Skip to main content
SpringerLink
Log in

Greibach Normal Form in Algebraically Complete Semirings

  • Conference paper
  • First Online:
Computer Science Logic (CSL 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2471))

Included in the following conference series:

Abstract

We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach’s normal form theorem depends only on a few equational properties of least pre-fixed-points in semirings, and elimination of chain- and deletion rules depend on their inequational properties (and the idempotency of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed-points.

This author was supported by BRICS (Aalborg) and the National Foundation of Hungary for Scientific Research, grant T35169.

This author was supported by a travel grant from the Humboldt-Foundation.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Aceto, Z. Ésik and A. Ingólfsdóttir. A fully equational proof of Parikh’s theorem. BRICS Report Series, RS-01-28, Aarhus, 2001.

    Google Scholar 

  2. J. W. de Bakker and D. Scott. A theory of programs. IBM Seminar, August, 1969.

    Google Scholar 

  3. H. Bekić. Definable operations in general algebra, and the theory of automata and flowcharts. Technical Report, IBM Laboratory, Vienna, 1969.

    Google Scholar 

  4. L. Bernátsky, S. L. Bloom, Z. Ésik, and Gh. Stefanescu. Equational theories of relations and regular sets, extended abstract. In Proceedings of the Conference on Words, Combinatorics and Semigroups,Kyoto, 1992, pages 40–48. World Scientific Publishing Co. Pte. Ltd., 1994.

    Google Scholar 

  5. S. L. Bloom and Z. Ésik. Iteration algebras, Int. J. Foundations of Computer Science, 3(1991), 245–302.

    Article  Google Scholar 

  6. S.L. Bloom and Z. Ésik. Iteration Theories, Springer, 1993.

    Google Scholar 

  7. M. Boffa. Une condition impliquant toutes les identités rationnelles. RAIRO Inform. Théor. Appl., 29 (1995), 515–518.

    MATH  MathSciNet  Google Scholar 

  8. J. H. Conway. Regular Algebra and Finite Machines. Chapman and Hall, London, 1971.

    MATH  Google Scholar 

  9. Z. Ésik. Completeness of Park induction, Theoretical Computer Science, 177 (1997), 217–283.

    Article  MATH  MathSciNet  Google Scholar 

  10. Z. Ésik and W. Kuich. Inductive *-semirings. To appear in Theoretical Computer Science.

    Google Scholar 

  11. S.A. Greibach. A new normal-form theorem for context-free, phrase-structure grammars. J. of the Association for Computing Machinery, 12 (1965), 42–52.

    MATH  MathSciNet  Google Scholar 

  12. M. Harrison. Introduction to Formal Languages. Addison Wesley, Reading, 1978.

    Google Scholar 

  13. M. W. Hopkins and D. Kozen. Parikh’s theorem in commutative Kleene algebra. In Proc. Symp. Logic in Computer Science (LICS’99), IEEE Press, 1999, 394–401.

    Google Scholar 

  14. D. Kozen. A completeness theorem for Kleene algebras and the algebra of regular events. In 6th Ann. Symp. on Logic in Computer Science, LICS’91. Computer Society Press, 1991, 214–225.

    Google Scholar 

  15. D. Kozen. On the complexity of reasoning in Kleene algebra. In Proc. 12th Symp. Logic in Computer Science, IEEE Press, 1997, 195–202.

    Google Scholar 

  16. D. Krob. Complete systems of B-rational identities. Theoret. Comput. Sci., 89 (1991), 207–343.

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Leiβ. Towards Kleene Algebra with Recursion. In Proc. 5th Workshop on Computer Science Logic, CSL’ 91. Springer LNCS 626, 242–256, 1991.

    Google Scholar 

  18. K. C. Ng and A. Tarski. Relation algebras with transitive closure. Notices of the American Math. Society, 24:A29–A30, 1977.

    Google Scholar 

  19. D. Niwinski. Equational µ-calculus. In Computation Theory (Zaborow, 1984), pages 169–176, Springer LNCS 208, 1984.

    Google Scholar 

  20. V. R. Pratt. Action Logic and Pure Induction. In Logics in AI: European Workshop JELIA’ 90. Springer LNCS 478, 97–120, 1990.

    Google Scholar 

  21. V. N. Redko. On the determining totality of relations for the algebra of regular events. (Russian) Ukrain. Mat. Ž., 16 (1964), 120–126.

    MathSciNet  Google Scholar 

  22. D.J. Rosenkrantz. Matrix equations and normal forms for context-free grammars. Journal of the Association for Computing Machinery, 14 (1967), 501–507.

    MATH  MathSciNet  Google Scholar 

  23. A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the Association for Computing Machinery, 13 (1966), 158–169.

    MATH  MathSciNet  Google Scholar 

  24. L. Santocanale. On the equational definition of the least prefixed point. In MFCS 2001, pages 645–656, Springer LNCS 2136, 2001.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, University of Szeged, Szeged, Hungary

    Zoltán Ésik

  2. Centrum für Informations- und Sprachverarbeitung, University of Munich, Munich, Germany

    Hans Leiβ

Authors
  1. Zoltán Ésik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans Leiβ
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratory for Foundations of Computer Science Division of Informatics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK

    Julian Bradfield

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ésik, Z., Leiβ, H. (2002). Greibach Normal Form in Algebraically Complete Semirings. In: Bradfield, J. (eds) Computer Science Logic. CSL 2002. Lecture Notes in Computer Science, vol 2471. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45793-3_10

Download citation

  • DOI: https://doi.org/10.1007/3-540-45793-3_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44240-0

  • Online ISBN: 978-3-540-45793-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation