Enumerations of the Kolmogorov function

Beigel, Richard; Buhrman, Harry; Fejer, Peter A.; Fortnow, Lance; Grabowski, Piotr; Longpré, Luc; Muchnik, Andrej; Stephan, Frank; Torenvliet, Leen

doi:10.2178/jsl/1146620156

Cited by 20 publications

(52 citation statements)

References 29 publications

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“…We show (1) ⇒ (2) ⇒ (3) ⇒ (1). Given an autocomplex set A, choose an A-recursive order g, where C(A n) ≥ g(n), and in order to obtain a function h as required by (2), let h(n) = min{l : g(l) ≥ n}. Given a function h as in (2), in order to obtain a function f as required by (3), simply let f (n) be equal to (an appropriate encoding of) the prefix of A of length h(n).…”

Section: Proposition 22mentioning

confidence: 99%

“…Given an autocomplex set A, choose an A-recursive order g, where C(A n) ≥ g(n), and in order to obtain a function h as required by (2), let h(n) = min{l : g(l) ≥ n}. Given a function h as in (2), in order to obtain a function f as required by (3), simply let f (n) be equal to (an appropriate encoding of) the prefix of A of length h(n). Finally, given an A-recursive function f as in (3), let u(n) be an A-recursive order such that some fixed oracle Turing machine M computes f with oracle A such that M queries on input n only bits A(m) of A where m ≤ u(n).…”

Section: Proposition 22mentioning

confidence: 99%

“…traceable. (2) There is a fixed recursive function z(n) such that for each f ≤ T A, and almost every n, the set {x : f (x) = ϕ x (x)} has at least n elements below z(n). The following statements are equivalent for any set A:…”

Section: Theorem 53 If a Set A Has High Turing Degree Then There Imentioning

confidence: 99%

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Kolmogorov complexity and the Recursion Theorem

Kjos-Hanssen

Merkle

Stephan

2011

Trans. Amer. Math. Soc.

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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 wtt-compute 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 A-recursive lower bound on the Kolmogorov complexity of the initial segments of A. A is PA-complete, 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 C-complexity 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 H-complexity 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 no longer permit the usage of the Recursion Theorem.

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Section: Proposition 22mentioning

confidence: 99%

Section: Proposition 22mentioning

confidence: 99%

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Kolmogorov complexity and the Recursion Theorem

Kjos-Hanssen

Merkle

Stephan

2011

Trans. Amer. Math. Soc.

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“…The next proof will use the following fact [3]: For every oracle A and every A-recursive function f , if there is a constant c with…”

Section: Remark 10mentioning

confidence: 99%

Index sets and universal numberings

Jain

Stephan

Teutsch

2011

Journal of Computer and System Sciences

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This paper studies the Turing degrees of various properties defined for universal numberings, that is, for numberings which list all partial-recursive functions. In particular properties relating to the domain of the corresponding functions are investigated like the set DEQ of all pairs of indices of functions with the same domain, the set DMIN of all minimal indices of sets and DMIN * of all indices which are minimal with respect to equality of the domain modulo finitely many differences. A partial solution to a question of Schaefer is obtained by showing that for every universal numbering with the Kolmogorov property, the set DMIN * is Turing equivalent to the double jump of the halting problem. Furthermore, it is shown that the join of DEQ and the halting problem is Turing equivalent to the jump of the halting problem and that there are numberings for which DEQ itself has 1-generic Turing degree.

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“…For a closed set [Q] (also denoted as Q) of 2 ω let Q = {ρ ∈ 2 <ω : [ρ] ∩ [Q] = ∅} (where [ρ] = {X ∈ 2 ω : X ⊃ ρ}). Definition 1.1 (Beigel et al [2]). Fix the canonical representation of finite sets, with each finite set denoted by D n , where n is the (canonical) index of this finite set.…”

Section: Introductionmentioning

confidence: 99%

Cone avoiding closed sets

Liu¹

2014

Trans. Amer. Math. Soc.

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We prove that for an arbitrary subtree T of 2 <ω with each element extendable to a path, a given countable class M closed under disjoint union, and any set A, if none of the members of M strongly k-enumerate T for any k, then there exists an infinite set contained in either A orĀ such that for every C ∈ M, C ⊕ G also does not strongly k-enumerate T . We give applications of this result, which include: (1) RT 2 2 doesn't imply WWKL 0 ; (2) [Ambos-Spies et al. [1]] DNR is strictly weaker than WWKL 0 ; (3) ] for any Martin-Löf random set A either A orĀ contains an infinite subset that does not compute any Martin-Löf random set; etc. We also discuss further generalizations of this result.2010 Mathematics Subject Classification. Primary 03B30; Secondary 03F35 03C62 68Q30 03D32 03D80 28A78.

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Enumerations of the Kolmogorov function

Cited by 20 publications

References 29 publications

Kolmogorov complexity and the Recursion Theorem

Kolmogorov complexity and the Recursion Theorem

Index sets and universal numberings

Cone avoiding closed sets

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