From Matrix Interpretations over the Rationals to Matrix Interpretations over the Naturals

  • Salvador Lucas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6167)


Matrix interpretations generalize linear polynomial interpretations and have been proved useful in the implementation of tools for automatically proving termination of Term Rewriting Systems. In view of the successful use of rational coefficients in polynomial interpretations, we have recently generalized traditional matrix interpretations (using natural numbers in the matrix entries) to incorporate real numbers. However, existing results which formally prove that polynomials over the reals are more powerful than polynomials over the naturals for proving termination of rewrite systems failed to be extended to matrix interpretations. In this paper we get deeper into this problem. We show that, under some conditions, it is possible to transform a matrix interpretation over the rationals satisfying a set of symbolic constraints into a matrix interpretation over the naturals (using bigger matrices) which still satisfies the constraints.


Matrix and Polynomial Interpretations Program Analysis Termination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alarcón, B., Lucas, S., Navarro-Marset, R.: Proving Termination with Matrix Interpretations over the Reals. In: Proc. of WST 2009, pp. 12–15 (2009)Google Scholar
  2. 2.
    Alarcón, B., Lucas, S., Navarro-Marset, R.: Using Matrix Interpretations over the Reals in Proofs of Termination. In: Proc. of PROLE 2009, pp. 255–264 (2009)Google Scholar
  3. 3.
    Arts, T., Giesl, J.: Termination of Term Rewriting Using Dependency Pairs. Theoretical Computer Science 236, 133–178 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  5. 5.
    Borralleras, C., Lucas, S., Navarro-Marset, R., Rodríguez-Carbonell, E., Rubio, A.: Solving Non-linear Polynomial Arithmetic via SAT Modulo Linear Arithmetic. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 294–305. Springer, Heidelberg (2009)Google Scholar
  6. 6.
    Borralleras, C., Rubio, A.: Orderings and Constraints: Theory and Practice of Proving Termination. In: Comon-Lundh, H., Kirchner, C., Kirchner, H. (eds.) Jouannaud Festschrift. LNCS, vol. 4600, pp. 28–43. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Contejean, E., Marché, C., Tomás, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. Journal of Automated Reasoning 34(4), 325–363 (2006)CrossRefGoogle Scholar
  8. 8.
    Dershowitz, N.: Termination of rewriting. Journal of Symbolic Computation 3, 69–115 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Endrullis, J.: Jambox, Automated Termination Proofs For String and Term Rewriting,
  10. 10.
    Endrullis, J., Waldmann, J., Zantema, H.: Matrix Interpretations for Proving Termination of Term Rewriting. Journal of Automated Reasoning 40(2-3), 195–220 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fuhs, C., Navarro-Marset, R., Otto, C., Giesl, J., Lucas, S., Schneider-Kamp, P.: Search Techniques for Rational Polynomial Orders. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) AISC 2008, Calculemus 2008, and MKM 2008. LNCS (LNAI), vol. 5144, pp. 109–124. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. With Applications. Academic Press, London (1985)zbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    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
  15. 15.
    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
  16. 16.
    Lucas, S.: Practical use of polynomials over the reals in proofs of termination. In: Proc. of PPDP 2007, pp. 39–50. ACM Press, New York (2007)Google Scholar
  17. 17.
    Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. In: Society for Industrial and Applied Mathematics. SIAM, Philadelphia (2000)Google Scholar
  18. 18.
    Zhang, F.: Matrix Theory. Springer, Berlin (1999)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Salvador Lucas
    • 1
  1. 1.ELP Group, DSICUniversidad Politécnica de ValenciaValenciaSpain

Personalised recommendations