π11-Martin-Löf random reals as measures of natural open sets
نویسنده
چکیده
We elaborate a recursive-theoretic method to define real numbers which are random to any jump of the halting problem. This is obtained by generalizing a result of V. Becher and G. Chaitin dealing with the class of cofinite sets. As in the work of V. Becher and S. Grigorieff (but this time in the context of open sets in Cantor space) appeal to completeness phenomena replaces machine arguments. Mathematics Subject Classification: 03D25, 03D32.
منابع مشابه
D ec 2 01 6 RANDOMNESS VIA INFINITE COMPUTATION AND EFFECTIVE DESCRIPTIVE SET THEORY
We study randomness beyond Π11-randomness and its Martin-Löf type variant, introduced in [HN07] and further studied in [BGM]. The class given by the infinite time Turing machines (ITTMs), introduced by Hamkins and Kidder, is strictly between Π11 and Σ 1 2. We prove that the natural randomness notions associated to this class have several desirable properties resembling those of the classical ra...
متن کاملAlgorithmically random closed sets and probability
by Logan M. Axon Algorithmic randomness in the Cantor space, 2 ω , has recently become the subject of intense study. Originally defined in terms of the fair coin measure, algorithmic randomness has since been extended, for example in Reimann and Slaman [22, 23], to more general measures. Others have meanwhile developed definitions of algorithmic randomness for different spaces, for example the ...
متن کاملRandomness and Recursive Enumerability
One recursively enumerable real α dominates another one β if there are nondecreasing recursive sequences of rational numbers (a[i] : i ∈ ω) approximating α and (b[i] : i ∈ ω) approximating β and a positive constant C such that for all n, C(α−a[n]) ≥ (β−b[n]). See [Solovay, 1975] and [Chaitin, 1977]. We show every recursively enumerable random real dominates all other recursively enumerable real...
متن کاملOn Schnorr and computable randomness, martingales, and machines
This paper falls within an overall program articulated in Downey, Hirschfeldt, Nies and Terwijn [8], and Downey and Hirschfeldt [4], of trying to calibrate the algorithmic randomness of reals. There are three basic approaches to algorithmic randomness. They are to characterize randomness in terms of algorithmic predictability (“a random real should have bits that are hard to predict”), algorith...
متن کاملNormal and Martin-Löf Random Numbers
We survey the current known relations between four classes of real numbers: Liouville numbers, computable reals, Borel absolutely normal numbers and Martin-Löf random reals. The expansions of the reals will play an important role. The paper refers to the original material and does not repeat the proofs; a characterisation of Liouville numbers in terms of their expansions will be proved. Finally...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Theor. Comput. Sci.
دوره 589 شماره
صفحات -
تاریخ انتشار 2015