On Infinite Words Determined by Indexed Languages

TL;DR

This paper characterizes infinite words determined by indexed languages, showing they are exactly the morphic words, and introduces a new pumping lemma for indexed languages that may have broader applications.

Contribution

It proves that infinite words determined by indexed languages are precisely the morphic words and introduces a novel pumping lemma for indexed languages.

Findings

Infinite words determined by indexed languages are exactly the morphic words.

A new pumping lemma for indexed languages is established.

The result bridges indexed languages and morphic words in formal language theory.

Abstract

We characterize the infinite words determined by indexed languages. An infinite language $L$ determines an infinite word $\alpha$ if every string in $L$ is a prefix of $\alpha$. If $L$ is regular or context-free, it is known that $\alpha$ must be ultimately periodic. We show that if $L$ is an indexed language, then $\alpha$ is a morphic word, i.e., $\alpha$ can be generated by iterating a morphism under a coding. Since the other direction, that every morphic word is determined by some indexed language, also holds, this implies that the infinite words determined by indexed languages are exactly the morphic words. To obtain this result, we prove a new pumping lemma for the indexed languages, which may be of independent interest.