On the Generation of Positivstellensatz Witnesses in Degenerate Cases

Monniaux, David; Corbineau, Pierre

doi:10.1007/978-3-642-22863-6_19

Cited by 23 publications

(36 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Among the eighteen symmetric moments considered, eight of them (z, x 2 , xy, z 2 , x 2 z, xyz, z 3 , xyz 2 ) appear to be maximized at the nonzero equilibria x ± , meaning that max x(t) x l y m z n = 1. These conjectures can be confirmed by sharp upper bounds on z, z 2 , z 3 , and x 2 z since the other four moments are proportional to these according to (21). Sharp bounds indeed have been proved for z by Malkus [20], for z 2 by Knobloch [15], and for z 3 in §5.4.…”

Section: Conjectured Extremal Trajectoriesmentioning

confidence: 73%

See 1 more Smart Citation

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

Goluskin

2017

J Nonlinear Sci

View full text Add to dashboard Cite

We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions. Nonnegativity is enforced by requiring the polynomials to be sums of squares, a condition which is then formulated as a semidefinite program (SDP) that can be solved computationally. Although such computations are subject to numerical error, we demonstrate two ways to obtain rigorous results: using interval arithmetic to control the error of an approximate SDP solution, and finding exact analytical solutions to relatively small SDPs. Previous formulations are extended to allow for bounds depending analytically on parametric variables. These methods are illustrated using the Lorenz equations, a system with three state variables (x, y, z) and three parameters (β, σ, r). Bounds are reported for infinite-time averages of all eighteen moments x l y m z n up to quartic degree that are symmetric under (x, y) → (−x, −y). These bounds apply to all solutions regardless of stability, including chaotic trajectories, periodic orbits, and equilibrium points. The analytical approach yields two novel bounds that are sharp: the mean of z 3 can be no larger than its value of (r − 1) 3 at the nonzero equilibria, and the mean of xy 3 must be nonnegative. The interval arithmetic approach is applied at the standard chaotic parameters to bound eleven average moments that all appear to be maximized on the shortest periodic orbit. Our best upper bound on each such average exceeds its value on the maximizing orbit by less than 1%. Many bounds reported here are much tighter than would be possible without computer assistance. *

show abstract

Section: Conjectured Extremal Trajectoriesmentioning

confidence: 73%

“…Moments grouped together (e.g. z, x 2 , xy) have identical normalized averages according to (21). Chaotic averages are obtained by numerical integration (cf.…”

Section: Discussionmentioning

confidence: 99%

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

Goluskin

2017

J Nonlinear Sci

View full text Add to dashboard Cite

show abstract

“…Because of numerical approximation errors, it is difficult to integrate this method into theorem provers. Recent developments in SOS address this issue by producing rational polynomial decompositions [23,30]. Proof producing strategies for proving real-number properties based on interval arithmetic and branch and bound methods are available in PVS [7], Coq [29], HOL Light [39].…”

Section: Related Workmentioning

confidence: 99%

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

2015

View full text Add to dashboard Cite

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.

show abstract

“…There exist improvements to this approach but they have not yet made their way into the procedures above (Monniaux and Corbineau 2011). PVS provides some strategies based on numerical computations: numerical performs interval arithmetic to verify inequalities involving transcendental functions (Daumas et al 2009); bernstein performs global optimization based on Bernstein polynomials to verify systems of polynomial inequalities (Muñoz and Narkawicz 2013).…”

Section: Linear Inequalitiesmentioning

confidence: 99%

Formalization of real analysis: a survey of proof assistants and libraries

Boldo¹,

Lelay²,

Melquiond³

2015

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

In the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis.

show abstract

On the Generation of Positivstellensatz Witnesses in Degenerate Cases

Cited by 23 publications

References 30 publications

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

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

Formalization of real analysis: a survey of proof assistants and libraries

Contact Info

Product

Resources

About