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Kolmogorov complexity and the Recursion Theorem. Manuscript, submitted for publication
, 2005
"... Abstract. Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wttcompute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of ..."
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Abstract. Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wttcompute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial Arecursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PAcomplete, that is, A can compute a {0, 1}valued DNR function, iff A can compute a function F such that F (n) is a string of length n and maximal Ccomplexity among the strings of length n. A ≥T K iff A can compute a function F such that F (n) is a string of length n and maximal Hcomplexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem. 1.
Uniform almost everywhere domination
 Journal of Symbolic Logic
, 2006
"... ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed t ..."
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Cited by 25 (1 self)
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ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the prooftheoretic strength of the regularity of Lebesgue measure for G δ sets. Our constructions essentially settle the reverse mathematical classification of this principle. 1.
An extension of the recursively enumerable Turing degrees
 Journal of the London Mathematical Society
, 2006
"... Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overco ..."
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Cited by 22 (16 self)
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Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with nonempty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally nonrecursive functions, and rn is the weak degree of the nrandom reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is onetoone, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1.
BINARY SUBTREES WITH FEW LABELED PATHS
"... Abstract. We prove several quantitative Ramseyan results involving ternary complete trees with {0, 1}labeled edges where we attempt to nd a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson's in computability theory; we show ..."
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Abstract. We prove several quantitative Ramseyan results involving ternary complete trees with {0, 1}labeled edges where we attempt to nd a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson's in computability theory; we show that there is a bounded Π 0 1 class of positive measure which is not strongly (Medvedev) reducible to DNR3; in fact, the class of 1random reals is not strongly reducible to DNR3. 1.
1 Bounded Limit Recursiveness
, 2007
"... Let X be a Turing oracle. A function f(n) issaidtobe boundedly limit recursive in X if it is the limit of an Xrecursive sequence of Xrecursive functions ˜f(n, s) such that the number of times ˜f(n, s) changes is bounded by a recursive function of n. Let us say that X is BLRlow if every function w ..."
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Let X be a Turing oracle. A function f(n) issaidtobe boundedly limit recursive in X if it is the limit of an Xrecursive sequence of Xrecursive functions ˜f(n, s) such that the number of times ˜f(n, s) changes is bounded by a recursive function of n. Let us say that X is BLRlow if every function which is boundedly limit recursive in X is boundedly limit recursive in 0. This is a lowness property in the sense of Nies. These notions were introduced by Joshua A. Cole and the speaker in a recently submitted paper on mass problems and hyperarithmeticity. The purpose of this talk is to compare BLRlowness to similar properties which have been considered in the recursiontheoretic literature. Among the properties discussed are: Ktriviality, superlowness, jumptraceability, weak jumptraceability, total ωrecursive enumerability, array recursiveness, array jumprecursiveness, and strong jumptraceability. 2 Definition. If X is a Turing oracle, let BLR(X) betheset of numbertheoretic functions f: ω → ω which are boundedly limit recursive in X. This means that there exist an Xrecursive approximating function ˜ f(n, s) and a recursive bounding function ̂ f(n) such that and for all n. f(n) = lims ˜ f(n, s) {s  ˜ f(n, s) ̸ = ˜ f(n, s +1)}  < ̂ f(n) In particular, BLR(0) = {f  f ≤ wtt 0 ′}. The BLR operator was introduced in Mass problems and hyperarithmeticity, by Joshua A. Cole and Stephen G. Simpson, 20 pages, submitted 2006 to JML. 3 Cole and Simpson used the BLR operator to construct a natural embedding of the hyperarithmetical hierarchy into P w. Namely, we proved that the Muchnik degrees inf(h ∗ α, 1) forα<ωCK 1 are distinct ∈Pw.
DIAGONALLY NONCOMPUTABLE FUNCTIONS AND BIIMMUNITY
"... Abstract. We prove that every diagonally noncomputable function computes a set A which is biimmune, meaning that neither A nor its complement has an infinite computably enumerable subset. 1. ..."
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Abstract. We prove that every diagonally noncomputable function computes a set A which is biimmune, meaning that neither A nor its complement has an infinite computably enumerable subset. 1.