Borel ranks and Wadge degrees of context free $\omega$-languages

OLIVIER FINKEL
2006 Mathematical Structures in Computer Science  
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 0 α -complete and some Π 0 α -complete ω-languages accepted by Büchi 1counter automata. And the supremum of the set of Borel ranks of context free
more » ... guages is an ordinal γ 1 2 which is strictly greater than the first non recursive ordinal ω CK 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.
doi:10.1017/s0960129506005597 fatcat:g3jri3nlt5aurozzy5neishngy