Acta Informatica

, Volume 40, Issue 5, pp 349–365

Self-embedded context-free grammars with regular counterparts


    • Faculty of Computer Science‘Al.I.Cuza’ University
    • Singapore-MIT AllianceNational University of Singapore, CS Programme
  • Wei-Ngan Chin
    • School of Computing, Department of Computer ScienceNational University of Singapore
  • Salvador Valerio Cavadini
    • Facultad de Matemática Aplicada, Centro de Investigación y Desarrollo de Software Universidad Católica de Santiago del Estero

DOI: 10.1007/s00236-003-0133-8

Cite this article as:
Andrei, S., Chin, W. & Cavadini, S.V. Acta Informatica (2004) 40: 349. doi:10.1007/s00236-003-0133-8


In general, it is undecidable if an arbitrary context-free grammar has a regular solution. Past work has focused on special cases, such as one-letter grammars, non self-embedded grammars and the finite-language grammars, for which regular counterparts have been proven to exist. However, little is known about grammars with the self-embedded property. Using systems of equations, we highlight a number of subclasses of grammars, with self-embeddedness terms, such as \(X \alpha X\) and \(\gamma X \gamma\), that can still have regular languages as solutions. Constructive proofs that allow these subclasses of context-free grammars to be transformed to regular expressions are provided. We also point out a subclass of context-free grammars that is inherently non-regular. Our latest results can help demarcate more precisely the known boundaries between the regular and non-regular languages, within the context-free domain.

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004