Finitary Winning in \omega-Regular Games [report]

Krishnendu Chatterjee, Thomas A. Henzinger, Florian Horn
2007 unpublished
Games on graphs with ω-regular objectives provide a model for the control and synthesis of reactive systems. Every ω-regular objective can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens "eventually." Two main strengths of the classical, infinite-limit formulation of liveness are robustness (independence from the granularity of transitions) and simplicity (abstraction of complicated time bounds). However, the classical liveness
more » ... al liveness formulation suffers from the drawback that the time until something good happens may be unbounded. A stronger formulation of liveness, so-called finitary liveness, overcomes this drawback, while still retaining robustness and simplicity. Finitary liveness requires that there exists an unknown, fixed bound b such that something good happens within b transitions. While for one-shot liveness (reachability) objectives, classical and finitary liveness coincide, for repeated liveness (Büchi) objectives, the finitary formulation is strictly stronger. In this work we study games with finitary parity and Streett (fairness) objectives. We prove the determinacy of these games, present algorithms for solving these games, and characterize the memory requirements of winning strategies. We show that finitary parity games can be solved in polynomial time, which is not known for infinitary parity games. For finitary Streett games, we give an EXPTIME algorithm and show that the problem is NP-hard. Our algorithms can be used, for example, for synthesizing controllers that do not let the response time of a system increase without bound. ⋆ The paper is a combined and extended version of the papers [5, 11]. 2 Games with ω-Regular Objectives 2.1 Game graphs Game graphs. A game graph G = ((S, E), (S 1 , S 2 )) consists of a directed graph (S, E) with a finite state space S and a set E of edges, and a partition (S 1 , S 2 ) of the state space S into two sets. The states in S 1 are player 1 states, and the states in S 2 are player 2 states. For a state s ∈ S, we write E(s) = {t ∈ S | (s, t) ∈ E} for the set of successor states of s. We assume that every state has at least one out-going edge, i.e., E(s) is non-empty for all states s ∈ S. Plays. A game is played by two players: player 1 and player 2, who form an infinite path in the game graph by moving a token along edges. They start by
doi:10.21236/ada637154 fatcat:yvhafnmblbd7te72kq5dt46ni4