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Information Theory for the source and Noisy-channel coding. I Information theory.

So we wish. Eugene M. This article is a brief guide to the field of algorithmic information theory AITits underlying philosophy, and the most important concepts. AIT arises by mixing information theory and computation theory to obtain an objective and absolute information theory and coding pdf of information in an individual object, and in so doing gives rise to an objective and robust notion of randomness of individual objects. This is in contrast to classical information theory that is based on random variables and communication, and has no bearing on information and randomness of individual objects.

After a brief overview, the major subfields, applications, history, and a map of the field are presented. Algorithmic information theory AIT is the information theory of individual objects, using computer scienceand concerns itself with the relationship between computation, information, and randomness. The information content or complexity of an object can be measured by the length of its shortest description.

For instance the string.

This Algorithmic "Solomonoff" Probability AP is key in addressing the old philosophical problem of induction in a formal way. The major drawback of AC and AP are their incomputability. Time-bounded "Levin" complexity penalizes a slow program by adding the logarithm of its running time to its length.

Information theory and coding pdf leads to computable variants of AC and AP, and Universal "Levin" Search US that solves all inversion problems in optimal apart from some unrealistically large multiplicative constant time. AC and AP also allow a formal and rigorous definition of randomness of individual strings that does not depend on physical information theory and coding pdf philosophical intuitions about nondeterminism or likelihood. Roughly, a string is Algorithmic "Martin-Loef" Random AR if it is incompressible in the sense that its algorithmic complexity is equal to its length.

It serves as the foundation of the Minimum Description Length MDL principle, can simplify proofs in computational complexity theory, has been used to define a universal similarity metric between objects, solves the Maxwell daemon problem, and many others. Algorithmic complexity formalizes the notions of simplicity and complexity. Intuitively, a string is simple if it can be described in a few words, like "the string of one million ones", and is complex if there is no such short description, like for a random string whose shortest description is specifying it literally bit by bit.

Typically one is only interested in descriptions or codes that are effective in the sense that decoders are algorithms on some computer.

This can be termed Kolmogorov's Thesis: Note that the shortest code is one for which there is a general decompressor: There are many variants, mainly for technical reasons: The historically first "plain" complexity, the now more important "prefix" complexity, and many others. Most of them coincide within an additive term in the length of the string. A prefix Turing machine has a separate input information theory and coding pdf which it reads from left-to-right without backing up, a separate worktape on which the computation takes place, and a separate output tape on which the output is written.

We define a [[halting program]] as the initial segment of the input that is scanned at the time when the machine halts, and the output is the string that has been written to the separate output tape at that time. It is based on. The definition of Algorithmic "Solomonoff" Probability APalso called universal a information theory and coding pdf probability, was made precise by L.

The continuous notion is suitable for applications to prediction of the next element in a growing time sequence, and it is this notion that was invented by Solomonoff. If married with sequential decision theoryit leads to an optimal reinforcement learning agent embedded in an arbitrary unknown environment Hutterand a formal definition and test of intelligence. Much has been written about which algorithm is better under which circumstances.

Consider the following alternative very simple solution to the problem: Universal "Levin" Search US extends this idea from two algorithms to all algorithms.

First, since there are infinitely many algorithms, computation time has to be assigned non-uniformly. How does this fit into AIT?

Time-bounded "Levin" complexity penalizes a slow program by adding the logarithm of its running time to its length: While universal search is nice in theory, it is not applicable in this form due to huge hidden multiplicative constants in the running time. Another restriction is that verification needs to be fast.

Hutter developed a more general asymptotically fastest algorithm, which removes the multiplicative constant and necessity of verification, unfortunately at the expense of an even larger additive constant. The mathematical formalization of the concept of probability or chance has a long intertwined history. The now standard axioms of probability, learned by all students, are due to Kolmogorov While mathematically convincing, the semantics is far from information theory and coding pdf.

Frequentists interpret probabilities as limits of observed relative information theory and coding pdf, objectivists think of them as real aspects of the world, subjectivists regard them as one's degree of belief often elicited from betting ratioswhile Cournot only assigns meaning to events of high probability, namely as happening for sure in our world.

None of these approaches answers the question of whether some specific individual object or observation, like the binary strings information theory and coding pdf, is random. Kolmogorov's axioms do not allow one to ask such questions.

Von Miseswith refinements to his approach by Waldand Church with various degrees of success attempted to formalize the intuitive notion of one string looking more random than another see the example in the introduction. For information theory and coding pdf " Unfortunately no sequence can satisfy all randomness tests. The Mises-Wald-Church approach seemed satisfactory until Ville showed that some sequences are random according to their definition and yet lack certain properties that are universally agreed to be satisfied by random sequences.

Martin-Loefrather than give a definition and check whether it satisfied all requirements, took the approach to formalize the notion of all effectively testable requirements in the form of tests for randomness.

The tests are constructive namely all and only lower semi-computable ones, which are typically all one ever cares about.

Since the tests are constructed from Turing machines, they can be effectively enumerated according to the effective enumeration of the Turing machines they derive from. Since the set of sequences satisfying a test having the randomness property the test verifies has measure one, and there are only countably many tests, the set of sequences satisfying all such tests also has measure one. The theory is developed for both finite strings and infinite sequences.

In the latter case the notion of test is more complicated and we speak of sequential tests. For infinite sequences one can show that these are exactly the sequences which are incompressible in the sense that the algorithmic prefix complexity of every initial segment is at least equal to their length. Chaitin and C. This notion makes intuitive sense: A string can be compressed iff there are some regularities in the string iff the string is non-random.

AIT helps to tackle many philosophical problems in the sense that it allows one to formalize and quantify many intuitive but vague concepts of great importance as we have seen above, and hence allows one to talk about them in a meaningful and rigorous way, thus leading to a deeper understanding than without AIT. Most importantly, AC formalizes and quantifies the concepts of simplicity and complexity in an essentially unique way.

A core scientific paradigm is Occam's razor, usually interpreted as "among two monata secawan madu rena kdi that describe the data equally well, the simpler one should be preferred. AIT is also useful in the foundations of thermodynamics and its second theorem about entropy increase, and in particular for solving the problem of Maxwell's demon.

By often crudely approximating the "ideal" concepts, AIT has been applied to various problems of practical interest, e. The major problem is that the approximation accuracy is hard to assess and most theorems in AIT break down. The universal similarity metric is probably the greatest practical success of AIT: A reasonable definition for the similarity between two objects is how difficult it is to transform them into each other.

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