On the Expressive Power of Regular Expressions with Backreferences

TL;DR

This paper explores the expressive power of regular expressions with backreferences, showing they generate exactly indexed languages, and clarifies their position within formal language classes, including stack and nonerasing stack languages.

Contribution

It establishes that rewbs are contained within indexed languages, improves the known upper-bound from context-sensitive languages, and clarifies their hierarchy within formal language classes.

Findings

Rewbs generate exactly indexed languages.

Some rewbs define languages outside stack languages.

Rewbs without captured references are in nonerasing stack languages.

Abstract

A rewb is a regular expression extended with a feature called backreference. It is broadly known that backreference is a practical extension of regular expressions, and is supported by most modern regular expression engines, such as those in the standard libraries of Java, Python, and more. Meanwhile, indexed languages are the languages generated by indexed grammars, a formal grammar class proposed by A$.$V$.$Aho. We show that these two models' expressive powers are related in the following way: every language described by a rewb is an indexed language. As the smallest formal grammar class previously known to contain rewbs is the class of context sensitive languages, our result strictly improves the known upper-bound. Moreover, we prove the following two claims: there exists a rewb whose language does not belong to the class of stack languages, which is a proper subclass of indexed languages, and the language described by a rewb without a captured reference is in the class of nonerasing stack languages, which is a proper subclass of stack languages. Finally, we show that the hierarchy investigated in a prior study, which separates the expressive power of rewbs by the notion of nested levels, is within the class of nonerasing stack languages.