Relative Prefix Distance Between Languages

The prefix distance between two words x and y is defined as the number of symbols in x and y that do not belong to their longest common prefix. The relative prefix distance from a language L1 to a language L2, if finite, is the smallest integer k such that for every word in L1, there is a word in L2 with prefix distance at most k. We study the prefix distance between regular, visibly pushdown, deterministic contextfree, and context-free languages. We show how to compute the distance between regular languages and determine whether the distance is bounded. For deterministic context-free languages and visibly pushdown languages, we show that the relative prefix distance to and from regular languages is decidable.