Monotonicity Criteria for Polynomial Interpretations over the Naturals

Friedrich Neurauter, Harald Zankl, and Aart Middeldorp

Proceedings of the 5th International Joint Conference on Automated Reasoning (IJCAR 2010), Lecture Notes in Artificial Intelligence 6173, pp. 502 – 517, 2010.


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.


  PDF |    doi:10.1007/978-3-642-14203-1_42  |  © Springer


author = "Friedrich Neurauter and Harald Zankl and Aart Middeldorp",
title = "Monotonicity Criteria for Polynomial Interpretations over the Naturals",
booktitle = "Proceedings of the 5th International Joint Conference on Automated Reasoning",
series = "Lecture Notes in Artificial Intelligence",
volume = 6173,
pages = "502--517",
publisher = "Springer-Verlag",
year = 2010
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