Every sequence is reducible to a random one

Gács, Péter

doi:10.1016/s0019-9958(86)80004-3

Cited by 95 publications

(85 citation statements)

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“…Results closely related to (i) implies (iii) were proved by Martin-Löf [20]. Condition (ii) is similar to a characterization that Gács [9] gave of 1-randomness in terms of length conditional Kolmogorov complexity. He proved that X ∈ 2 ω is 1-random iff (∀n)…”

Section: Initial Segment Complexity and Degrees Of Randomnesssupporting

confidence: 61%

On initial segment complexity and degrees of randomness

Miller

2008

Trans. Amer. Math. Soc.

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Abstract. One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They. The equivalence classes under this relation are the K-degrees. We prove that if X ⊕ Y is 1-random, then X and Y have no upper bound in the K-degrees (hence, no join). We also prove that n-randomness is closed upward in the K-degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the vL-degrees. Unlike the K-degrees, many basic properties of the vL-degrees are easy to prove. We show that X ≤ K Y implies X ≤ vL Y , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for ≤ C , the analogue of ≤ K for plain Kolmogorov complexity.Two other interesting results are included. First, we prove that for any Z ∈ 2 ω , a 1-random real computable from a 1-Z-random real is automatically 1-Z-random. Second, we give a plain Kolmogorov complexity characterization of 1-randomness. This characterization is related to our proof that X ≤ C Y implies X ≤ vL Y .

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Section: Initial Segment Complexity and Degrees Of Randomnesssupporting

confidence: 61%

On initial segment complexity and degrees of randomness

Miller

2008

Trans. Amer. Math. Soc.

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“…Fundamental results by Gacs [2] and Kučera [6] showed that every set is Turing reducible to a Martin-Löf random one. Since a Martin-Löf random set has effective dimension 1, it follows from (1) that every Turing upper cone is of effective dimension 1.…”

Section: Effective Dimension Of Cones and Degreesmentioning

confidence: 99%

A LOWER CONE IN THE wtt DEGREES OF NON-INTEGRAL EFFECTIVE DIMENSION

Nies¹,

Nies²,

Reimann³

2008

Computational Prospects of Infinity

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“…Kučera [13] and Gács [8] both proved that every sequence is computable from a Martin-Löf random. We need the version due to Gács, who proved that for any sequence X, there are a Martin-Löf random Z and a functional Φ such that Φ Z = X with use ϕ(n) = n + o(n).…”

Section: Proposition 42 (1) ⇒ (3)mentioning

confidence: 99%

Lowness for effective Hausdorff dimension

Lempp

Miller

et al. 2014

J. Math. Log.

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Abstract. We examine the sequences A that are low for dimension, i.e., those for which the effective (Hausdorff) dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit ofWe show that there is a perfect Π 0 1 -class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order n ε , for any ε > 0.

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Every sequence is reducible to a random one

Cited by 95 publications

References 1 publication

On initial segment complexity and degrees of randomness

On initial segment complexity and degrees of randomness

A LOWER CONE IN THE wtt DEGREES OF NON-INTEGRAL EFFECTIVE DIMENSION

Lowness for effective Hausdorff dimension

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