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Random reals and possibly infinite computations Part I: Randomness in ∅′

Published online by Cambridge University Press:  12 March 2014

Verónica Becher  and
Serge Grigorieff
Verónica Becher
Affiliation:
Departamento de Computación, Fceyn Universidad de Buenos Aires, ArgentinaE-mail:, vbecher@dc.uba.ar
Serge Grigorieff
Affiliation:
Liafa, Université Paris7, 2, Pl. Jussieu 75251 Paris Cedex 05, FranceE-mail:, seg@liafa.jussieu.fr
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Abstract

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅′ of the probability that the output be in some set under complexity assumptions about .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 891 - 913
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Becher, V. and Chaitin, G., Another example of higher order randomness, Fundamenta Informaticae, vol. 51 (2002), no. 4, pp. 325–338.Google Scholar
[2]Becher, V., Chaitin, G., and Daicz, S., A highly random number, Proceedings of the third discrete mathematics and theoretical computer science conference (DMTCS '01) (Calude, C. S., Dineen, M. J., and Sburlan, S., editors), Springer-Verlag, 2001, pp. 55–68.Google Scholar
[3]Becher, V. and Grioorieff, S., Recursion and topology on 2≤ω-m for possibly infinite computations, Theoretical Computer Science, vol. 322 (2004), pp. 85–136.CrossRefGoogle Scholar
[4]Becher, V., Random reals and possibly infinite computations, Part II: Higher order randomness, In preparation.Google Scholar
[5]Becher, V., Wadge reducibility and spectral continuous maps into 2≤ω, In preparation.Google Scholar
[6]Boasson, L. and Nivat, M., Adherences of languages, Journal of Computer and System Sciences, vol. 20 (1980), pp. 285–309.CrossRefGoogle Scholar
[7]Calude, C., Information and randomness, Springer, 1994.CrossRefGoogle Scholar
[8]Calude, C. S., Hertling, P. H., and Khoussainov, B.Wang, Y., Recursively enumerable reals and Chaitin Ω numbers, STACS 98 (Paris, 1998), Lecture Notes in Computer Science, no. 1373, Springer-Verlag, 1998, pp. 596–606.CrossRefGoogle Scholar
[9]Chaitin, G., A theory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329–340, Available on Chaitin's home page.CrossRefGoogle Scholar
[10]Chaitin, G., Algorithmic entropy of sets, Computers & Mathematics with Applications, vol. 2 (1976), pp. 233–245. Available on Chaitin's home page.CrossRefGoogle Scholar
[11]Chaitin, G., Algorithmic information theory, 1st ed., Cambridge University Press, 1987.CrossRefGoogle Scholar
[12]Downey, R., Some computability-theoretical aspects of reals and randomness, Notes from lectures given at the University Notre Dame. Available at MSCS, University of Wellington, NZ, 2000.Google Scholar
[13]Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity, Springer, 2005, To appear. Preliminary version, November 30th 2004, available on Downey's home page.Google Scholar
[14]Downey, R., Hirschfeldt, D., and Nies, A., Randomness, computability and density, SIAM Journal on Computing, vol. 31 (2002), pp. 1169–1183, Extended abstract in Proceedings of the STACS 2001, LNCS 2010.CrossRefGoogle Scholar
[15]Downey, R. and Laforte, G. L., Presentations of computably enumerable reals, Theoretical Computer Science, vol. 284 (2002), no. 2, pp. 539–555.CrossRefGoogle Scholar
[16]Head, T., The adherences of languages as topological spaces, Automata andinfinite words (Nivat, M. and Perrin, D., editors), Lecture Notes in Computer Science, vol. 192, 1985, pp. 147–163.CrossRefGoogle Scholar
[17]Head, T., The topological structure of adherence of regular languages, RAIRO, Theoretical Informatics and Applications, vol. 20 (1986), pp. 31–41.CrossRefGoogle Scholar
[18]Kechris, A. S., Classical descriptive set theory, Springer, 1995.CrossRefGoogle Scholar
[19]Levin, L., On the notion of random sequence, Soviet Math. Dokl., vol. 14 (1973), no. 5, pp. 1413–1416.Google Scholar
[20]Li, M. and Vitanyi, P., An introduction to Kolmogorov complexity and its applications, 2d ed., Springer, 1997.CrossRefGoogle Scholar
[21]Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[22]Miller, J., Personal communication.Google Scholar
[23]Moschovakis, Y. N., Descriptive set theory, North Holland, 1980.Google Scholar
[24]Muchnik, An. A., Personal communication.Google Scholar
[25]Odifreddi, P., Classical recursion theory, Studies in Logic, vol. 125, North-Holland, 1989.Google Scholar
[26]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, 1967.Google Scholar
[27]Sacks, G. E., Degrees of unsolvahility, Annals of mathematical studies, Princeton University Press, 1966.Google Scholar
[28]Schnorr, C. P., Process complexity and effective random tests, Journal of Computer and System Sciences, vol. 7 (1973), pp. 376–388.CrossRefGoogle Scholar
[29]Schnorr, C. P., A survey of the theory of random sequences, Basic problems in methodology and linguistics (Butts, R. E. and Hintikka, J., editors), D. Reidel, 1977, pp. 193–210.CrossRefGoogle Scholar
[30]Soare, R., Recursion theory and Dedekind cuts, Transactions of the American Mathematical Society, vol. 140 (1969), pp. 271–294.Google Scholar
[31]Solovay, R. M., Draft of a paper (or a series of papers) on Chaitin's work, Unpublished manuscript, IBM Research Center, NY, 1975.Google Scholar
[32]Solovay, R. M., On random R.E. sets, Non-classical logics, model theory and computability (Arruda, A. I., da Costa, N. C. A., and Chuaqui, R., editors), North-Holland, 1977, pp. 283–307.CrossRefGoogle Scholar
[33]Turing, A., On computable numbers, with an application to the Entscheidungsproblem., Proceedings of the London Mathematical Society, 2nd series, vol. 42 (1936), pp. 230–265. Correction, A. Turing, On computable numbers, with an application to the Entscheidungsproblem., Proceedings of the London Mathematical Society, 2nd series, vol. 43 (1937) pp. 544–546.Google Scholar
[34]Wadge, W. W., Degrees of complexity of subsets of the Baire space, Notices of the American Mathematical Society, (1972), pp. A–714.Google Scholar
[35]Wadge, W. W., Degrees of complexity of subsets of the Baire space, Ph.D. thesis, University of Berkeley, 1984.Google Scholar