A weak inverse of language neighborhoods and its properties

Abstract

While the edit distance neighborhood is useful for approximate pattern matching, it is not suitable for the negative lookahead feature for the practical regex matching engines. This motivates us to introduce a new operation. We define the edit distance interior operation on a language L, which is to compute the largest subset I(L) of L such that the edit distance neighborhood of I(L) is in L. In other words, L includes the edit distance neighborhood of the largest edit distance interior language. Given an edit distance value r, we show that the radius-r edit distance interior operation is a weak inverse of the radius-r edit distance neighborhood operation, and vice versa. A characterization of the edit distance interior languages and their proper hierarchy with respect to the radius are presented. Closure properties up to basic Boolean operations and decision properties of such languages are also discussed. In addition, we demonstrate that the regular languages are closed under the edit distance interior operation whereas the context-free languages are not.