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On subcreative sets and S-reducibility

Published online by Cambridge University Press:  12 March 2014

John T. Gill III  and
Paul H. Morris
John T. Gill III
Affiliation:
Stanford University, Stanford, California 94305
Paul H. Morris
Affiliation:
University of California, Irvine, California 92664
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Abstract

Subcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets with respect to the complete sets of other reducibilities. It is shown that q-cylindrification is an order-preserving map from the r.e. T-degrees to the r.e. S-degrees. Consequently, T-complete sets are precisely the r.e. sets whose q-cylindrifications are S-complete.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 4 , December 1974 , pp. 669 - 677
Copyright
Copyright © Association for Symbolic Logic 1974

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References

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