Presentations of K-Trivial Reals and Kolmogorov Complexity

Stephan, Frank; Wu, Guohua

doi:10.1007/11494645_57

Cited by 7 publications

(10 citation statements)

References 11 publications

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“…We show that in this result the assumption that the real number should be left-computable can be omitted. As any weakly 1-generic real is hyperimmune [13] and not Martin-Löf random we obtain a similar strengthening of a result by Stephan and Wu [22], who had shown that any nearly computable number that is left computable is hyperimmune (and, therefore, not Martin-Löf random). Finally, also in their result that any nearly computable number that is left computable is strongly Kurtz random (and, therefore, not K-trivial) one can omit the assumption that the real number should be left-computable.…”

Section: Introductionsupporting

confidence: 86%

“…We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (2017) and by Stephan and Wu (2005) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (2002) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.…”

supporting

confidence: 79%

“…The equivalence of the conditions ( 4) and (6) in the following proposition is essentially the content of Theorem 5 in [22]. Proposition 7.2.…”

Section: Left Computable Nearly Computable Numbersmentioning

confidence: 89%

“…For example, an argument by Downey, Hirschfeldt, and LaForte [5,Pages 105,106] shows that any strongly left-computable and nearly computable real number is even computable. In Section 8 we discuss three further results of the same type, one by Hoyrup [10] and two by Stephan and Wu [22]. Hoyrup [10] has shown that any nearly computable and left-computable number that is not computable is weakly 1-generic.…”

Section: Introductionmentioning

confidence: 94%

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Nearly Computable Real Numbers

Hertling¹,

Janicki²

2023

Preprint

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In this article we call a sequence (a n ) n of elements of a metric space nearly computably Cauchy if for every strictly increasing computable function r : N → N the sequence (d(a r(n+1) , a r(n) )) n converges computably to 0. We show that there exists a strictly increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence (a n ) n of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (2002) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (2017) and by Stephan and Wu (2005) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (2002) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.

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Section: Introductionsupporting

confidence: 86%

supporting

confidence: 79%

“…The equivalence of the conditions ( 4) and (6) in the following proposition is essentially the content of Theorem 5 in [22]. Proposition 7.2.…”

Section: Left Computable Nearly Computable Numbersmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 94%

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Nearly Computable Real Numbers

Hertling¹,

Janicki²

2023

Preprint

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“…set A of finite binary strings satisfying w∈A 2 −|w| = x is actually a computable set. A corollary of a result of Stephan and Wu [SW05] is that any such real is weakly 1-random, i.e. it must belong to every effective open set of measure one.…”

Section: Genericity For Left-ce Realsmentioning

confidence: 93%

Genericity of Weakly Computable Objects

Hoyrup

2016

Theory Comput Syst

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In computability theory many results state the existence of objects that in many respects lack algorithmic structure but at the same time are effective in some sense. Friedberg and Muchnik's answer to Post's problem is one of the most celebrated results in this form. The main goal of the paper is to develop a general result that embodies a large number of these particular constructions, capturing the essential idea that is common to all of them, and expressing it in topological terms. To do so, we introduce the effective topological notions of irreversible function and directional genericity and provide two main results that identify situations when such constructions are possible, clarifying the role of topology in many arguments from computability theory. We apply these abstract results to particular situations, illustrating their strength and deriving new results. This paper is an extended version of the conference paper [Hoy14] with detailed proofs and new results.

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Initial Segment Complexities of Randomness Notions

Hölzl

Kräling

Stephan

et al. 2010

IFIP Advances in Information and Communication Technology

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Abstract. Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of A's initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with Martin-Löf randomness relative to the halting problem K; that is, a set A is Martin-Löf random relative to K iff there is no function f such that for all m andIn the present work it is shown that characterisations of this style can also be given for other randomness criteria like strongly random, Kurtz random relative to K, PA-incomplete Martin-Löf random and strongly Kurtz random; here one does not just quantify over all functions f but over functions f of a specific form. For example, A is Martin-Löf random and PA-incomplete iff there is no A-recursive function f such that for all m and all n > f(m) it holds that C(A(0)A(1) . . . A(n)) ≤ n − m. The characterisation for strong randomness relates to functions which are the concatenation of an A-recursive function executed after a Krecursive function; this solves an open problem of Nies.In addition to this, characterisations of a similar style are also given for Demuth randomness and Schnorr randomness relative to K. Although the unrelativised versions of Kurtz randomness and Schnorr randomness do not admit such a characterisation in terms of plain Kolmogorov complexity, Bienvenu and Merkle gave one in terms of Kolmogorov complexity defined by computable machines.

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Presentations of K-Trivial Reals and Kolmogorov Complexity

Cited by 7 publications

References 11 publications

Nearly Computable Real Numbers

Nearly Computable Real Numbers

Genericity of Weakly Computable Objects

Initial Segment Complexities of Randomness Notions

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