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Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences
, 2007
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Every sequence is decompressible from a random one
 In Logical Approaches to Computational Barriers, Proceedings of the Second Conference on Computability in Europe, Springer Lecture Notes in Computer Science, volume 3988 of Computability in Europe
, 2006
"... ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of b ..."
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Cited by 10 (6 self)
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ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. It is shown that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, a new characterization of constructive dimension is given in terms of Turing reduction compression ratios.
EXTRACTING THE KOLMOGOROV COMPLEXITY OF STRINGS AND SEQUENCES FROM SOURCES WITH LIMITED INDEPENDENCE
"... An infinite binary sequence has randomness rate σ if, for almost every n, the Kolmogorov complexity of its prefix of length n is at least σn. It is known that for every rational σ ∈ (0, 1), on one hand, there exists sequences with randomness rate σ that can not be effectively transformed into a sequ ..."
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Cited by 8 (5 self)
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An infinite binary sequence has randomness rate σ if, for almost every n, the Kolmogorov complexity of its prefix of length n is at least σn. It is known that for every rational σ ∈ (0, 1), on one hand, there exists sequences with randomness rate σ that can not be effectively transformed into a sequence with randomness rate higher than σ and, on the other hand, any two independent sequences with randomness rate σ can be transformed into a sequence with randomness rate higher than σ. We show that the latter result holds even if the two input sequences have linear dependency (which, informally speaking, means that all prefixes of length n of the two sequences have in common a constant fraction of their information). The similar problem is studied for finite strings. It is shown that from any two strings with sufficiently large Kolmogorov complexity and sufficiently small dependence, one can effectively construct a string that is random even conditioned by any one of the input strings.
Infinite subsets of random sets of integers
 Math. Res. Lett
, 2009
"... Abstract. There is an infinite subset of a MartinLöf random set of integers that does not compute any MartinLöf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a MartinLöf random set of integers, and apply a re ..."
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Cited by 7 (0 self)
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Abstract. There is an infinite subset of a MartinLöf random set of integers that does not compute any MartinLöf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a MartinLöf random set of integers, and apply a result of Miller. 1.
Π 0 1 CLASSES WITH COMPLEX ELEMENTS.
"... Abstract. An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π 0 1 class P contains a complex element if and only if it contains a wttcover for the Cantor set. That is, if and only if for every real ..."
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Cited by 3 (0 self)
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Abstract. An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π 0 1 class P contains a complex element if and only if it contains a wttcover for the Cantor set. That is, if and only if for every real Y there is an X in the P such that X �wtt Y. We show that this is also equivalent to the Π 0 1 class’s being large in some sense. We give an example of how this result can be used in the study of scattered linear orders. §1. Introduction. There has been interest in the literature over many years in studying various notions of the size of subclasses of 2 ω. In this paper we have tried to generalise and consolidate some of these ideas. We investigate a notion of size that has appeared independently in [1] and [5], namely the notion of a computable perfect class (computably growing in [5] and nonuphi in [1]). It is
TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION
"... Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show tha ..."
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Cited by 3 (2 self)
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Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. 1.
Algorithmically Independent Sequences
, 2008
"... Two objects are independent if they do not affect each other. Independence is wellunderstood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite bi ..."
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Two objects are independent if they do not affect each other. Independence is wellunderstood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite binary sequences and studies their properties. Our two proposed notions of independence have some of the intuitive properties that one naturally expects. For example, for every sequence x, the set of sequences that are independent (in the weaker of the two senses) with x has measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be effectively constructed via Turing reductions (from one or more input sequences). In this respect, we prove several impossibility results. For example, it is shown that there is no effective way of producing from an arbitrary sequence with positive constructive Hausdorff dimension two sequences that are independent (even in the weaker type of independence) and have superlogarithmic complexity. Finally, a few conjectures and open questions are discussed.
Effective dimension of points visited by Brownian motion
, 2007
"... We consider the individual points on a MartinLöf random path of Brownian motion. We show (1) that Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension < 1. The proof of (1) amounts es ..."
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Cited by 2 (0 self)
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We consider the individual points on a MartinLöf random path of Brownian motion. We show (1) that Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension < 1. The proof of (1) amounts essentially to showing for almost all times t, the path f is random relative to t and the dimension of (t, f(t)) is 2. 1
Symbolic dynamics: entropy = dimension = complexity
"... Let d be a positive integer. Let G be the additive monoid N d or the additive group Z d. Let A be a finite set of symbols. The shift action of G on A G is given by S g (x)(h) = x(g + h) for all g,h ∈ G and all x ∈ A G. A Gsubshift is defined to be a nonempty closed set X ⊆ A G such that S g (x) ∈ ..."
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Let d be a positive integer. Let G be the additive monoid N d or the additive group Z d. Let A be a finite set of symbols. The shift action of G on A G is given by S g (x)(h) = x(g + h) for all g,h ∈ G and all x ∈ A G. A Gsubshift is defined to be a nonempty closed set X ⊆ A G such that S g (x) ∈ X for all g ∈ G and all x ∈ X. Given a Gsubshift X, the topological entropy ent(X) is defined as usual [31]. The standard metric on A G is defined by ρ(x,y) = 2 −Fn  where n is as large as possible such that x↾Fn = y↾Fn. Here Fn = {0,1,...,n} d if G = N d, and Fn = {−n,...,−1,0,1,...,n} d if G = Z d. For any X ⊆ A G the Hausdorff dimension dim(X) and the effective Hausdorff dimension effdim(X) are defined as usual [14, 26, 27] with respect to the standard metric. It is well known that effdim(X) = sup x∈X liminfnK(x↾Fn)/Fn  where K denotes Kolmogorov complexity [9]. If X is a Gsubshift, we prove that ent(X) = dim(X) = effdim(X), and ent(X) ≥ limsup n K(x↾Fn)/Fn  for all x ∈ X, and ent(X) = limnK(x↾Fn)/Fn  for some x ∈ X.
LIMIT COMPUTABILITY AND CONSTRUCTIVE MEASURE
"... Abstract. In this paper we study constructive measure and dimension in the class ∆0 2 of limit computable sets. We prove that the lower cone of any Turingincomplete set in ∆0 2 has ∆0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in ∆0 2 always has ∆0 2measure ..."
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Abstract. In this paper we study constructive measure and dimension in the class ∆0 2 of limit computable sets. We prove that the lower cone of any Turingincomplete set in ∆0 2 has ∆0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in ∆0 2 always has ∆0 2measure 0, upper cones in ∆0 2 have nonzero ∆0 2dimension. In particular the ∆0 2dimension of the Turing degree of ∅ ′ (the Halting Problem) is 1. Finally, it is proved that the low sets do not have ∆0 2measure 0, which means that the low sets do not form a small subset of ∆0 2. This result has consequences for the existence of biimmune sets. 1.