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The Kolmogorov-Loveland stochastic sequences are not closed under selecting subsequences

Published online by Cambridge University Press:  12 March 2014

Wolfgang Merkle
Affiliation:
Ruprecht-Karls-Universität Heidelberg, Institut Für Informatik, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany, E-mail: merkle@math.uni-heidelberg.de
Corresponding

Abstract

It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the question whether the Kolmogorov-Loveland stochastic sequences are closed under selecting sequences by Kolmogorov-Loveland selection rules, i.e., by not necessarily monotonic, partial computable selection rules. The following previously known results are obtained as corollaries. The Mises-Wald-Church stochastic sequences are not closed under computable permutations, hence in particular they form a strict superclass of the class of Kolmogorov-Loveland stochastic sequences. The Kolmogorov-Loveland selection rules are not closed under composition.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2003

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