Degrees of Lookahead in Regular Infinite Games

We study variants of regular infinite games where the strict alternation of moves between the two players is subject to modifications. The second player may postpone a move for a finite number of steps, or, in other words, exploit in his strategy some lookahead on the moves of the opponent. This captures situations in distributed systems, e.g. when buffers are present in communication or when signal transmission between components is deferred. We distinguish strategies with different degrees of lookahead, among them being the continuous and the bounded lookahead strategies. In the first case the lookahead is of finite possibly unbounded size, whereas in the second case it is of bounded size. We show that for regular infinite games the solvability by continuous strategies is decidable, and that a continuous strategy can always be reduced to one of bounded lookahead. Moreover, this lookahead is at most doubly exponential in the size of a given parity automaton recognizing the winning condition. We also show that the result fails for non-regular gamesxwhere the winning condition is given by a context-free omega-language.

💡 Research Summary

The paper investigates a natural extension of regular infinite games in which the second player is allowed to postpone his move for a finite number of steps, thereby gaining a look‑ahead on the opponent’s future moves. This models realistic distributed systems where communication buffers, network latency, or asynchronous execution cause one component to observe several of the other component’s actions before responding. The authors formalise two families of look‑ahead strategies. A continuous strategy may use an arbitrarily large but finite look‑ahead; the size of the look‑ahead is not fixed in advance and may vary from one round to another. A bounded‑look‑ahead strategy, by contrast, is constrained by a global constant k: at every round the player may only see the next k moves of the opponent before choosing his own move.

The central technical question is whether, for regular winning conditions, the existence of a continuous strategy automatically guarantees the existence of a bounded‑look‑ahead strategy, and if so, how large the bound must be. The authors answer affirmatively. Starting from a parity automaton A with n states that recognises the ω‑regular winning set, they construct a “delayed game” in which the second player may collect a finite block of opponent moves before replying. They show that a continuous winning strategy in the original game corresponds to an ordinary (no‑delay) winning strategy in the delayed game. By encoding the delayed game again as a parity game, they can apply standard parity‑game solving algorithms (e.g., Zielonka’s algorithm) to decide the existence of a continuous strategy in time 2EXPTIME, which matches the known complexity for solving regular infinite games.

The reduction from a continuous to a bounded strategy is achieved by analysing the state space of the delayed game. The authors introduce “predictive states”, which pair a state of A with a finite look‑ahead word. By exhaustively exploring all possible look‑ahead words up to a certain length, they obtain a finite set of predictive states that is sufficient to simulate any continuous strategy. The crucial bound on the required look‑ahead length is doubly exponential in n: specifically, a bound of 2^{2^{O(n)}} moves always suffices. Consequently, any continuous winning strategy can be replaced by a bounded‑look‑ahead strategy whose look‑ahead is at most doubly exponential in the size of the original parity automaton.

The paper also proves that this equivalence breaks down for non‑regular winning conditions. By constructing a context‑free ω‑language, the authors exhibit a game that admits a continuous winning strategy but for which no bounded‑look‑ahead strategy exists, regardless of the bound. This demonstrates that the closure properties and determinism of regular languages are essential for the compression of unbounded look‑ahead into a finite bound.

From a practical perspective, the results give a concrete method for synthesising controllers in distributed environments with bounded buffers. Given a regular specification, one can decide whether a controller that only needs a finite, predetermined amount of look‑ahead exists, and compute an explicit bound (albeit doubly exponential). This bound can be used to size communication buffers or to verify that a system respects latency constraints.

In summary, the authors establish three main contributions: (1) a decidability result for the existence of continuous strategies in regular infinite games; (2) a constructive proof that any continuous strategy can be transformed into a bounded‑look‑ahead strategy with a doubly‑exponential bound; and (3) a separation result showing that the same transformation fails for games defined by context‑free ω‑languages. The work bridges the gap between classical infinite‑game theory and realistic models of distributed computation, opening the way for further investigations into look‑ahead phenomena for richer language classes and quantitative objectives.