Difference between revisions of "Algorithmic complexity"

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{{wikilink|Kolmogorov complexity}}
 
{{wikilink|Kolmogorov complexity}}
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 far<ref>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. </ref>. 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. <ref> FORTNOW, Lance. "Kolmogorov Complexity" Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.120.4949&rep=rep1&type=pdf </ref>
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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.
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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 far<ref>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. </ref>. If the data has few regular patterns, it is difficult to compress it or describe it shortly, giving it a high Kolmogorow 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. <ref> FORTNOW, Lance. "Kolmogorov Complexity" Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.120.4949&rep=rep1&type=pdf </ref>
  
 
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. <ref> LI, Ming. & VITANY, Paul. “Algorithmic Complexity”. Available at: http://homepages.cwi.nl/~paulv/papers/020608isb.pdf </ref>
 
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. <ref> LI, Ming. & VITANY, Paul. “Algorithmic Complexity”. Available at: http://homepages.cwi.nl/~paulv/papers/020608isb.pdf </ref>
  
This notion can be used to state many important results in computational theory. The most famous, perhaps, is [http://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin.27s_incompleteness_theorem Chaitin's incompleteness theorem], a version of Gödel’s incompleteness theorem.
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This notion can be used to state many important results in computational theory. Possibly the most famous is [http://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin.27s_incompleteness_theorem Chaitin's incompleteness theorem], a version of Gödel’s incompleteness theorem.
  
 
==Blog Posts==
 
==Blog Posts==

Revision as of 22:00, 13 September 2012

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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.

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 far[1]. If the data has few regular patterns, it is difficult to compress it or describe it shortly, giving it a high Kolmogorow 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]

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. Possibly the most famous is Chaitin's incompleteness theorem, a version of Gödel’s incompleteness theorem.

Blog Posts

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


See also