On Promptness in Parity Games

Parity games are infinite-duration two-player turn-based games that provide powerful 6 formal-method techniques for the automatic synthesis and verification of distributed and reactive 7 systems. This kind of game emerges as a natural evaluation technique for the solution of the μ8 calculus model-checking problem and is closely related to alternating ω-automata. Due to these strict 9 connections, parity games are a well-established environment to describe liveness properties such 10 as “every request that occurs infinitely often is eventually responded”. Unfortunately, the classical 11 form of such a condition suffers from the strong drawback that there is no bound on the effective 12 time that separates a request from its response, i.e., responses are not promptly provided. Recently, to 13 overcome this limitation, several variants of parity game have been proposed, in which quantitative 14 requirements are added to the classic qualitative ones. In this paper, we make a general study of the 15 concept of promptness in parity games that allows to put under a unique theoretical framework several 16 of the cited variants along with new ones. Also, we describe simple polynomial reductions from all 17 these conditions to either Büchi or parity games, which simplify all previous known procedures. In 18 particular, they allow to lower the complexity class of cost and bounded-cost parity games recently 19 introduced. Indeed, we provide solution algorithms showing that determining the winner of these 20 games is in UPTIME ∩ COUPTIME. 21