Counting dependent and independent strings

• Authors: Marius Zimand
• Venue: MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
• Year: 2010

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 ≠ 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 log n, 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 log n, the size of same set is at least (1/C)2n-α - poly(n)2α, for some positive constant C. - There exists a set of n-bit strings A of size poly(n)2α 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)2α. 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)) ċ 2α (for all α ≥ 5 log n). - If the tuple of n-bit strings (x1,...,xt) is mutually α-independent, then t ≤ poly(n)2α (for all α ≥ 7 log n + 6).