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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 105 (26 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, relativization, and Turing degrees
 J. Symbolic Logic
, 2005
"... We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompre ..."
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Cited by 63 (23 self)
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We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompressible: C(x) c. The `only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of timebounded Ccomplexity.
Relativizing Chaitin’s halting probability
 J. Math. Log
"... Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way of p ..."
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Cited by 34 (9 self)
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Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way of producing an Arandom real for every A ∈ 2 ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, Ω A U can be vastly different for different choices of U. Even for a fixed U, there are oracles A = ∗ B such that Ω A U and Ω B U are 1random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. 1.
Reals which Compute Little
, 2002
"... We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is superlow if A # # ,andA is jumptraceable if the values of (e) can be e#ectively approximated in a sense to be specified. We investigate those properties, in particular showing that superlownes ..."
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Cited by 33 (13 self)
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We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is superlow if A # # ,andA is jumptraceable if the values of (e) can be e#ectively approximated in a sense to be specified. We investigate those properties, in particular showing that superlowness and jumptraceability coincide within the r.e. sets but none of the properties implies the other within the #r.e. sets. Finally we prove that, for any low r.e. set B, there is is a Ktrivial set A ##T B. 1
Almost everywhere domination and superhighness
 Mathematical Logic Quarterly
"... Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almo ..."
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Cited by 29 (9 self)
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Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2 ω, each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of KjosHanssen, KjosHanssen/Miller/Solomon, and others concerning LRreducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 ′′ is
Lowness properties and approximations of the jump
 Proceedings of the Twelfth Workshop of Logic, Language, Information and Computation (WoLLIC 2005). Electronic Lecture Notes in Theoretical Computer Science 143
, 2006
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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 18 (9 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} ∗.
On Schnorr and computable randomness, martingales, and machines
 Mathematical Logic Quarterly
, 2004
"... examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines. ..."
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Cited by 17 (6 self)
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examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines.
Benign cost functions and lowness properties
"... Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. ..."
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Cited by 17 (10 self)
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Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LRhard random degrees, and all ωc.e. random degrees. The last result implies recent results of Diamondstone’s and Ng’s regarding cupping with supwerlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. 1.