Skip to main content
SpringerLink
Log in

Confluence of Non-Left-Linear TRSs via Relative Termination

  • Conference paper
Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7180))

Abstract

We present a confluence criterion for term rewrite systems by relaxing termination requirements of Knuth and Bendix’ confluence criterion, using joinability of extended critical pairs. Because computation of extended critical pairs requires equational unification, which is undecidable, we give a sufficient condition for testing joinability automatically.

This work is supported by the Grant-in-Aids for Young Scientists (B) 22700009 and Scientific Research (B) 23300005 of the Japan Society for the Promotion of Science.

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. Aoto, T., Toyama, Y.: Persistency of confluence. Journal of Universal Computer Science 3(11), 1134–1147 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Aoto, T., Toyama, Y.: A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems. In: Proc. 22nd RTA. LIPIcs, vol. 10, pp. 91–106 (2011)

    Google Scholar 

  3. Aoto, T., Yoshida, J., Toyama, Y.: Proving Confluence of Term Rewriting Systems Automatically. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 93–102. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  4. Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press (1998)

    Google Scholar 

  5. Felgenhauer, B., Zankl, H., Middeldorp, A.: Layer systems for proving confluence. In: Proc. 31st FSTTCS. LIPIcs, vol. 13, pp. 288–299 (2011)

    Google Scholar 

  6. Geser, A.: Relative Termination. PhD thesis, Universität Passau, Available as technical report 91-03 (1990)

    Google Scholar 

  7. Giesl, J., Thiemann, R., Schneider-Kamp, P.: Proving and Disproving Termination of Higher-Order Functions. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 216–231. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  8. Gomi, H., Oyamaguchi, M., Ohta, Y.: On the Church-Rosser property of non-E-overlapping and strongly depth-preserving term rewriting systems. Trans. IPSJ 37(12), 2147–2160 (1996)

    MathSciNet  Google Scholar 

  9. Gomi, H., Oyamaguchi, M., Ohta, Y.: On the Church-Rosser property of root-E-overlapping and strongly depth-preserving term rewriting systems. Trans. IPSJ 39(4), 992–1005 (1998)

    MathSciNet  Google Scholar 

  10. Hirokawa, N., Middeldorp, A.: Decreasing diagrams and relative termination. Journal of Automated Reasoning 47, 481–501 (2011)

    Article  Google Scholar 

  11. Huet, G.: Confluent reductions: Abstract properties and applications to term rewriting systems: Abstract properties and applications to term rewriting systems. Journal of the ACM 27, 797–821 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jouannaud, J.P.: Confluent and Coherent Equational Term Rewriting Systems: Application to Proofs in Abstract Data Types. In: Protasi, M., Ausiello, G. (eds.) CAAP 1983. LNCS, vol. 159, pp. 269–283. Springer, Heidelberg (1983)

    Chapter  Google Scholar 

  13. Jouannaud, J.P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM Journal on Computing 15(4), 1155–1194 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klein, D., Hirokawa, N.: Maximal completion. In: Proc. 22nd RTA. LIPIcs, vol. 10, pp. 71–80 (2011)

    Google Scholar 

  15. Klop, J.: Combinatory reduction systems. PhD thesis, Utrecht University (1980)

    Google Scholar 

  16. Knuth, D.E., Bendix, P.: Simple word problems in universal algebras. In: Computational Problems in Abstract Algebra, pp. 263–297 (1970)

    Google Scholar 

  17. Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean Termination Tool 2. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  18. Ogawa, M.: Chew’s Theorem Revisited -Uniquely Normalizing Property of Nonlinear Term Rewriting Systems. In: Ibaraki, T., Iwama, K., Yamashita, M., Inagaki, Y., Nishizeki, T. (eds.) ISAAC 1992. LNCS, vol. 650, pp. 309–318. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  19. Ohlebusch, E.: Modular properties of composable term rewriting systems. Journal of Symbolic Computation 20, 1–41 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Stump, A., Kimmell, G., Omar, R.E.H.: Type preservation as a confluence problem. In: Proc. 22nd RTA. LIPIcs, vol. 10, pp. 345–360 (2011)

    Google Scholar 

  21. TeReSe: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press (2003)

    Google Scholar 

  22. Toyama, Y.: On the Church-Rosser property for the direct sum of term rewriting systems. Journal of the ACM 34(1), 128–143 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Toyama, Y.: Commutativity of term rewriting systems. In: Programming of Future Generation Computers II, pp. 393–407. North-Holland (1988)

    Google Scholar 

  24. van Oostrom, V.: Developing developments. Theoretical Computer Science 175(1), 159–181 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. van Oostrom, V.: Confluence by Decreasing Diagrams. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 306–320. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  26. Yamamoto, A.: Completeness of Extending Unification Based on Basic Narrowing. In: Fujisaki, T., Nakata, I., Tanaka, H. (eds.) Logic Programming 1988. LNCS, vol. 383, pp. 1–10. Springer, Heidelberg (1989)

    Google Scholar 

  27. Zankl, H., Felgenhauer, B., Middeldorp, A.: CSI – A Confluence Tool. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 499–505. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  28. Zankl, H., Felgenhauer, B., Middeldorp, A.: Labelings for decreasing diagrams. In: Proc. 22nd RTA. LIPIcs, pp. 377–392 (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Science, Japan Advanced Institute of Science and Technology, Japan

    Dominik Klein & Nao Hirokawa

Authors
  1. Dominik Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nao Hirokawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Microsoft Research, One Microsoft Way, 98052-6399, Redmond, WA, USA

    Nikolaj Bjørner

  2. School of computer Science, University of Manchester, Kilburn Building, Oxford Road, MP13 9PL, Manchester, UK

    Andrei Voronkov

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Klein, D., Hirokawa, N. (2012). Confluence of Non-Left-Linear TRSs via Relative Termination. In: Bjørner, N., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2012. Lecture Notes in Computer Science, vol 7180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28717-6_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-28717-6_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28716-9

  • Online ISBN: 978-3-642-28717-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation