Kolmogorov complexity and computably enumerable sets

Barmpalias, George; Li, Angsheng

doi:10.1016/j.apal.2013.06.007

Cited by 6 publications

(6 citation statements)

References 58 publications

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“…Moreover, it turned out that the above condition is equivalent to ∀n, C(A ↾ n ) ≥ log n − c which is a well known property that was studied in [Bar68]. It follows from (1.5) and [Bar05] (see [BL13] for more details) that the existence of a maximum degree is an elementary difference between the K-degrees of c.e. sets and the K-degrees of c.e.…”

Section: 5)mentioning

confidence: 95%

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Universal computably enumerable sets and initial segment prefix-free complexity

Barmpalias

2013

Information and Computation

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We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set A with non-trivial prefixfree initial segment complexity, there exists a Turing complete computably enumerable set B with complexity strictly less than the complexity of A. On the other hand it is known that sets with trivial initial segment prefix-free complexity are not Turing complete.Moreover we give a generalization of this result for any finite collection of computably enumerable sets A i , i < k with non-trivial initial segment prefixfree complexity. An application of this gives a negative answer to a question from [DH10, Section 11.12] and [MS07] which asked for minimal pairs in the structure of the c.e. reals ordered by their initial segment prefix-free complexity.Further consequences concern various notions of degrees of randomness. For example, the Solovay degrees and the K-degrees of computably enumerable reals and computably enumerable sets are not elementarily equivalent. Also, the degrees of randomness of c.e. reals based on plain and prefix-free complexity are not elementarily equivalent; the same holds for the degrees of c.e. sets.

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Section: 5)mentioning

confidence: 95%

“…sets lead to more general questions regarding the c.e. sets in the K-degrees and the C-degrees which were articulated in a research proposal that was presented (along with several related results) in [BL13]. An interesting product of this project is the following result from [BHLM13].…”

Section: Introductionmentioning

confidence: 91%

Universal computably enumerable sets and initial segment prefix-free complexity

Barmpalias

2013

Information and Computation

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“…traceability. 7 The main result from [21] is the following characterization of the compressibility of computably enumerable (c.e. from now on) sets in terms of asymptotic bounds on the oracle use in relative computations.…”

Section: Previous Work On Kobayashi Compressibilitymentioning

confidence: 99%

Kobayashi compressibility

Barmpalias¹,

Downey²

2016

Preprint

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Kobayashi [21] introduced a uniform notion of compressibility of infinite binary sequences X in terms of relative Turing computations with sub-identity use of the oracle. Given f : N → N we say that X is f -compressible if there exists Y such that for each n we compute X ↾ n using at most the first f (n) bits of the oracle Y. Kobayashi compressibility has remained a relatively obscure notion, with the exception of some work on resource bounded Kolmogorov complexity. The main goal of this note is to show that it is relevant to a number of topics in current research on algorithmic randomness.We prove that Kobayashi compressibility can be used in order to define Martin-Löf randomness, a strong version of finite randomness and Kurtz randomness, strictly in terms of Turing reductions. Moreover these randomness notions naturally correspond to Turing reducibility, weak truth-table reducibility and truth-table reducibility respectively. Finally we discuss Kobayashi's main result from [21] regarding the compressibility of computably enumerable sets, and provide additional related original results.

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“…This can be seen as evidence that this class is robust and plays an important role in computability and randomness. The following characterization is from [BL11b]. If A is a c.e.…”

Section: (24)mentioning

confidence: 99%

Algorithmic randomness and measures of complexity

Barmpalias¹

2013

Bull. Symbolic Logic

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We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

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Kolmogorov complexity and computably enumerable sets

Cited by 6 publications

References 58 publications

Universal computably enumerable sets and initial segment prefix-free complexity

Universal computably enumerable sets and initial segment prefix-free complexity

Kobayashi compressibility

Algorithmic randomness and measures of complexity

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