scholarly journals Measures and their random reals

2015 ◽  
Vol 367 (7) ◽  
pp. 5081-5097 ◽  
Author(s):  
Jan Reimann ◽  
Theodore A. Slaman
Keyword(s):  
Random Reals

CHAITIN’S Ω AS A CONTINUOUS FUNCTION

2019 ◽  
Vol 85 (1) ◽  
pp. 486-510
Author(s):  
RUPERT HÖLZL ◽  
WOLFGANG MERKLE ◽  
JOSEPH MILLER ◽  
FRANK STEPHAN ◽  
LIANG YU
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Universal Machine ◽  
Image Position ◽  
Turing Complete ◽  
Random Reals

AbstractWe prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$. For example, we show that the maximal value of ${\rm{\hat{\Omega }}}$ must be random, the minimal value must be Turing complete, and that ${\rm{\hat{\Omega }}}\left( X \right) \oplus X{ \ge _T}\emptyset \prime$ for every X. We also obtain some machine-dependent results, including that for every $\varepsilon > 0$, there is a universal machine V such that ${{\rm{\hat{\Omega }}}_V}$ maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that $X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left( X \right)$; and that there is a real X and a universal machine V such that ${{\rm{\Omega }}_V}\left( X \right)$ is rational.

From index sets to randomness in ∅n: random reals and possibly infinite computations part II

2009 ◽  
Vol 74 (1) ◽  
pp. 124-156 ◽  
Author(s):  
Verónica Becher ◽  
Serge Grigorieff
Keyword(s):  
Large Class ◽  
Turing Machine ◽  
Index Sets ◽  
Random Reals

AbstractWe obtain a large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle ∅(n−1)) à la Chaitin. Any such real is defined as the probability that a universal monotone Turing machine performing possibly infinite computations on infinite (resp. finite large enough, resp. finite self-delimited) inputs produces an output in a given set . In particular, we develop methods to transfer many-one completeness results of index sets to n-randomness of associated probabilities.

Perfect sets of random reals

1993 ◽  
Vol 83 (1-2) ◽  
pp. 153-176 ◽  
Author(s):  
Jörg Brendle ◽  
Haim Judah
Keyword(s):  
Random Reals

scholarly journals Random reals, the rainbow Ramsey theorem, and arithmetic conservation

2013 ◽  
Vol 78 (1) ◽  
pp. 195-206 ◽  
Author(s):  
Chris J. Conidis ◽  
Theodore A. Slaman
Keyword(s):  
Ramsey Theorem ◽  
First Order ◽  
Base Theory ◽  
Random Reals

AbstractWe investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 + 2-RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA0 for arithmetic sentences. Thus, from the Csima–Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-RAN has non-trivial arithmetic consequences. In Section 4, we show that 2-RAN is conservative over RCA0 + BΣ2 for -sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P− + IΣ1 and no stronger than P− + BΣ2.

scholarly journals Denjoy, Demuth and density

2014 ◽  
Vol 14 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Laurent Bienvenu ◽  
Rupert Hölzl ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Computable Function ◽  
Density Theorem ◽  
Positive Density ◽  
Random Real ◽  
Open Problems ◽  
Covering Problems ◽  
Closed Class ◽  
Random Reals ◽  
Upper And Lower Derivatives ◽  
Computable Functions

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class 𝒞 ⊆ [0, 1] containing z has positive density at z. Under the stronger assumption that z is not LR-hard, we show that every such class has density one at z. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Löf random reals and K-trivial sets: the noncupping and covering problems. We say that f : [0, 1] → ℝ satisfies the Denjoy alternative at z ∈ [0, 1] if either the derivative f′(z) exists, or the upper and lower derivatives at z are +∞ and -∞, respectively. The Denjoy–Young–Saks theorem states that every function f : [0, 1] → ℝ satisfies the Denjoy alternative at almost every z ∈ [0, 1]. We answer a question posed by Kučera in 2004 by showing that a real z is computably random if and only if every computable function f satisfies the Denjoy alternative at z. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real zDA-random if every Markov computable function satisfies the Denjoy alternative at z. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin.24(3) (1983) 391–406) by showing that every Turing incomplete Martin-Löf random real is DA-random. The proof involves the notion of nonporosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Löf randomness.

scholarly journals The Kolmogorov complexity of random reals

2004 ◽  
Vol 129 (1-3) ◽  
pp. 163-180 ◽  
Author(s):  
Liang Yu ◽  
Decheng Ding ◽  
Rodney Downey
Keyword(s):  
Kolmogorov Complexity ◽  
Random Reals
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