Current discrete randomness and information conservation inequalities are over total recursive functions, i.e. restricted to deterministic processing. This restriction implies that an algorithm can break algorithmic randomness conservation inequalities. We address this issue by proving tight bounds of randomness and information conservation with respect to recursively enumerable transformations, i.e. processing by algorithms. We also show conservation of randomness of finite strings with respect to enumerable distributions, i.e. semicomputable semi-measures.
This work is resultant from my trip to Montpellier with Alexander Shen and P ' e ter G ' a cs. Kolmogorov complexity was introduced independently in @cite_2 @cite_0 @cite_3 . For a detailed history of Algorithmic Information Theory, we refer to @cite_6 . @cite_5 introduced laws of information non-growth over deterministic functions and later revisited in @cite_1 . The definition of @math and theorem relies on modified arguments of Section 2 in @cite_1 . An extension of rarity to semi-measures can be found in the recent work of @cite_4 and also can seen in the work of @cite_1 . @cite_8 contains an extended survey of randomness conservation inequalities and also describes properties of the rarity term @math used in this article.
@cite_4 @cite_8 @cite_1 @cite_6 @cite_3 @cite_0 @cite_2 @cite_5
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The notion of Kolmogorov-Martin-Lof Random sequences is extended from computable to enumerable distributions. This allows definitions of various other properties, such as mutual information in infinite sequences. Enumerable distributions (as well as distributions faced in some finite multi-party settings) are semi measures, handling those requires care. The article further develops Kolmogorov's algorithmic complexity theory. The definition of randomness is modified to satisfy strong invariance properties (conservation inequalities). This allows definitions of concepts such as mutual information in individual infinite sequences. Applications to several areas, like probability theory, theory of algorithms, intuitionistic logic are considered. These theories are simplified substantially with the postulate that the objects they consider are independent of (have small mutual information with) any sequence specified by a mathematical property. The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others. A new definition of program-size complexity is made. H(A,B C,D) is defined to be the size in bits of the shortest self-delimiting program for calculating strings A and B if one is given a minimal-size self-delimiting program for calculating strings C and D. This differs from previous definitions: (1) programs are required to be self-delimiting, 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 properties of the entropy concept of information theory. For example, H(A,B) = H(A) + H(B A) - 0(1). Also, if a program of length k is assigned measure 2 -k, then H(A) = -log2 (the probability that the standard universal computer will calculate A) - - 0(1). A method and apparatus for cutting resilient foamed synthetic plastics filter material by pressing a die against a block of such material to provide cuts which are interleaved extending from opposite faces of the block the cuts at each such face being joined by a curved cut conforming to the desired exterior surface shape of the cut block when it is stretched to provide a corrugated configuration.