Computing the edit distance of a regular language

The edit distance (or Levenshtein distance) between two words is the smallest number of substitutions, insertions, and deletions of symbols that can be used to transform one of the words into the other. In this paper we consider the problem of computing the edit distance of a regular language (also known as constraint system), that is, the set of words accepted by a given finite automaton. This quantity is the smallest edit distance between any pair of distinct words of the language. We show that the problem is of polynomial time complexity. We distinguish two cases depending on whether the given automaton is deterministic or nondeterministic. In the latter case the time complexity is higher. Incidentally, we also obtain an upper bound on the edit distance of a regular language in terms of the automaton accepting the language.