Scattered one-counter languges have rank less than $\omega^2$

TL;DR

This paper proves that scattered one-counter languages have an ordinal rank less than 52, confirming a conjecture and extending understanding of the structure of such languages in formal language theory.

Contribution

It establishes that scattered one-counter languages have a rank less than 52, advancing the classification of language orderings.

Findings

Scattered one-counter languages have rank less than 52.

Confirms the conjecture about the rank of one-counter languages.

Extends the known bounds for context-free and regular languages.

Abstract

A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that scattered context-free (regular, resp.) orderings have rank less than $\omega^\omega$ ($\omega$, resp). In this paper we confirm the conjecture that one-counter languages have rank less than $\omega^2$.