Borel Ranks and Wadge Degrees of Context Free omega-Languages

We show that the Borel hierarchy of the class of context free ω-languages, or even of the class of ω-languages accepted by Büchi 1-counter automata, is the same as the Borel hierarchy of the class of ω-languages accepted by Turing machines with a Büchi acceptance condition. In particular, for each recursive non null ordinal α, there exist some Σ α -complete and some Π α -complete ω-languages accepted by Büchi 1counter automata. And the supremum of the set of Borel ranks of context free ω-languages is an ordinal γ 2 which is strictly greater than the first non recursive ordinal ω 1 . We then extend this result, proving that the Wadge hierarchy of context free ω-languages, or even of ω-languages accepted by Büchi 1-counter automata, is the same as the Wadge hierarchy of ω-languages accepted by Turing machines with a Büchi or a Muller acceptance condition.