On winning conditions of high borel complexity in pushdown games

In a recent paper [19, 20] Serre has presented some decidable winning conditions Omega(A1 >... >An >An+1) of arbitrarily high finite Borel complexity for games on finite graphs or on push-down graphs. We answer in this paper several questions which were raised by Serre in [19, 20]. We study classes C-n (A), defined in [20], and show that these classes are included in the class of non-ambiguous context free omega-languages. Moreover from the study of a larger class C-n(lambda) (A) we infer that the complements of languages in C-n (A) are also non-ambiguous context free w-languages. We conclude the study of classes C-n (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 Omega(A1>A2), 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.