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.
This research is supported by FWF (Austrian Science Fund) project P18763.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. TCS 236(1-2), 133–178 (2000)
Contejean, E., Marché, C., Tomás, A.P., Urbain, X.: Mechanically proving termination using polynomial interpretations. JAR 34(4), 325–363 (2005)
Dershowitz, N.: A note on simplification orderings. IPL 9(5), 212–215 (1979)
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)
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)
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)
Hirokawa, N., Middeldorp, A.: Automating the dependency pair method. I&C 199(1-2), 172–199 (2005)
Hirokawa, N., Middeldorp, A.: Tyrolean Termination Tool: Techniques and features. I&C 205(4), 474–511 (2007)
Hong, H., Jakuš, D.: Testing positiveness of polynomials. JAR 21(1), 23–38 (1998)
Lankford, D.: On proving term rewrite systems are noetherian. Tech. Rep. MTP-3, Louisiana Technical University, Ruston (1979)
Lucas, S.: Polynomials over the reals in proofs of termination: From theory to practice. TIA 39(3), 547–586 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Neurauter, F., Middeldorp, A., Zankl, H. (2010). Monotonicity Criteria for Polynomial Interpretations over the Naturals . In: Giesl, J., Hähnle, R. (eds) Automated Reasoning. IJCAR 2010. Lecture Notes in Computer Science(), vol 6173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14203-1_42
Download citation
DOI: https://doi.org/10.1007/978-3-642-14203-1_42
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14202-4
Online ISBN: 978-3-642-14203-1
eBook Packages: Computer ScienceComputer Science (R0)