On the dissimilarity operation on finite languages

The distinguishability language of a regular language L is the set of words distinguishing between pairs of words under the Myhill-Nerode equivalence induced by L, i.e., between pairs of distinct left quotients of L. The similarity relation induced by a language L is a similarity relation inspired by the Myhill-Nerode equivalence and it was used to obtain compact representation of automata for a finite language L, i.e., deterministic finite cover automata, which are deterministic finite automata accepting all the words of L and possibly some other words that are longer than any word of L. The dissimilarity language of a finite language L is defined as the set of words that separate a pair of words which are not similar w.r.t. to a (finite) language L. In this paper we extend the study of distinguishability operation on regular languages to l-dissimilarity, for l ∈ N, and the dissimilarity operation on finite languages. We examine their properties, the state complexity, and relations that can be established between these operations.