Generating all polynomial invariants in simple loops

Abstract

This paper presents a method for automatically generating all polynomial invariants in simple loops. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Based on this connection, a fixpoint procedure using operations on ideals and Gröbner basis constructions is proposed for finding all polynomial invariants. Most importantly, it is proved that the procedure terminates in at most m + 1 iterations, where m is the number of program variables. The proof relies on showing that the irreducible components of the varieties associated with the ideals generated by the procedure either remain the same or increase their dimension at every iteration of the fixpoint procedure. This yields a correct and complete algorithm for inferring conjunctions of polynomial equalities as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover non-trivial invariants for several examples to illustrate the power of the technique.1 If the field K is algebraically closed then IV(I) = Rad(I), the radical of I, which is the set of polynomials p such that there exists k ∈ N satisfying p k ∈ I.

Verification as Learning Geometric Concepts

“…Existing sound tools for non-linear invariant generation can produce invariants that are conjunctions of polynomial equalities [51,38,50,14,46,53,17]. However, by imposing strict restrictions on syntax (such as no nested loops) [51,38] do not need to assume the degree of polynomials as the input.…”

“…However, by imposing strict restrictions on syntax (such as no nested loops) [51,38] do not need to assume the degree of polynomials as the input. Bagnara et al [4] introduce new variables for monomials and generate linear invariants over them by abstract interpretation over convex polyhedra.…”

Using dynamic analysis to discover polynomial and array invariants

“…The details of NLA and AES are given in Tables II and III, respectively. The NLA test suite consists of 24 programs from various sources collected by Rodríguez-Carbonell and Kapur [23], [24]. These programs implement classic arithmetic algorithms that are widely used in programming, such as mult, div, pow, mod, sqrt, gcd, lcm.…”

Invariant and Type Inference for Matrices

“…Since one ingredient of our method is numeric invariant generation, we also compare it with other polynomial invariant generation techniques [22,25,24]. Papers [22,25] compute polynomial equalities of fixed degree as invariants using the polynomial ideal theory.…”

“…Our algorithm thus returns a finite representation of the polynomial invariant ideal, whereas [22,25] may only iteratively increase the polynomial degree to infer such a basis. Paper [24] derives polynomial invariants for loops with positive rational eigenvalues, by iteratively approximating the polynomial invariant ideal using Gröbner basis computation [3]. In contrast to [24], our approach generates polynomial invariants over scalars for polynomial loops with algebraic, and not just rational, eigenvalues.…”