Derivatives of Approximate Regular Expressions

TL;DR

This paper introduces Approximate Regular Expressions (AREs) with similarity operators, enabling the construction of finite automata recognizing words within a bounded distance from a regular language, and provides derivative-based methods for their recognition.

Contribution

The paper presents the first definition of Approximate Regular Expressions with similarity operators and derivative computations for recognition.

Findings

Defined similarity operators for regular expressions.

Derived formulas for Brzozowski and Antimirov derivatives of AREs.

Constructed recognizers for approximate regular expression languages.

Abstract

Our aim is to construct a finite automaton recognizing the set of words that are at a bounded distance from some word of a given regular language. We define new regular operators, the similarity operators, based on a generalization of the notion of distance and we introduce the family of regular expressions extended to similarity operators, that we call AREs (Approximate Regular Expressions). We set formulae to compute the Brzozowski derivatives and the Antimirov derivatives of an ARE, which allows us to give a solution to the ARE membership problem and to provide the construction of two recognizers for the language denoted by an ARE. As far as we know, the family of approximative regular expressions is introduced for the first time in this paper. Classical approximate regular expression matching algorithms are approximate matching algorithms on regular expressions. Our approach is rather to process an exact matching on approximate regular expressions.