On the complexity of ω-type Turing acceptors

Abstract

Turing machines are considered as recognizers of sets of infinite (ω-type) sequences, so called ω-languages. The basic results on such ω-type Turing acceptors were presented in a preceding paper. This paper focuses on the theory of deterministic ω-type Turing acceptors (ω-DTA's) which turns out to be crucially different from the ‘classical’ theory of Turing machines. It is shown that there exists no ω-DTA which is universal for all ω-DTA's. Two infinite complexity hierarchies for ω-DTA's are established, the ‘states hierarchy’, corresponding to the number of states in the machine, and the ‘designated sets hierarchy’, corresponding to the number of designated sets of states used in the recognition. Concrete examples of ω-languages characterizing each of the complexity classes are exhibited. Two additional examples of interesting ω-languages are presented: 1. (i) An ω-language which is ‘inherently non-deterministic’, i.e. can be recognized by a non-deterministic Turing acceptor but by no deterministic acceptor. 2. (ii) An ω-language which cannot be recognized even by a non-deterministic Turing acceptor. The above examples are constructed without using diagonalization. Oscillating ω-DTA's, i.e. ω-DTA's which are allowed to oscillate on ω-inputs, are also considered and are shown to be strictly more powerful than non-oscillating ω-DTA's, yet strictly less powerful than non-deterministic ω-Turing acceptors.