Counting Dependent and Independent Strings

Abstract

We derive quantitative results regarding sets of n-bit strings that have different dependency or independency properties. Let Cx be the Kolmogorov complexity of the string x. A string y has α dependency with a string x if Cy-Cy | x ≥ α. A set of strings {x1,..., xt} is pairwise α-independent if for all i ≠ j, Cxi-Cxi | xj Cx1+...+Cxt-α, for every permutation π of [t]. We show that: • For every n-bit string x with complexity Cx ≥ α + 7 log n, the set of n-bit strings that have α dependency with x has size at least 1/polyn2n-α. In case α is computable from n and Cx ≥ α + 12 log n, the size of the same set is at least 1/C2n-α-polyn2α, for some positive constant C. • There exists a set of n-bit strings A of size polyn2α such that any n-bit string has α-dependency with some string in A. • If the set of n-bit strings {x1,..., xt} is pairwise α-independent, then t ≤ polyn2α. This bound is tight within a polyn factor, because, for every n, there exists a set of n-bit strings {x1,..., xt} that is pairwise α-dependent with t = 1/polyn · 2α for all α ≥ 5 log n. • If the tuple of n-bit strings x1,..., xt is mutually α-independent, then t ≤ polyn2α for all α ≥ 7 log n + 6.