Abstract
The notion of immunity is useful to classify degrees of noncomputability. Meanwhile, the notion of immunity for topological spaces can be thought of as an opposite notion of density. Based on this viewpoint, we introduce a new degree-theoretic invariant called layer density which assigns a value n to each subset of Cantor space. Armed with this invariant, we shed light on an interaction between a hierarchy of density/immunity and a mechanism of type-two computability.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Binns, S.: Hyperimmunity in 2ℕ. Notre Dame Journal of Formal Logic 48(2), 293–316 (2007)
Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic 17(1), 73–117 (2011)
Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theor. Comput. Sci. 305(1-3), 43–76 (2003)
Cenzer, D., Kihara, T., Weber, R., Wu, G.: Immunity and non-cupping for closed sets. Tbilisi Math. J. 2, 77–94 (2009)
Demuth, O., Kučera, A.: Remarks on 1-genericity, semigenericity and related concepts. Comment. Math. Univ. Carolinae 28, 85–94 (1987)
Higuchi, K., Kihara, T.: Inside the Muchnik degrees: Discontinuity, learnability, and constructivism (preprint)
Lewis, A.E.M., Shore, R.A., Sorbi, A.: Topological aspects of the Medvedev lattice. Arch. Math. Log. 50(3-4), 319–340 (2011)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, xVIII+437 pages. Springer, Heidelberg (1987)
Weihrauch, K.: Computable Analysis: An Introduction. Texts in Theoretical Computer Science, 285 pages. Springer (2000)
Ziegler, M.: Real computation with least discrete advice: A complexity theory of nonuniform computability with applications to effective linear algebra. Annals of Pure and Applied Logic 163(8), 1108–1139 (2012), http://www.sciencedirect.com/science/article/pii/S016800721100203X
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kihara, T. (2012). A Hierarchy of Immunity and Density for Sets of Reals. In: Cooper, S.B., Dawar, A., Löwe, B. (eds) How the World Computes. CiE 2012. Lecture Notes in Computer Science, vol 7318. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30870-3_39
Download citation
DOI: https://doi.org/10.1007/978-3-642-30870-3_39
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30869-7
Online ISBN: 978-3-642-30870-3
eBook Packages: Computer ScienceComputer Science (R0)