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- Title
FINITE CHOICE, CONVEX CHOICE AND FINDING ROOTS.
- Authors
LE ROUX, STÉPHANE; PAULY, ARNO
- Abstract
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.
- Subjects
CONVEX sets; LATTICE theory; DIMENSIONS; COMPUTER algorithms; MATHEMATICAL proofs; REAL numbers
- Publication
Logical Methods in Computer Science (LMCS), 2015, Vol 11, Issue 4, p1
- ISSN
1860-5974
- Publication type
Article
- DOI
10.2168/LMCS-11(4:6)2015