Abstract
We investigate the notion of K-triviality for closed sets and continuous functions. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial \(\Pi^0_1\) class with no computable elements. For any K-trivial degree d, there is a K-trivial continuous function of degree d.
This research was partially supported by NSF grants DMS 0532644 and 0554841. Remmel was also partially supported by NSF grant DMS 0400507.
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References
Barmpalias, G., Brodhead, P., Cenzer, D., Remmel, J.B., Weber, R.: Algorithmic Randomness of Continuous Functions (in preparation)
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)
Barmpalias, G., Brodhead, P., Cenzer, D., Dashti, S., Weber, R.: Algorithmic randomness of closed sets, J. Logic and Computation (to appear)
Brodhead, P., Cenzer, D., Remmel, J.B.: Random continuous functions (Information Berichte, FernUniversität (2006). In: Cenzer, D., Dillhage, R., Grubb, T., Weihrauch, K. (eds.) CCA 2006. Third International Conference on Computability and Complexity in Analysis. Electronic Notes in Computer Science, pp. 79–89. Springe, Heidelberg (2006)
Cenzer, D.: \(\Pi^{0}_{1}\) Classes, ASL Lecture Notes in Logic (to appear)
Cenzer, D., Downey, R., Jockusch, C.G., Shore, R.: Countable thin \(\Pi^{0}_{1}\) classes. Ann. Pure Appl. Logic 59, 79–139 (1993)
Cenzer, D., Hinman, P.G.: Density of the Medvedev lattice of \(\Pi^0_1\) classes. Archive for Math. Logic 42, 583–600 (2003)
Cenzer, D., Remmel, J. B.: \(\Pi^0_1\) classes, In: Ersov, Y., Goncharov, S., Marek, V., Nerode, A., Remmel, J. (eds.): Handbook of Recursive Mathematics, Vol. 2: Recursive Algebra, Analysis and Combinatorics, Elsevier Studies in Logic and the Foundations of Mathematics, Vol. 139 pp. 623–821 (1998)
Dekker, J., Myhill, J.: Retraceable sets. Canad. J. Math. 10, 357–373 (1985)
Downey, R.: Five Lectures on Algorithmic Randomness. In: Chong, C.T. (ed.) Computational Prospects of Infinity Proc. 2005 Singapore meeting (to appear 2005)
Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity, in preparation. Current draft available at http://www.mcs.vuw.ac.nz/~downey/ .
Downey, R., Hirschfeldt, D., Nies, A., Stephan, F.: Trivial reals. In: Proc. 7th and 8th Asian Logic Conference, pp. 101–131. World Scientific Press, Singapore (2003)
Downey, R.: Some computability-theoretic aspects of reals and randomness. In: Cholak, P. (ed.) The Notre Dame Lectures ASL Lecture Notes in Logic (2005)
Kučera, A., Terwijn, S.: Lowness for the class of random sets. Journal of Symbolic Logic 64, 1396–1402 (1999)
Nies, A.: Lowness properties and randomness. Advances in Mathematics 197, 274–305 (2005)
Nies, A.: Computability and Randomness, in preparation. Current draft available at http://www.cs.auckland.ac.nz/~nies .
Ohashi, K.: A stronger form of a theorem of Friedberg. Notre Dame J. Formal Logic 5, 10–12 (1964)
Simpson, S.G.: Mass problems and randomness. Bull. Symbolic Logic 11, 1–27 (2005)
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Barmpalias, G., Cenzer, D., Remmel, J.B., Weber, R. (2007). K-Trivial Closed Sets and Continuous Functions. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_14
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DOI: https://doi.org/10.1007/978-3-540-73001-9_14
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