Kolmogorov complexity theory

  • What is the chain rule for Kolmogorov complexity?

    The chain rule for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states: That is, the combined randomness of two sequences X and Y is the sum of the randomness of X plus whatever randomness is left in Y once we know X..

  • What is the coding theorem in Kolmogorov complexity?

    The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then K(x) ≤ log(1/δ) + O(1)..

  • What is the generalized Kolmogorov complexity?

    The algorithmic information or Kolmogorov complexity of a bitstring x is the length of the shortest program that computes x and halts.
    This is one of the fundamental concepts of theoretical computer science.
    Follow this link to the Kolmogorov mailing list home page at IDSIA..

  • What is the Kolmogorov complexity in information theory?

    In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output..

  • What is the Kolmogorov complexity programming language?

    We define the Kolmogorov complexity relative to a given language as the length of the shortest program p plus input y, such that, when given input y, p computes (outputs) x and then halts..

  • Why is Kolmogorov complexity important?

    Kolmogorov Complexity can be viewed as the ultimate compressor—producing for any arbitrary string (or file or image), a minimum description of that string, given some form of description language..

  • The algorithmic information or Kolmogorov complexity of a bitstring x is the length of the shortest program that computes x and halts.
    This is one of the fundamental concepts of theoretical computer science.
    Follow this link to the Kolmogorov mailing list home page at IDSIA.
  • We define the Kolmogorov complexity relative to a given language as the length of the shortest program p plus input y, such that, when given input y, p computes (outputs) x and then halts.
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output.
In algorithmic information theory the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program that  DefinitionHistory and contextBasic resultsKolmogorov randomness
Kolmogorov complexity of an object or algorithm is the length of its optimal specification. In some sense, it could be thought of as algorithmic entropy, in the sense that it is the amount of information contained in the object.
We define the Kolmogorov complexity relative to a given language as the length of the shortest program p plus input y, such that, when given input y, p computes (outputs) x and then halts.
Kolmogorov complexity theory
Kolmogorov complexity theory

Soviet mathematician

Andrey Nikolaevich Kolmogorov was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.
Effective complexity is a measure of complexity defined in a 1996 paper by Murray Gell-Mann and Seth Lloyd that attempts to measure the amount of non-random information in a system.
It has been criticised as being dependent on the subjective decisions made as to which parts of the information in the system are to be discounted as random.

Statistical function

In 1973, Andrey Kolmogorov proposed a non-probabilistic approach to statistics and model selection.
Let each datum be a finite binary string and a model be a finite set of binary strings.
Consider model classes consisting of models of given maximal Kolmogorov complexity.
The Kolmogorov structure function of an individual data string expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data.
The structure function determines all stochastic properties of the individual data string: for every constrained model class it determines the individual best-fitting model in the class irrespective of whether the true model is in the model class considered or not.
In the classical case we talk about a set of data with a probability distribution, and the properties are those of the expectations.
In contrast, here we deal with individual data strings and the properties of the individual string focused on.
In this setting, a property holds with certainty rather than with high probability as in the classical case.
The Kolmogorov structure function precisely quantifies the goodness-of-fit of an individual model with respect to individual data.

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