Algorithms for Omega-Regular Games with Imperfect Information

TL;DR

This paper develops algorithms for omega-regular games with imperfect information, enabling the computation of winning states for both deterministic and randomized strategies, advancing controller synthesis under partial observation.

Contribution

It introduces fixed-point algorithms for computing winning states in omega-regular games with imperfect information, including for randomized strategies, with proven optimality.

Findings

Algorithms for deterministic observation-based strategies.

Algorithms for randomized strategies with Buechi objectives.

Proven optimality of the algorithms.

Abstract

We study observation-based strategies for two-player turn-based games on graphs with omega-regular objectives. An observation-based strategy relies on imperfect information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixed-point algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for any omega-regular objective. The fixed point is computed in the lattice of antichains of state sets. This algorithm has the advantages of being directed by the objective and of avoiding an explicit subset construction on the game graph. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomized observation-based strategy for a Buechi objective. This set is of interest because in the absence of perfect information, randomized strategies are more powerful than deterministic ones. We show that our algorithms are optimal by proving matching lower bounds.