State Complexity of Prefix Distance of Subregular Languages

The neighbourhood of a regular language of constant radius with respect to the prefix distance is always regular. We give upper bounds and matching lower bounds for the size of the minimal deterministic finite automaton (DFA) needed for the radius k prefix distance neighbourhood of an n state DFA that recognizes, respectively, a finite, a prefix-convex, a prefix-closed, a prefix-free, and a right ideal language. For prefix-closed languages the lower bound automata are defined over a binary alphabet. For finite and prefix-convex regular languages the lower bound constructions use an alphabet that depends on the size of the DFA and it is shown that the size of the alphabet is optimal.