Beyond Absolute Positiveness for Universally Quantified Non-Linear Polynomial Constraints

2026-06-29

AI Summary

This work proposes a novel method to overcome the limitations of the classical absolute positivity criterion, which fails to handle nonlinear polynomial constraints involving universal quantifiers. Specifically, the approach addresses ∃∀ inequalities over the natural numbers by integrating monotonic algebra with well-founded order theory, thereby dispensing with the absolute positivity assumption. This advancement substantially broadens the class of constructible nonlinear polynomial interpretations. Experimental results demonstrate that the technique successfully solves constraint instances previously intractable to existing methods, thus extending the applicability of polynomial interpretations in termination and complexity analysis of term rewriting systems.

Abstract

Polynomial interpretations from function symbols to natural numbers induce a prominent class of monotone algebras and corresponding well-founded orders on terms, used, e.g., for termination analysis and complexity analysis of term rewrite systems. Finding such polynomial interpretations for a given set of term constraints involves solving a set of $\exists\forall$ inequalities over the natural numbers. Conventionally, the absolute positiveness criterion is used to reduce $\exists\forall$ inequalities to $\exists$ inequalities. This extended abstract reports on work in progress to go beyond absolute positiveness, allowing for finding non-linear polynomial interpretations that were outside the reach of existing techniques.

Problem

• polynomial interpretations
• non-linear constraints
• universally quantified inequalities
• absolute positiveness
• term rewrite systems

Innovation

• polynomial interpretations
• non-linear constraints
• absolute positiveness
• termination analysis
• exists-forall inequalities