Beyond Absolute Positiveness for Universally Quantified Non-Linear Polynomial Constraints

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.