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COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS

Published online by Cambridge University Press:  12 August 2016

DENIS R. HIRSCHFELDT,
CARL G. JOCKUSCH JR.,
RUTGER KUYPER and
PAUL E. SCHUPP
DENIS R. HIRSCHFELDT
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CHICAGO CHICAGO, IL, USA E-mail: drh@math.uchicago.edu
CARL G. JOCKUSCH JR.
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN CHAMPAIGN, IL, USA E-mail: jockusch@math.uiuc.edu
RUTGER KUYPER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON MADISON, WI, USA E-mail: mail@rutgerkuyper.com
PAUL E. SCHUPP
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN CHAMPAIGN, IL, USA E-mail: schupp@illinois.edu
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Abstract

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A coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if $A \le _{{\rm{T}}} \emptyset {\rm{'}}$ is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.

Keywords

Primary 03D30Secondary 03D2803D32Coarse reducibilityalgorithmic randomnessK-triviality

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016, pp. 1028 - 1046
Copyright
Copyright © The Association for Symbolic Logic 2016 

References

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