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Measures and their random reals
 IN PREPARATION
"... We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of ..."
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Cited by 11 (2 self)
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We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every nonhyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for a continuous measure can be found throughout the hyperarithmetical Turing degrees.
EXTRACTING INFORMATION IS HARD: A TURING DEGREE OF NONINTEGRAL EFFECTIVE HAUSDORFF DIMENSION
"... Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ..."
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Cited by 8 (0 self)
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Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ∈ 2ω is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density. 1.
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.
Fixed point theorems on partial randomness
, 2009
"... In our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008], we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T, such as free energy F (T), energy E(T), and statistical m ..."
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Cited by 4 (4 self)
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In our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008], we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T, such as free energy F (T), energy E(T), and statistical mechanical entropy S(T), into the theory. These quantities are real functions of real argument T> 0. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by programsize complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity. Namely, the computability of the value of partition function Z(T) gives a sufficient condition for T ∈ (0, 1) to be a fixed point on partial randomness. In this paper, we show that the computability of each of all the thermodynamic quantities above gives the sufficient condition also. Moreover, we show that the computability of F (T) gives completely different fixed points from the computability of Z(T).
Centre for Discrete Mathematics and Theoretical Computer ScienceFixed point theorems on partial randomness ∗
, 2009
"... Abstract. In our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008], we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T, such as free energy F (T), energy E(T), and stati ..."
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Abstract. In our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008], we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T, such as free energy F (T), energy E(T), and statistical mechanical entropy S(T), into the theory. These quantities are real functions of real argument T> 0. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by programsize complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity. Namely, the computability of the value of partition function Z(T) gives a sufficient condition for T ∈ (0, 1) to be a fixed point on partial randomness. In this paper, we show that the computability of each of all the thermodynamic quantities above gives the sufficient condition also. Moreover, we show that the computability of F (T) gives completely different fixed points from the computability of Z(T). Key words: algorithmic randomness, fixed point theorem, partial randomness, Chaitin Ω number, algorithmic information theory, thermodynamic quantities AMS subject classifications (2000) 68Q30, 26E40, 03D80, 82B30, 82B03 1
Propagation of partial randomness
"... Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongl ..."
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Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongly frandom relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including nonKtriviality and autocomplexity. We prove that frandomness relative to a PAdegree implies strong frandomness, but frandomness does not imply frandomness relative to a PAdegree. Keywords: partial randomness, effective Hausdorff dimension, MartinLöf randomness, Kolmogorov complexity, models of arithmetic.
Cone avoidance and randomness preservation
"... Let X be an infinite sequence of 0’s and 1’s. Let f be a computable function. Recall that X is strongly frandom if and only if the a priori Kolmogorov complexity of each finite initial segment τ of X is bounded below by f(τ) minus a constant. We study the problem of finding a PAcomplete Turing orac ..."
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Let X be an infinite sequence of 0’s and 1’s. Let f be a computable function. Recall that X is strongly frandom if and only if the a priori Kolmogorov complexity of each finite initial segment τ of X is bounded below by f(τ) minus a constant. We study the problem of finding a PAcomplete Turing oracle which preserves the strong frandomness of X while avoiding a Turing cone. In the context of this problem, we prove that the cones which cannot always be avoided are precisely the Ktrivial ones. We also prove: (1) If f is convex and X is strongly frandom and Y is MartinLöf random relative to X, then X is strongly frandom relative to Y. (2) X is complex relative to some oracle if and only if X is random with respect to some continuous probability measure.