Limit complexities revisited

The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from (Vereshchagin, 2002) saying that $\limsup_n\KS(x|n)$ (here $\KS(x|n)$ is conditional (plain) Kolmogorov complexity of $x$ when $n$ is known) equals $\KS^{\mathbf{0’}(x)$, the plain Kolmogorov complexity with $\mathbf{0’$-oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of (Muchnik, 1987) about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a by-product, we get a criterion of $\mathbf{0’}$ Martin-L"of randomness (called also 2-randomness) proved in (Miller, 2004): a sequence $\omega$ is 2-random if and only if there exists $c$ such that any prefix $x$ of $\omega$ is a prefix of some string $y$ such that $\KS(y)\ge |y|-c$. (In the 1960ies this property was suggested in (Kolmogorov, 1968) as one of possible randomness definitions; its equivalence to 2-randomness was shown in (Miller, 2004) while proving another 2-randomness criterion (see also (Nies et al. 2005)): $\omega$ is 2-random if and only if $\KS(x)\ge |x|-c$ for some $c$ and infinitely many prefixes $x$ of $\omega$. Finally, we show that the low-basis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.

💡 Research Summary

The paper “Limit Complexities Revisited” presents a unified and streamlined treatment of several classical results concerning Kolmogorov complexity, its conditional and prefix variants, limit frequencies, a‑priori probabilities, and effectively open sets, and shows how they all fit into a single framework based on a simple “finite‑part replacement” argument.

The first main theorem establishes that the lim sup of the plain conditional complexity C(x | n) as n grows equals the plain complexity C^{0′}(x) computed with a 0′‑oracle, up to an additive constant. The proof proceeds in two directions. In the easy direction, any program of length k that computes x with a 0′‑oracle actually uses only a finite initial segment 0_n of the oracle; therefore for all sufficiently large n the conditional complexity C(x | n) is bounded by k+O(1). The converse direction is more delicate: assuming lim sup C(x | n) < k, one defines the finite sets U_n = {u | C(u | n) < k}. Each U_n contains fewer than 2^k strings. Using a 0′‑computable “horizontal‑line addition” operation, the authors iteratively enlarge the sets while preserving the size bound, eventually producing a 0′‑enumerable set that contains exactly those strings that belong to U_n for all sufficiently large n. This construction eliminates the extra existential quantifier in the definition of the lim inf set, thereby making the set 0′‑enumerable.

The same technique is applied to prefix complexity K(x) and to the universal a‑priori semimeasure m(x). Since K(x) = −log m(x) up to a constant, showing that lim inf m(x | n) equals m^{0′}(x) up to a Θ(1) factor immediately yields the analogous result for K: lim sup K(x | n) = K^{0′}(x)+O(1). The proof for the semimeasures again uses a “increase operation”: for each triple (r, N, u) the algorithm tries to raise all values m_n(u) for n > N to at least r, provided the semimeasure condition remains satisfied. Whether such an operation is admissible is decidable with a 0′‑oracle, and by processing all triples in a computable order one defines a new lower‑semicomputable semimeasure m′ that dominates the lim inf of the original sequence.

The paper then turns to limit frequencies. For a (partial) computable function f:N→N the limit frequency q_f(x) is defined as the lim inf of the relative occurrence of x in the initial segment of f. The authors prove that for any partial computable f, q_f is bounded above by a 0′‑lower‑semicomputable semimeasure. This extends earlier work that considered only total computable functions, showing that allowing partiality does not increase the maximal possible limit frequency.

Next, the authors address effectively open subsets of Cantor space Ω equipped with the standard Bernoulli (uniform) measure. Given an enumerable family {U_n} of open sets each of measure at most ε, they construct a uniformly 0′‑effectively open set V of measure ≤ ε that contains the interior of the lim inf of the U_n’s. The construction mirrors the earlier finite‑set argument: for each string x and integer N an operation adds the basic open set Ω_x to all U_n with n > N, provided the total measure stays ≤ ε. Whether this operation is admissible can be decided with a 0′‑oracle, and by performing all admissible operations in a computable order one obtains V as a 0′‑effectively open set. This yields a computable analogue of the classical measure‑theoretic fact that the lim inf of a sequence of sets of measure ≤ ε also has measure ≤ ε.

Finally, the paper revisits the characterization of 2‑randomness (Martin‑Löf randomness relative to 0′) originally proved by Miller. Using the previously developed tools, the authors give a short proof of the equivalence: a sequence ω is 2‑random iff there exists a constant c such that every prefix x of ω can be extended to a string y with C(y) ≥ |y| − c. The forward direction follows from the fact that if ω were not 2‑random, one could construct a 0′‑effectively open set of arbitrarily small measure covering ω, contradicting the assumed complexity bound on extensions. The reverse direction is obtained by constructing, for each c, a 0′‑effectively open set of measure ≤ 2^{−c} that covers all sequences whose prefixes fail the complexity condition, again using the finite‑part addition technique.

The paper also shows how the low‑basis theorem can be employed to obtain alternative proofs of the above results and to strengthen the statement about effectively open sets, thereby providing a more robust version of the 2‑randomness criterion.

Overall, the work demonstrates that a simple combinatorial construction—adding finite “horizontal” or “vertical” slices while checking admissibility with a 0′‑oracle—suffices to derive a wide range of limit‑complexity theorems, to relate limit frequencies to a‑priori probabilities, and to give clean characterizations of higher‑order randomness. This unifies several strands of algorithmic information theory and offers a clearer conceptual picture of how oracle information, limit operations, and effective measure interact.