Kolmogorov Complexity for Possibly Infinite Computations
 Verónica Becher,
 Santiago Figueira
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
In this paper we study the Kolmogorov complexity for noneffective computations, that is, either halting or nonhalting computations on Turing machines. This complexity function is defined as the length of the shortest input that produce a desired output via a possibly nonhalting computation. Clearly this function gives a lower bound of the classical Kolmogorov complexity. In particular, if the machine is allowed to overwrite its output, this complexity coincides with the classical Kolmogorov complexity for halting computations relative to the first jump of the halting problem. However, on machines that cannot erase their output –called monotone machines–, we prove that our complexity for non effective computations and the classical Kolmogorov complexity separate as much as we want. We also consider the prefixfree complexity for possibly infinite computations. We study several properties of the graph of these complexity functions and specially their oscillations with respect to the complexities for effective computations.
 Becher, V., Daicz, S., and Chaitin, G., 2001, “A highly random number,” pp. 55–68 in Combinatorics, Computability and Logic: Proceedings of the Third Discrete Mathematics and Theoretical Computer Science Conference (DMTCS’01), C.S. Calude, M.J. Dineen, and S. Sburlan, eds., London: SpringerVerlag.
 Chaitin, G.J., 1975, A theory of programsize formally identical to information theory, Journal of the ACM 22, 329–340. CrossRef
 Chaitin, G., 1976a, “Algorithmic entropy of sets,” Computers & Mathematics with Applications 2, 233–245.
 Chaitin, G.J., 1976b, “Informationtheoretical characterizations of recursive infinite strings,” Theoretical Computer Science 2: 45–48. CrossRef
 FerbusZanda, M. and Grigorieff, S., 2004, “Kolmogorov complexities K _{max}, K _{min}” (submitted).
 Katseff, H.P. and Sipser, M., 1981, “Several results in programsize complexity,” Theoretical Computer Science 15, 291–309. CrossRef
 Kolmogorov, A.N., 1965, “Three approaches to the quantitative definition of information,” Problems of Information Transmission 1, 1–7.
 Levin, L.A., 1974, “Laws of information conservation (nongrowth) and aspects of the foundations of probability theory,” Problems of Information Transmission 10, 206– 210.
 Li, M. and Vitányi, P. 1997, An Introduction to Kolmogorov Complexity and its Applications (2nd edition), Amsterdam: Springer.
 Loveland, D.W., 1969, A Variant of the Kolmogorov Concept of Complexity, Information and Control (15), 510–526.
 Shoenfield, J.R.M., 1959, “On degrees of unsovability,” Annals of Mathematics 69, 644–653.
 Solovay, R.M., 1977, “On random r.e. sets,” pp. 283–307 in NonClassical Logics, Model Theory, and Computability, A.I. Arruda, N.C.A. da Costa, and R. Chuaqui, eds., Amsterdam: NorthHolland.
 Title
 Kolmogorov Complexity for Possibly Infinite Computations
 Journal

Journal of Logic, Language and Information
Volume 14, Issue 2 , pp 133148
 Cover Date
 20050301
 DOI
 10.1007/s1084900522556
 Print ISSN
 09258531
 Online ISSN
 15729583
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 infinite computations
 Kolmogorov complexity
 monotone machines
 noneffective computations
 programsize complexity
 Turing machines
 Authors

 Verónica Becher ^{(1)}
 Santiago Figueira ^{(1)}
 Author Affiliations

 1. Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina