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An incomplete set of shortest descriptions

Published online by Cambridge University Press:  12 March 2014

Frank Stephan  and
Jason Teutsch
Frank Stephan
Affiliation:
National University of Singapore, Department of Mathematics,10 Lower Kent Ridge Road, Singapore119076, Republic of Singapore, E-mail: fstephan@comp.nus.edu.sg
Jason Teutsch
Affiliation:
Ruprecht-Karls-Universität Heidelberg, Institut Für Informatik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany, E-mail: teutsch@math.uni-heidelberg.de
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Abstract

The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete degree by identifying an acceptable set of domain-random strings within each degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 291 - 307
Copyright
Copyright © Association for Symbolic Logic 2012

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