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Some decidable winning conditions of arbitrarily high finite Borel complexityfor games on finite graphs or on pushdown graphs have been recently presentedby O. Serre in [ Games with Winning Conditions of High Borel Complexity, in theProceedings of the International Conference ICALP 2004, LNCS, Volume 3142, p.1150-1162 ]. We answer in this paper several questions which were raised bySerre in the above cited paper. We first show that, for every positive integern, the class C_n(A), which arises in the definition of decidable winningconditions, is included in the class of non-ambiguous context free omegalanguages, and that it is neither closed under union nor under intersection. Weprove also that there exists pushdown games, equipped with such decidablewinning conditions, where the winning sets are not deterministic context freelanguages, giving examples of winning sets which are non-deterministicnon-ambiguous context free languages, inherently ambiguous context freelanguages, or even non context free languages.

On Winning Conditions of High Borel Complexity in Pushdown Games