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A Theory of Program Size Formally Identical to Information Theory
, 1975
"... A new definition of programsize complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest selfdelimiting program for calculating strings A and B if one is given a minimalsize selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) ..."
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A new definition of programsize complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest selfdelimiting program for calculating strings A and B if one is given a minimalsize selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) programs are required to be selfdelimiting, i.e. no program is a prefix of another, and (2) instead of being given C and D directly, one is given a program for calculating them that is minimal in size. Unlike previous definitions, this one has precisely the formal 2 G. J. Chaitin properties of the entropy concept of information theory. For example, H(A;B) = H(A) + H(B=A) + O(1). Also, if a program of length k is assigned measure 2 \Gammak , then H(A) = \Gamma log 2 (the probability that the standard universal computer will calculate A) +O(1). Key Words and Phrases: computational complexity, entropy, information theory, instantaneous code, Kraft inequality, minimal program, probab...
AN ANALYSIS OF OPTIMAL RETRIEVAL SYSTEMS WITH UPDATES
, 1974
"... The performance of computerimplemented systems for data storage, retrieval, and update is investigated. A data structure is modeled by a set D = {d 1, d. d D of data bases. A set of questions A = {Xlk 2."' about any d E D may be answered. A memory that is bitaddressable by an algorit ..."
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The performance of computerimplemented systems for data storage, retrieval, and update is investigated. A data structure is modeled by a set D = {d 1, d. d D of data bases. A set of questions A = {Xlk 2.&quot;' about any d E D may be answered. A memory that is bitaddressable by an algorithm or an automaton models a computer. A retrieval system is composed of a particular mapping of data bases onto memory representations and a particular algorithm or automaton. By accessing bits of memory the algorithm can answer any X E A about the d represented in memory and can update memory to represent a new d * E D. Lower bounds are derived for the performance measures of storage efficiency, retrieval efficiency, and update efficiency. The minima are simultaneously
A. OPTICAL CHANNELS 1. UNIQUENESS OF THE OPTIMUM RECEIVER FOR THE MARY PURE STATE PROBLEM National Aeronautics and Space Administration (Grant NGL 22009013)
"... We have seen1 that the necessary and sufficient conditions for a set of operators 7i T.> 0 1 (1) S r. = I 1 to be the optimum decision operators are that the operator X defined by X = pip 7Ti satisfy X = X and X pip i> 0 all i. We have further shown that, on the space spanned by the m linearl ..."
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We have seen1 that the necessary and sufficient conditions for a set of operators 7i T.> 0 1 (1) S r. = I 1 to be the optimum decision operators are that the operator X defined by X = pip 7Ti satisfy X = X and X pip i> 0 all i. We have further shown that, on the space spanned by the m linearly independent message vectors (u i, the optimum 7ri are orthogonal projectors. 1 That is, 7T. = 6.... (3) We now claim that the optimum 7. on that space are unique.1 PROOF: Let us suppose to the contrary that there are two optimum measurements {i} and {i} each satisfying Eqs. 1 and 2. Then any measurement { Tr} with