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On the construction of effectively random sets

Published online by Cambridge University Press:  12 March 2014

Wolfgang Merkle  and
Nenad Mihailović
Wolfgang Merkle
Affiliation:
Institut für Informatik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, D–69120 Heidelberg, Germany, E-mail: merkle@math.uni-heidelberg.de
Nenad Mihailović
Affiliation:
Institut für Informatik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, D–69120 Heidelberg, Germany, E-mail: mihailovic@math.uni-heidelberg.de
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Abstract.

We present a comparatively simple way to construct Martin-Löf random and rec-random sets with certain additional properties, which works by diagonalizing against appropriate martingales. Reviewing the result of Gács and Kučera, for any given set X we construct a Martin-Löf random set from which X can be decoded effectively.

By a variant of the basic construction we obtain a rec-random set that is weak truth-table autoreducible and we observe that there are Martin-Löf random sets that are computably enumerable self-reducible. The two latter results complement the known facts that no rec-random set is truth-table autoreducible and that no Martin-Löf random set is Turing-autoreducible [8, 24].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 3 , September 2004 , pp. 862 - 878
Copyright
Copyright © Association for Symbolic Logic 2004

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