138 research outputs found
The cohesive principle and the Bolzano-Weierstra{\ss} principle
The aim of this paper is to determine the logical and computational strength
of instances of the Bolzano-Weierstra{\ss} principle (BW) and a weak variant of
it.
We show that BW is instance-wise equivalent to the weak K\"onig's lemma for
-trees (-WKL). This means that from every bounded
sequence of reals one can compute an infinite -0/1-tree, such that
each infinite branch of it yields an accumulation point and vice versa.
Especially, this shows that the degrees d >> 0' are exactly those containing an
accumulation point for all bounded computable sequences.
Let BW_weak be the principle stating that every bounded sequence of real
numbers contains a Cauchy subsequence (a sequence converging but not
necessarily fast). We show that BW_weak is instance-wise equivalent to the
(strong) cohesive principle (StCOH) and - using this - obtain a classification
of the computational and logical strength of BW_weak. Especially we show that
BW_weak does not solve the halting problem and does not lead to more than
primitive recursive growth. Therefore it is strictly weaker than BW. We also
discuss possible uses of BW_weak.Comment: corrected typos, slightly improved presentatio
Rates of asymptotic regularity for Halpern iterations of nonexpansive mappings
In this paper we obtain new effective results on the Halpern iterations of
nonexpansive mappings using methods from mathematical logic or, more
specifically, proof-theoretic techniques. We give effective rates of asymptotic
regularity for the Halpern iterations of nonexpansive self-mappings of nonempty
convex sets in normed spaces. The paper presents another case study in the
project of {\em proof mining}, which is concerned with the extraction of
effective uniform bounds from (prima-facie) ineffective proofs.Comment: in C.S. Calude, G. Stefanescu, and M. Zimand (eds.), Combinatorics
and Related Areas. A Collection of Papers in Honour of the 65th Birthday of
Ioan Tomesc
Effective results on compositions of nonexpansive mappings
This paper provides uniform bounds on the asymptotic regularity for
iterations associated to a finite family of nonexpansive mappings. We obtain
our quantitative results in the setting of -convex spaces, a class
of geodesic spaces which generalizes metric spaces with a convex geodesic
bicombing
The strength of countable saturation
We determine the proof-theoretic strength of the principle of countable
saturation in the context of the systems for nonstandard arithmetic introduced
in our earlier work.Comment: Corrected typos in Lemma 3.4 and the final paragraph of the
conclusio
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