State Complexity of Neighbourhoods and Approximate Pattern Matching

The neighbourhood of a language L with respect to an additive distance consists of all strings that have distance at most the given radius from some string of L. We show that the worst case deterministic state complexity of a radius r neighbourhood of a language recognized by an n state nondeterministic finite automaton A is (r + 2)n − 1. In the case where A is deterministic we get the same lower bound for the state complexity of the neighbourhood if we use an additive quasi-distance. The lower bound constructions use an alphabet of size linear in n. We show that the worst case state complexity of the set of strings that contain a substring within distance r from a string recognized by A is (r + 2)n−2 + 1.