Complexity of Complexity and Maximal Plain versus Prefix-Free Kolmogorov Complexity

Abstract

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.