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 Buechi 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.
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