On the Length of the Wadge Hierarchy of Omega Context Free Languages

TL;DR

This paper establishes that the Wadge hierarchy of omega context free languages surpasses the ordinal epsilon_omega in length, indicating a highly complex structure and extending understanding of their position within the Borel hierarchy.

Contribution

It proves that the Wadge hierarchy of omega context free languages exceeds epsilon_omega, and demonstrates the existence of omega context free languages that are Sigma^0_omega-complete Borel sets.

Findings

Wadge hierarchy length exceeds epsilon_omega

Existence of Sigma^0_omega-complete omega context free languages

Improved understanding of the Borel complexity of omega context free languages

Abstract

We prove in this paper that the length of the Wadge hierarchy of omega context free languages is greater than the Cantor ordinal epsilon_omega, which is the omega-th fixed point of the ordinal exponentiation of base omega. The same result holds for the conciliating Wadge hierarchy, defined by J. Duparc, of infinitary context free languages, studied by D. Beauquier. We show also that there exist some omega context free languages which are Sigma^0_omega-complete Borel sets, improving previous results on omega context free languages and the Borel hierarchy.