Kolmogorov Complexity for Possibly Infinite Computations
Tóm tắt
In this paper we study the Kolmogorov complexity for non-effective computations, that is, either halting or non-halting computations on Turing machines. This complexity function is defined as the length of the shortest input that produce a desired output via a possibly non-halting 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 prefix-free 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.
Tài liệu tham khảo
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: Springer-Verlag.
Chaitin, G.J., 1975, A theory of program-size formally identical to information theory, Journal of the ACM 22, 329–340.
Chaitin, G., 1976a, “Algorithmic entropy of sets,” Computers & Mathematics with Applications 2, 233–245.
Chaitin, G.J., 1976b, “Information-theoretical characterizations of recursive infinite strings,” Theoretical Computer Science 2: 45–48.
Ferbus-Zanda, M. and Grigorieff, S., 2004, “Kolmogorov complexities Kmax, Kmin” (submitted).
Katseff, H.P. and Sipser, M., 1981, “Several results in program-size complexity,” Theoretical Computer Science 15, 291–309.
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 (non-growth) 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 Non-Classical Logics, Model Theory, and Computability, A.I. Arruda, N.C.A. da Costa, and R. Chuaqui, eds., Amsterdam: North-Holland.