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18
Trivial Reals
"... Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivi ..."
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Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivial real. We also analyze various computabilitytheoretic properties of the Htrivial reals, showing for example that no Htrivial real can compute the halting problem. Therefore, our construction of an Htrivial computably enumerable set is an easy, injuryfree construction of an incomplete computably enumerable set. Finally, we relate the Htrivials to other classes of &quot;highly nonrandom &quot; reals that have been previously studied.
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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examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines.
Computably enumerable sets in the Solovay and the strong weak truth table degrees
 in New Computational Paradigms: First Conference on Computability in Europe, CiE 2005
, 2005
"... Abstract. The strong weak truth table reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky a ..."
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Abstract. The strong weak truth table reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky and Weinberger on applications of computability to differential geometry. Yu and Ding showed that the relevant degree structure restricted to the c.e. reals has no greatest element, and asked for maximal elements. We answer this question for the case of c.e. sets. Using a doubly nonuniform argument we show that there are no maximal elements in the sw degrees of the c.e. sets. We note that the same holds for the Solovay degrees of c.e. sets. 1
Random noncupping revisited
 J. Complexity
, 2006
"... Abstract. Say that Y has the strong random anticupping property if there is a set A such that for every MartinLöf random set R Y ≤T A ⊕ R ⇒ Y ≤T R (in this case A is an anticupping witness for Y). Nies has shown that every random ∆ 0 2 set has the strong random anticupping property via a promptly s ..."
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Abstract. Say that Y has the strong random anticupping property if there is a set A such that for every MartinLöf random set R Y ≤T A ⊕ R ⇒ Y ≤T R (in this case A is an anticupping witness for Y). Nies has shown that every random ∆ 0 2 set has the strong random anticupping property via a promptly simple anticupping witness. We show that every ∆ 0 2 set has the random anticupping property via a promptly simple anticupping witness. Moreover, we prove the following stronger statement: for every noncomputable Y ≤T ∅ ′ there exists a promptly simple A such that Y ≤T A ⊕ R ⇒ A ≤T R for all MartinLöf random sets R. 1.
Algorithmic Randomness and Computability
"... Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. ..."
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Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.
Degrees of monotone complexity
 J. Symbolic Logic
, 2006
"... Abstract. Levin and Schnorr (independently) introduced the monotone complexity, Km(α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2 ω by α ≤Km β iff there is a constant c such that Km(α n) ..."
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Abstract. Levin and Schnorr (independently) introduced the monotone complexity, Km(α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2 ω by α ≤Km β iff there is a constant c such that Km(α n) ≤ Km(β n) + c for all n. The monotone degree of α is the set of all β such that α ≤Km β and β ≤Km α. We show the monotone degrees contain an antichain of size 2ℵ0, a countable dense linear ordering (of degrees of cardinality 2ℵ0), and a minimal pair. Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the Kdegrees, where K is the prefixfree Kolmogorov complexity. A minimal pair of Kdegrees was constructed by Csima and Montalbán. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We define a real to be (Km,K)trivial if for some constant c, Km(α n) ≤ K(n) + c for all n. It is not known whether there is a Kmminimal real, but we show that any such real must be (Km,K)trivial. Finally, we consider the monotone degrees of the computably enumerable (c.e.) and
RANDOMNESS AND THE LINEAR DEGREES OF COMPUTABILITY
"... Abstract. We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤ℓ β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasimaximality property. As a corollary we may conclude that there exists no ℓcomplete ∆2 real. Upon r ..."
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Abstract. We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤ℓ β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasimaximality property. As a corollary we may conclude that there exists no ℓcomplete ∆2 real. Upon realizing that quasimaximality does not characterize the random reals—there exist reals which are not random but which are of quasimaximal ℓdegree—it is then natural to ask whether maximality could provide such a characterization. Such hopes, however, are in vain since no real is of maximal ℓdegree. 1. introduction In the process of computing a real α given an oracle for β it is natural to consider the condition that for the computation of the first n bits of α we are only allowed to use the information in the first n bits of β. It is not difficult to see that this notion of oracle computation is complexity sensitive in many ways. We can then generalize this definition in a straightforward
Computability and randomness: Five questions
"... 1 How were you initially drawn to the study of computation and randomness? My first contact with the area was in 1996 when I still worked at the University of Chicago. Back then, my main interest was in structures from computability theory, such as the Turing degrees of computably enumerable sets. I ..."
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1 How were you initially drawn to the study of computation and randomness? My first contact with the area was in 1996 when I still worked at the University of Chicago. Back then, my main interest was in structures from computability theory, such as the Turing degrees of computably enumerable sets. I analyzed them via coding with firstorder formulas. During a visit to New Zealand, Cris Calude in Auckland introduced me to algorithmic information theory, a subject on which he had just finished a book [3]. We wrote a paper [4] showing that a set truthtable above the halting problem is not MartinLöf random (in fact the proof showed that it is not even weakly random [33, 4.3.9]). I also learned about Solovay reducibility, which is a way to gauge the relative randomness of real numbers with a computably enumerable left cut. These topics, and many more, were studied either in Chaitin’s work [6] or in Solovay’s visionary, but never published, manuscript [35], of which Cris possessed a copy. l In April 2000 I returned to New Zealand. I worked with Rod Downey and Denis Hirschfeldt on the Solovay degrees of real numbers with computably enumerable left cut. We proved that this degree structure is dense, and that the top degree, the degree of Chaitin’s Ω, cannot be split into two lesser degrees [9]. During this visit I learned about Ktriviality, a notion formalizing the intuitive idea of a set of natural numbers that is far from random. To understand Ktriviality, we first need a bit of background. Sets of natural numbers (simply called sets below) are a main topic of study in computability theory. Sets can be “identified ” with infinite sequences of bits. Given a set A, the bit in position n has value 1 if n is in A, otherwise its value is 0. A string is a finite sequence of bits, such as 11001110110. Let K(x) denote the length of a shortest prefixfree description of a string x (sometimes called the prefixfree Kolmogorov complexity of x even though Kolmogorov didn’t introduce it). We say that K(x) is the prefixfree complexity of x. Chaitin [6] defined a set A ⊆ N to be Ktrivial if each initial segment of A has prefixfree complexity no greater than the prefixfree complexity of its length. That is, there is b ∈ N such that, for each n,
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"... L1computability, layerwise computability and Solovay reducibility ..."
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