Measuring the expressive power of practical regular expressions by classical stacking automata models

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 four claims: (1) there exists a rewb whose language does not belong to the class of stack languages, which is a proper subclass of indexed languages, (2) 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, (3) there exists a rewb that describes a stack language but not a nonerasing stack language, and (4) a rewb extended with another practical extension called lookaheads can describe a non-indexed language. 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. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).