Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T17:15:41.088Z Has data issue: false hasContentIssue false
  • English
  • Français

MARTIN-LÖF RANDOMNESS IN SPACES OF CLOSED SETS

Published online by Cambridge University Press:  22 April 2015

LOGAN M. AXON
LOGAN M. AXON*
Affiliation:
DEPARTMENT OF MATHEMATICS GONZAGA UNIVERSITY 502 E. BOONE AVE. SPOKANE, WA 99258, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Algorithmic randomness was originally defined for Cantor space with the fair-coin measure. Recent work has examined algorithmic randomness in new contexts, in particular closed subsets of 2ɷ ([2] and [8]). In this paper we use the probability theory of closed set-valued random variables (RACS) to extend the definition of Martin-Löf randomness to spaces of closed subsets of locally compact, Hausdorff, second countable topological spaces. This allows for the study of Martin-Löf randomness in many new spaces, but also gives a new perspective on Martin-Löf randomness for 2ɷ and on the algorithmically random closed sets of [2] and [8]. The first half of this paper is devoted to developing the machinery of Martin-Löf randomness for general spaces of closed sets. We then prove some general results and move on to show how the algorithmically random closed sets of [2] and [8] fit into this new framework.

Keywords

Algorithmic randomnessrandom closed sets
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 2 , June 2015 , pp. 359 - 383
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Axon, Logan M., Algorithmically random closed sets and probability, Ph.D Thesis, ProQuest LLC, University of Notre Dame, Ann Arbor, MI, 2010.Google Scholar
Barmpalias, George, Brodhead, Paul, Cenzer, Douglas, Dashti, Seyyed, and Weber, Rebecca, Algorithmic randomness of closed sets. Journal of Logic and Computation, vol. 17 (2007), no. 6, pp. 1041–1062.Google Scholar
Brodhead, Paul, Computable aspects of closed sets, Ph.D Thesis, ProQuest LLC, University of Florida, Ann Arbor, MI, 2008.Google Scholar
Brodhead, Paul, Cenzer, Douglas, Toska, Ferit, and Wyman, Sebastian, Algorithmic randomness and capacity of closed sets. Logical Methods in Computer Science, vol. 7 (2011), no. 3:16, Special issue: 7th International Conference on Computability and Complexity in Analysis (CCA 2010).Google Scholar
Choquet, Gustave, Theory of capacities. Annales de l’Institut Fourier, Grenoble, vol. 5 (1953–1955), pp. 131–295.CrossRefGoogle Scholar
Mauldin, R. Daniel and McLinden, Alexander P., Random closed sets viewed as random recursions. Archive for Mathematical Logic, vol. 48 (2009), no. 3–4, pp. 257–263.Google Scholar
Day, Adam R. and Miller, Joseph S., Randomness for non-computable measures. Transactions of the American Mathematical Society, vol. 365 (2013), pp. 3575–3591.Google Scholar
Diamondstone, David and Kjos-Hanssen, Bjørn, Members of random closed sets, Mathematical theory and computational practice, Lecture Notes in Computer Science, vol. 5635, Springer, Berlin, 2009, pp. 144–153.Google Scholar
Downey, Rod, Hirschfeldt, Denis R., Nies, André, and Terwijn, Sebastiaan A., Calibrating randomness. Bulletin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 411–491.Google Scholar
Fouché, Willem, Arithmetical representations of Brownian motion. I, this Journal, vol. 65 (2000), no. 1, pp. 421–442.Google Scholar
Fouché, Willem, The descriptive complexity of Brownian motion. Advances in Mathematics, vol. 155 (2000), no. 2, pp. 317–343.Google Scholar
Hertling, Peter and Weihrauch, Klaus, Random elements in effective topological spaces with measure. Information and Computation, vol. 181 (2003), no. 1, pp. 32–56.Google Scholar
Kjos-Hanssen, Bjørn, The probability distribution as a computational resource for randomness testing. Journal of Logic and Analysis, vol. 2 (2010), p. 10, 13.Google Scholar
Kjos-Hanssen, Bjørn and Nerode, Anil, The law of the iterated logarithm for algorithmically random Brownian motion, Logical foundations of computer science, Lecture Notes in Computer Science, vol. 4514, Springer, Berlin, 2007, pp. 310–317.CrossRefGoogle Scholar
Matheron, G., Random sets and integral geometry, Wiley Series in Probability and Mathematical Statistics, Wiley, New York–London–Sydney, 1975.Google Scholar
Molchanov, Ilya, Theory of random sets, Probability and its Applications (New York), Springer-Verlag, London, 2005.Google Scholar
Nguyen, Hung T., An introduction to random sets, Chapman & Hall/CRC, Boca Raton, FL, 2006.Google Scholar
Nies, André, Computability and randomness, first ed., Oxford Logic Guides, Oxford University Press, New York, 2009.Google Scholar
Reimann, Jan and Slaman, Theodore A., Measures and their random reals, Transactions of the American Mathematical Society, to appear.Google Scholar