Linear Termination over N is Undecidable

TL;DR

This paper proves that determining termination of single-rule term rewrite systems using linear polynomial interpretations over natural numbers, rationals, or reals is undecidable, simplifying previous proofs and extending undecidability results.

Contribution

It establishes undecidability of polynomial termination with linear interpretations over various domains for single-rule systems, providing a simpler proof and extending known results.

Findings

Undecidability of polynomial termination over natural numbers with linear interpretations

Undecidability extends to rationals and reals

Simpler proof of undecidability for single-rule systems

Abstract

Recently it was shown that it is undecidable whether a term rewrite system can be proved terminating by a polynomial interpretation in the natural numbers. In this paper we show that this is also the case when restricting the interpretations to linear polynomials, as is often done in tools, and when only considering single-rule rewrite systems. What is more, the new undecidability proof is simpler than the previous one. We further show that polynomial termination over the rationals/reals is undecidable.