Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited

TL;DR

This paper revisits polynomial interpretations over various number domains, clarifying their relationships and limitations in proving termination of term rewrite systems, extending previous results to provide a complete understanding.

Contribution

It extends prior work by fully characterizing the relationships between polynomial interpretations over integers, rationals, and reals in termination proofs.

Findings

Polynomial interpretations with real or rational coefficients do not subsume integer coefficient interpretations.

The results hold for incremental termination proofs as well.

Complete relationships between the different polynomial interpretation variants are established.

Abstract

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect to termination proving power, Lucas managed to prove in 2006 that there are rewrite systems that can be shown polynomially terminating by polynomial interpretations with real (algebraic) coefficients, but cannot be shown polynomially terminating using polynomials with rational coefficients only. He also proved the corresponding statement regarding the use of rational coefficients versus integer coefficients. In this article we extend these results, thereby giving the full picture of the relationship between the aforementioned variants of polynomial interpretations. In particular, we show that polynomial interpretations with real or rational coefficients do not subsume polynomial interpretations with integer coefficients. Our results hold also for incremental termination proofs with polynomial interpretations.