NDLI: On C-Degrees, H-Degrees and T-Degrees

Content Provider	IEEE Xplore Digital Library
Author	Merkle, W. Stephan, F.
Copyright Year	2007
Description	Author affiliation: Ruprecht-Karls-Univ. Heidelberg, Heidelberg (Merkle, W.)
Abstract	Following a line of research that aims at relating the computation power and the initial segment complexity of a set, the work presented here investigates into the relations between Turing reducibility, defined in terms of computation power, and C-reducibility and H-reducibility, defined in terms of the complexity of initial segments. The global structures of all C-degrees and of all H-degrees are rich and allows to embed the lattice of the powerset of the natural numbers under inclusion. In particular, there are C-degrees, as well as H-degrees, that are different from the least degree and are the meet of two other degrees, whereas on the other hand there are pairs of sets that have a meet neither in the C-degrees nor in the H-degrees; these results answer questions in a survey by Nies and Miller. There are r.e. sets that form a minimal pair for C-reducibility and $Sigma_{2}$ $^{0}$ sets that form a minimal pair for H-reducibility, which answers questions by Downey and Hirschfeldt. Furthermore, the following facts on the relation between C-degrees, H-degrees and Turing degrees hold. Every C-degree contains at most one Turing degree and this bound is sharp since there are C-degrees that do contain a Turing degree. For the comprising class of complex sets, neither the C-degree nor the H-degree of such a set can contain a Turing degree, in fact, the Turing degree of any complex set contains infinitely many C-degrees. Similarly the Turing degree of any set that computes the halting problem contains infinitely many H-degrees, while the H-degree of any 2-random set R is never contained in the Turing degree of R. By the latter, H-equivalence of Martin-Lof random sets does not imply their Turing equivalence. The structure of the Cdegrees contained in the Turing degree of a complex sets is rich and allows to embed any countable distributive lattice; a corresponding statement is true for the structure of H-degrees that are contained in the Turing degree of a set that computes the halting problem.
Starting Page	60
Ending Page	69
File Size	228279
Page Count	10
File Format	PDF
ISBN	0769527809
ISSN	10930159
DOI	10.1109/CCC.2007.17
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2007-06-13
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Lattices Mathematics Distributed computing Embedded computing Length measurement Turing machines Books Computational complexity
Content Type	Text
Resource Type	Article

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