An incomplete set of shortest descriptions
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.
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1327068704
Zentralblatt MATH identifier: 06025971
Mathematical Reviews number (MathSciNet): MR2951642
Digital Object Identifier: doi:10.2178/jsl/1327068704