episciences.org_1607_1642970311
1642970311
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
Logical Methods in Computer Science
1860-5974
12
02
2015
Volume 11, Issue 4
Finite choice, convex choice and finding roots
Stéphane Le
Roux
Arno
Pauly
We investigate choice principles in the Weihrauch lattice for finite sets on
the one hand, and convex sets on the other hand. Increasing cardinality and
increasing dimension both correspond to increasing Weihrauch degrees. Moreover,
we demonstrate that the dimension of convex sets can be characterized by the
cardinality of finite sets encodable into them. Precisely, choice from an n+1
point set is reducible to choice from a convex set of dimension n, but not
reducible to choice from a convex set of dimension n-1. Furthermore we consider
searching for zeros of continuous functions. We provide an algorithm producing
3n real numbers containing all zeros of a continuous function with up to n
local minima. This demonstrates that having finitely many zeros is a strictly
weaker condition than having finitely many local extrema. We can prove 3n to be
optimal.
12
02
2015
1607
arXiv:1302.0380
10.2168/LMCS-11(4:6)2015
https://lmcs.episciences.org/1607