Weak Inverse Neighborhoods of Languages

Hyunjoon Cheon, Yo-Sub Han

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 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. In addition, we demonstrate that regular languages are closed under the edit-distance interior operation whereas context-free languages are not. Then, we characterize the edit-distance interior languages and present a proper hierarchy with respect to the radius of operations. The family of edit-distance interior languages is closed under intersection, but not closed under union, complement and catenation.