Skip to main content
SpringerLink
Log in

Self-embedded context-free grammars with regular counterparts

  • Published:
Acta Informatica Aims and scope Submit manuscript

Abstract.

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auteberg J, Berstel J, Boasson L (1997) Context-Free Languages and Pushdown Automata. In: Rozenberg G, Salomaa A (eds) Handbook of Formal Languages. Word, Language, Grammar. Vol. 1, pp 111-174. Springer, Berlin Heidelberg New York

  2. Andrei Ş (2000) Bidirectional Parsing. Ph.D Thesis, Fachbereich Informatik, Universität Hamburg, Hamburg http://www.sub.uni-hamburg.de/ disse/134/inhalt.html

  3. Andrei Ş, Cavadini S, Chin WN (2003) A New Algorithm for Regularizing One-Letter Context-Free Grammars. Theoretical Computer Science 306 (1-3): 113-122

    Google Scholar 

  4. Arden DN (1960) Delayed logic and finite state machines. Theory of computing machine design. University of Michigan Press, Ann Arbor

  5. Bar-Hillel Y, Perles M, Shamir E (1961) On formal properties of simple phrase structure grammars. Z. Phonetik. Sprachwiss. Kommunikationsforsch. 14: 143-172

    Google Scholar 

  6. Chomsky N (1959) On Certain Formal Properties of Grammars. Information and Control 2: 137-167

    MATH  Google Scholar 

  7. Chomsky N (1959) A Note on Phrase Structure Grammars. Information and Control 2: 393-395

    MATH  Google Scholar 

  8. Chomsky N, Schützenberger MP (1963) The algebraic theory of context-free languages. In: Braffort P, Hirschberg D (eds) Computer Programming and Formal Systems, pp 118-161. Amsterdam, North-Holland

  9. Chrobak M (1986) Finite automata and unary languages. Theoretical Computer Science 47: 149-158

    Article  MathSciNet  MATH  Google Scholar 

  10. Cohen DIA (1997) Introduction to Computer Theory. 2nd edn. Wiley

  11. Ginsburg S, Rice HG (1962) Two families of languages related to ALGOL. Journal of the Association for Computing Machinery 9: 350-371

    Article  MATH  Google Scholar 

  12. Hosoya H, Pierce BC (2001) Regular expression pattern matching for XML. ACM SIG-PLAN Notices 36 (3): 67-80

    Google Scholar 

  13. Leiss EL (1994) Language equations over a one-letter alphabet with union, concatenation and star: a complete solution. Theoretical Computer Science 131: 311-330

    Article  MathSciNet  MATH  Google Scholar 

  14. Leiss, EL (1997) Solving systems of explicit language relations. Theoretical Computer Science 186: 83-105

    Article  MathSciNet  Google Scholar 

  15. Nederhof MJ (2000) Practical Experiments with Regular Approximation of Context-Free Languages. Computational Linguistics 26 (1): 17-44

    Article  MathSciNet  Google Scholar 

  16. Parikh RJ (1961) Language-generating devices. Quarterly Progress Report, Research Laboratory of Electronics, M.I.T. 60: 199-212

  17. Parikh RJ (1966) On context-free languages. Journal of the Association for Computing Machinery 13: 570-581

    Article  MATH  Google Scholar 

  18. Salomaa A (1969) Theory of Automata. Pergamon Press, Oxford

  19. Simovici DA, Tenney RL (1999) Theory of Formal Languages with Applications. World Scientific, Singapore

Download references

Author information

Authors and Affiliations

  1. Faculty of Computer Science, ‘Al.I.Cuza’ University, Str. Berthelot, nr.16, 6600, Iaşi, România

    Stefan Andrei

  2. Singapore-MIT Alliance, National University of Singapore, CS Programme, 3 Science Drive 2, 117543, Singapore

    Stefan Andrei

  3. School of Computing, Department of Computer Science, National University of Singapore, 3 Science Drive 2, 117543, Singapore

    Wei-Ngan Chin

  4. Facultad de Matemática Aplicada, Centro de Investigación y Desarrollo de Software , Universidad Católica de Santiago del Estero, Avda, Alsina y Dalmasio Velez Sarfield, Santiago del Estero, Argentina

    Salvador Valerio Cavadini

Authors
  1. Stefan Andrei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei-Ngan Chin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Salvador Valerio Cavadini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Andrei.

Additional information

Received: 17 January 2003, Published online: 17 February 2004

Stefan Andrei: stefan@infoiasi.ro

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andrei, S., Chin, WN. & Cavadini, S.V. Self-embedded context-free grammars with regular counterparts. Acta Informatica 40, 349–365 (2004). https://doi.org/10.1007/s00236-003-0133-8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00236-003-0133-8

Keywords

Search

Navigation