Skip to main content
SpringerLink
Log in

Two-Way Equational Tree Automata for AC-Like Theories: Decidability and Closure Properties

  • Conference paper
  • First Online:
Rewriting Techniques and Applications (RTA 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2706))

Included in the following conference series:

Abstract

We study two-way tree automata modulo equational theories. We deal with the theories of Abelian groups (ACUM), idempotent commutative monoids (ACUI), and the theory of exclusive-or (ACUX), as well as some variants including the theory of commutative monoids (ACU). We show that the one-way automata for all these theories are closed under union and intersection, and emptiness is decidable. For two-way automata the situation is more complex. In all these theories except ACUI, we show that two-way automata can be effectively reduced to one-way automata, provided some care is taken in the definition of the so-called push clauses. (The ACUI case is open.) In particular, the two-way automata modulo these theories are closed under union and intersection, and emptiness is decidable. We also note that alternating variants have undecidable emptiness problem for most theories, contrarily to the non-equational case where alternation is essentially harmless.

Partially supported by the ACI “cryptologie” PSI-Robuste, ACI VERNAM, the RNTL project EVA and the ACI jeunes chercheurs “Sécurité informatique, protocoles cryptographiques et détection d’intrusions”.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. T. Colcombet. Rewriting in the partial algebra of typed terms modulo AC. In A. Kucera and R. Mayr, editors, Electronic Notes in Theoretical Computer Science, volume 68. Elsevier Science Publishers, 2002.

    Google Scholar 

  2. H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. Tree automata techniques and applications. http://www.grappa.univ-lille3.fr/tata, 1997.

  3. T. Frühwirth, E. Shapiro, M. Y. Vardi, and E. Yardeni. Logic programs as types for logic programs. In LICS’91, 1991.

    Google Scholar 

  4. F. Gécseg and M. Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 1–68. Springer Verlag, 1997.

    Google Scholar 

  5. S. Ginsburg and E. H. Spanier. Semigroups, Presburger formulas and languages. Pacific Journal of Mathematics, 16(2):285–296, 1966.

    MATH  MathSciNet  Google Scholar 

  6. J. Goubault-Larrecq. A method for automatic cryptographic protocol verification. In FMPPTA’2000, 15th IPDPS Workshops, pages 977–984. Springer-Verlag LNCS 1800, 2000.

    Google Scholar 

  7. J. Goubault-Larrecq and K. N. Verma. Alternating two-way AC-tree automata. In preparation.

    Google Scholar 

  8. D. Lugiez. A good class of tree automata. Application to inductive theorem proving. In ICALP’98, pages 409–420. Springer-Verlag LNCS 1443, 1998.

    Google Scholar 

  9. D. Lugiez. Counting and equality constraints for multitree automata. In FOSSACS’03. Springer-Verlag LNCS, 2003.

    Google Scholar 

  10. M. L. Minsky. Recursive unsolvability of Post’s problem of “tag” and other topics in the theory of Turing machines. Annals of Mathematics, Second Series, 74(3):437–455, 1961.

    MathSciNet  Google Scholar 

  11. D. Monniaux. Abstracting cryptographic protocols with tree automata. In SAS’99, pages 149–163. Springer-Verlag LNCS 1694, 1999.

    Google Scholar 

  12. H. Ohsaki. Beyond regularity: Equational tree automata for associative and commutative theories. In CSL’01, pages 539–553. Springer-Verlag LNCS 2142, 2001.

    Google Scholar 

  13. H. Ohsaki and T. Takai. Decidability and closure properties of equational tree languages. In RTA’02, pages 114–128. Springer-Verlag LNCS 2378, 2002.

    Google Scholar 

  14. R. J. Parikh. On context-free languages. Journal of the ACM, 13(4):570–581, 1966.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Y. Vardi. Reasoning about the past with two-way automata. In ICALP’98, pages 628–641. Springer Verlag LNCS 1443, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LSV/CNRS UMR 8643 & INRIA Futurs projet SECSI & ENS Cachan, France

    Kumar Neeraj Verma

Authors
  1. Kumar Neeraj Verma
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dpt. Lenguajes y Sistemas Informáticos (LSI), Technical University of Catalonia (UPC), Jordi Girona 1, 08034, Barcelona, Spain

    Robert Nieuwenhuis

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Verma, K.N. (2003). Two-Way Equational Tree Automata for AC-Like Theories: Decidability and Closure Properties. In: Nieuwenhuis, R. (eds) Rewriting Techniques and Applications. RTA 2003. Lecture Notes in Computer Science, vol 2706. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44881-0_14

Download citation

  • DOI: https://doi.org/10.1007/3-540-44881-0_14

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40254-1

  • Online ISBN: 978-3-540-44881-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation