Abstract
A concept of randomness for infinite time register machines (ITRMs), resembling Martin-Löf-randomness, is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies computability and that an analogue of van Lambalgen’s theorem holds.
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References
Carl, M.: The distribution of ITRM-recognizable reals. To appear in: Annals of Pure and Applied Logic. In: Special Issue from CiE (2012)
Carl, M.: Optimal Results on ITRM-recognizability. arXiv:1306.5128v1 (preprint)
Carl, M., Schlicht, P.: Infinite Computations with Random Oracles. arXiv:1307.0160v3 (submitted)
Cutland, N.: Computability - An introduction to recursive function theory. Cambridge University Press (1980)
Downey, R.G., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer LLC (2010)
Hamkins, J., Lewis, A.: Infinite Time Turing Machines. Journal of Symbolic Logic 65(2), 567–604 (2000)
Hjorth, G., Nies, A.: Randomness in effective descriptive set theory. Journal of the London Mathematical Society (to appear)
Koepke, P., Miller, R.: An enhanced theory of infinite time register machines
Carl, M., Fischbach, T., Koepke, P., Miller, R., Nasfi, M., Weckbecker, G.: The basic theory of infinite time register machines
Kanamori, A.: The higher infinite. Springer (2005)
Kechris, A.: Measure and category in effective descriptive set theory. Annals of Mathematical Logic 5(4), 337–384 (1973)
Koepke, P.: Turing computations on ordinals. Bulletin of Symbolic Logic 11, 377–397 (2005)
Koepke, P.: Ordinal Computability. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 280–289. Springer, Heidelberg (2009)
Mathias, A.R.D.: Provident sets and rudimentary set forcing (preprint), https://www.dpmms.cam.ac.uk/~ardm/fifofields3.pdf
Sacks, G.: Higher recursion theory. Springer (1990)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer (1987)
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Carl, M. (2014). Algorithmic Randomness for Infinite Time Register Machines. In: Beckmann, A., Csuhaj-Varjú, E., Meer, K. (eds) Language, Life, Limits. CiE 2014. Lecture Notes in Computer Science, vol 8493. Springer, Cham. https://doi.org/10.1007/978-3-319-08019-2_9
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DOI: https://doi.org/10.1007/978-3-319-08019-2_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08018-5
Online ISBN: 978-3-319-08019-2
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