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 $D_2({\mathrm{\Sigma }}_1^1)$-complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_2({\mathrm{\Sigma }}_1^1)$ is the class of 2-differences of ${\mathrm{\Sigma }}_1^1$-sets. Secondly, it is ${\mathrm{\Sigma }}_1^1$-complete to determine whether the omega-language of a given Turing machine is uncountable.