Abstract
The possible interactions between a controller and its environment can naturally be modelled as the arena of a two-player game, and adding an appropriate winning condition permits to specify desirable behavior. The classical model here is the positional game, where both players can (fully or partially) observe the current position in the game graph, which in turn is indicative of their mutual current states. In practice, neither sensing and actuating the environment through physical devices nor data forwarding to and from the controller and signal processing in the controller are instantaneous. The resultant delays force the controller to draw decisions before being aware of the recent history of a play and to submit these decisions well before they can take effect asynchronously. It is known that existence of a winning strategy for the controller in games with such delays is decidable over finite game graphs and with respect to omega -regular objectives. The underlying reduction, however, is impractical for non-trivial delays as it incurs a blow-up of the game graph which is exponential in the magnitude of the delay. For safety objectives, we propose a more practical incremental algorithm successively synthesizing a series of controllers handling increasing delays and reducing the game-graph size in between. It is demonstrated using benchmark examples that even a simplistic explicit-state implementation of this algorithm outperforms state-of-the-art symbolic synthesis algorithms as soon as non-trivial delays have to be handled. We furthermore address the practically relevant cases of non-order-preserving delays and bounded message loss, as arising in actual networked control, thereby considerably extending the scope of regular game theory under delay.
Highlights
Two-player games played over game arenas are established models for the synthesis of correct-by-construction reactive controllers [10,23]
An orthogonal direction has been exploited by Kupferman and Vardi in [22] to consider the problem of synthesis with incomplete information for branching-time logics captured by computational tree logic (CTL) and CTL∗
As playing a game under delay δ amounts to pre-deciding actions δ steps in advance, the problem of finding a winning strategy for the controller in G that wins under delay δ can be reduced to the problem of finding an undelayed winning strategy for the controller in a related safety game which is obtained by building a synchronous product of the original game graph with a shift register implementing the delay by queueing the latent actions in FIFO manner
Summary
Two-player games played over game arenas are established models for the synthesis of correct-by-construction reactive controllers [10,23]. An orthogonal direction has been exploited by Kupferman and Vardi in [22] to consider the problem of synthesis with incomplete information for branching-time logics captured by computational tree logic (CTL) and CTL∗ In their setting, the current position in the game is known up to certain observability constraints, that is, the state space of the game graph is partitioned into finite observations while the controller cannot distinguish states pertaining to the same observation. Recent state information is well available in these settings, as no restriction concerning the minimum age of observable state information is imposed As the latter is an increasingly relevant problem in, e.g., networked control with its non-trivial end-to-end communication latencies, we here address the problem of two-player safety games played over an explicit game arena subject to delayed observation and delayed action of the controlled process, obtaining a specific and practically extremely relevant case of imperfect information amenable to optimized synthesis algorithms. The model of delay in [1] is different from ours as it leads to non-regular languages
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