From Nerode’s congruence to suffix automata with mismatches

In this paper we focus on the minimal deterministic finite automaton S_k that recognizes the set of suffixes of a word w up to k errors. As first results we give a characterization of the Nerode''s right-invariant congruence that is associated with S_k. This result generalizes the classical characterization described in [5]. As second result we present an algorithm that makes use of S_k to accept in an efficient way the language of all suffixes of w up to k errors in every window of size r of a text, where r is the repetition index of w. Moreover, we give some experimental results on some well-known words, like prefixes of Fibonacci and Thue-Morse words. Finally, we state a conjecture and an open problem on the size and the construction of the suffix automaton with mismatches.