Decision Problems For Turing Machines

We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_2(\Sigma_1^1)$-complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_2(\Sigma_1^1)$ is the class of 2-differences of $\Sigma_1^1$-sets. Secondly, it is $\Sigma_1^1$-complete to determine whether the omega-language of a given Turing machine is uncountable.Comment: To appear in Information Processing Letter

On Winning Conditions of High Borel Complexity in Pushdown Games

In a recent paper [19, 20] Serre has presented some decidable winning conditions ΩA1⊲...⊲An⊲An+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 Cλ n(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 ΩA1⊲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