# Separating many words by counting occurrences of factors

## TL;DR

This paper investigates the existence of finite factor-counting languages that can distinguish all pairs of distinct words within a given language, providing comprehensive results for regular and infinite-word languages.

## Contribution

It characterizes which languages admit a finite set of factors that can separate all distinct words by counting their occurrences.

## Key findings

- Finite separating languages exist for all regular languages.
- Criteria established for infinite words' factor sets.
- Complete classification of languages with finite separating factor sets.

## Abstract

For a given language $L$, we study the languages $X$ such that for all distinct words $u, v \in L$, there exists a word $x \in X$ that appears a different number of times as a factor in $u$ and in $v$. In particular, we are interested in the following question: For which languages $L$ does there exist a finite language $X$ satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.