Every 2-random real is Kolmogorov random
نویسنده
چکیده
We study reals with infinitely many incompressible prefixes. Call A ∈ 2 Kolmogorov random if (∃∞n) C(A n) > n − O(1), where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by Martin-Löf, Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random.1 Together with the converse—proved by Nies, Stephan and Terwijn [11]—this provides a natural characterization of 2-randomness in terms of plain complexity. We finish with a related characterization
منابع مشابه
The Dimensions of Individual Strings and Sequences
A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that ar...
متن کاملLecture Notes on Randomness for Continuous Measures
Most studies on algorithmic randomness focus on reals random with respect to the uniform distribution, i.e. the (1/2, 1/2)-Bernoulli measure, which is measure theoretically isomorphic to Lebesgue measure on the unit interval. The theory of uniform randomness, with all its ramifications (e.g. computable or Schnorr randomness) has been well studied over the past decades and has led to an impressi...
متن کاملKolmogorov-Loveland Randomness and Stochasticity
An infinite binary sequence X is Kolmogorov-Loveland (or KL) random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X . A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence. One of the major open ...
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 کاملOn the Complexity of Random Strings
Abst rac t . We show that the set R of Kolmogorov random strings is truth-table complete. This improves the previously known Turing completeness of R and shows how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of non-random strings. As an application we obtain that Post's simple set is truth-table complete in every Kolmogorov nu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Symb. Log.
دوره 69 شماره
صفحات -
تاریخ انتشار 2004