Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

Abstract

Sturm's theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semi-open interval, not counting multiplicity. A generalization of Sturm's theorem is known as Tarski's theorem, which provides a linear relationship between functions known as Tarski queries and cardinalities of certain sets. The linear system that results from this relationship is in fact invertible and can be used to explicitly count the number of roots of a univariate polynomial on a set defined by a system of polynomial relations. This paper presents a formalization of these results in the PVS theorem prover, including formal proofs of Sturm's and Tarski's theorems. These theorems are at the basis of two decision procedures, which are implemented as computable functions in PVS. The first, based on Sturm's theorem, determines satisfiability of a single polynomial relation over an interval. The second, based on Tarski's theorem, determines the satisfiability of a system of polynomial relations over the real line. The soundness and completeness properties of these decision procedures are formally verified in PVS. The procedures and their correctness properties enable the implementation of PVS strategies for automatically proving existential and universal statements on polynomial systems. Since the decision procedures are formally verified in PVS, the soundness of the strategies depends solely on the internal logic of PVS rather than on an external oracle.

A Decision Procedure for Univariate Real Polynomials in Isabelle/HOL

“…Lastly, the previously-mentioned work of Anthony Narkawicz and César Muñoz [11] is the most similar to ours. Their work was carried out in parallel to and independently of ours.…”

“…It must be noted that a very similar approach, albeit in the PVS system, was developed in parallel and independently by Anthony Narkawicz and César Muñoz. [11] Such a decision procedure has many applications in the context of theorem proving, since non-linear real inequalities are often difficult to prove by hand.…”

Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOL

“…As both HOL Light and Isabelle/HOL have a relatively comprehensive library of complex analysis (i.e., at least including Cauchy's integral theorem), our evaluation tactic could be useful when deriving analytical proofs in these two proof assistants. The ability to count the real roots of a polynomial only requires Sturm's theorem, so this capability is widely available among major proof assistants including PVS [18], Coq [15], HOL Light [17] and Isabelle [5,10,13]. However, as far as we know, our procedures to count complex roots are novel, as they require a formalisation of the argument principle [14], which is only available in Isabelle at the time of writing.…”

Counting polynomial roots in Isabelle/HOL: a formal proof of the Budan-Fourier theorem

“…Counting distinct real roots with Sturm's theorem has been widely implemented among major proof assistants including PVS [25], Coq [22], HOL Light [24] and Isabelle [11,15,18]. In contrast, our previous complex-root-counting procedure [21] seems to be the only one that counts complex roots, since counting complex roots usually requires a formal proof of the argument principle in complex analysis, which (to the best of our knowledge) is only available in Isabelle/HOL [20].…”