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 result 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 [A. Blumer, J. Blumer, D. Haussler, A. Ehrenfeucht, M. Chen, J Seiferas, The smallest automaton recognizing the subwords of a text, Theoretical Computer Science, 40, 1985. 31-55]. 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.