Algorithmic Complexity

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Maha (+10/-10) just standardizing the "Kolmogorov" spelling.
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joaolkf moved [[Kolmogorov complexity]] to [[Algorithmic complexity]]: same concept
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The Algorithmic Complexity or Kolmogorov Complexity of a set of data is the size of the shortest possible description of the data.

See also: Solomonoff Induction, AIXI

See also


  1. SIPSER, M. (1983) "A complexity theoretic approach to randomness". In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330{335. ACM, New York.
  2. FORTNOW, Lance. "Kolmogorov Complexity" Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.120.4949&rep=rep1&type=pdf
  3. LI, Ming. & VITANY, Paul. “Algorithmic Complexity”. Available at: http://homepages.cwi.nl/~paulv/papers/020608isb.pdf

Algorithmic complexity is an inverse measure of compressibility. If the data is complex and random, the shortest possible description of it becomes longer. This is also one of the best definitions of randomness so far1. If the data has few regular patterns, it is difficult to compress it or describe it shortly, giving it a high KolmogorowKolmogorov complexity and randomness. If there isn't any way to describe the data so that the description is shorter than the data itself, the data is incompressible. 2

The Algorithmic Complexity or Kolmogorov Complexity of a set of data is the size of the shortest possible description of thisthe data. It’s

Algorithmic complexity is an inverse measure of compressibility. How much moreIf the data is needed to makecomplex and random, the shortest description, more complex and random the data is.possible description of it becomes longer. This is also one of the best accountsdefinitions of randomness so far1. AsIf the numbers of patterns in the date increases, more ways to compress it, describedata has few regular patterns, it shortly, and there are less Kolmogorov Complexity and less randomness. As the number of patterns in the date decreases, less waysis difficult to compress it or describe it shortly, so there are more Kolmogorovgiving it a high Kolmogorow complexity and randomness. In the limit situation,If there isn’isn't a singleany way to describe the data which takes less spaceso that the description is shorter than the data itself, the data is incompressible. 2

This notion can be used to state many important results in computational theory. ThePossibly the most famous, perhaps,famous is Chaitin's incompleteness theorem, a version of Gödel’s incompleteness theorem.

The Algorithmic Complexity or Kolmogorov Complexity of a set of data is the size of the shortest possible description of this data. It’s an inverse measure of compressibility. How much more data is needed to make the shortest description, more complex and random the data is. This is also one of the best accounts of randomness so far1. As the numbers of patterns in the date increases, more ways to compress it, describe it shortly, and there are less Kolmogorov Complexity and less randomness. As the number of patterns in the date decreases, less ways to compress it or describe it shortly, so there are more Kolmogorov complexity and randomness. In the limit situation, there isn’t a single way to describe the data which takes less space than the data itself, the data is incompressible. 2

The Algorithmic Complexity or Kolmogorov Complexity of a set of data is the size of the shortest possible description of data. It’s an inverse measure of compressibility. How much more data is needed to make the shortest description, more complex and random the data is. This is also,also one of the best accounts of randomness so far1. As the numbers of patterns in the date increases, more ways to compress it, describe it shortly there are,shortly, and there are less Kolmogorov Complexity and less randomness. As the number of patterns in the date decreases, less ways to compress it or describe it shortly, so there are more Kolmogorov complexity and randomness. In the limit situation, there isn’t a single way to describe the data which takes less space than the data itself, the data is incompressible. 2

This notion can be used to state many important results in computational theory. The most famous, perhaps, areis Chaitin's incompleteness theorem, a version of Gödel’s incompleteness theorem.

The KolmogorovAlgorithmic Complexity or AlgorithmicKolmogorov Complexity of a set of data is the shortest possible description of data. It’s an inverse measure of compressibility. How much more data is needed to make the shortest description, more complex and random the data is. This is also, one of the best accounts of randomness so far1. As the numbers of patterns in the date increases, more ways to compress it, describe it shortly there are, and there are less Kolmogorov Complexity and less randomness. As the number of patterns in the date decreases, less ways to compress it or describe it shortly, so there are more Kolmogorov complexity and randomness. In the limit situation, there isn’t a single way to describe the data which takes less space than the data itself, the data is incompressible. 2

The Kolmogorov Complexity or Algorithmic Complexity of a set of data is the shortest possible description of data. It’s an inverse measure of compressibility. How much more data is needed to make the shortest description, more complex and random the data is. This is also, one of the best accounts of randomness so far1. As the numbers of patterns in the date increases, more ways to compress it, describe it shortly there are, and there are less Kolmogorov Complexity and less randomness. As the number of patterns in the date decreases, less ways to compress it or describe it shortly, so there are more Kolmogorov complexity and randomness. In the limit situation, there isn’t a single way to describe the data which takes less space than the data itself, the data is incompressible. 2

More formally, the Kolmogorov complexity C(x) of a set x, is the size in bits of the shortest binary program (in a fixed programming language) that prints the set x as its only output. If C(x) is equal or greater than the size of x in bits, x is incompressible. 3

This notion can be used to state many important results in computational theory. The most famous, perhaps, are Chaitin's incompleteness theorem , a version of Gödel’s incompleteness theorem.

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References


  1. SIPSER, M. (1983) "A complexity theoretic approach to randomness". In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330{335. ACM, New York.
  2. FORTNOW, Lance. "Kolmogorov Complexity" Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.120.4949&rep=rep1&type=pdf
  3. LI, Ming. & VITANY, Paul. “Algorithmic Complexity”. Available at: http://homepages.cwi.nl/~paulv/papers/020608isb.pdf