Algebraic and Algorithmic Methods for Computing Polynomial Loop Invariants

TL;DR

This paper introduces algebraic and algorithmic techniques to compute polynomial loop invariants for complex loops, extending beyond linear cases using algebraic geometry and efficient algorithms for specific invariant forms.

Contribution

It presents two algorithms for generating polynomial invariants in general polynomial loops and a more efficient method for specific invariant forms, advancing the analysis of complex program loops.

Findings

Algorithms successfully generate polynomial invariants for complex loops.

The new methods outperform existing approaches in efficiency.

Applicable to a broader class of loops with polynomial assignments.

Abstract

Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, generating invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and the assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case, where the polynomials can have arbitrary degrees. Using tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants within a given vector subspace, for a branching loop with nondeterministic conditional statements. These algorithms combine linear algebraic subroutines with computations on polynomial ideals. They differ depending on whether the initial values of the loop variables are specified or treated as parameters. Additionally, we present a much more efficient algorithm for generating polynomial invariants of a specific form, applicable to all initial values. This algorithm avoids expensive ideal computations.