Abstract
The aim of this expository paper is to present a nice series of results, obtained in the papers of Chaitin [3], Solovay [8], Calude et al. [2], Ku\(\mathrm{\check{c}}\)era and Slaman [5]. This joint effort led to a full characterization of lower semicomputable random reals, both as those that can be expressed as a “Chaitin Omega” and those that are maximal for the Solovay reducibility. The original proofs were somewhat involved; in this paper, we present these results in an elementary way, in particular requiring only basic knowledge of algorithmic randomness. We add also several simple observations relating lower semicomputable random reals and busy beaver functions.
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References
Bienvenu, L., Downey, R.: Kolmogorov complexity and Solovay functions. In: Symposium on Theoretical Aspects of Computer Science (STACS 2009). Dagstuhl Seminar Proceedings, vol. 09001, pp. 147–158. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany (2009), http://drops.dagstuhl.de/opus/volltexte/2009/1810
Calude, C., Hertling, P., Khoussainov, B., Wang, Y.: Recursively Enumerable Reals and Chaitin Omega Numbers. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 596–606. Springer, Heidelberg (1998)
Chaitin, G.: Information-theoretical characterizations of recursive infinte strings. Theoretical Computer Science 2, 45–48 (1976)
Hölzl, R., Kräling, T., Merkle, W.: Time-Bounded Kolmogorov Complexity and Solovay Functions. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 392–402. Springer, Heidelberg (2009)
Kučera, A., Slaman, T.: Randomness and recursive enumerability. SIAM Journal on Computing 31, 199–211 (2001)
Levin, L.: Forbidden information. In: The 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), p. 761 (2002)
Shen, A.: Algorithmic Information theory and Kolmogorov complexity. Technical report TR2000-034. Technical report, Uppsala University (2000)
Solovay, R.: Draft of a paper (or series of papers) on Chaitin’s work. Unpublished notes, 215 pages (1975)
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Bienvenu, L., Shen, A. (2012). Random Semicomputable Reals Revisited. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_3
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DOI: https://doi.org/10.1007/978-3-642-27654-5_3
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