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21
Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 34 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
Lowness for the Class of Random Sets
, 1998
"... A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the noti ..."
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Cited by 31 (3 self)
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A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the notion of randomness as originally defined by P. MartinLof in [8]. A set is MartinLofrandom, or 1random for short, if it cannot be approximated in measure by recursive means. These sets have played a central role in the study of algorithmic randomness. One can relativize the definition of randomness to an arbitrary oracle. Relativized randomness has been studied by several authors. The intuitive meaning of "A is 1random relative to B" is that A is independent of B. A justification for this interpretation is given by M. van Lambalgen [7]. In this introduction we review some of the basic properties of sets which are 1random and we state the main problem. We work in the Cantor space 1 The fi...
Algorithmic randomness of closed sets
 J. LOGIC AND COMPUTATION
, 2007
"... We investigate notions of randomness in the space C[2 N] of nonempty closed subsets of {0, 1} N. A probability measure is given and a version of the MartinLöf test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that any random 4 clos ..."
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Cited by 11 (8 self)
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We investigate notions of randomness in the space C[2 N] of nonempty closed subsets of {0, 1} N. A probability measure is given and a version of the MartinLöf test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that any random 4 closed set is perfect, has measure 0, and has box dimension log2. A 3 random closed set has no nc.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If Tn = T ∩ {0, 1} n, then for any random closed set [T] where T has no dead ends, K(Tn) ≥ n − O(1) but for any k, K(Tn) ≤ 2 n−k + O(1), where K(σ) is the prefixfree complexity of σ ∈ {0, 1} ∗.
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.
Randomness, lowness and degrees
 J. of Symbolic Logic
, 2006
"... Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally a ..."
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Cited by 9 (4 self)
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Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2 ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. 1.
KOLMOGOROV COMPLEXITY AND SOLOVAY FUNCTIONS
, 2009
"... Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinit ..."
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Cited by 8 (3 self)
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Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as MartinLöf randomness and Ktriviality.
On the construction of effective random sets
 MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2002, LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... We give a direct and rather simple construction of MartinLöf random and recrandom sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a MartinLöf random set R from which X can be decoded effectively. Second, by essentially the same ..."
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Cited by 7 (0 self)
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We give a direct and rather simple construction of MartinLöf random and recrandom sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a MartinLöf random set R from which X can be decoded effectively. Second, by essentially the same construction we obtain a MartinLöf random set R that is computably enumerable selfreducible. Alternatively, using the observation that a set is computably enumerable selfreducible if and only if its associated real is computably enumerable, the existence of such a set R follows from the known fact that every Chaitin real is MartinLöf random and computably enumerable. Third, by a variant of the basic construction we obtain a recrandom set that is weak truthtable autoreducible. The mentioned results on self and autoreducibility complement work of Ebert, Merkle, and Vollmer [79], from which it follows that no MartinLöf random set is Turingautoreducible and that no recrandom set is truthtable autoreducible.
MartinLöf Random and PAcomplete Sets
 In [4
, 2002
"... A set A is MartinLof random iff the class fAg does not have \Sigma 1 measure 0. A set A is PAcomplete if one can compute relative to A a consistent and complete extension of Peano Arithmetic. It is shown that every MartinLof random set either permits to solve the halting problem K or is not ..."
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Cited by 5 (0 self)
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A set A is MartinLof random iff the class fAg does not have \Sigma 1 measure 0. A set A is PAcomplete if one can compute relative to A a consistent and complete extension of Peano Arithmetic. It is shown that every MartinLof random set either permits to solve the halting problem K or is not PAcomplete. This result implies a negative answer to the question of AmbosSpies and Kucera whether there is a MartinLof random set not above K which is also PAcomplete.
A random degree with strong minimal cover
 the Bulletin of the London Mathematical Society
, 2006
"... We show that there exists a MartinLöf random degree which has a strong minimal cover. 1. ..."
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Cited by 4 (4 self)
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We show that there exists a MartinLöf random degree which has a strong minimal cover. 1.