Finite-state strategies in delay games

Abstract

What is a finite-state strategy in a delay game? We answer this surprisingly non-trivial question by presenting a very general framework that allows to remove delay: finite-state strategies exist for all winning conditions where the resulting delay-free game admits a finite-state strategy. The framework is applicable to games whose winning condition is recognized by an automaton with an acceptance condition that satisfies a certain aggregation property. Our framework also yields upper bounds on the complexity of determining the winner of such delay games and upper bounds on the necessary lookahead to win the game. In particular, we cover all previous results of that kind as special cases of our uniform approach. ⋆ Supported by the projects "TriCS" (ZI 1516/1-1) and (LO 1174/3-1) of the German Research Foundation (DFG). The work presented here was carried out while the second author was a member of the Reactive Systems Group at Saarland University. 3 The models of Gale-Stewart games and arena-based games are interreducible, but delay games are naturally presented as a generalization of Gale-Stewart games. This is the reason we prefer this model here.

Partially Observable Games for Secure Autonomy

“…However, since POSGs simply extend POMDPs to multiple players, computing optimal strategies requires infinite memory [6]. To circumvent this difficulty, we represent observation-based strategies with finite memory and we use finite-state strategies (FSSs) (see also FSSs in Delay Games [16]). If such an FSS has n memory states, we say the memory size for the underlying strategy σ is n.…”

Approximating the Minimal Lookahead Needed to Win Infinite Games

“…Note that we do not consider the computation of a strategy realizing the approximation, as the notion of finite-state strategies for delay games comes with some technical complications [8].…”

“…The construction of G k is a refinement of a similar game used to prove Proposition 1 [6,8]. For a detailed explanation of the construction, we refer the reader to these works.…”

“…Here, we consider the problem of determining the minimal lookahead that is sufficient for Player O to win an ω-regular delay game. It is trivial to determine this value in doubly-exponential-time by hardcoding the exponential lookahead into the game, thereby turning the delay game into a classical, i.e., delay-free, game (see [8], Section 3.1 for details). As the resulting games can be solved in doublyexponential time, one obtains the minimal lookahead in doubly-exponential time by exhaustive search.…”

Safety Shielding under Delayed Observation

“…We base our shields on their proposed algorithm for solving delayed safety games. Note that the delayed games discussed in (Winter and Zimmermann 2020) follow a different concept. In their setting, a delay is a lookahead granted by the input player as an advantage to the delayed player: the delayed player P1 lags behind input player P0 in that P1 has to produce the i-th action when i + j inputs are available.…”