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On the computability of a construction of Brownian motion†

Published online by Cambridge University Press:  17 May 2013

GEORGE DAVIE  and
WILLEM L. FOUCHÉ
GEORGE DAVIE
Affiliation:
Department of Decision Sciences, School of Economic Sciences, University of South Africa, PO Box 392, 0003 Pretoria, South Africa Email: davieg@unisa.ac.za; fouchwl@gmail.com
WILLEM L. FOUCHÉ
Affiliation:
Department of Decision Sciences, School of Economic Sciences, University of South Africa, PO Box 392, 0003 Pretoria, South Africa Email: davieg@unisa.ac.za; fouchwl@gmail.com
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Abstract

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Issue 6 , December 2013 , pp. 1257 - 1265
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

†

The research in this paper was supported by the National Research Foundation (NRF) of South Africa and by the European Union grant agreement PIRSES-GA-2011-2011-294962 in Computable Analysis (COMPUTAL).

References

Asarin, E. A. (1988) Individual random signals: an approach based on complexity, doctoral dissertation, Moscow State University.Google Scholar
Asarin, E. A. and Prokovskii, A. V. (1986) Use of the Kolmogorov complexity in analysing control system dynamics. Automation and Remote Control 47 21–28. (Translated from: Primeenenie kolmogorovskoi slozhnosti k anlizu dinamiki upravlemykh sistem. Automatika i Telemekhanika (Automation Remote Control) 1 25–33.)Google Scholar
Calude, C. S. (2002) Information and randomness, Texts in Theoretical Computer Science, Springer-Verlag.CrossRefGoogle Scholar
Chaitin, G. (1987a) A theory of program size formally identical to information theory. Journal of the ACM 22 329–340.CrossRefGoogle Scholar
Chaitin, G. A. (1987b) Algorithmic information theory, Cambridge University Press.CrossRefGoogle Scholar
Davie, G. (2001) The Borel–Cantelli lemmas, probability laws and Kolmogorov complexity. Annals of Probability 29 (4)1426–1434.CrossRefGoogle Scholar
Downey, R. G. and Hirschfeldt, D. R. (2010) Algorithmic randomness and complexity, Springer.CrossRefGoogle Scholar
Fouché, W. L. (2000a) Arithmetical representations of Brownian motion I. Journal of Symbolic Logic 65 421–442.CrossRefGoogle Scholar
Fouché, W. L. (2000b) The descriptive complexity of Brownian motion. Advances in Mathematics 155 317–343.CrossRefGoogle Scholar
Fouché, W. L. (2008) Dynamics of a generic Brownian motion: Recursive aspects. In: Beckmann, A. (ed.) From Gödel to Einstein: Computability between Logic and Physics. Theoretical Computer Science 394 175–186.CrossRefGoogle Scholar
Fouché, W. L. (2009) Fractals generated by algorithmically random Brownian motion. In: Ambos-Spies, K., Löwe, B. and Merkle, W. (eds.) Proceedings CiE 2009. Springer-Verlag Lecture Notes in Computer Science 5635 208–217.CrossRefGoogle Scholar
Fouché, W. L. (2013) Kolmogorov complexity and the geometry of Brownian motion (in preparation).Google Scholar
Gács, P. (2005) Uniform test of algorithmic randomness over a general space. Theoretical Computer Science 341 91–137.CrossRefGoogle Scholar
Hoyrup, M. and Rojas, C. (2009a) Computability of probability measures and Martin-Löf randomness over metric spaces. Information and Computation 207 830–847.CrossRefGoogle Scholar
Hoyrup, M. and Rojas, C. (2009b) An application of Martin-Löf randomness to effective probability theory. Mathematical theory and computational practice. Springer-Verlag Lecture Notes in Computer Science 5635 260–269.CrossRefGoogle Scholar
Hoyrup, M. (2011) Randomness and the ergodic decomposition. In: Löwe, B., Normann, D., Soskov, I. and Soskova, A. A. (eds.) Proceedings CiE 2011. Springer-Verlag Lecture Notes in Computer Science 6735 122–131.CrossRefGoogle Scholar
Kjos-Hanssen, B. and Nerode, A. (2007) The law of the iterated logarithm for algorithmically random Brownian motion. In: Artemov, S. N. and Nerode, A. (eds.) Proceedings on Logical Foundations of Computer Science, LFCS 2007. Springer-Verlag Lecture Notes in Computer Science 4514 310–317.CrossRefGoogle Scholar
Kjos-Hanssen, B. and Nerode, A. (2009) Effective dimension of points visited by Brownian motion. Theoretical Computer Science 410 347–354.CrossRefGoogle Scholar
Kjos-Hanssen, B. and Szabados, T. (2011) Kolmogorov complexity and strong approximation of Brownian motion. Proceedings of the American Mathematical Society 139 3307–3316.CrossRefGoogle Scholar
Kolmogorov, A. N. (1965) Three approaches to the quantitative definition of information. Problems of Information Transmission 1 1–7.Google Scholar
Levin, L. (1971) Some theorems on the algorithmic approach to probability theory and information theory, Dissertation in Mathematics Moscow University (in Russian).Google Scholar
Levin, L. (1974) Laws of information conservation (non-growth) and aspects of the foundation of probability theory. Problems of Information Transmission 10 206–210.Google Scholar
Li, M. and Vitanyi, P. (2008) An Introduction to Kolmogorov Complexity and Its Applications, 3rd edition, Texts in Computer Science, Springer.CrossRefGoogle Scholar
Martin-Löf, P. (1966) The definition of random sequences. Information and Control 9 602–619.CrossRefGoogle Scholar
Nies, A. (2008) Computability and randomness, Oxford Logic Guides 51, Clarendon Press.Google Scholar
Solomonoff, R. (1964a) A formal theory of inductive inference, I. Information and Control 7 1–22.CrossRefGoogle Scholar
Solomonoff, R. (1964b) A formal theory of inductive inference, II. Information and Control 7 224–254.CrossRefGoogle Scholar
Weihrauch, K. (1999) Computability on the probability measures on the Borel sets of the unit interval. Theoretical Computer Science 219 421–437.CrossRefGoogle Scholar
Weihrauch, K. (2000) Computable Analysis, Springer.CrossRefGoogle Scholar