Usable Rules for Context-Sensitive Rewrite Systems

  • Raúl Gutiérrez
  • Salvador Lucas
  • Xavier Urbain
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5117)


Recently, the dependency pairs (DP) approach has been generalized to context-sensitive rewriting (CSR). Although the context-sensitive dependency pairs (CS-DP) approach provides a very good basis for proving termination of CSR, the current developments basically correspond to a ten-years-old DP approach. Thus, the task of adapting all recently introduced dependency pairs techniques to get a more powerful approach becomes an important issue. In this direction, usable rules are one of the most interesting and powerful notions. Actually usable rule have been investigated in connection with proofs of innermost termination of CSR. However, the existing results apply to a quite restricted class of systems. In this paper, we introduce a notion of usable rules that can be used in proofs of termination of CSR with arbitrary systems. Our benchmarks show that the performance of the CS-DP approach is much better when such usable rules are considered in proofs of termination of CSR.


Dependency pairs term rewriting termination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alarcón, B., Gutiérrez, R., Iborra, J., Lucas, S.: Proving Termination of Context-Sensitive Rewriting with MU-TERM. ENTCS 188, 105–115 (2007)Google Scholar
  2. 2.
    Alarcón, B., Gutiérrez, R., Lucas, S.: Context-Sensitive Dependency Pairs. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 297–308. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Alarcón, B., Gutiérrez, R., Lucas, S.: Improving the Context-Sensitive Dependency Graph. ENTCS 188, 91–103 (2007)Google Scholar
  4. 4.
    Alarcón, B., Lucas, S.: Termination of Innermost Context-Sensitive Rewriting Using Dependency Pairs. In: Konev, B., Wolter, F. (eds.) FroCos 2007. LNCS (LNAI), vol. 4720, pp. 73–87. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Arts, T., Giesl, J.: Proving Innermost Normalisation Automatically. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 157–171. Springer, Heidelberg (1997)Google Scholar
  6. 6.
    Arts, T., Giesl, J.: Termination of Term Rewriting Using Dependency Pairs. Theoretical Computer Science 236(1–2), 133–178 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Borralleras, C.: Ordering-Based Methods for Proving Termination Automatically. PhD thesis, Departament de Llenguatges i Sistemes Informàtics, UPC (2003)Google Scholar
  8. 8.
    Durán, F., Lucas, S., Meseguer, J., Marché, C., Urbain, X.: Proving Operational Termination of Membership Equational Programs. Higher-Order and Symbolic Computation (to appear, 2008)Google Scholar
  9. 9.
    Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: SAT Solving for Termination Analysis with Polynomial Interpretations. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 340–354. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Giesl, J., Arts, T., Ohlebusch, E.: Modular Termination Proofs for Rewriting Using Dependency Pairs. Journal of Symbolic Computation 34(1), 21–58 (2002)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Giesl, J., Middeldorp, A.: Innermost Termination of Context-Sensitive Rewriting. In: Ito, M., Toyama, M. (eds.) DLT 2002. LNCS, vol. 2450, pp. 231–244. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Giesl, J., Middeldorp, A.: Transformation Techniques for Context-Sensitive Rewrite Systems. Journal of Functional Programming 14(4), 379–427 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Giesl, J., Thiemann, R., Schneider-Kamp, P.: The Dependency Pair Framework: Combining Techniques for Automated Termination Proofs. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 301–331. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and Improving Dependency Pairs. Journal of Automatic Reasoning 37(3), 155–203 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Gramlich, B.: Generalized Sufficient Conditions for Modular Termination of Rewriting. Applicable Algebra in Engineering, Communication and Computing 5, 131–151 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Gutiérrez, R., Lucas, S., Urbain, X.: Usable Rules for Context-Sensitive Rewrite System, DSIC-II/03/08. Technical report, UPV (2008)Google Scholar
  17. 17.
    Hirokawa, N., Middeldorp, A.: Tyrolean Termination Tool: Techniques and Features. Information and Computation 205(4), 474–511 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Lucas, S.: Context-Sensitive Computations in Functional and Functional Logic Programs. Journal of Functional and Logic Programming 1998(1), 1–61 (1998)Google Scholar
  19. 19.
    Lucas, S.: Termination of on-demand rewriting and termination of OBJ programs. In: Proc. of PPDP 2001, pp. 82–93. ACM Press, New York (2001)CrossRefGoogle Scholar
  20. 20.
    Lucas, S.: Context-Sensitive Rewriting Strategies. Information and Computation 178(1), 293–343 (2002)MathSciNetGoogle Scholar
  21. 21.
    Lucas, S.: MU-TERM: A Tool for Proving Termination of Context-Sensitive Rewriting. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 200–209. Springer, Heidelberg (2004), Google Scholar
  22. 22.
    Lucas, S.: Proving Termination of Context-Sensitive Rewriting by Transformation. Information and Computation 204(12), 1782–1846 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  24. 24.
    Urbain, X.: Modular & Incremental Automated Termination Proofs. Journal of Automated Reasoning 32(4), 315–355 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Raúl Gutiérrez
    • 1
  • Salvador Lucas
    • 1
  • Xavier Urbain
    • 2
  1. 1.DSICUniversidad Politécnica de ValenciaSpain
  2. 2.Cédric-CNAM, ENSIIEFrance

Personalised recommendations