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Lowness Properties and Randomness
 ADVANCES IN MATHEMATICS
"... The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2 ..."
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Cited by 95 (24 self)
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The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2], showing that each low for MartinLof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering
Randomness and reducibility
 J. Comput. System Sci
, 2001
"... How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of ..."
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Cited by 28 (4 self)
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How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as
A Characterization of C.E. Random Reals
 THEORETICAL COMPUTER SCIENCE
, 1999
"... A real # is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals; # is random if its binary expansion is a random sequence. Our aim is to offer a selfcontained proof, based on the papers [7, 14, 4, 13], of the following theorem: areal is c.e. and ra ..."
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Cited by 13 (0 self)
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A real # is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals; # is random if its binary expansion is a random sequence. Our aim is to offer a selfcontained proof, based on the papers [7, 14, 4, 13], of the following theorem: areal is c.e. and random if and only if it a Chaitin# real, i.e., the halting probability of some universal selfdelimiting Turing machine.
Eliminating concepts
 Proceedings of the IMS workshop on computational prospects of infinity
, 2008
"... Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdo ..."
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Cited by 7 (2 self)
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Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdoor access to the proofs. 1. Outline of the results All sets will be subsets of N unless otherwise stated. K(x) denotes the prefix free complexity of a string x. A set A is Ktrivial if, within a constant, each initial segment of A has minimal prefix free complexity. That is, there is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished). Note that the particular effective epresentation of a number n by a string (unary here) is irrelevant, since up to a constant K(n) is independent from the representation. A is low for MartinLöf randomness if each MartinLöf random set is already MartinLöf random relative to A. This class was defined in Zambella [28], and studied by Kučera and Terwijn [17]. In this survey we will see that the two classes are equivalent [24]. Further concepts have been introduced: to be a basis for MLrandomness (Kučera [16]), and to be low for K (Muchnik jr, in a seminar at Moscow State, 1999). They will also be eliminated, by showing equivalence with Ktriviality. All
Interactions of Computability and Randomness
"... We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktrivialit ..."
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Cited by 4 (2 self)
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We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktriviality. We include some interactions of randomness with computable analysis. Mathematics Subject Classification (2010). 03D15, 03D32. Keywords. Algorithmic randomness, lowness property, Ktriviality, cost function.
An Enumerable Undecidable Set With Low Prefix Complexity: A Simplified Proof
"... e). To do so we start with n k = 2k (say) and with A = ;. Then we enumerate the graph of the function (p k ; n) 7! p k (n). If (for some k) we nd that p k (n k ) is dened and dierent from 0 we add n k to A. In this way we will obtain an enumerable undecidable set. However it may not satisfy the ineq ..."
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Cited by 1 (0 self)
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e). To do so we start with n k = 2k (say) and with A = ;. Then we enumerate the graph of the function (p k ; n) 7! p k (n). If (for some k) we nd that p k (n k ) is dened and dierent from 0 we add n k to A. In this way we will obtain an enumerable undecidable set. However it may not satisfy the inequality KP(A 1:n ) KP(n) +O(1). To ensure this inequality let us rst rewrite it using a priori distribution m(z) as follows: m(A 1:n ) m(n)=c for some positive c and all<F1
UNIFIED CHARACTERIZATIONS OF LOWNESS PROPERTIES VIA KOLMOGOROV
"... Abstract. Consider a randomness notion C. A uniform test in the sense of C is a total computable procedure that each oracle X produces a test relative to X in the sense of C. We say that a binary sequence Y is Crandom uniformly relative to X if Y passes all uniform C tests relative to X. Suppose no ..."
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Abstract. Consider a randomness notion C. A uniform test in the sense of C is a total computable procedure that each oracle X produces a test relative to X in the sense of C. We say that a binary sequence Y is Crandom uniformly relative to X if Y passes all uniform C tests relative to X. Suppose now we have a pair of randomness notions C and D where C D, for instance MartinLöf randomness and Schnorr randomness. Several authors have characterized classes of the form Low(C;D) which consist of the oracles X that are so feeble that C DX. Our goal is to do the same when the randomness notion D is relativized uniformly: denote by Low⋆(C;D) the class of oracles X such that every Crandom is uniformly Drandom relative to X. (1) We show that X 2 Low⋆(MLR; SR) if and only if X is c.e. tttraceable if and only if X is anticomplex if and only if X is MartinLöf packing measure zero with respect to all computable dimension functions. (2) We also show that X 2 Low⋆(SR;WR) if and only if X is computably i.o. tttraceable if and only ifX is not totally complex if and only ifX is Schnorr Hausdorff measure zero with respect to all computable dimension functions. 1.
Real Numbers: From Computable to Random
, 2000
"... A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging s ..."
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A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence (equivalently, if its expansion in base b ≥ 2 is random). The aim of this paper is to review some recent results on computable, c.e. and random reals. In particular, we will present a complete characterization of the class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we will show that every c.e. and random real is the halting probability of some Solovay machine, that is, a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block of 1 bits. A few open problems will be also discussed. 1 Notation and Background We will use notation that is standard in computability theory and algorithmic information theory; we will assume familiarity with Turing machine computations, computable and computably enumerable (c.e.) sets (see, for example, Soare [48] or Odifreddi [40]) and elementary algorithmic information theory (see,
The Classes of Algorithmically Random Reals
, 2003
"... in fulfilment of the requirements for the degree of ..."
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