State complexity of prefix distance

The prefix distance between strings x and y is the number of symbol occurrences in the strings that do not belong to the longest common prefix of x and y. The suffix and the substring distances are defined analogously in terms of the longest common suffix and longest common substring, respectively, of two strings. We show that the set of strings within prefix distance k from an n state DFA (deterministic finite automaton) language can be recognized by a DFA with (k+1)⋅n−k(k+1)2 states and that this number of states is needed in the worst case. Also we give tight bounds for the nondeterministic state complexity of the set of strings within prefix, suffix or substring distance k from a regular language.