Borel Ranks and Wadge Degrees of Context Free Omega Languages

TL;DR

This paper demonstrates that 1-counter B"uchi automata have the same topological acceptance power as Turing machines with B"uchi conditions, revealing complex hierarchies and ranks in omega context free languages.

Contribution

It establishes the equivalence in topological complexity between 1-counter B"uchi automata and Turing machines with B"uchi acceptance, and determines the supremum of Borel ranks for omega context free languages.

Findings

Existence of Sigma^0_alpha-complete omega context free languages for all non null recursive ordinals alpha.

The supremum of Borel ranks of context free omega languages is the ordinal gamma^1_2.

1-counter B"uchi automata can recognize languages with the same topological complexity as Turing machines.

Abstract

We show that, from a topological point of view, considering the Borel and the Wadge hierarchies, 1-counter B\"uchi automata have the same accepting power than Turing machines equipped with a B\"uchi acceptance condition. In particular, for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free languages accepted by 1-counter B\"uchi automata, and the supremum of the set of Borel ranks of context free omega languages is the ordinal gamma^1_2 which is strictly greater than the first non recursive ordinal. This very surprising result gives answers to questions of H. Lescow and W. Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", LNCS 803, Springer, 1994, p. 583-621].