Relating and contrasting plain and prefix Kolmogorov complexity

In [3] a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. The proof uses Levin's symmetry of information, Levin's formula relating plain and prefix complexity and Gacs' theorem that complexity of complexity given the string can be high. We argue that the proof technique and results mentioned above are useful to simplify existing proofs and to solve open questions. We present a short proof of Solovay's result [21] relating plain and prefix complexity: $K (x) = C (x) + CC (x) + O(CCC (x))$ and $C (x) = K (x) - KK (x) + O(KKK (x))$, (here $CC(x)$ denotes $C(C(x))$, etc.). We show that there exist $\omega$ such that $\liminf C(\omega_1\dots \omega_n) - C(n)$ is infinite and $\liminf K(\omega_1\dots \omega_n) - K(n)$ is finite, i.e. the infinitely often C-trivial reals are not the same as the infinitely often K-trivial reals (i.e. [1,Question 1]). Solovay showed that for infinitely many $x$ we have $|x| - C (x) \le O(1)$ and $|x| + K (|x|) - K (x) \ge \log^{(2)} |x| - O(\log^{(3)} |x|)$, (here $|x|$ denotes the length of $x$ and $\log^{(2)} = \log\log$, etc.). We show that this result holds for prefixes of some 2-random sequences. Finally, we generalize our proof technique and show that no monotone relation exists between expectation and probability bounded randomness deficiency (i.e. [6, Question 1]).Comment: 20 pages, 1 figur

Kolmogorov complexity and computably enumerable sets

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity of $A$ is uniformly computable from the halting problem

ALGORITHMIC RANDOMNESS AND MEASURES OF COMPLEXITY

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability

ALGORITHMIC RANDOMNESS AND MEASURES OF COMPLEXITY

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability