Distance calculation on strings

Abstract

A monoid of strings (words) over a finite alphabet is considered. The notion of distance on strings is important in the problem of inductive learning related to artificial intelligence, in cryptography, and in some other fields of mathematics. The distance is defined as a minimum length of the transformation path that transforms one string into another. One example is the Levenstein distance, with the transformations being insertions, deletions, and substitutions of letters. A quadratic algorithm for calculating this distance is known to exist. In this paper, a more general case—insertion and deletion of words of arbitrary length—is considered. For this case, the problem of distance calculation turns out to be unsolvable. The basic results of this work are the formulation of the condition of computability of distance and the algorithm for distance calculation, which is polynomial in string length.