Abstract
A fundamental goal of computability theory is to understand the way that objects relate to each other in terms of their information content. We wish to understand the relative information content between sets of natural numbers, how one subset of the natural numbers Y can be used to specify another one X. This specification can be computational, or arithmetic, or even by the application of a countable sequence of Borel operations. Each notion in the spectrum gives rise to a different model of relative computability. Which of these models best reflects the real world computation is under question.
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.
References
Ahmad, S.: Embedding the diamond in the Σ2 enumeration degrees. J. Symbolic Logic 56, 195–212 (1991)
Arslanov, M.M., Cooper, S.B., Kalimullin, I.S.: Splitting properties of total enumeration degrees. Algebra and Logic 42, 1–13 (2003)
Coles, R., Downey, R., Slaman, T.: Every set has a least jump enumeration. Bulletin London Math. Soc. 62, 641–649 (2000)
Cooper, S.B.: Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense. J. Symbolic Logic 49, 503–513 (1984)
Cooper, S.B., Soskova, M.I.: How enumeration reducibility yields extended Harrington non-splitting. J. Symbolic Logic 73, 634–655 (2008)
Friedberg, R.M., Rogers Jr., H.: Reducibility and completeness for sets of integers. Z. Math. Logik Grundlag. Math. 5, 117–125 (1959)
Ganchev, H.A., Soskova, M.I.: Definability via \(\mathcal{K}\)-pairs (submitted)
Ganchev, H.A., Soskova, M.I.: Cupping and definability in the local structure of the enumeration degrees. J. Symbolic Logic 77(1), 133–158 (2012)
Ganchev, H.A., Soskova, M.I.: Embedding distributive lattices in the \(\Sigma^0_2\) enumeration degrees. J. Logic Comput. 22, 779–792 (2012)
Ganchev, H.A., Soskova, M.I.: Interpreting true arithmetic in the local structure of the enumeration degrees. To Appear in J. Symbolic Logic (2012)
Giorgi, M., Sorbi, A., Yang, Y.: Properly \(\Sigma^0_2\) enumeration degrees and the high/low hierarchy. J. Symbolic Logic 71, 1125–1144 (2006)
Goncharov, S., Harizanov, V., Knight, J., McCoy, C., Miller, R., Solomon, R.: Enumerations in computable structure theory. Ann. Pure Appl. Logic 136, 219–236 (2005)
Harrington, L.: Understanding Lachlan’s monster paper, Handwritten notes (1980)
Harrington, L., Shelah, S.: The undecidability of the recursively enumerable degrees. Bull. Symb. Logic 6(1), 79–80 (1982)
Jockusch, C.G.: Semirecursive sets and positive reducibility. Trans. Amer. Math. Soc. 131, 420–436 (1968)
Kalimullin, I.S.: Definability of the jump operator in the enumeration degrees. Journal of Mathematical Logic 3, 257–267 (2003)
Lachlan, A.H.: A recursively enumerable degree which will not split over all lesser ones. Ann. Math. Logic 9, 307–365 (1975)
Nies, A., Shore, R.A., Slaman, T.A.: Interpretability and definability in the recursively enumerable degrees. Proc. London Math. Soc. 77, 241–249 (1998)
Richter, L.J.: Degree structures: Local and global investigations. J. Symbolic Logic 46, 723–731 (1981)
Rogers Jr., H.: Theory of recursive functions and effective computability. McGraw-Hill Book Company, New York (1967)
Selman, A.L.: Arithmetical reducibilities I. Z. Math. Logik Grundlag. Math. 17, 335–350 (1971)
Shore, R.A.: Biinterpretability up to double jump in the degrees below 0 (to appear)
Shore, R.A., Slaman, T.A.: Defining the Turing jump. Math. Res. Lett. 6, 711–722 (1999)
Simpson, S.G.: First order theory of the degrees of recursive unsolvability. Annals of Mathematics 105, 121–139 (1977)
Slaman, T.A., Woodin, W.: Definability in the enumeration degrees. Arch. Math. Logic 36, ñ255–ñ267 (1997)
Slaman, T.A., Woodin, W.H.: Definability in degree structures (2005), http://math.berkeley.edu/slaman/talks/sw.pdf
Soskov, I.: A note on ω jump inversion of degree spectra of structures, This proceedings.
Soskov, I.N.: A jump inversion theorem for the enumeration jump. Arch. Math. Logic 39, 417–437 (2000)
Soskov, I.N.: Degree spectra and co-spectra of structures. Ann. Univ. Sofia 96, 45–68 (2004)
Soskova, M.I.: The automorphism group of the enumeration degrees, in preparation
Soskova, M.I.: A non-splitting theorem in the enumeration degrees. Ann. Pure Appl. Logic 160, 400–418 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Soskova, M.I. (2013). The Turing Universe in the Context of Enumeration Reducibility. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_44
Download citation
DOI: https://doi.org/10.1007/978-3-642-39053-1_44
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39052-4
Online ISBN: 978-3-642-39053-1
eBook Packages: Computer ScienceComputer Science (R0)