Generalized Homogeneous Polynomials for Efficient Template-Based Nonlinear Invariant Synthesis

TL;DR

This paper introduces a novel approach to algebraic invariant synthesis by focusing on generalized homogeneous invariants, reducing template size and improving efficiency in synthesizing invariants for programs.

Contribution

It formally defines generalized homogeneous algebraic invariants and modifies existing synthesis methods to generate only these invariants, resulting in more efficient invariant synthesis.

Findings

Outperforms previous methods on high-degree template programs

Reduces template size for algebraic invariants

Proves the soundness of the modified synthesis approach

Abstract

The template-based method is one of the most successful approaches to algebraic invariant synthesis. In this method, an algorithm designates a template polynomial p over program variables, generates constraints for p=0 to be an invariant, and solves the generated constraints. However, this approach often suffers from an increasing template size if the degree of a template polynomial is too high. We propose a technique to make template-based methods more efficient. Our technique is based on the following finding: If an algebraic invariant exists, then there is a specific algebraic invariant that we call a generalized homogeneous algebraic invariant that is often smaller. This finding justifies using only a smaller template that corresponds to a generalized homogeneous algebraic invariant. Concretely, we state our finding above formally based on the abstract semantics of an imperative program proposed by Cachera et al. Then, we modify their template-based invariant synthesis so that it generates only generalized homogeneous algebraic invariants. This modification is proved to be sound. Furthermore, we also empirically demonstrate the merit of the restriction to generalized homogeneous algebraic invariants. Our implementation outperforms that of Cachera et al. for programs that require a higher-degree template.