On Winning Conditions of High Borel Complexity in Pushdown Games

In a recent paper [19, 20] Serre has presented some decidable winning conditions ΩA1B...BAnBAn+1 of arbitrarily high finite Borel complexity for games on finite graphs or on pushdown graphs. We answer in this paper several questions which were raised by Serre in [19, 20]. We study classes Cn(A), defined in [20], and show that these classes are included in the class of non-ambiguous context free ω-languages. Moreover from the study of a larger class Cn(A) we infer that the complements of languages in Cn(A) are also non-ambiguous context free ω-languages. We conclude the study of classes Cn(A) by showing that they are neither closed under union nor under intersection. We prove also that there exists pushdown games, equipped with winning conditions in the form ΩA1BA2 , where the winning sets are not deterministic context free languages, giving examples of winning sets which are non-deterministic non-ambiguous context free languages, inherently ambiguous context free languages, or even non context free languages.