canonical form to express ω-regular objectives [Tho97]. The construction of a winning strategy for player 1 in such games corresponds to the controllersynthesis problem for real-time systems [DM02, MPS95,WH91] with respect to achieving a desired ω-regular objective. Timed automaton games proceed in an infinite sequence of rounds. In each round, both players simultaneously propose moves, with each move consisting of an action and a time delay after which the player wants the proposed action to take place. Of the two proposed moves, the move with the shorter time delay “wins” the round and determines the next state of the game. Let a set Φ of runs be the desired objective for player 1. Then player 1 has a winning strategy for Φ if it has a strategy to ensure that, no matter what player 2 does, one of the following two conditions holds: (1) time diverges and the resulting run belongs to Φ, or (2) time does not diverge but player-1’s moves are chosen only finitely often (and thus it is not to be blamed for the convergence of time) [dAFH + 03, HP06]. This definition of winning is equivalent to restricting both players to play according to receptive strategies [AH97, SGSAL98], which do not allow a player to win by blocking time.
In timed automaton games, there are cases where a player can win by proposing a certain strategy of moves, but where moves that deviate in the timing by an arbitrarily small amount from the winning strategy result in a strategy that does not ensure winning any more. If this is the case, then the synthesized controller needs to work with infinite precision in order to achieve the control objective. As this requirement is unrealistic, we propose two notions of robust winning strategies. In the first robust model, each move of player 1 (the “controller”) must allow some jitter when the action of the move is taken. The jitter may be arbitrarily small, but it must be greater than 0. We call such strategies limit-robust. In the second robust model, we give a lower bound on the jitter, i.e., every move of player 1 must allow for a fixed jitter, which is specified as a parameter of the game. We call these strategies bounded-robust. The strategies of player 2 (the “plant”) are left unrestricted (apart from being receptive). We show that (1) general strategies are strictly more powerful than limit-robust strategies; and (2) limit-robust strategies are strictly more powerful than bounded-robust strategies for any lower bound on the jitter, i.e., there are games in which player 1 can win with a limit-robust strategy, but there does not exist any nonzero bound on the jitter for which player 1 can win with a bounded-robust strategy.
The following example illustrates this issue.
Example 1.1. Consider the timed automaton T in Fig. 1. The edges denoted a k 1 for k ∈ {1, 2, 3, 4} are controlled by player 1, and the edges denoted a j 2 for j ∈ {1, 2, 3} are controlled by player 2. The objective of player 1 is ✷(¬l 3 ), i.e., to avoid the location l 3 . The important part of the automaton is the cycle l 0 , l 1 . The only way to avoid l 3 in a time divergent run is to cycle between l 0 and l 1 infinitely often. In addition, player 1 may choose to also cycle between l 0 and l 2 , but that does not help (or harm) it. Due to strategies being required to be receptive, player 1 cannot just cycle between l 0 and l 2 forever, it must also cycle between l 0 and l 1 ; that is, to satisfy ✷(¬l 3 ) player 1 must ensure (✷✸l 0 ) ∧ (✷✸l 1 ), where ✷✸ denotes “infinitely often”. But note that player 1 may cycle between l 0 and l 2 any finite number of times as it wants between an l 0 , l 1 cycle.
In our analysis below, we omit such l 0 , l 2 cycles for simplicity. Let the game start from the location l 0 at time 0, and let l 1 be visited at time t 0 for the first time. Also, let α j denote the difference between times when l 0 is visited for the (j + 1)-th time, and when l 1 is visited for the j-th time. We can have at most 1 time unit between two successive visits to l 0 , and we must have strictly more than 1 time unit elapse between two successive visits to l 1 . Thus, α j must be in a strictly decreasing sequence. Also, for player 1 to cycle between
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