A Hierarchy of Immunity and Density for Sets of Reals
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.
KeywordsOpen Ball Computable Function Computable Point Winning Strategy Learnable Function
Unable to display preview. Download preview PDF.
- 6.Higuchi, K., Kihara, T.: Inside the Muchnik degrees: Discontinuity, learnability, and constructivism (preprint)Google Scholar
- 8.Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, xVIII+437 pages. Springer, Heidelberg (1987)Google Scholar
- 9.Weihrauch, K.: Computable Analysis: An Introduction. Texts in Theoretical Computer Science, 285 pages. Springer (2000)Google Scholar
- 10.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