Algebraic Tools for Computing Polynomial Loop Invariants

Abstract

Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, the generation of invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and 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 exhibit arbitrary degrees. Applying tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants for a while loop, up to a specified degree. These algorithms differ based on whether the initial values of the loop variables are given or treated as parameters. Furthermore, we introduce various methods to address cases where the algebraic problem exceeds the computational capabilities of our methods. In such instances, we identify alternative approaches to generate specific polynomial invariants.