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Symmetry of information states that C(x) + C(y|x) = C(x, y) + O(logC(x)). In [3] an online variant of Kolmogorov complexity is introduced and we show that a similar relation does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring x1x2 . . . xn be the length of a shortest program that computes x2 on input x1, computes x4 on input x1x2x3, etc; and similar for odd complexity. ...

It is shown that from two strings that are partially random and independent (in the sense of Kolmogorov complexity) it is possible to effectively construct polynomially many strings that are random and pairwise independent. If the two initial strings are random, then the above task can be performed in polynomial time. It is also possible to construct in polynomial time a random string, from two...

1 From probability theory to Kolmogorov complexity 3 1.1 Randomness and Probability theory . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Intuition of finite random strings and Berry’s paradox . . . . . . . . . . . . 5 1.3 Kolmogorov complexity relative to a function . . . . . . . . . . . . . . . . . 6 1.4 Why binary programs? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5...

We define a new discrete version of scaled dimension and we find connections between the scaled dimension of a string and its Kolmogorov complexity and predictability. We give a new characterization of constructive scaled dimension by Kolmogorov complexity, and prove a new result about scaled dimension and prediction.

We study the properties of the set of binary strings with the relation \the Kolmogorov complexity of x conditional to y is small". We prove that there are pairs of strings which have no greatest common lower bound with respect to this pre-order. We present several examples when the greatest common lower bound exists but its complexity is much less than mutual information (extending G acs and K...

Exact constructive dimension as a generalisation of Lutz’s [Lut00, Lut03] approach to constructive dimension was recently introduced in [Sta11]. It was shown that it is in the same way closely related to a priori complexity, a variant of Kolmogorov complexity, of infinite sequences as their constructive dimension is related to asymptotic Kolmogorov complexity. The aim of the present paper is to...