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

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. We show that for all n there exists an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence x2x1x4x3 . . . , decreases the sum of odd and even complexity to C(x). Our result is related to the problem of inferrence of causality in timeseries. 1998 ACM Subject Classification E.4 Coding and Information Theory