Breaking the Mold: Nonlinear Ranking Function Synthesis Without Templates

TL;DR

This paper introduces a novel method for synthesizing nonlinear polynomial ranking functions for loops, overcoming undecidability issues by working within linear integer/real rings, and guarantees termination analysis success for a broad class of loops.

Contribution

It develops a complete synthesis approach for nonlinear polynomial ranking functions using linear integer/real rings, surpassing template-based methods and handling unbounded dimensions and degrees.

Findings

Guarantees success if a loop admits a polynomial ranking function.

Effectively subsumes linear lexicographic ranking function synthesis.

Works within linear integer/real rings to bypass undecidability.

Abstract

This paper studies the problem of synthesizing (lexicographic) polynomial ranking functions for loops that can be described in polynomial arithmetic over integers and reals. While the analogous ranking function synthesis problem for linear arithmetic is decidable, even checking whether a given function ranks an integer loop is undecidable in the nonlinear setting. We side-step the decidability barrier by working within the theory of linear integer/real rings (LIRR) rather than the standard model of arithmetic. We develop a termination analysis that is guaranteed to succeed if a loop (expressed as a formula) admits a (lexicographic) polynomial ranking function. In contrast to template-based ranking function synthesis in real arithmetic, our completeness result holds for lexicographic ranking functions of unbounded dimension and degree, and effectively subsumes linear lexicographic ranking function synthesis for linear integer loops.