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INTRINSIC SMALLNESS

Part of: Computability and recursion theory

Published online by Cambridge University Press:  05 October 2020

JUSTIN MILLER
JUSTIN MILLER*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAMENOTRE DAME, IN, USAE-mail: jmille74@nd.edus
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Abstract

Recent work in computability theory has focused on various notions of asymptotic computability, which capture the idea of a set being “almost computable.” One potentially upsetting result is that all four notions of asymptotic computability admit “almost computable” sets in every Turing degree via coding tricks, contradicting the notion that “almost computable” sets should be computationally close to the computable sets. In response, Astor introduced the notion of intrinsic density: a set has defined intrinsic density if its image under any computable permutation has the same asymptotic density. Furthermore, introduced various notions of intrinsic computation in which the standard coding tricks cannot be used to embed intrinsically computable sets in every Turing degree. Our goal is to study the sets which are intrinsically small, i.e. those that have intrinsic density zero. We begin by studying which computable functions preserve intrinsic smallness. We also show that intrinsic smallness and hyperimmunity are computationally independent notions of smallness, i.e. any hyperimmune degree contains a Turing-equivalent hyperimmune set which is “as large as possible” and therefore not intrinsically small. Our discussion concludes by relativizing the notion of intrinsic smallness and discussing intrinsic computability as it relates to our study of intrinsic smallness.

Keywords

intrinsic computabilityintrinsic densityasymptotic computationhyperimmunityweakly computably traceable

MSC classification

Primary: 03D30: Other degrees and reducibilities
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 2 , June 2021 , pp. 558 - 576
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Copyright
© The Association for Symbolic Logic 2020

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References

Arslanov, M. M., Some generalizations of a fixed-point theorem . Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika , vol. 5 (1981), pp. 9–16. English translation in Soviet Mathematics (Iz. VUZ) , vol. 25 (1981), no. 5, pp. 1–10.Google Scholar
Astor, E. P., Asymptotic density, immunity, and randomness . Computability , vol. 4 (2015), no. 2, pp. 141–158.10.3233/COM-150040CrossRefGoogle Scholar
Astor, E. P., The computational content of intrinsic density, preprint 2017, arXiv:1708.04267.Google Scholar
Astor, E. P., Hirschfeldt, D. R., and Jockusch, C. G. Jr, Dense computability, upper cones, and minimal pairs . Computability , vol. 8 (2019), no. 2, pp. 155–177.10.3233/COM-180231CrossRefGoogle Scholar
Downey, R. G. and Hirschfeldt, D. R., Algorithmic randomness and complexity, Theory and Applications of Computability , Springer, New York, 2010.10.1007/978-0-387-68441-3CrossRefGoogle Scholar
Downey, R. G., Jockusch, C. G. Jr., and Schupp, P. E., Asymptotic density and computably enumerable sets . Journal of Mathematical Logic , vol. 13 (2013), no. 2, 1350005.10.1142/S0219061313500050CrossRefGoogle Scholar
Jockusch, C. G. Jr, Upward closure and cohesive degrees . Israel Journal of Mathematics , vol. 15 (1973), no. 3, pp. 332–335.10.1007/BF02787575CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Schupp, P. E., Generic computability, Turing degrees, and asymptotic density . Journal of the London Mathematical Society , vol. 85 (2012), no. 2, pp. 472–490.10.1112/jlms/jdr051CrossRefGoogle Scholar
Kapovich, I., Myasnikov, A., Schupp, P. E., 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.CrossRefGoogle Scholar
Kjos-Hanssen, B., Merkle, W., and Stephan, F., Kolmogorov complexity and the Recursion theorem . Transactions of the American Mathematical Society , vol. 363 (2011), no. 10, pp. 5465–5480.10.1090/S0002-9947-2011-05306-7CrossRefGoogle Scholar
Sacks, G. E., On the degrees less than 0 . Annals of Mathematics, vol. 77 (1963), no. 2, pp. 211–231.10.2307/1970214CrossRefGoogle Scholar
Soare, R. I., Turing computability: theory and applications, Theory and Applications of Computability, Springer, New York, 2016.Google Scholar