Generating Loop Invariants by Computing Vanishing Ideals of Sample Points

TL;DR

This paper introduces a novel method for generating polynomial loop invariants by computing vanishing ideals of sample points, avoiding complex quantifier elimination techniques, and capable of handling high-degree invariants.

Contribution

The paper presents a new approach using vanishing ideals and rational function interpolation to generate polynomial invariants efficiently for loop programs with symbolic initial values.

Findings

Successfully generates invariants with degrees up to 15.

Avoids first-order quantifier elimination and CAD.

Demonstrates effectiveness on various loop programs.

Abstract

Loop invariants play a very important role in proving correctness of programs. In this paper, we address the problem of generating invariants of polynomial loop programs. We present a new approach, for generating polynomial equation invariants of polynomial loop programs through computing vanishing ideals of sample points. We apply rational function interpolation, based on early termination technique, to generate invariants of loop programs with symbolic initial values. Our approach avoids first-order quantifier elimination and cylindrical algebraic decomposition(CAD). An algorithm for generating polynomial invariants is proposed and some examples are given to illustrate the algorithm. Furthermore, we demonstrate on a set of loop programs with symbolic initial values that our algorithm can yield polynomial invariants with degrees high up to 15.