Complexity of Complexity and Strings with Maximal Plain and Prefix Kolmogorov Complexity

Peter Gacs showed [1] 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) ≥ log n− log n−O(1). Here log(i) = log log i etc. Following Elena Kalinina [3], we provide a gametheoretic proof of this result; modifying her argument, we get a better (and tight) bound log n − O(1). We also show the same bound for prefix-free complexity. Robert Solovay’s showed [10] 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 |x| − c log |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.