From matrix interpretations over the rationals to matrix interpretations over the naturals

TL;DR

This paper investigates the relationship between matrix interpretations over the rationals and naturals, showing how rational matrix interpretations can be transformed into natural ones under certain conditions, impacting termination proofs in rewriting systems.

Contribution

It demonstrates that rational matrix interpretations satisfying certain constraints can be converted into natural matrix interpretations with larger matrices, extending the applicability of termination proof techniques.

Findings

Rational matrix interpretations can be transformed into natural ones with larger matrices.

Under specific conditions, the transformation preserves the constraints.

This extends the theoretical foundation for termination proofs in rewriting systems.

Abstract

Matrix interpretations generalize linear polynomial interpretations and have been proved useful in the implementation of tools for automatically proving termination of Term Rewriting Systems. In view of the successful use of rational coefficients in polynomial interpretations, we have recently generalized traditional matrix interpretations (using natural numbers in the matrix entries) to incorporate real numbers. However, existing results which formally prove that polynomials over the reals are more powerful than polynomials over the naturals for proving termination of rewrite systems failed to be extended to matrix interpretations. In this paper we get deeper into this problem. We show that, under some conditions, it is possible to transform a matrix interpretation over the rationals satisfying a set of symbolic constraints into a matrix interpretation over the naturals (using bigger matrices) which still satisfies the constraints.