Resource bounded symmetry of information revisited

The information contained in a string x about a string y is the difference between the Kolmogorov complexity of y and the conditional Kolmogorov complexity of y given x, i.e., I(x : y) = C(y) − C(y |x). The Kolmogorov–Levin Theorem says that I(x : y) is symmetric up to a small additive term. We investigate if this property also holds for several versions of polynomial time bounded Kolmogorov complexity. We study symmetry of information for some variants of distinguishing complexity CD where CD(x) is the length of a shortest program which accepts x and only x. We show relativized worlds where symmetry of information does not hold in a strong way for deterministic and nondeterministic polynomial time distinguishing complexities CD and CND. On the other hand, for nondeterministic polynomial time distinguishing complexity with randomness, CAMD, we show that symmetry of information holds for most pairs of strings in any set in NP. Our techniques extend work of Buhrman et al. (CCC 2004) on language compression by AM algorithms, and have the following application to the compression of samplable sources, introduced in Trevisan et al. (CCC 2004): any element x in the support of a polynomial time samplable source X can be given a description of size − log Pr[X = x] +O(log n), from which x can be recovered by an AM algorithm.