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

Original · EN

Symmetry of information states that C(x) + C(y|x) = C(x,y) + O(C(x)). We show that a similar relation for online Kolmogorov complexity 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 exist 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).