On Promptness in Parity Games*†

Abstract

Abstract. Parity games are a powerful formalism for the automatic synthesis and verification of reactive systems. They are closely related to alternating ω-automata and emerge as a natural method for the solution of the µ-calculus model checking problem. Due to these strict connections, parity games are a well-established environment to describe liveness properties such as "every request that occurs infinitely often is eventually responded". Unfortunately, the classical form of such a condition suffers from the strong drawback that there is no bound on the effective time that separates a request from its response, i.e., responses are not promptly provided. Recently, to overcome this limitation, several parity game variants have been proposed, in which quantitative requirements are added to the classic qualitative ones. In this paper, we make a general study of the concept of promptness in parity games that allows to put under a unique theoretical framework several of the cited variants along with new ones. Also, we describe simple polynomial reductions from all these conditions to either Büchi or parity games, which simplify all previous known procedures. In particular, they improve the complexity results of cost and bounded-cost parity games. Indeed, we provide solution algorithms showing that determining the winner of these games lies in UPTime ∩ CoUPTime.

Parity and Streett Games with Costs

“…Here, n is the number of vertices, m is the number of edges, and d is the number of colors in the game. Furthermore, since solving parity games is in NP ∩ coNP and the blowup in our reduction is polynomial, we obtain the following remark (note that this was recently improved to UP ∩ coUP [27]). …”

“…For parity games with costs, we proved half-positional determinacy and that solving these games is not harder than solving parity games. The decision problem is in NP ∩ coNP (this was recently improved to UP ∩ coUP [27]). …”

“…Here, we rely on the characterization of the winning region W 0 (G) of a parity game with costs G as computed by Algorithm 1: the sets X j can be determined in NP (respectively in coNP) due to Remark 3.6 and the attractors can be computed in (deterministic) linear time (note that this was recently improved to UP ∩ coUP [27]). …”

Solving Parity Games in Scala

“…Hence, every parity game solver can be used in practice as a model checker for a µ-calculus specification (and vice-versa). Using this approach, liveness and safety properties can be addressed in a very elegant and easy way [29]. Also, this offers a very powerful machinery to check for component software reliability [1,3].…”

Prompt Interval Temporal Logic

“…They make use of the concept of distance between positions in a play that refers to the number of edges traversed in the game arena; the classical parity winning condition is then reformulated to take into consideration only those states occurring with a bounded distance. Such an idea has been generalised to deal with more involved prompt parity conditions [13,19]. In the field of formal languages, promptness comes into play in [6], where ωB-regular languages and their automata counterpart, known as ωB-automata, are studied.…”