Geodesic
Kolmogorov complexity, pseudorandom generators and statistical models testing
Kybernetika, Tome 38 (2002) no. 6, pp. 747-759 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

An attempt to formalize heuristic concepts like strings (sequences resp.) “typical” for a probability measure is stated in the paper. Both generating and testing of such strings is considered. Kolmogorov complexity theory is used as a tool. Classes of strings “typical” for a given probability measure are introduced. It is shown that no pseudorandom generator can produce long strings from the classes. The time complexity of pseudorandom generators with oracles capable to recognize “typical” strings is shown to be at least exponential with respect to the length of the output. Tests proclaiming some strings “typical” are introduced. We show that the problem of testing strings to be “typical” is undecidable. As a consequence, the problem of correspondence between probability measures and data is undecidable too. If the Lebesgue measure is considered, then the conditional probability of failure of a test is shown to exceed a positive lower bound almost surely.
An attempt to formalize heuristic concepts like strings (sequences resp.) “typical” for a probability measure is stated in the paper. Both generating and testing of such strings is considered. Kolmogorov complexity theory is used as a tool. Classes of strings “typical” for a given probability measure are introduced. It is shown that no pseudorandom generator can produce long strings from the classes. The time complexity of pseudorandom generators with oracles capable to recognize “typical” strings is shown to be at least exponential with respect to the length of the output. Tests proclaiming some strings “typical” are introduced. We show that the problem of testing strings to be “typical” is undecidable. As a consequence, the problem of correspondence between probability measures and data is undecidable too. If the Lebesgue measure is considered, then the conditional probability of failure of a test is shown to exceed a positive lower bound almost surely.
Classification : 60A10, 65C10, 68Q30
Keywords: Kolmogorov complexity; typical string; pseudorandom generator 
@article{KYB_2002_38_6_a5,
     author = {\v{S}indel\'a\v{r}, Jan and Bo\v{c}ek, Pavel},
     title = {Kolmogorov complexity, pseudorandom generators and statistical models testing},
     journal = {Kybernetika},
     pages = {747--759},
     year = {2002},
     volume = {38},
     number = {6},
     mrnumber = {1954395},
     zbl = {1265.68083},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2002_38_6_a5/}
}
TY  - JOUR
AU  - Šindelář, Jan
AU  - Boček, Pavel
TI  - Kolmogorov complexity, pseudorandom generators and statistical models testing
JO  - Kybernetika
PY  - 2002
SP  - 747
EP  - 759
VL  - 38
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/KYB_2002_38_6_a5/
LA  - en
ID  - KYB_2002_38_6_a5
ER  - 
%0 Journal Article
%A Šindelář, Jan
%A Boček, Pavel
%T Kolmogorov complexity, pseudorandom generators and statistical models testing
%J Kybernetika
%D 2002
%P 747-759
%V 38
%N 6
%U http://geodesic.mathdoc.fr/item/KYB_2002_38_6_a5/
%G en
%F KYB_2002_38_6_a5
Šindelář, Jan; Boček, Pavel. Kolmogorov complexity, pseudorandom generators and statistical models testing. Kybernetika, Tome 38 (2002) no. 6, pp. 747-759. http://geodesic.mathdoc.fr/item/KYB_2002_38_6_a5/

[1] Calude C.: Theories of Computational Complexity. North–Holland, Amsterdam 1988 | MR | Zbl

[2] Fine T. L.: Theories of Probability – an Examination of Foundations. Academic Press, New York 1973 | MR | Zbl

[3] Kolmogorov A. N.: Three approaches to the quantitative definition of information. Problems Inform. Transmission 1 (1965), 1, 1–7 | MR

[4] Kramosil I., Šindelář J.: A note on the law of iterated logarithm from the viewpoint of Kolmogorov program complexity. Problems Control Inform. Theory 16 (1987), 6, 399–409 | MR

[5] Kramosil I., Šindelář J.: On pseudo-random sequences and their relation to a class of stochastical laws. Kybernetika 28 (1991), 6, 383–391 | MR

[6] Li M., Vitayi P.: Introduction to Kolmogorov Complexity and its Applications. Springer, New York 1997 | MR

[7] Martin-Löf P.: The definition of random sequences. Inform. and Control 9 (1966), 602–619 | DOI | MR

[8] Rogers H., Jr.: Theory of Recursive Functions and Effective Computability. McGraw–Hill, New York 1967 | MR | Zbl

[9] Šindelář J., Boček P.: Kolmogorov complexity and probability measures. Kybernetika 38 (2002), 729–745 | MR