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Randomness, Unpredictability and Absence of Order: The Identification by the Theory of Recursivity of the Mathematical Notion of Random Sequence

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Philosophy of Probability

Part of the book series: Philosophical Studies Series ((PSSP,volume 56))

Abstract

The theory of recursivity which was initiated by Gödel, Church, Turing and Post between 1930 and 1936 leads 30 years later to an absolute definition of randomness that seems to fulfil the main objectives stated by von Mises. The definition of random sequences by Martin-Löf in 1965 and the other works on the so-called ‘algorithmic theory of information’ by Kolmogorof, Chaitin, Schnorr and Levin (among others) may be understood as the formulation of a thesis similar to the Church-Turing’s Thesis about the notion of algorithmic calculability. Here is this new thesis we call the Martin-Löf-Chaitin’s Thesis: the intuitive informal concept of random sequences (of 0 and 1) is satisfactorily defined by the notion of Martin-Löf-Chaitin random sequences (MLC-random sequences) that is, sequences which do not belong to any recursively null set. In this paper (a short version of [Delahaye 1990]), we first recall and explain shortly the notion of MLC-random sequences; and propose afterwards a comparison between the Church-Turing’s Thesis and the Martin-Löf-Chatin’s Thesis. Our conclusion is that there is a huge similarity between the two thesis, but that today the Martin-Löf-Chaitin’s Thesis is more problematic and more complex than the Church-Turing’s Thesis.

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Bibliography

  1. C. H. Bennett. Logical Depth and Physical Complexity. In “The Universal Turing Machine: A Half-Century Survey”. Edited by R. Herken. Oxford University Press. 1988. pp. 227–257.

    Google Scholar 

  2. E. Borel. Presque tous les nombres réels sont normaux. Rend. Cir. Mat. Palermo. 27. 1909. pp. 247–271.

    Article  Google Scholar 

  3. G. J. Chaitin. On the Length of Programs for Computing Finite Binary Sequences. J.A.C.M. 13. 1966. pp. 547–569. Also in [Chaitin 1987a].

    Google Scholar 

  4. G. J. Chaitin. On the Length of Programs for Computing Finite Binar Sequences, Statistical Considerations. J.A.C.M. 16. 1969. pp. 145–159. Also in [Chaitin 1987a].

    Google Scholar 

  5. G. L. Chaitin. Information Theoretic Limitations of Formal Systems. J.A.C.M. 1974. pp. 403–424. Also in [Chaitin 1987a].

    Google Scholar 

  6. G. L. Chaitin. A theory of program size formally identical to information theory. J.A.C.M. 22. 1975. pp. 329–340. Also in [Chaitin 1987a].

    Google Scholar 

  7. G. J. Chaitin. Randomness and Mathematical Proof. Scientific American. 232. May 1975. pp. 47–52.

    Article  Google Scholar 

  8. G. J. Chaitin. Algorithmic Information Theory. IBM Journal of Research and Development. 31. 1977. pp. 350–359. Also in [Chaitin 1987a].

    Article  Google Scholar 

  9. G. J. Chaitin. Toward a mathematical definition of “life”. In “The Maximum Entropy Formalism”. R. D. Levine and M. Tribus (eds.). MIT Press. 1979. pp. 477–498. Also in [Chaitin 1987a].

    Google Scholar 

  10. G. J. Chaitin. Randomness and Gödel’s Theorem. IBM Research R. RC 11582. 1985. Also in [Chaitin 1987a].

    Google Scholar 

  11. G. J. Chaitin. Information, Randomness and Incompleteness: Papers on Algorithmic Information Theory. World Scientific, Singapore, 1987.

    Book  Google Scholar 

  12. G. J. Chaitin. Algorithmic information theory. Cambridge Tracts in Theoretical Computer Science 1. Cambridge University Press, New York, 1987.

    Book  Google Scholar 

  13. G. J. Chaitin. Incompleteness Theorems for Random Reals. Advances in Applied Mathematics. 8. 1987. pp. 119–146. Also in [Chaitin 1987a].

    Article  Google Scholar 

  14. G. J. Chaitin. Randomness in Arithmetic. Scientific American. July 1988. pp. 80–85.

    Google Scholar 

  15. G. J. Chaitin and J. T. Schwartz. A note on Monte Carlo Primality Tests and Algorithmic Information Theory. Communication rcn Pure and Applied Mathematics. 31. 1978. pp.521–527. Also in [Chaitin 1987a].

    Article  Google Scholar 

  16. A. Church. On the concept of a random sequence. Bulletin Amer. Math. Soc. 46. 1940. pp. 130–135.

    Google Scholar 

  17. T. M. Cover, P. Gacs, R. M. Gray. Kolmogorof s contributions to information theory and algorithmic complexity. The Annals of Probability, Vol 17, n°3, 1989, pp. 840–865.

    Article  Google Scholar 

  18. R. P. Daley. Minimal-Program Complexity of Pseudo-Recursive and Pseudo-Random Sequences. Mathematical System Theory, Vol. 9, n°l, 1975. pp. 83–94.

    Article  Google Scholar 

  19. R. P. Daley. Noncomplex Sequences: Characterizations and Examples. J. Symbol. Logic. 41. 1976. pp. 626–636.

    Article  Google Scholar 

  20. J. P. Delahaye. Une Extension Spectaculaire duThéorème de Gödel. La Recherche n°200, juin 1988. pp. 860–862.

    Google Scholar 

  21. J. P. Delahaye. Cinq Classes d’Idées. Rapport Laboratoire d’Informatique Fondamentale de Lille. Univ. Sc. Lille, Bât M3, 59655 Villeneuve d’Ascq. Avril 1989.

    Google Scholar 

  22. J.-P. Delahaye. Chaitin’s Equation: An Extension of GödePs Theorem. Notices of The American Mathematical Society. October 1989. pp. 984–987.

    Google Scholar 

  23. J.-P. Delahaye. Le hasard comme imprevisibilité et comme absence d’ordre. Rapport du Laboratoire d’Informatique Fondamentale de Lille. Université de Sci. et Tech, de Lille 59655 Villeneuve d’Ascq cédex F, 1990.

    Google Scholar 

  24. J. E. Dies. Information et Complexité. Annales de L’Institut Henri Poincaré, Section B, Calcul des Probabilités et Statistiques. Nouvelle Série Vol. 12. 1976. pp. 365–390

    Google Scholar 

  25. J. E. Dies. Information et Complexité. Annales de L’Institut Henri Poincaré, Section B, Calcul des Probabilités et Statistiques. Nouvelle Série Vol. 14. 1978. pp. 113–118.

    Google Scholar 

  26. P. Gacs. On the symmetry of algorithmic information. Dokl. Akad. Nauk SSSR. 218. 1974. pp. 1477–1480.

    Google Scholar 

  27. P. Gacs. Exact Expressions for Some Randomness Tests. Z. Math. Logik. Grundl. Math. 26. 1980.

    Google Scholar 

  28. P. Gacs. On the relation between descriptional complexity and probability. Theo. Comp. 22. 1983. pp. 71–93.

    Article  Google Scholar 

  29. P. Gacs. Every Sequence Is Reducible to a Random One. Information and Control. 70. 1986. pp. 186–192.

    Article  Google Scholar 

  30. H. Gaifman, M. Snir. Probabilities over rich languages, randomness and testing. Journal of Symbolic Logic. Vol. 47. 1982. pp. 495–548.

    Article  Google Scholar 

  31. M. Gardner. The random number 0 bids fair to hold the mysteries of the universe. Scientific American 241, No. 5 Nov. 1979. pp. 20–34.

    Article  Google Scholar 

  32. A. N. Kolmogorof. Osnovye ponyatiya teorii veroyatnostei. ONTI. Moscow, 1936. Traduction anglaise: Foundations of the theory of probability. Chelsea, New York. 1950.

    Google Scholar 

  33. A. N. Kolmogorof. On table of random numbers. Sankhya The Indian Journal of Statistics. A25. 369. 1963. pp. 369–376.

    Google Scholar 

  34. A. N. Kolmogorof. Three approaches for defining the concept of information quantity. Information Transmission. V. 1. 1965. pp. 3–11.

    Google Scholar 

  35. A. N. Kolmogorof. Logical basis for Information Theory and Probability Theory. IEEE Transaction on Information Theory. Vol. IT14, n°5. 1968. pp. 662–664.

    Article  Google Scholar 

  36. A. N. Kolmogorof. Some Theorems on algorithmic entropy and the algorithmic quantity of information. Uspeki Mat. Nauk, Vol. 23:2. 1968. pp. 201.

    Google Scholar 

  37. A. N. Kolmogorof. Combinatorial foundations of information theory and the calculus of probabilities. Russian Mathematical Surveys. Vol. 38.4. 1983. pp. 29–40.

    Article  Google Scholar 

  38. A. N. Kolmogorof. On logical foundations of probability theory. In “Probability Theory and Mathematical Statistics. Lecture Notes in Mathematics.” Ed. K. Ito and J. V. Prokhorov. Vol 1021. Springer-Verlag., Berlin. 1983. pp. 1–5.

    Chapter  Google Scholar 

  39. A. N. Kolmogorof and V. A. Uspenskii. Algorithms and Randomness. SIAM Theory Probab. Appl. Vol. 32. 1987. pp. 389–412.

    Article  Google Scholar 

  40. L. A. Levin. On the notion of random sequence. Dokl. Akad. Nauk SSSR 212, 3. 1973.

    Google Scholar 

  41. L. A. Levin. Laws of information conservation (non-growth) and aspects of the foundation of probability theory. Problems Inform. Transmission. 10 n°3. 1974. pp. 206–210.

    Google Scholar 

  42. L. A. Levin. On the principle of conservation of information in intuitionistic mathematics. Dokl. Akad. Nauk. SSSR Tom 227 n°6. 1976.

    Google Scholar 

  43. L. A. Levin. On the principle of conservation of information in intuitionistic mathematics. Soviet Math. Dok. 17 n°2. 1976. pp. 601–605.

    Google Scholar 

  44. L. A. Levin. Various measures of complexity for finite objects (axiomatic descriptions). Soviet Math. Dokl. 17.n°2. 1976. pp. 522–526.

    Google Scholar 

  45. L. A. Levin. Uniform tests of randomness. Soviet Math. Dokl. 17, n°2. 1976. pp. 337–340.

    Google Scholar 

  46. L. A. Levin. Randomness conservative inequalities: Information and independence in mathematical theories. Inf. Contr. 61. 1984. pp. 15–37.

    Article  Google Scholar 

  47. L. A. Levin and V. V. V’Yugin. Invariant Properties of Informational Bulks. Lecture Notes in Computer Science n°53. Springer, Berlin. 1977. pp. 359–364.

    Google Scholar 

  48. M. Li, P.M.B. Vitanyi. A New Approach to Formal Language Theory by Kolmogorof Complexity. Proc 16th International Colloquium on Automata Languages and Programming. 1989.

    Google Scholar 

  49. M. Li, P.M.B. Vitanyi. Inductive Reasoning and Kolmogorof Complexity. Proc. 4th Annual IEEE Structure in Complexity Theory Conference. 1989.

    Google Scholar 

  50. M. Li, P.M.B. Vitanyi. Kolmogoroff Complexity and Its Applications. Handbook of Theoretical Computer Science. J. van Leeuwen Editor. North-Holland. 1990.

    Google Scholar 

  51. M. Li, P.M.B. Vitanyi. Introduction to Kolmogorof Complexity and Its Applications. Addison-Wesley, Reading, Mass. To appear.

    Google Scholar 

  52. D. W. Loveland. A new interpretation of the von Mises’ Concept of random sequence. Zeitschr. F. Math. Logik und Grundlagen d. Math. Bd l2. 1966. pp. 279–294.

    Google Scholar 

  53. D. W. Loveland. The Kleene Hierarchy Classification of Recusively Random Sequences. Trans. Amer. Math. Soc. 125. 1966. pp. 487–510.

    Article  Google Scholar 

  54. D. W. Loveland. Minimal Program Complexity Measure. Conference Record ACM Symposium on Theory of Computing. May 1968. pp. 61–65.

    Google Scholar 

  55. D. W. Loveland. A variant of the Kolmogorof concept of complexity. Information and Control. 15. 1969. pp. 510–526.

    Article  Google Scholar 

  56. P. Martin-Löf. On the Concept of a Random Sequence. Theory Probability Appl. Vol 11 1966. pp. 177–179.

    Google Scholar 

  57. P. Martin-Lof. The Definition of Random Sequences. Information and Control. 9. 1966. pp. 602–619.

    Article  Google Scholar 

  58. P. Martin-Löf. Algorithms and Randomness. Intl. Stat. Rev. 37, 265. 1969. pp. 265–272.

    Article  Google Scholar 

  59. P. Martin-Löf. The Literature on von Mises’ Kollektivs Revisited. Theoria, XXXV. 1969. pp. 12–37.

    Google Scholar 

  60. P. Martin-Löf. On the notion of Randomness, in “Intuitionism and Proof Theory”. A. Kino, J. Myhill and R. E. Vesley, eds. North-Holland Publishing Co. Amsterdam. 1970, pp. 73–78.

    Google Scholar 

  61. P. Martin-Löf. Complexity Oscillations in Infinite Binary Sequences. Zeitschrift fur Wahrscheinlichkeitstheory und Vervandte Gebiete. 19. 1971. pp. 225–230.

    Article  Google Scholar 

  62. P. Martin-Löf. The notion of redundancy and its use as a quantitative measure of the discrepancy between statistical hypothesis and a set of observational data. Scand. J. Stat. Vol. 1. 1974. pp. 3–18.

    Google Scholar 

  63. M. O’Connor. An Unpredictibility Approach to Finite-State Randomness. Journal of Computer and System Sciences. 37. 1988. pp. 324–336.

    Article  Google Scholar 

  64. K. R. Popper. Logik der Forschund. Springer. 1935. Traduction Francaise: La Logique de la Découverte Scientifique. Payot, Paris. 1978.

    Google Scholar 

  65. J. Rissanen. Stochastic Complexity in Statistical Inquiry. World Scientific. Series in Computer Science. Vol 15. 1989.

    Google Scholar 

  66. C. P. Schnorr. A unified approach to the definition of random sequence. Math. Systems Theory. 5. 1971. pp. 246–258.

    Article  Google Scholar 

  67. C. P. Schnorr. Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics. Vol 218. Berlin-Heidelberg-New York. Springer, 1971.

    Book  Google Scholar 

  68. C. P. Schnorr. The process complexity and effective random tests. Proc. ACM Conf. on Theory of Computing. 1972. pp. 168–176.

    Google Scholar 

  69. C. P. Schnorr. Process complexity and effective random tests. J. Comput. Syst. Sci. 7. 1973. pp. 376–388.

    Article  Google Scholar 

  70. C. P. Schnorr. A survey of the theory of random sequences. In “Basic Problems in Mathodology and Linguistics.” Ed. R. E. Butts, J. Hintikka. D. Reidel, Dordrecht. 1977. pp. 193–210.

    Google Scholar 

  71. A. Kh. Shen’. The concept of (α, β)-stochasticity in the Kolmogorof sense, and its properties. Soviet Math. Dokl. Vol. 28. 1983. pp. 295–299.

    Google Scholar 

  72. A. Kh. Shen\ On relation between different algorithmic definitions of randomness. Soviet Math. Dokl. Vol. 38. 1989. pp. 316–319.

    Google Scholar 

  73. R. J. Solomonoff. A formal theory of inductive Inference. Information and Control. 7. 1964. pp. 1–22.

    Article  Google Scholar 

  74. A. M. Turing. On Computable Numbers, with an application to the Entscheidungsproblem. Proceeding of the London Mathematical Society. 2, 42, 1936–7 pp. 230–265.

    Google Scholar 

  75. A. M. Turing. On Computable Numbers, with an application to the Entscheidungsproblem. Corrections 43. 1937. pp. 544–546.

    Google Scholar 

  76. M. van Lambalgen. Von Mises’ definition of random sequences reconsidered. Journal of Symbolic Logic. Vol 52. 1987. pp. 725–755.

    Article  Google Scholar 

  77. J. Ville. Sur la notion de collectif. C. R. Acad. Scien. Paris. 203. 1936. pp. 26–27.

    Google Scholar 

  78. J. Ville. Sur les suites indifférentes. C. R. Acad. Scien. Paris. 202. 1936. p. 1393.

    Google Scholar 

  79. J. Ville. Etude critique de la notion de collectif. Gauthier-Villars. Paris. 1939.

    Google Scholar 

  80. R. von Mises. Grundlagen der Wahrscheinlichkietsrecnnung. Math. Z. 5. 100. 1919.

    Article  Google Scholar 

  81. R. von Mises. On the foundation of probability and statistics. Am. Math. Statist. 12. 1941. pp. 191–205.

    Article  Google Scholar 

  82. R. von Mises. Selected papers of Richard von Mises. Providence, Rhode Island, Amer. Math. Soc. 1964.

    Google Scholar 

  83. A. Wald. Die Widerspruchsfreiheit des Kollektivbegriffes der Wahrscheinlichkeitsrechnung. Ergebnisse eines Mathetischen Kolloquiums. 8. 1937. pp. 38–72.

    Google Scholar 

  84. A. Wald. Die Widerspruchsfreiheit des Kollektivgriffes. Actualités Sci. Indust. 735. 1938. pp. 79–99.

    Google Scholar 

  85. A. K. Zvonkin, L. A. Levin. The Complexity of finite object and the development of the concepts of information and randomness by means of the theory of algorithms. Russ. Math. Survey. 25, 6. 1970. pp. 83–124.

    Article  Google Scholar 

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  1. Lille University, France

    Jean-Paul Delahaye

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  1. Jean-Paul Delahaye
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  1. University of Paris I — Sorbonne and C.N.R.S., France

    Jacques-Paul Dubucs

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Delahaye, JP. (1993). Randomness, Unpredictability and Absence of Order: The Identification by the Theory of Recursivity of the Mathematical Notion of Random Sequence. In: Dubucs, JP. (eds) Philosophy of Probability. Philosophical Studies Series, vol 56. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8208-7_8

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  • DOI: https://doi.org/10.1007/978-94-015-8208-7_8

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