非线性不等式的Taylor区间逼近形式化验证
Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations
摘要：
本文提出了一个多元非线性不等式的形式化验证工具。我们的验证方法是基于泰勒近似的区间算术。我们的工具在HOL Light证明助手中实现，它能够验证矩形域上的多元非线性多项式和非多项式不等式。我们的工作的主要特点之一是一个有效的实现的验证过程，可以证明非平凡的高维不等式在几秒钟内。我们开发了验证工具作为Flyspeck项目（开普勒猜想的正式证明）的一部分。Flyspeck项目包括大约1000个非线性不等式。我们成功地测试了我们的方法对100多个Flyspeck不等式，并估计，形式化验证过程是约3000倍，比非正式的验证方法在C++中实现慢。我们还描述了未来的工作和前瞻性的优化我们的方法。
英文摘要：
We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.
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