The Complexity of Complexity

Allender, Eric

doi:10.1007/978-3-319-50062-1_6

Cited by 13 publications

(33 citation statements)

References 38 publications

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“…The following surprising result establishes a fairly precise relationship between and . Let (1) ( ) = ( ) and ( +1) ( ) = ( ( ) ( )).…”

Section: Definitionmentioning

confidence: 99%

“…It is also true that if is computable then ( ↾ ) ≤ ( ) + (1). Chaitin (1977) considered sequences with this property, which are now called -trivial.…”

Section: Theorem 11 (Chaitin (1976)) ( ↾ ) ≤ ( ) + (1) Iff Is Computmentioning

confidence: 99%

“…Effective dimensions. If = 0 1 … is random, then we might expect a sequence such as 0 00 1 00 2 00 … to be " 1 3 -random." Making precise sense of the idea of partial algorithmic randomness has led to significant applications.…”

Section: Some Applicationsmentioning

confidence: 99%

“…So, for instance, for the class P of polynomial-time computable functions, P is the class of functions that can be computed in polynomial time with as an oracle. (For references to the articles in this and the following theorem, see [1].) The choice of universal machine does have some effect on efficient computations, but we can quantify over all universal machines.…”

Section: Theorem 14 (Lutz and Lutzmentioning

confidence: 99%

“…The idea of normality, which goes back to Borel (1909), was extended by von Mises (1919), who suggested the following definition of randomness for individual binary sequences. 1 A selection function is an increasing function ∶ ℕ → ℕ. We think of ( ) as the th place selected in forming a subsequence of a given sequence.…”

mentioning

confidence: 99%

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Computability and Randomness

Downey¹,

Hirschfeldt²

2019

Notices Amer. Math. Soc.

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Historical Roots Von Mises. Around 1930, Kolmogorov and others founded the theory of probability, basing it on measure theory. Probability theory is concerned with the distribution of outcomes in sample spaces. It does not seek to give any meaning to the notion of an individual object, such as a single real number or binary string, being random, but rather studies the expected values of random variables.

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“…The following surprising result establishes a fairly precise relationship between and . Let (1) ( ) = ( ) and ( +1) ( ) = ( ( ) ( )).…”

Section: Definitionmentioning

confidence: 99%

“…It is also true that if is computable then ( ↾ ) ≤ ( ) + (1). Chaitin (1977) considered sequences with this property, which are now called -trivial.…”

Section: Theorem 11 (Chaitin (1976)) ( ↾ ) ≤ ( ) + (1) Iff Is Computmentioning

confidence: 99%

Section: Some Applicationsmentioning

confidence: 99%

Section: Theorem 14 (Lutz and Lutzmentioning

confidence: 99%

mentioning

confidence: 99%

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