Counting Dependent and Independent Strings

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