Advertisement

Search Techniques for Rational Polynomial Orders

  • Carsten Fuhs
  • Rafael Navarro-Marset
  • Carsten Otto
  • Jürgen Giesl
  • Salvador Lucas
  • Peter Schneider-Kamp
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5144)

Abstract

Polynomial interpretations are a standard technique used in almost all tools for proving termination of term rewrite systems (TRSs) automatically. Traditionally, one applies interpretations with polynomials over the naturals. But recently, it was shown that interpretations with polynomials over the rationals can be significantly more powerful. However, searching for such interpretations is considerably more difficult than for natural polynomials. Moreover, while there exist highly efficient SAT-based techniques for finding natural polynomials, no such techniques had been developed for rational polynomials yet. In this paper, we tackle the two main problems when applying rational polynomial interpretations in practice: (1) We develop new criteria to decide when to use rational instead of natural polynomial interpretations. (2) Afterwards, we present SAT-based methods for finding rational polynomial interpretations and evaluate them empirically.

Keywords

termination term rewriting SAT solving dependency pairs 

Topics

computer algebra systems and automated theorem provers  implementation and performance issues 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 133–178 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Contejean, E., Marché, C., Monate, B., Urbain, X.: CiME, http://cime.lri.fr
  3. 3.
    Contejean, E., Marché, C., Tomás, A.P., Urbain, X.: Mechanically proving termination using polynomial interpretations. Journal of Automated Reasoning 34(4), 325–363 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dershowitz, N.: Termination of rewriting. Journal of Symbolic Computation 3, 69–116 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Endrullis, J.: Jambox, http://joerg.endrullis.de
  6. 6.
    Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 574–588. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Fuhs, C., Giesl, J., Middeldorp, A., Thiemann, R., Schneider-Kamp, P., 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
  8. 8.
    Gebhardt, A., Hofbauer, D., Waldmann, J.: Matrix Evolutions. In: Proc. WST 2007 (2007)Google Scholar
  9. 9.
    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
  10. 10.
    Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: Automatic termination proofs in the DP framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. Journal of Automated Reasoning 37(3), 155–203 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hirokawa, N., Middeldorp, A.: Automating the dependency pair method. Information and Computation 199(1,2), 172–199 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hirokawa, N., Middeldorp, A.: Tyrolean Termination Tool: Techniques and features. Information and Computation 205(4), 474–511 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hong, H., Jakuš, D.: Testing positiveness of polynomials. Journal of Automated Reasoning 21(1), 23–38 (1998)CrossRefMathSciNetGoogle Scholar
  15. 15.
    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
  16. 16.
    Lucas, S.: Polynomials over the reals in proofs of termination: From theory to practice. RAIRO Theoretical Informatics and Applications 39(3), 547–586 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lucas, S.: On the relative power of polynomials with real, rational, and integer coefficients in proofs of termination of rewriting. Applicable Algebra in Engineering, Communication and Computing 17(1), 49–73 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lucas, S.: Practical use of polynomials over the reals in proofs of termination. In: Proc. PPDP 2007, pp. 39–50. ACM Press, New York (2007)Google Scholar
  19. 19.
    Marché, C., Zantema, H.: The termination competition. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 303–313. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  20. 20.
    Thiemann, R., Middeldorp, A.: Innermost termination of rewrite systems by labeling. In: Proc. WRS 2007. ENTCS 204, pp. 3–19 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Carsten Fuhs
    • 1
  • Rafael Navarro-Marset
    • 2
  • Carsten Otto
    • 1
  • Jürgen Giesl
    • 1
  • Salvador Lucas
    • 2
  • Peter Schneider-Kamp
    • 1
  1. 1.LuFG Informatik 2RWTH Aachen UniversityGermany
  2. 2.DSICUniversidad Politécnica de ValenciaSpain

Personalised recommendations