Abstract
The notion of immune sets is extended to closed sets and \(\Pi^0_1\) classes in particular. We construct a \(\Pi^0_1\) class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A,B) of separating sets for A and B is immune. We show that every perfect thin \(\Pi^0_1\) class is immune. We define the stronger notion of prompt immunity and construct an example of a \(\Pi^0_1\) class of positive measure which is promptly immune. We show that the immune degrees in the Medvedev lattice of closed sets forms a filter. We show that for any \(\Pi^0_1\) class P with no computable element, there is a \(\Pi^0_1\) class Q which is not immune and has no computable element, and which is Medvedev reducible to P. We show that any random closed set is immune.
This research was partially supported by NSF grants DMS-0554841 and DMS 652372.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ambos-Spies, K., Jockusch, C.G., Shore, R.A., Soare, R.I.: An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Trans. Amer. Math. Soc. 281, 109–128 (1984)
Barmpalias, G., Brodhead, P., Cenzer, D., Dashti, S., Weber, R.: Algorithmic randomness of closed sets. J. Logic Comp. 17, 1041–1062 (2007)
Brodhead, P., Cenzer, D., Dashti, S.: Random closed sets. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 55–64. Springer, Heidelberg (2006)
Binns, S.: A splitting theorem for the Medvedev and Muchnik lattices. Math. Logic Q. 49, 327–335 (2003)
Binns, S.: Small \(\Pi^0_1\) classes. Arch. Math. Logic 45, 393–410 (2006)
Binns, S.: Hyperimmunity in 2ℕ. Notre Dame J. Formal Logic 4, 293–316 (2007)
Cenzer, D., Downey, R., Jockusch, C.J.: Countable Thin \(\Pi^0_1\) Classes. Ann. Pure Appl. Logic 59, 79–139 (1993)
Cenzer, D., Nies, A.: Initial segments of the lattice of \(\Pi^0_1\) classes. Journal of Symbolic Logic 66, 1749–1765 (2001)
Cholak, P., Coles, R., Downey, R., Hermann, E.: Automorphisms of the Lattice of \(\Pi^0_1\) Classes. Transactions Amer. Math. Soc. 353, 4899–4924 (2001)
Cenzer, D., Remmel, J.B.: \(\Pi^0_1\) classes in mathematics. In: Ershov, Y., Goncharov, S., Nerode, A., Remmel, J. (eds.) Handbook of Recursive Mathematics, Part Two. Elsevier Studies in Logic, vol. 139, pp. 623–821 (1998)
Cenzer, D., Remmel, J.B.: Effectively Closed Sets. ASL Lecture Notes in Logic (in preparation)
Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity (in preparation), http://www.mcs.vuw.ac.nz/~downey/
Jockusch, C.G.: \(\Pi^0_1\) classes and boolean combinations of recursively enumerable sets. J. Symbolic Logic 39, 95–96 (1974)
Jockusch, C.G., Soare, R.: Degrees of members of \(\Pi^0_1\) classes. Pacific J. Math. 40, 605–616 (1972)
Jockusch, C.G., Soare, R.: \(\Pi^0_1\) classes and degrees of theories. Trans. Amer. Math. Soc. 173, 35–56 (1972)
Medvedev, Y.T.: Degrees of difficulty of the mass problem. Dokl. Akad. Nauk SSSR (N.S.) 104, 501–504 (1955) (in Russian)
Muchnik, A.A.: On strong and weak reducibilities of algorithmic problems. Sibirsk. Mat. Ž. 4, 1328–1341 (1963) (in Russian)
Simpson, S.: Mass problems and randomness. Bull. Symb. Logic 11, 1–27 (2005)
Simpson, S.: \(\Pi^0_1\) classes and models of \(\text{WKL}_0\). In: Simpson, S. (ed.) Reverse Mathematics 2001. Association for Symbolic Logic: Lecture Notes in Logic, vol. 21, pp. 352–378 (2005)
Simpson, S.: An extension of the recursively enumerable Turing degrees. J. London Math. Soc. 75, 287–297 (2007)
Soare, R.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cenzer, D., Weber, R., Wu, G. (2009). Immunity for Closed Sets. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-03073-4_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03072-7
Online ISBN: 978-3-642-03073-4
eBook Packages: Computer ScienceComputer Science (R0)