Skip to main content
SpringerLink
Log in

Some Recent Progress in Algorithmic Randomness

  • Conference paper
Mathematical Foundations of Computer Science 2004 (MFCS 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3153))

Abstract

Recently there has been exciting progress in our understanding of algorithmic randomness for reals, its calibration, and its connection with classical measures of complexity such as degrees of unsolvability.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allender, E., Buhrman, H., Koucký, M.: What Can be Efficiently Reduced to the K-Random Strings? In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 584–595. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  2. Ambos-Spies, K.: Algorithmic randomness revisited, in Language, Logic and Formailzation of Knowledge. Bibliotheca, 33–52 (1998)

    Google Scholar 

  3. Ambos-Spies, K., Kučera, A.: Randomness in computability theory. In: Cholak, P., et al. (eds.) Computability Theory: Current Trends and Open Problems, Contemporary Mathematics, American Mathematical Society, vol. 257, pp. 1–14 (2000)

    Google Scholar 

  4. Ambos-Spies, K., Mayordomo, E.: Resource bounded measure and randomness. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory, pp. 1–48. Marcel-Decker, New York (1997)

    Google Scholar 

  5. Ambos-Spies, K., Mayordomo, E., Wang, Y., Zheng, X.: Resource bounded balanced genericity, stochasticity and weak randomness. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 63–74. Springer, Heidelberg (1996)

    Google Scholar 

  6. Ambos-Spies, K., Merkle, W., Reimann, J., Stephan, F.: Hausdorff Dimension in Exponential Time. In: Computational Complexity 2001, pp. 210–217. IEEE Computer Society, Los Alamitos (2001)

    Google Scholar 

  7. Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. J. Complexity 16(4), 676–690 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Barzdins, J.: Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set. Soviet Mathematics Doklady 9, 1251–1254 (1968)

    Google Scholar 

  9. Ñies, A., Bedregal, B.: Lowness properties of reals and hyper-immunity. In: WOLLIC 2003. Electronic Notes in Theoretical Computer Science, vol. 84, Elsevier, Amsterdam (2003), http://www.elsevier.nl/locate/entcs/volume84.html

    Google Scholar 

  10. Calude, C.: Information and Randomness, An Algorithmic Perspective. Springer, Heidelberg (1994) Second edition 2002

    Google Scholar 

  11. Calude, C., Coles, R.: Program size complexity of initial segments and domination relation reducibility. In: Karhümaki, J., Mauer, H., Paǔn, G., Rozenberg, G. (eds.) Jewels are Forever, pp. 225–237. Springer, Heidelberg (1999)

    Google Scholar 

  12. Calude, C., Coles, R., Hertling, P., Khoussainov, B.: Degree-theoretic aspects of computably enumerable reals. In: Cooper, Truss (eds.) Models and Computability, Cambridge University Press, Cambridge (1999)

    Google Scholar 

  13. Calude, C., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin Ω numbers. In: Proc. STACS 1998, pp. 596–606. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  14. Calude, C., Nies, A.: Chaitin’s Ω numbers and strong reducibilities. Journal of Universal Computer Science (1997)

    Google Scholar 

  15. Calude, C., Staiger, L., Terwijn, S.A.: On partial randomness (in preparation)

    Google Scholar 

  16. Chaitin, G.: A theory of program size formally identical to information theory. Journal of the ACM 22, 329–340 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chaitin, G.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  18. Chernov, A., Muchnik, A.A., Romashchenko, A., Shen, A., Vereshchagin, N.: Upper semi-lattice of binary strings with the relation ”x is simple conditional to y”. Theor. Comput. Sci. 271(1-2), 69–95 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Church, A.: On the concept of a random number. Bulletin of the AMS 46, 130–135 (1940)

    Article  MathSciNet  Google Scholar 

  20. Coles, R., Cholak, P., Downey, R., Herrmann, E.: Automorphisms of the lattice of \(\Pi^{1}_{0}\) classes: perfect thin classes and a.n.c. degrees. Trans. Amer. Math. Soc. 353, 4899–4924 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Downey, R.: Some computability-theoretical aspects of reals and randomness (to appear), Lecture Notes in Logic, volume edited by Peter Cholak

    Google Scholar 

  22. Downey, R., Griffiths, E.: On Schnorr randomness. Journal of Symbolic Logic 69(2), 533–554 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Downey, R., Griffiths, E., LaForte, G.: On Schnorr and computable randomness, martingales, and machines. To appear Mathematical Logic Quarterly

    Google Scholar 

  24. Downey, R., Hirschfeldt, D., Miller, J., Nies, A.: On ΩA (in preparation)

    Google Scholar 

  25. Downey, R., Griffiths, E., Reid, S.: On Kurtz randomness (submitted)

    Google Scholar 

  26. Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, Heidelberg (to appear)

    Google Scholar 

  27. Downey, R., Hirschfeldt, D., LaForte, G.: Randomness and reducibility. To appear Journal of Computing and System Sciences, extended abstract appeared in Mathematical Foundations of Computer Science (2001); In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 316–327. Springer, Heidelberg (2001)

    Google Scholar 

  28. Downey, R., Hirschfeldt, D., LaForte, G.: Undecidability of Solovay and other degree structures for c.e. reals (in preparation)

    Google Scholar 

  29. Downey, R., Hirschfeldt, D.R., Lempp, S., Solomon, R.: A Δ02 set with no infinite low subset in either it or its complement. Journal of Symbolic Logic 66, 1371–1381 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Downey, R.G., Hirschfeldt, D.R., Nies, A.: Randomness, computability, and density. SIAM J. Comput. 31, 1169–1183 (2002), Extended abstract appeared In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 195–201. Springer, Heidelberg (2001)

    Google Scholar 

  31. Downey, R., Hirschfeldt, D.R., Nies, A., Stephan, F.: Trivial reals. Electronic Notes in Theoretical Computer Science, ENTCS (2002) Final version appears in Downey, et al. (eds.)Proceedings Asian Logic Conferences. World Scientific (2003)

    Google Scholar 

  32. Downey, R., Hirschfeldt, D.R., Nies, A., Terwijn, S.: Calibrating randomness. Bull. Symb Logic (to appear)

    Google Scholar 

  33. Downey, R., Jockusch, C., Stob, M.: Array recursive sets and multiple permitting arguments. In: Proceedings Oberwolfach. Lecture Notes in Mathematics 1990, pp. 141–174. Springer, Heidelberg (1989)

    Google Scholar 

  34. Downey, R., Jockusch, C., Stob, M.: Array nonrecursive degrees and genericity. In: Cooper, Slaman, Wainer (eds.) Computability, Enumerability, Unsolvability. London Mathematical Society Lecture Notes Series, vol. 224, pp. 93–105. Cambridge University Press, Cambridge (1996)

    Chapter  Google Scholar 

  35. Downey, R., LaForte, G.: Presentations of computably enumerable reals. Theoretical Computer Science 284, 539–555 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Downey, R., Miller, J.: A basis theorem for \(\Pi^{0}_{1}\) classes and jumps of random reals (in preparation)

    Google Scholar 

  37. Downey, R., Miller, J., Reimann, J.: Finite randomness (in preparation)

    Google Scholar 

  38. Downey, R., Reimann, J.: On Schnorr dimension (in preparation)

    Google Scholar 

  39. Downey, R., Terwijn, S.A.: Computably enumerable reals and uniformly presentable ideals. Mathematical Logic Quarterly 48(1), 29–40 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  40. Downey, R., Yue, Y.: Relative computability vs relative complexity (in preparation)

    Google Scholar 

  41. Downey, R., Wu, G., Zheng, X.: Degrees of d.c.e. reals. Mathematical Logic Quarterly (to appear)

    Google Scholar 

  42. Falconer, K.: Fractal geometry, mathematical foundations & applications. Wiley & Sons, Chichester (1992)

    Google Scholar 

  43. Fortnow, L., Freivalds, R., Gasarch, W., Kummer, M., Kurtz, S., Smith, C., Stephan, F.: On the relative size of learnable sets. Theoretical Computer Science 197, 139–156 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  44. Gács, P.: Every sequence is reducible to a random one. Information and Control 70, 186–192 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  45. Gaifmann, H., Snir, M.: Probabilities over rich languages. J. Symb. Logic 47, 495–548 (1982)

    Article  Google Scholar 

  46. Hausdorff, F.: Dimension und äußeres Maß. Mathematische Annalen 79, 157–179 (1919)

    Article  MathSciNet  Google Scholar 

  47. Hitchcock, J., Lutz, J., Terwijn, S.: The arithmetical complexity of dimension and randomness, manuscript (January 2003)

    Google Scholar 

  48. Ho, C.-K.: Relatively recursive reals and real functions. Theoretical Computer Science 219, 99–120 (1999)

    Article  Google Scholar 

  49. Ishmukhametov, S.: Weak recursive degrees and a problem of Spector. In: Arslanov, M., Lempp, S. (eds.) Recursion Theory and Complexity, de Gruyter, Berlin, pp. 81–88 (1999)

    Google Scholar 

  50. Jech, T.: Set theory. Academic Press, London (1978)

    Google Scholar 

  51. Jockusch, C.

    Google Scholar 

  52. Kamae, T.: Subsequences of normal sequences. Israel J. Math. 16, 121–149 (1973)

    Article  MathSciNet  Google Scholar 

  53. Katseff, H.: Complexity dips in random infinite binary sequences. Information and Control 38, 258–263 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  54. Kautz, S.: Degrees of Random Sets, PhD thesis, Cornell University (1991)

    Google Scholar 

  55. Khoussainov, B.: Randomness, computability, and algebraic specifications. Annals of Pure and Applied Logic 91, 1–15 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  56. Kjos-Hanssen, B., Stephan, F., Nies, A.: On a question of Ambos-Spies and Kučera (to appear)

    Google Scholar 

  57. Ko, K.-I.: On the notion of infinite pseudorandom sequences. Theoretical Computer Science 48, 9–33 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  58. Kolmogorov, A.: Three Approaches to the Quantitative Definition of Information, in Problems of Information Transmission (Problemy Peredachi Informatsii), vol. 1, pp. 1–7 (1965)

    Google Scholar 

  59. Kraft, L.: A device for quantizing, grouping, and coding amplitude modulated pulses. M.Sc. thesis, MIT (1949)

    Google Scholar 

  60. Kučera, A.: Measure, \(\Pi^{0}_{1}\) -classes and complete extensions of PA. In: Ebbinghaus, H.-D., Müller, G.H., Sacks, G.E. (eds.) Recursion theory week. Lect. Notes in Math., vol. 1141, pp. 245–259. Springer, Heidelberg (1985)

    Google Scholar 

  61. Kučera, A.: On the use of diagonally nonrecursive functions. In: Logic Colloquium 1987, pp. 219–239. Elsevier, Amsterdam (1987)

    Google Scholar 

  62. Kučera, A.: On relative randomness. Annals of Pure and Applied Logic 63, 61–67 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  63. Kučera, A., Slaman, T.A.: Randomness and recursive enumerability. SIAM J. Comput. 31, 199–211 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  64. Kučera, A., Terwijn, S.A.: Lowness for the class of random sets. Journal of Symbolic Logic 64(4), 1396–1402 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  65. Kummer, M.: Kolmogorov complexity and instance complexity of recursively enumerable sets. SIAM Journal of Computing 25, 1123–1143 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  66. Kurtz, S.: Randomness and Genericity in the Degrees of Unsolvability, PhD thesis, University of Illinois (1981)

    Google Scholar 

  67. van Lambalgen, M.: Random sequences, PhD thesis, University of Amsterdam (1987)

    Google Scholar 

  68. van Lambalgen, M.: The axiomatization of randomness. Journal of Symbolic Logic 55(3), 1143–1167 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  69. Lathrop, J., Lutz, J.: Recursive computational depth. Information and Computation 153, 139–172 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  70. Levin, L.: On the notion of a random sequence. Soviet Math. Dokl. 14, 1413–1416 (1973)

    MATH  Google Scholar 

  71. Li, M., Vitányi, P.: An introduction to Kolmogorov complexity and its applications. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  72. Loveland, D.: A variant of the Kolmogorov concept of complexity. Information and Control 15, 510–526 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  73. Lutz, J.: Category and measure in complexity classes. SIAM J. Comput. 19, 1100–1131 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  74. Lutz, J.: Almost everywhere high nonuniform complexity. J. Comput. System Sci. 44, 220–258 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  75. Lutz, J.: Dimension in complexity classes. In: Proc. 15th Conference on Computational Complexity, pp. 158–169. IEEE Computer Society, Los Alamitos (2000)

    Chapter  Google Scholar 

  76. Lutz, J.: The dimensions of individual strings and sequences, Information and Computation. To appear in Preliminary version: Gales and the constructive dimension of individual sequences, Proc. 27th International Colloquium on Automata, Languages, and Programming, pp. 902–913. Springer, Heidelberg (2000)

    Google Scholar 

  77. Martin-Löf, P.: The definition of random sequences. Information and Control 9, 602–619 (1966)

    Article  MathSciNet  Google Scholar 

  78. Mayordomo, E.: A Kolmogorov complexity characterization of constructive Hausdorff dimension. Information Processing Letters 84, 1–3 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  79. Merkle, W.: The complesity of stochastic sequences. In: Computational Complexity, 2003, pp. 230–235. IEEE, Los Alamitos (2003)

    Google Scholar 

  80. Merkle, W., Miller, J., Nies, A., Reimann, J., Stephan, F.: Nonmonotonic randomness (in preparation)

    Google Scholar 

  81. Merkle, W., Mihailović, N.: On the construction of effective random sets. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, p. 568. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  82. Merkle, W., Mihailovic, N., Slaman, T.: Some results on effective randomness (to appear)

    Google Scholar 

  83. Muchnik, A.A., Semenov, A., Uspensky, V.: Mathematical metaphysics of randomness. Theor. Comput. Sci. 207(2), 263–317 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  84. Miller, J.: Kolmogorov random reals are 2-random (to appear)

    Google Scholar 

  85. Miller, J.: More on randomness (in preparation)

    Google Scholar 

  86. Miller, J., Yu, L.: On initial segment complexity and degrees of randomness (in preparation) (This may turn into two papers)

    Google Scholar 

  87. Miller, W., Martin, D.A.: The degree of hyperimmune sets. Z. Math. Logik Grundlag. Math. 14, 159–166 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  88. von Mises, R.: Grundlagen der Wahrscheinlichkeitsrechnung. Math. Z. 5, 52–99 (1919)

    Article  MathSciNet  Google Scholar 

  89. Moschovakis, Y.N.: Descriptive set theory. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  90. Meng, N.K.: MSc Thesis, National University of Singapore (in preparation)

    Google Scholar 

  91. Nies, A.: Reals which compute little (to appear)

    Google Scholar 

  92. Nies, A.: Lowness properties and randomness (to appear)

    Google Scholar 

  93. Nies, A.: Each Low(CR) set is computable, typeset manuscript (January 2003)

    Google Scholar 

  94. Nies, A.: Low for random sets: the story, preprint, available on http://www.cs.auckland.ac.nz/~nies

  95. Nies, A.: Ideals in the recursively enumerable degrees (in preparation)

    Google Scholar 

  96. Nies, A., Stephan, F., Terwijn, S.A.: Randomness, relativization, and Turing degrees (in preparation)

    Google Scholar 

  97. Odifreddi, P.: Classical recursion theory, vol. 1, Studies in logic and the foundations of mathematics, vol. 125. North-Holland, Amsterdam (1989)

    Google Scholar 

  98. Odifreddi, P.: Classical recursion theory, vol. 2, Studies in logic and the foundations of mathematics, vol. 143. North-Holland, Amsterdam (1999)

    Google Scholar 

  99. Oxtoby, J.: Measure and Category, 2nd edn. Springer, Heidelberg (1980)

    MATH  Google Scholar 

  100. Raichev, A.: Ph.D. thesis, University of Wisconsin-Madison (in progress)

    Google Scholar 

  101. Raisonnier, J.: A mathematical proof of S. Shelah’s theorem on the measure problem and related results. Israel J. Math. 48, 48–56 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  102. Rogers Jr., H.: Theory of recursive functions and effective computability. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  103. Sacks, G.E.: Degrees of unsolvability. Annals of Mathematics Studies 55. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  104. Schnorr, C.: Zufälligkeit und Wahrscheinlichkeit. Lect. Notes in Math., vol. 218. Springer, Heidelberg (1971)

    Book  MATH  Google Scholar 

  105. Shannon, C.: The mathematical theory of communication. Bell System Tech. J. 27, 379–423, 623–656 (1948)

    Google Scholar 

  106. Shen, A.K.: The frequency approach to the definition of the notion of a random sequence (in Russian). Semiotika i Informatika 18, 14–41 (1982)

    MATH  Google Scholar 

  107. Shen, A.K.: On relations between different algorithmic definitions of randomness. Soviet Mathematics Doklady 38, 316–319 (1988)

    Google Scholar 

  108. Soare, R.: Recursively enumerable sets and degrees. Springer, Heidelberg (1987)

    Google Scholar 

  109. Soare, R.: Computability Theory and Differential Geometry, bulletin Symbolic Logic (to appear)

    Google Scholar 

  110. Solomonoff, R.J.: A preliminary report on a general theory of inductive inference, Tech. Report ZTB-138, Zator Company, Cambridge, Mass. (November 1960)

    Google Scholar 

  111. Solovay, R.: A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics 92, 1–56 (1970)

    Article  MathSciNet  Google Scholar 

  112. Solovay, R.: Draft of a paper (or series of papers) on Chattin’s work, unpublished manuscript. IBM Thomas J. Watson Research Center, New York, p. 215 (May 1975)

    Google Scholar 

  113. Staiger, L.: Constructive dimension equals Kolmogorov complexity, Research Report CDMTCS-210, University of Auckland (January 2003)

    Google Scholar 

  114. Stephan, F.: personal communication

    Google Scholar 

  115. Stephan, F.: Martin-Löf random and PA-complete sets, Forschungsbericht Mathematische Logik 58/2002, Universit”at Heidelberg (2002)

    Google Scholar 

  116. Terwijn, S.: Computability and Measure, PhD thesis, University of Amsterdam/ ILLC (1998)

    Google Scholar 

  117. Terwijn, S.: On the quantitative structure of \(\Delta^{0}_{2}\). In: Berger, U., Osswald, H., Schuster, P. (eds.) Reuniting the Antipodes — Constructive and Nonstandard Views of the Continuum, Synthèse Library, pp. 271–284. Kluwer, Dordrecht (2000)

    Google Scholar 

  118. Terwijn, S.: Complexity and Randomness, Notes for a course given at the University of Auckland (March 2003), report CDMTCS-212, University of Auckland

    Google Scholar 

  119. Terwijn, S., Zambella, D.: Computational randomness and lowness. Journal of Symbolic Logic 66(3), 1199–1205 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  120. Tadaki, K.: A generalization of Chaitin’s halting probability Ω and halting selfsimilar sets. Hokkaido Math. J. 32, 219–253 (2002)

    MathSciNet  Google Scholar 

  121. Ville, J.: Étude critique de la notion de collectif, Gauthier-Villars (1939)

    Google Scholar 

  122. Wang, Y.: Randomness and complexity, PhD thesis, University of Heidelberg (1996)

    Google Scholar 

  123. Wang, Y.: A separation of two randomness concepts. Information Processing Letters 69(3), 115–118 (1999)

    Article  MathSciNet  Google Scholar 

  124. Liang, Y., Decheng, D.: There is no sw-complete c.e. real. Journal of Symbolic Logic (to appear)

    Google Scholar 

  125. Liang, Y., Decheng, D.: The initial segment complexity of random reals. Proc. Amer. math. Soc. (to appear)

    Google Scholar 

  126. Liang, Y., Decheng, D., Downey, R.: The Kolmogorov complexity of random reals. Annals of Pure and Applied Logic (to appear)

    Google Scholar 

  127. Zambella, D.: On sequences with simple initial segments, ILLC technical report ML-1990-05, University of Amsterdam (1990)

    Google Scholar 

  128. Zheng, X.: On the Turing degrees of weakly computable real numbers. Journal of Logic and Computation 13, 159–172 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  129. Zvonkin, A., Levin, L.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Math. Surveys 25(6), 83–124 (1970)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical and Computing Sciences, Victoria University, PO Box 600, Wellington, New Zealand

    Rod Downey

Authors
  1. Rod Downey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Theoretical Computer Science and Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Jiří Fiala  & Jan Kratochvíl  & 

  2. Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Malostranske nam. 25, 118 00, Praha1, Czech Republic

    Václav Koubek

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Downey, R. (2004). Some Recent Progress in Algorithmic Randomness. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds) Mathematical Foundations of Computer Science 2004. MFCS 2004. Lecture Notes in Computer Science, vol 3153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28629-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-28629-5_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22823-3

  • Online ISBN: 978-3-540-28629-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation