Complexity of complexity and maximal plain versus prefix-free kolmogorov complexity

• Authors: Bruno Bauwens
• Venue: ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
• Year: 2012

Peter Gacs showed [2] that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x, i.e., C(C(x)|x)≥logn−log(2)n−O(1). Here log2(i)=loglogi etc. Following Elena Kalinina [4], we provide a game-theoretic proof of this result; modifying her argument, we get a better (and tight) bound logn−O(1). We also show the same bound for prefix-free complexity. Robert Solovay's showed [11] that infinitely many strings x have maximal plain complexity but not maximal prefix-free complexity (among the strings of the same length); i.e. for some c: |x|−C(x)≤c and |x|+K(|x|)−K(x)≥log(2) |x|−clog(3) |x|. Using the result above, we provide a short proof of Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n−C(x)≤c, and n+K(n)−K(x)≥K(K(n)|n)−3K( K(K(n)|n) |n)−c . This is very close to the upperbound K(K(n)|n)+O(1) proved by Solovay.