Property and Equivalence Testing on Strings

We investigate property testing and related questions, where instead of the usual Hamming and edit distances between input strings, we consider the more relaxed edit distance with moves. Using a statistical embedding of words which has similarities with the Parikh mapping, we first construct a tolerant tester for the equality of two words, whose complexity is independent of the string size, and we derive an approximation algorithm for the normalized edit distance with moves. We then consider the question of testing if a string is a member of a given language. We develop a method to compute, in polynomial time in the representation, a geometric approximate description of a regular language by a finite union of polytopes. As an application, we have a new tester for regular languages given by their nondeterministic finite automaton (or regular expressions), whose complexity does not depend on the automaton, except for a polynomial time preprocessing step. Furthermore, this method allows us to compare languages and validates the new notion of equivalent testing that we introduce. Using the geometrical embedding we can distinguish between a pair of automata that compute the same language, and a pair of automata whose languages are not ε-equivalent in an appropriate sense. Our equivalence tester is deterministic and has polynomial time complexity, whereas the non-approximated version is PSPACE-complete. Last, we extend the geometric embedding, and hence the tester algorithms, to infinite regular languages and to context-free grammars as well. For context-free grammars the equivalence test has now exponential time complexity, but in comparison, the non-approximated version is not even recursively decidable.