Context-free ordinals

TL;DR

This paper investigates the order types of context-free languages under lexicographic ordering, establishing bounds on their Hausdorff rank and characterizing which ordinals can be represented by well-ordered context-free languages.

Contribution

It proves that scattered lexicographic orders of context-free languages have Hausdorff rank below , and characterizes ordinals representable by such languages as those less than .

Findings

Hausdorff rank of scattered context-free lexicographic orders is less than 

An ordinal is the order type of a well-ordered context-free language iff it is less than 

Provides bounds on the complexity of lexicographic orderings of context-free languages.

Abstract

We consider context-free languages equipped with the lexicographic ordering. We show that when the lexicographic ordering of a context-free language is scattered, then its Hausdorff rank is less than $\omega^\omega$. As a corollary of this result we obtain that an ordinal is the order type of a well-ordered context-free language iff it is less than $\omega^{\omega^\omega}$.