Solving Counter Parity Games Dietmar Berwanger 1 , Lukasz Kaiser 2 , and Simon Leßenich 1,3⋆ 1 LSV, CNRS & ENS Cachan, France 2 LIAFA, CNRS & Universit´ e Paris Diderot – Paris 7, France 3 Mathematische Grundlagen der Informatik, RWTH Aachen, Germany Abstract. We study a class of parity games equipped with counters that evolve according to arbitrary non-negative affine functions. These games capture several cost models for dynamic systems from the litera- ture. We present an elementary algorithm for computing the exact value of a counter parity game, which both generalizes previous results and improves their complexity. To this end, we introduce a class of ω-regular games with imperfect information and imperfect recall, solve them using automata-based techniques, and prove a correspondence between finite- memory strategies in such games and strategies in counter parity games. 1 Introduction Games with ω-regular winning conditions, and especially parity games, are a fundamental model for program verification and synthesis [15]. Such winning conditions allow to express reachability, safety, and liveness properties. However, when specifying, for instance, that for each request Q i there will be finally a response R i , one is often interested not only in the existence of a response R i –a qualitative property, but also that the response will occur in at most k seconds after the request – a quantitative constraint. Quantitative questions about reactive systems have been approached in sev- eral ways. One possibility is to extend a temporal logic with new, quantitative operators as, for instance, the “prompt” operator for LTL proposed in [12]. While the existence of a response R i is formulated in LTL by FR i , the Prompt-LTL formula F p R i expresses that the waiting time is bounded. Realizability for this logic was solved in [12] and optimal bounds on the waiting time for Prompt- LTL formulas were established in [17]. Another possibility is to consider formulas which evaluate to numbers rather than truth values. The quantitative version of CTL with discounts studied in [6], and the quantitative μ-calculus investigated in [8] follow this direction. Both model-checking and realizability problems for most of these logics are reduced to solving games with additional quantitative features. Several classes of such games have therefore been investigated [3,4,5,10], to provide better al- gorithms for existing logics and to suggest new formalisms with good algorith- mic properties. One relevant example is the synthesis of optimal strategies in ⋆ This author was supported by the ESF Research Networking Programme GAMES.