Separating many words by counting occurrences of factors

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.

16 pages. Full version of an article to appear in the proceedings of the 23rd International Conference on Developments in Language Theory (DLT 2019)