arXiv 1402.08582014-02-04cs.CC

方程组的鲁棒可满足性

Robust Satisfiability of Systems of Equations

Peter Franek, Marek Krcal

摘要：

本文研究非线性方程组的\emph{robust satisfiability}问题，即对于给定的有限单纯复形$K$和$α>0$上的连续函数$f:\,K\to\mathbb{R}^n$，是否每个函数$g:\,K\to\mathbb{R}^n$使得$\|g-f\|_\infty \leq α$在$K$中有根.通过将映射的扩张问题归结为球面扩张问题，我们证明了对于每一个固定的$n$，在假设$\dim K \le 2n-3$的条件下，这个问题在多项式时间内是可判定的.这是对\emph{topological degree}先前计算应用以及数值和区间分析中相关概念的实质性扩展。通过逆约化，我们证明了当$\dim K\ge 2n-2$，其中的阈值来自同伦理论中的\emph{stable range}时，该问题是不可判定的。为了使我们的论述清晰，我们将重点放在$f$为分段线性时的设置上。这类函数可以逼近一般的连续函数，从而得到了在其他可能情况下的逼近方案和鲁棒可满足性的不可判定性。

英文摘要：

We study the problem of \emph{robust satisfiability} of systems of nonlinear equations, namely, whether for a given continuous function $f:\,K\to\mathbb{R}^n$ on a~finite simplicial complex $K$ and $α>0$, it holds that each function $g:\,K\to\mathbb{R}^n$ such that $\|g-f\|_\infty \leq α$, has a root in $K$. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed $n$, assuming $\dim K \le 2n-3$. This is a substantial extension of previous computational applications of \emph{topological degree} and related concepts in numerical and interval analysis. Via a reverse reduction we prove that the problem is undecidable when $\dim K\ge 2n-2$, where the threshold comes from the \emph{stable range} in homotopy theory. For the lucidity of our exposition, we focus on the setting when $f$ is piecewise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.

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