Alternation and ω-type Turing acceptors

Abstract

Wagner and Staiger (1977) characterized in the recursion-theoretic hierarchies the classes of ω-languages accepted by deterministic (D) and nondeterministic (N) ω-Turing acceptors under various acceptance conditions. In this paper their results are extended to alternating (A) ω-Turing acceptors. It is shown that under a certain acceptance condition alternating ω-Turing acceptors accept precisely the arithmetical ω-languages. On the other hand, the class of ω-languages accepted by alternating ω-Turing acceptors under Muller's condition lies properly between Σ¹₁ and Δ¹₂ in the analytical hierarchy. In terms of acceptional power each of the following situations is possible, depending on the acceptance condition chosen: D = N = A, D < N = A, D = N < A, and D < N < A.