## Journal of Symbolic Logic

### An incomplete set of shortest descriptions

#### 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.

#### Article information

Source
J. Symbolic Logic Volume 77, Issue 1 (2012), 291-307.

Dates
First available in Project Euclid: 20 January 2012

http://projecteuclid.org/euclid.jsl/1327068704

Digital Object Identifier
doi:10.2178/jsl/1327068704

Mathematical Reviews number (MathSciNet)
MR2951642

Zentralblatt MATH identifier
06025971

#### Citation

Stephan, Frank; Teutsch, Jason. An incomplete set of shortest descriptions. J. Symbolic Logic 77 (2012), no. 1, 291--307. doi:10.2178/jsl/1327068704. http://projecteuclid.org/euclid.jsl/1327068704.

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