Skip to main content
SpringerLink
Log in

Elimination Transformations for Associative–Commutative Rewriting Systems

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

To simplify the task of proving termination and AC-termination of term rewriting systems, elimination transformations have been vigorously studied since the 1990s. Dummy elimination, distribution elimination, general dummy elimination, and improved general dummy elimination are examples of elimination transformations. In this paper we clarify the essence of elimination transformations based on the notion of dependency pairs. We first present a theorem that gives a general and essential property for elimination transformations, making them sound with AC-termination. Based on the theorem, we design an elimination transformation called the argument filtering transformation. Next, we clarify the relation among various elimination transformations by comparing them with a corresponding restricted argument filtering transformation. Finally, we compare the AC-dependency pair method with the argument filtering transformation.

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. Arts, T.: Automatically proving termination and innermost normalization of term rewriting systems. Ph.D. thesis, Utrecht University, Utrecht, The Netherlands (1997)

  2. Arts, T., Giesl, J.: Automatically proving termination where simplification orderings fail. In: Proc. 7th Int. Joint Conf. on Theory and Practice of Software Development, LNCS 1214 (TAPSOFT’97), pp. 261–272. Springer, Berlin Heidelberg New York (1997)

    Google Scholar 

  3. Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comp. Sci. 236, 133–178 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, New York (1998)

    Google Scholar 

  5. Bachmair, T., Plaisted, D.A.: Termination orderings for associative–commutative rewriting systems. J. Symb. Comput. 1, 329–349 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  6. Contejean, E., Marché C., Monate, B., Urbain, X.: CiME version 2, 2000. (Available at http://cime.lri.fr/)

  7. Dershowitz, N.: Orderings for term-rewriting systems. Theor. Comp. Sci. 17, 279–301 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ferreira, M., Zantema, H.: Dummy elimination: making termination easier. In: Proc. 10th Int. Conf. on Fundamentals of Computation Theory, LNCS 965 (FCT’95), pp. 243–252. Dresden, Germany. Springer, Berlin Heidelberg New York (1995)

    Google Scholar 

  9. Ferreira, M.: Termination of term rewriting, well-foundedness, totality and transformations. Ph.D. thesis, Utrecht University, Utrecht, The Netherlands (1995)

  10. Ferreira, M., Kesner, D., Puel, L.: Reducing AC-termination to termination. In: Proc. 23rd Int. Symp. on Mathematical Foundations of Computer Science, LNCS 1450 (MFCS’98), pp. 239–247 (1998)

  11. Giesl, J., Middeldorp, A.: Eliminating dummy elimination. In: Proc. 17th Int. Conf. on Automated Deduction, LNAI 1831 (CADE2000), pp. 309–323. Pittsburgh, PA. Springer, Berlin Heidelberg New York (2000)

    Google Scholar 

  12. Giesl, J., Kapur, D.: Dependency pairs for equational rewriting. In: Proc. 12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051 (RTA’01), pp. 93–107. Utrecht, The Netherlands. Springer, Berlin Heidelberg New York (2001)

    Chapter  Google Scholar 

  13. Giesl, J., Arts, T., Ohlebusch, E.: Modular termination proofs for rewriting using dependency pairs. J. Symb. Comput. 34(1), 21–58 (2002)

    Article  MathSciNet  Google Scholar 

  14. Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Improving dependency pairs. In: Proc. 10th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning, LNAI 2850 (LPAR03), pp. 165–179. Almaty, Kazakhstan. Springer, Berlin Heidelberg New York (2003)

    Google Scholar 

  15. Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Automated termination proofs with AProVE. In: Proc. 15th Int. Conf. on Rewriting Techniques and Applications, LNCS 3091 (RTA04), pp. 210–220. Aachen, Germany. Springer, Berlin Heidelberg New York (2004)

    Google Scholar 

  16. Hirokawa, N., Middeldorp, A.: Automating the dependency pair method. Inf. Comput. 199(1,2), 172–199 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Klop, J.W.: Term rewriting systems. Handbook of Logic in Computer Science II, pp. 1–112. Oxford University Press, London, UK (1992)

    Google Scholar 

  18. Kurihara, M., Ohuchi, A.: Modularity of simple termination of term rewriting systems with shared constructors. Theor. Comp. Sci., 103, 273–282 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kusakari, K., Toyama, Y.: On proving AC-termination by argument filtering method. IPSJ Trans. Program. 41, No. SIG 4 (PRO 7), 65–78 (2000)

    Google Scholar 

  20. Kusakari, K., Toyama, Y.: On proving AC-termination by AC-dependency pairs. IEICE Trans. Info. Syst. E84-D(5), 604–612 (2001)

    Google Scholar 

  21. Marché, C., Urbain, X.: Termination of associative–commutative rewriting by dependency pairs. In: Proc. 9th Int. Conf. on Rewriting Techniques and Applications, LNCS 1379 (RTA’98), pp. 241–255. Tsukuba, Japan (1998)

  22. Marché, C., Urbain, X.: Modular and incremental proofs of AC-termination. J. Symb. Comput. 38(1), 873–897 (2004)

    Article  Google Scholar 

  23. Middeldorp, A., Ohsaki, H., Zantema, H.: Transforming termination by self-labeling. In: Proc. 13th Int. Conf. on Automated Deduction, LNCS 1104 (CADE-13), pp. 373–387 (1996)

  24. Middeldorp, A.: Approximating dependency graphs using tree automata techniques. In: Proc. Int. Joint Conf. on Automated Reasoning, LNAI 2083 (IJCAR01), pp. 593–610. Siena, Italy (2001)

  25. Nakamura, M., Kusakari, K., Toyama, Y.: On proving termination by general dummy elimination. IEICE Trans. Info. Syst. J82-D-I(10), 1225–1231 (1999) (in Japanese)

    Google Scholar 

  26. Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, Berlin Heidelberg New York (2002)

    MATH  Google Scholar 

  27. Ohsaki, H., Middeldorp, A., Giesl, J.: Equational termination by semantic labeling. In: Proc. 14th Annual Conf. of the European Association for Computer Science Logic, LNCS 1862 (CSL’00), pp. 457–471, Fischbachau/Munich, Germany. Springer, Berlin Heidelberg New York (2000)

    Google Scholar 

  28. Terese. Term rewriting systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge, UK (2003)

    Google Scholar 

  29. Thiemann, R., Giesl, J., Schneider-Kamp, P.: Improved modular termination proofs using dependency pairs. In: Proc. 2nd Int. Joint Conf. on Automated Reasoning, LNAI 3097 (IJCAR2004), pp. 75–90 (2004)

  30. Urbain, X.: Automated incremental termination proofs for hierarchically defined term rewriting systems. In: Proc. 10th Int. Joint Conf. on Automated Reasoning, LNAI 2083 (IJCAR’01), pp. 485–498 (2001)

  31. Urbain, X.: Approche incrémentale des Preuves automatiques de Termination. Ph.D. Thesis, Université Paris-Sud, France (2002)

  32. Urbain, X.: Modular & incremental automated termination proofs. J. Autom. Reason. 32(4), 315–355 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zantema, H.: Termination of term rewriting: Interpretation and type elimination. J. Symb. Comput. 17, 23–50 (1994)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan

    Kusakari Keiichirou

  2. School of Information Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa, 923-1292, Japan

    Nakamura Masaki

  3. Research Institute of Electrical Communication, Tohoku University, Katahira, Aoba-ku, Sendai, 980-8577, Japan

    Toyama Yoshihito

Authors
  1. Kusakari Keiichirou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nakamura Masaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Toyama Yoshihito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kusakari Keiichirou.

Additional information

Parts of this work were done while K. Kusakari was completing his Ph.D. thesis, “Termination, AC-Termination and Dependence Pairs of Term Rewriting Systems,” at Japan Advanced Institute of Science and Technology, School of Information Science (March, 2000). A preliminary version of parts of this article appeared in K. Kusakari, M. Nakamura, Y. Toyama, Argument filtering transformation, Proc. of Int. Conf. on Principles and Practice of Declarative Programming, LNCS 1702 (Springer-Verlag, 1999) pp.47–61.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Keiichirou, K., Masaki, N. & Yoshihito, T. Elimination Transformations for Associative–Commutative Rewriting Systems. J Autom Reasoning 37, 205–229 (2006). https://doi.org/10.1007/s10817-006-9053-y

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-006-9053-y

Key words

Search

Navigation