Kolmogorov Complexity and Computational Complexity by Osamu Watanabe

By Osamu Watanabe

The mathematical idea of computation has given upward push to 2 very important ap­ proaches to the casual thought of "complexity": Kolmogorov complexity, usu­ best friend a complexity degree for a unmarried item reminiscent of a string, a series etc., measures the volume of knowledge essential to describe the article. Compu­ tational complexity, frequently a complexity degree for a suite of items, measures the compuational assets essential to realize or produce parts of the set. The relation among those complexity measures has been thought of for greater than 20 years, and will attention-grabbing and deep observations were got. In March 1990, the Symposium on concept and alertness of minimum­ size Encoding was once held at Stanford collage as part of the AAAI 1990 Spring Symposium sequence. a few periods of the symposium have been devoted to Kolmogorov complexity and its kinfolk to the computational complexity the­ ory, and perfect expository talks got there. Feeling that, as a result significance of the cloth, a way will be came upon to proportion those talks with researchers within the computing device technological know-how group, I requested the audio system of these periods to jot down survey papers in response to their talks within the symposium. In reaction, 5 audio system from the periods contributed the papers which seem during this book.

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Extra resources for Kolmogorov Complexity and Computational Complexity

Example text

Ko. On the notion of infinite pseudorandom sequences: Theoret. Comput. Sci. 39:9-33, 1986. K. Ko and U. Schoning. On circuit-size complexity and the low hierarchy in N P. SIAM J. Computing 14:41-51, 1985. S. Kurtz. On the random oracle hypothesis. Info. and Control 57:40-47, 1983. R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial-time reducibilities. Theoret. Comput. Sci. 1:103-123, 1975. M. Li and P. Vitanyi. Applications of Kolmogorov complexity in the theory of computation. Complexity Theory Retrospective, A.

Comput. Sci. 39:9-33, 1986. K. Ko and U. Schoning. On circuit-size complexity and the low hierarchy in N P. SIAM J. Computing 14:41-51, 1985. S. Kurtz. On the random oracle hypothesis. Info. and Control 57:40-47, 1983. R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial-time reducibilities. Theoret. Comput. Sci. 1:103-123, 1975. M. Li and P. Vitanyi. Applications of Kolmogorov complexity in the theory of computation. Complexity Theory Retrospective, A. ), Springer-Verlag Publ. Co. 147-203,1990.

A set A is self-p-printable if and only if A E K A[log, poly], that is, there is a universal oracle machine U and constants c and k with the property On Sets with Small Information Content 29 that for almost every x, x is in A if and only if x is in K UA [c . log n, n k) where n= Ixl. The idea of a "self-p-printable set" is easily generalized. For sets A and B, A is P(B )-printable if there is a deterministic oracle machine that computes relative to B the function enumA and that runs in polynomial time.

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