Asymmetry of the Kolmogorov complexity of online predicting odd and even bits

Bruno Bauwens
Symmetry of information states that C(x) + C(y|x) = C(x, y) + O(log C(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 x 1 x 2 . . . x n be the length of a shortest program that computes x 2 on input x 1 , computes x 4 on input x 1 x 2 x 3 , etc; and similar for odd complexity. We show that for all n there exists an n-bit x such that both odd and even complexity
more » ... e almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence x 2 x 1 x 4 x 3 . . . , decreases the sum of odd and even complexity to C(x). Our result is related to the problem of inferrence of causality in timeseries.