A Perfect Set of Reals with Finite Self-Information

Herbert, Ian

doi:10.2178/jsl.7804130

Cited by 4 publications

(6 citation statements)

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“…If A is also ∆ 0 2 , it is therefore low, in the usual computability theoretic sense. However, Herbert [8] has recently shown that there is a perfect Π 0 1 class of sets that have finite self-information, which in particular answers a question in a previous version of this paper by showing the existence of non-∆ 0 2 sets with finite self-information. Lutz [16] and others have developed an effective version of Hausdorff dimension based on martingales in inflationary environments, called s-gales.…”

Section: Relationships To Other Weakness Notionssupporting

confidence: 66%

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Finite Self-Information

Hirschfeldt

Weber

2012

Computability

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We present a definition, due to Levin, of mutual information I(A : B) for infinite sequences. We say that a set A has finite self-information if I(A : A) < ∞. It is easy to see that every K-trivial set has finite self-information. We answer a question of Levin by showing that the converse does not hold. Finally, we investigate the connections between having finite self-information and other notions of weakness such as jump-traceability. In particular, we show that our proof can be adapted to produce a set that is low for both effective Hausdorff dimension and effective packing dimension, but not K-trivial.

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Section: Relationships To Other Weakness Notionssupporting

confidence: 66%

“…In the forthcoming paper [11], Lempp, Miller, Ng, Turetsky, and Weber show that there is in fact a perfect Π 0 1 class of sets that are low for effective Hausdorff dimension, a result that also follows by the methods of Herbert [8].…”

Section: Relationships To Other Weakness Notionsmentioning

confidence: 91%

Finite Self-Information

Hirschfeldt

Weber

2012

Computability

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“…In this section, we prove the latter fact by constructing a perfect Π 0 1 -class containing only lowish for K sequences. This result was also recently proved by Herbert [9]. He built a perfect Π 0 1 -class Q ⊆ 2 ω of sequences with finite self-information (see Section 7) and observed that, as a consequence of his construction, every element of Q is low for dimension.…”

Section: A Perfect Class Of Low For Dimension Setsmentioning

confidence: 58%

“…They observed that their sequence is also low for dimension, so lowness for dimension does not imply lowness for randomness. We can also see this as a corollary to Theorem 5.1, where we prove that there are continuum many sequences that are low for dimension (see also Herbert [9]). Chaitin [4] proved that every K-trivial is ∆ 0 2 , so there are only countably many low for random sequences.…”

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confidence: 56%

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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“…Another closure property we get for KT p∆ 0 2 q, in contrast to LKp∆ 0 2 q as in [8], is that KT p∆ 0 2 q is closed under effective join (the effective join of reals A and B, denoted A' B, is the real whose binary expansion is given by A' Bp2nq " Apnq and A ' Bp2n`1q " Bpnq). The proof follows closely the proof that KT p0q is closed under effective join that was given by Downey, Hirschfeldt, Nies, and Stephan [7].…”

Section: Other Closuresmentioning

confidence: 95%

On Reals With -Bounded Complexity and Compressive Power

Herbert

2016

J. symb. log.

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The (prefix-free) Kolmogorov complexity of a finite binary string is the length of the shortest description of the string. This gives rise to some 'standard' lowness notions for reals: A is K-trivial if its initial segments have the lowest possible complexity and A is low for K if using A as an oracle does not decrease the complexity of strings by more than a constant factor. We weaken these notions by requiring the defining inequalities to hold up only up to all ∆ 0 2 orders, and call the new notions ∆ 0 2 -bounded K-trivial and ∆ 0 2 -bounded low for K. Several of the 'nice' properties of K-triviality are lost with this weakening. For instance, the new weaker definitions both give uncountable set of reals. In this paper we show that the weaker definitions are no longer equivalent, and that the ∆ 0 2 -bounded K-trivials are cofinal in the Turing degrees. We then compare them to other previously studied weakenings, namely infinitely-often K-triviality and weak lowness for K (in each, the defining inequality must hold up to a constant, but only for infinitely many inputs). We show that ∆ 0 2bounded K-trivial implies infinitely-often K-trivial, but no implication holds between ∆ 0 2 -bounded low for K and weakly low for K.

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A Perfect Set of Reals with Finite Self-Information

Abstract: We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information is

Cited by 4 publications

References 7 publications

Finite Self-Information

Finite Self-Information

Lowness for effective Hausdorff dimension

On Reals With -Bounded Complexity and Compressive Power

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