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Natural Halting Probabilities, Partial Randomness, and Zeta Functions

Authors Christian S. Calude, Michael A. Stay



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Christian S. Calude
Michael A. Stay

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Christian S. Calude and Michael A. Stay. Natural Halting Probabilities, Partial Randomness, and Zeta Functions. In Kolmogorov Complexity and Applications. Dagstuhl Seminar Proceedings, Volume 6051, p. 1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2006)
https://doi.org/10.4230/DagSemProc.06051.10

Abstract

We introduce the {it natural halting probability} and the {it natural complexity} of a Turing machine and we relate them to program-size complexity and Chaitin's halting probability. A classification of Turing machines according to their natural (Omega) halting probabilities is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on randomness and partial randomness are proved. For example, we show that the natural halting probability of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness---which cannot be characterised in terms of plain complexity---various types of partial randomness admit such characterisations.
Keywords
  • Natural halting probability
  • natural complexity

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