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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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Cited by 60 (31 self)
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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.
Some ComputabilityTheoretical Aspects of Reals and Randomness
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. Then we study such objects in terms of algorithmic randomness, culminating in some recent work of the author with Hirschfeldt, Laforte, and Nies conce ..."
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Cited by 21 (7 self)
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. Then we study such objects in terms of algorithmic randomness, culminating in some recent work of the author with Hirschfeldt, Laforte, and Nies concerning methods of calibrating randomness.
Presentations of computably enumerable reals
 Theoretical Computer Science
, 2002
"... Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings. ..."
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Cited by 9 (5 self)
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Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings.
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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Cited by 2 (2 self)
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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.
Computably enumerable reals and uniformly presentable ideals
 Mathematical Logic Quarterly
"... We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefixfree set of strings such that α = ∑ σ∈A 2−σ . Note that ∑ σ∈A 2−σ  is precisely the measure of the set of reals that have a ..."
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We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefixfree set of strings such that α = ∑ σ∈A 2−σ . Note that ∑ σ∈A 2−σ  is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate ∑ σ∈A 2−σ  by µ(A). It is known that whenever A so presents α then A ≤wtt α, where ≤wtt denotes weak truth table reducibility, and that the wtt degrees of presentations form an ideal I(α) in the computably enumerable wtt degrees. We prove that any such ideal is Σ 0 3, and conversely that if I is any Σ 0 3 ideal in the computably enumerable wtt degrees then there is a computable enumerable real α such that I = I(α). 1
The dynamics of cellular automata in shiftinvariant topologies
"... Abstract. We study the dynamics of cellular automata, and more specifically their transitivity and expansivity, when the set of configurations is endowed with a shiftinvariant (pseudo)distance. We first give an original proof of the nontransitivity of cellular automata when the set of configurati ..."
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Abstract. We study the dynamics of cellular automata, and more specifically their transitivity and expansivity, when the set of configurations is endowed with a shiftinvariant (pseudo)distance. We first give an original proof of the nontransitivity of cellular automata when the set of configurations is endowed with the Besicovitch pseudodistance. We then show that the Besicovitch pseudodistance induces a distance on the set of shiftinvariant measures and on the whole space of measures, and we prove that in these spaces also, cellular automata cannot be expansive nor transitive. 1