Contrasting Plain and Prefix-free Kolmogorov Complexity

Let SCRc = {σ ∈ 2n : K(σ) ≥ n + K(n) − c}, where K denotes prefix-free Kolmogorov complexity. These are the strings with essentially maximal prefix-free complexity. We prove that SCRc is not a Π1 set for sufficiently large c. This implies Solovay’s result that strings with maximal plain Kolmogorov complexity need not have maximal prefix-free Kolmogorov complexity, even up to a constant. We show that if Q ⊆ SCRc is an infinite Π1 set, then Q is hyperimmune. Furthermore, assuming that Q ∈ Π1 contains strings of every length, we derive a bound on the least element of Qr SCRc, matching the bound Solovay gave for Q = KRk = {σ ∈ 2n : C(σ) ≥ n− k}. We also give short derivations of Solovay’s formulae relating plain and prefix-free complexity and An. A. Muchnik’s result that these two complexity measures can disagree on the relative complexity of strings.