Mathematical metaphysics of randomness

Theoretical Computer Science

Volumn 207, Issue 2, 1998, Pages 263-317

  Muchnik, Andrei A ^a   Semenov, Alexei L ^a   Uspensky, Vladimir A ^b  

^a Institute of New Technologies   (Russian Federation)

^b MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Chaotic sequence; Lawless sequence; Random sequences; Stochastic sequence; Unpredictable sequence

Indexed keywords

EID: 0009893308     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(98)00069-3     Document Type: Review

References (31)

1. P.J. Cohen, Set Theory and the Continuum Hypothesis, W.A. Benjamin, New York, 1966.
2. P. Gács, Exact expressions for some randomness tests, Zeitschr. f. math. Logik und Grundlagen d. Math. 26 (1980) 385-394.
3. P. Gács, Every sequence is reducible to a random one, Inform. and Control 70 (2/3) (1986) 186-192.
4. S. Kakutani, On the equivalence of infinite product measures, Ann. Math. 49 (1948) 214-224.
5. A.N. Kolmogorov, On tables of random numbers, Sankhyā. The Indian J. Statist. Ser. A. 25 (4) (1963) 369-376.
6. A.N. Kolmogorov, Logical basis for information theory and probability theory, IEEE Trans. Inform. Theory IT-14 (1968) 662-664.
7. A.N. Kolmogorov, On the logical foundations of information theory and probability theory, Problems of Inform. Transmission. 5 (3) (1969) 1-4.
8. A.N. Kolmogorov, V.A. Uspensky, Algorithms and randomness, Proc. 1st World Congr. of the Bernoulli Soc. Tashkent, 1986. Vol 1: Probability Theory and Applications, VNU Science Press, Utrecht, 1987, pp. 3-53.
9. A.N. Kolmogorov, V.A. Uspensky, Algorithms and randomness, Theory of Probability and its Applications 32 (1987) 389-412.
10. L.A. Levin, On the notion of a random sequence, Soviet Math. Doklady 14 (1973) 1413-1416.
11. P. Martin-Löf, The definition of random sequences, Inform. and Control 9 (6) (1966) 602-619.
12. P. Martin-Löf, On the notion of randomness, in: A. Kino, J. Myhill, R.E. Vesley (Eds.), Intuitionism and Proof Theory, New York, 1968, pp. 73-78.
13. P. Martin-Löf, Notes on Constructive Mathematics, Almqvist & Wiksell, Stockholm, 1970.
14. R. von Mises, Grundlagen der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 5 (1919) 52-89.
15. R. von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, Springer, Wien, 1928.
16. R. von Mises, Probability, Statistics and Truth, 2d rev., English ed., Prep by H. Geiringer, Allen London; Macmillan, New York, 1957.
17. A.Kh. Shen', The concept of (α, α)-stochasticity in the Kolmogorov sense, and its properties, Soviet Math. Dokl. 28 (1) (1983) 295-299.
18. C.P. Schnorr, Process complexity and effective random tests, J. Comput. System Sci. 7 (1973) 376-388.
19. V.A. Uspensky, Complexity and entropy: an introduction to the theory of Kolmogorov complexity, in: Watanabe (Ed.), Kolmogorov Complexity and Computational Complexity, Springer, Berlin, 1992, 85-102.
20. V.A. Uspensky, A.L. Semenov, Theory of Algorithms: Main Discoveries and Applications, "Nauka", Moscow, 1987 (Russian). English translation titled "Algorithms: Main Ideas and Applications" is published by Kluwer Academic Publishers, Dordrecht, 1993.
21. V.A. Uspensky, A.L. Semenov, Theory of Algorithms: Main Discoveries and Applications, "Nauka", Moscow, 1987 (Russian). English translation titled "Algorithms: Main Ideas and Applications" is published by Kluwer Academic Publishers, Dordrecht, 1993.
22. V.A. Uspensky, A.L. Semenov, A.Kh. Shen', Can an individual sequence of zeros and ones be random?, Russian Math. Surveys 45 (1) (1990) 121-189.
23. V.A. Uspensky, A.Kh. Shen', Relations between varieties of Kolmogorov complexities, Math. Systems Theory 29 (1996) 271-292. (Also CWI Tech. Rep. CS-R9329.)
24. V.V. V'yugin, Algorithmic entropy (complexity) of finite objects and its applications to defining randomness and amount of information, Semiotica i Informatika 16 VINITI, Moscow (1981) 14-43 (in Russian).
25. V.V. V'yugin, Algorithmic entropy (complexity) of finite objects and its applications to defining randomness and amount of information, Selecta Mathematica formerly Sovietica 13 (4) (1994) 357-389.
26. V.V. V'yugin, Nonstochastic objects, Problems of Information Transmission 21 (2) (1985) 77-83.
27. A. Wald, Sur la notion de collectif dans le calcul des probabilités, C. R. Acad. Sci. Paris. 202 (1936) 180-183.
28. A. Wald, Die Widerspruchsfreiheit des Kollektivbegriffes der Wahrscheinlichkeitsrechnung, Ergebnisse eines Math. Kolloquiuns Vienna 8 (1937) 38-72.
29. A. Wald, Die widerspruchsfreiheit des kollektivbegriffes, Actualités Sci. Indust. 735 (1938) 79-99.
30. O. Watanabe (Ed.), Kolmogorov Complexity and Computational Complexity, Springer, Berlin, 1992.
31. A.K. Zvonkin, L.A. Levin, 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) (1970) 83-124.