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Journal of Logic and Computation Advance Access originally published online on August 14, 2008
Journal of Logic and Computation 2009 19(1):3-16; doi:10.1093/logcom/exn021
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

This article appears in the following Journal of Logic and Computation issue: Special Issue: Logic and Computation in the Real World: CiE 2007 [View the issue table of contents]

Original Articles

K-Triviality of Closed Sets and Continuous Functions

George Barmpalias

School of Mathematics, University of Leeds, Leeds LS2 9JT, England
E-mail: barmpalias{at}gmail.com

Douglas Cenzer

Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611
E-mail: cenzer{at}math.ufl.edu

Jeffrey B. Remmel

Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA
E-mail: jremmel{at}ucsd.edu

Rebecca Weber

Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551
E-mail: Rebecca.A.Weber{at}Dartmouth.edu

Received 10 October 2007.


  Abstract

We investigate the notion of K-triviality for closed sets and continuous functions in 2N. For every K-trivial degree d, there exists a closed set of degree d and a continuous function of degree d. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial Formula class with no computable elements. A closed set is K-trivial if and only if it is the set of zeroes of some K-trivial continuous function. We give a density result for the Medvedev degrees of K-trivial Formula sets. If W ≤TA', then W can compute a path through every A'-decidable random closed set if and only if W {equiv}TA'.

Keywords: Computability; randomness; {Pi}Formula classes


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