Verifying Nonlinear Real Formulas Via Sums of Squares

Abstract

Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable 'sum of squares' certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. Verifying nonlinear formulas over the realsOver the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary first-order formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root:The first quantifier elimination algorithm for this theory was developed by Tarski [32], 1 who actually demonstrated completeness and quantifier elimination just for the theory of real-closed fields, which can be characterized as ordered fields where all nonnegative elements have square roots (∀x. 0 ≤ x ⇒ ∃y. x = y 2 ) and all non-trivial polynomials of odd degree have a root. There are several interesting models of these axioms besides the reals (e.g. the algebraic reals, the computable reals, the hyperreals) yet Tarski's result shows that these different models satisfy exactly the same properties in the first-order language under consideration.However, Tarski's procedure is complicated and inefficient. Many alternative decision methods were subsequently proposed; two that are significantly simpler were given by Seidenberg [30] and Cohen [8], while the CAD algorithm [9], apparently the first ever to be implemented, is significantly more efficient, though relatively complicated. Cohen's ideas were recast by Hörmander [17] into a relatively simple algorithm. However, even CAD has poor worst-case complexity (doubly exponential), and the CohenHörmander algorithm is generally still slower. Thus, there has been limited progress on 1 Tarski actually discovered the procedure in 1930, but it remained unpublished for many years afterwards.

Certified, Efficient and Sharp Univariate Taylor Models in COQ

“…Nevertheless, our next priority is to complete our validation by providing an automatic tool within COQ to bound a polynomial on an interval. Different standard techniques exist that have already been applied in a formal setting, such as sum of squares [17] or Bernstein polynomials [18]. Combined with our Taylor models which prove |f (x) − TM f (x)| < 1 for x ∈ I, we could also derive that |TM f (x) − P (x)| < 2 .…”

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

“…Examples of numerical issues in SOS programming for Lyapunov function synthesis are noted in our previous work [50], and will not be reproduced here. Consequently, much work has focused on the problem of finding rational feasible points for sum-of-squares to generate polynomial positivity proofs that in exact arithmetic [24,38,45]. Recently, a self-validated SDP solver VSDP has been proposed by Lange et al [25].…”

Lyapunov Function Synthesis using Handelman Representations.

“…This is aided by the fact that SOS programming approach provides a positive definite polynomial as well as a numerical decomposition as a sum of squares ( Harrison [2007], Platzer et al [2009]). The difficulty of such an approach to Lyapunov function synthesis is two-fold.…”