Linear Termination is Undecidable

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Abstract
By means of a simple reduction from Hilbert's 10th problem we prove the somewhat surprising result that termination of one-rule rewrite systems by a linear interpretation in the natural numbers is undecidable. The very same reduction also shows the undecidability of termination of one-rule rewrite systems using the Knuth-Bendix order with subterm coefficients. The linear termination problem remains undecidable for one-rule rewrite systems that can be shown terminating by a (non-linear) polynomial interpretation. We further show the undecidability of the problem whether a one-rule rewrite system can be shown terminating by a polynomial interpretation with rational or real coefficients. Several of our results have been formally verified in the Isabelle/HOL proof assistant.

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