Monotonicity Criteria for Polynomial Interpretations over the Naturals

  • Friedrich Neurauter
  • Aart Middeldorp
  • Harald Zankl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6173)

Abstract

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. In an automated setting, termination tools are concerned with parametric polynomials whose coefficients (i.e., the parameters) are initially unknown and have to be instantiated suitably such that the resulting concrete polynomials satisfy certain conditions. We focus on monotonicity and well-definedness, the two main conditions that are independent of the respective term rewrite system considered, and provide constraints on the abstract coefficients for linear, quadratic and cubic parametric polynomials such that monotonicity and well-definedness of the resulting concrete polynomials are guaranteed whenever the constraints are satisfied. Our approach subsumes the absolute positiveness approach, which is currently used in many termination tools. In particular, it allows for negative numbers in certain coefficients. We also give an example of a term rewrite system whose termination proof relies on the use of negative coefficients, thus showing that our approach is more powerful.

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. TCS 236(1-2), 133–178 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Contejean, E., Marché, C., Tomás, A.P., Urbain, X.: Mechanically proving termination using polynomial interpretations. JAR 34(4), 325–363 (2005)MATHCrossRefGoogle Scholar
  3. 3.
    Dershowitz, N.: A note on simplification orderings. IPL 9(5), 212–215 (1979)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    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
  5. 5.
    Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: Maximal termination. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 110–125. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Hirokawa, N., Middeldorp, A.: Automating the dependency pair method. I&C 199(1-2), 172–199 (2005)MATHMathSciNetGoogle Scholar
  8. 8.
    Hirokawa, N., Middeldorp, A.: Tyrolean Termination Tool: Techniques and features. I&C 205(4), 474–511 (2007)MATHMathSciNetGoogle Scholar
  9. 9.
    Hong, H., Jakuš, D.: Testing positiveness of polynomials. JAR 21(1), 23–38 (1998)CrossRefGoogle Scholar
  10. 10.
    Lankford, D.: On proving term rewrite systems are noetherian. Tech. Rep. MTP-3, Louisiana Technical University, Ruston (1979)Google Scholar
  11. 11.
    Lucas, S.: Polynomials over the reals in proofs of termination: From theory to practice. TIA 39(3), 547–586 (2005)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Friedrich Neurauter
    • 1
  • Aart Middeldorp
    • 1
  • Harald Zankl
    • 1
  1. 1.Institute of Computer ScienceUniversity of InnsbruckAustria

Personalised recommendations