Affiliation:
1. Institute of Computer Science, ASCR Prague
2. IST Austria,Klosterneuburg
Abstract
We study the problem of
robust satisfiability
of systems of nonlinear equations, namely, whether for a given continuous function
f
:
K
→ Rn on a finite simplicial complex
K
and α>0, it holds that each function
g
:
K
→ Rn such that ║
g−f
║∞ ≤
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
≤ 2
n
−3. This is a substantial extension of previous computational applications of
topological degree
and related concepts in numerical and interval analysis.
Via a reverse reduction, we prove that the problem is undecidable when dim
K
≥ 2
n
−2, where the threshold comes from the
stable range
in homotopy theory.
For the lucidity of our exposition, we focus on the setting when
f
is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.
Funder
GA ČR
Center of Excellence -- Institute for Theoretical Computer Science, Prague
ERCCZ CORES
institutional support
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Cited by
6 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献