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DENSITY-1-BOUNDING AND QUASIMINIMALITY IN THE GENERIC DEGREES

Published online by Cambridge University Press:  08 May 2017

PETER CHOLAK  and
GREGORY IGUSA
PETER CHOLAK
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46556-5683, USAE-mail:[email protected]: http://www.nd.edu/∼cholak
GREGORY IGUSA
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46556-5683, USAE-mail:[email protected]: http://www3.nd.edu/∼gigusa
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Abstract

We consider the question “Is every nonzero generic degree a density-1-bounding generic degree?” By previous results [8] either resolution of this question would answer an open question concerning the structure of the generic degrees: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees.

We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide class of assumptions is sufficient to prove density-1-bounding.

We also consider a historic difficulty in constructing a potential counterexample: By previous results [7] any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal.

Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them.

Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp [4], this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees.

Keywords

Primary 03D30252832generic computabilitycoarse computabilitygeneric reducibilitycoarse reducibilitydensity 1 boundingquasiminimal
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 931 - 957
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Astor, E., Hirschfeldt, D., and Jockusch, C., Dense computability, upper cones, and minimal pairs (tentative title), in preparation.Google Scholar
Downey, R. G., Nies, A., Weber, R., and Yu, L., Lowness and ${\rm{\Pi }}_2^0$ nullsets, this Journal, vol. 71 (2006), pp. 1044–1052.Google Scholar
Dzhafarov, D. and Igusa, G., Notions of robust information coding , Computability, to appear.Google Scholar
Hirschfeldt, D., Jockusch, C., Kuyper, R., and Schupp, P., Coarse reducibility and algorithmic randomness , this Journal, to appear.Google Scholar
Hirschfeldt, D., Jockusch, C., McNicholl, T. H., and Schupp, P., Asymptotic density and the coarse computability bound . Computability, to appear.Google Scholar
Hirschfeldt, D. R., Nies, A., and Stephan, F., Using random sets as oracles . Journal of the London Mathematical Society, vol. 75 (2007), pp. 610–622.Google Scholar
Igusa, G., Nonexistence of minimal pairs for generic computability , this Journal, vol. 78 (2013), no. 2, pp. 511–522.Google Scholar
Igusa, G., The generic degrees of density-1 sets, and a characterization of the hyperarithmetic reals , this Journal, vol. 80 (2015), no. 4, pp. 1290–1314.Google Scholar
Jockusch, C. and Schupp, P., Generic computability, Turing degrees, and asymptotic density . Journal of the London Mathematical Society, vol. 85 (2012), no. 2, pp. 472–490.CrossRefGoogle Scholar
Kapovich, I., Miasnikov, A., Schupp, P., and Shpilrain, V., Generic-case complexity, decision problems in group theory, and random walks . Journal of Algebra, vol. 264 (2003), no. 2, pp. 665–694.Google Scholar
van Lambalgen, M., The axiomatization of randomness , this Journal, vol. 55 (1990), pp. 1143–1167.Google Scholar
Yu, L., Lowness for genericity . Archive for Mathematical Logic, vol. 45 (2006), pp. 233–238.CrossRefGoogle Scholar