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On the Expressive Power of Regular Expressions with Backreferences

Authors Taisei Nogami, Tachio Terauchi

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LIPIcs.MFCS.2023.71.pdf
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Author Details

Taisei Nogami
  • Waseda University, Tokyo, Japan
Tachio Terauchi
  • Waseda University, Tokyo, Japan

Funding

This work was supported by JSPS KAKENHI Grant Numbers JP20H04162, JP20K20625, and JP22H03570.

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Taisei Nogami and Tachio Terauchi. On the Expressive Power of Regular Expressions with Backreferences. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.71

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Regular expressions
  • Backreferences
  • Expressive power

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References

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