On Information Conservation and Algorithmic Complexity

In this paper we investigate standard prefix-free Kolmogorov complexity in the context of Zellner’s information conservation principle (ICP). We show that prefix-free Kolmogorov complexity K is not efficient in this sense. We introduce Information Conserving Algorithmic complexity K∗, defined on a kernel space of random strings. We prove that this version is efficient in a weak sense. We prove that universal Turing machines do not conserve information in a strong sense, but we conjecture the existence of at least one such machine U. Because K∗ conserves information, the prefix-free aspect of the program code can be ignored as an internal aspect of the representation. This leads to a variant of the universal distribution m∗ using a uniform density estimator ξU for the distribution of the random strings. This distribution is shown to be smoother than the standard Solomonoff distribution. Of course ξU is unknown, but since it ’absorbs’ our uncertainty about the distribution m∗ uniformly, it leads to a theory that can be applied to small data sets without the intervention of a O(1) factor. keywords: Kolmogorov Complexity, Zellner’s Information Conservation Principle, Universal Distribution