In a lossless compression system with target lengths, a compressor 𝒞 maps an integer m and a binary string x to an m -bit code p , and if m is sufficiently large, a decompressor 𝒟 reconstructs x from p . We call a pair ( m,x ) achievable for (𝒞,𝒟) if this reconstruction is successful. We introduce the notion of an optimal compressor 𝒞 opt by the following universality property: For any compressor-decompressor pair (𝒞,𝒟), there exists a decompressor 𝒟 ′ such that if (m,x) is achievable for (𝒞,𝒟), then ( m + Δ , x ) is achievable for (𝒞 opt , 𝒟 ′ ), where Δ is some small value called the overhead. We show that there exists an optimal compressor that has only polylogarithmic overhead and works in probabilistic polynomial time. Differently said, for any pair (𝒞,𝒟), no matter how slow 𝒞 is, or even if 𝒞 is non-computable, 𝒞 opt is a fixed compressor that in polynomial time produces codes almost as short as those of 𝒞. The cost is that the corresponding decompressor is slower. We also show that each such optimal compressor can be used for distributed compression, in which case it can achieve optimal compression rates as given in the Slepian–Wolf theorem and even for the Kolmogorov complexity variant of this theorem.

We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitanyi, and Zurek [1] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. It was claimed by Mahmud [11] that the prefix version of information distance equals max(K(x|y), K(y|) + O(1) (this equality with logarithmic precision was one of the main results of the paper by Bennett, Gács, Li, Vitanyi, and Zurek). We show that this claim is false, but does hold if the information distance is at least super logarithmic.

Motivation: Bioinformatics studies often rely on similarity measures between sequence pairs, which often pose a bottleneck in large-scale sequence analysis. Results: Here, we present a new convolutional kernel function for protein sequences called the Lempel-Ziv-Welch (LZW)-Kernel. It is based on code words identified with the LZW universal text compressor. The LZW-Kernel is an alignment-free method, it is always symmetric, is positive, always provides 1.0 for self-similarity and it can directly be used with Support Vector Machines (SVMs) in classification problems, contrary to normalized compression distance, which often violates the distance metric properties in practice and requires further techniques to be used with SVMs. The LZW-Kernel is a one-pass algorithm, which makes it particularly plausible for big data applications. Our experimental studies on remote protein homology detection and protein classification tasks reveal that the LZW-Kernel closely approaches the performance of the Local Alignment Kernel (LAK) and the SVM-pairwise method combined with Smith-Waterman (SW) scoring at a fraction of the time. Moreover, the LZW-Kernel outperforms the SVM-pairwise method when combined with Basic Local Alignment Search Tool (BLAST) scores, which indicates that the LZW code words might be a better basis for similarity measures than local alignment approximations found with BLAST. In addition, the LZW-Kernel outperforms n-gram based mismatch kernels, hidden Markov model based SAM and Fisher kernel and protein family based PSI-BLAST, among others. Further advantages include the LZW-Kernel's reliance on a simple idea, its ease of implementation, and its high speed, three times faster than BLAST and several magnitudes faster than SW or LAK in our tests. Availability and implementation: LZW-Kernel is implemented as a standalone C code and is a free open-source program distributed under GPLv3 license and can be downloaded from https://github.com/kfattila/LZW-Kernel. Supplementary information: Supplementary data are available at Bioinformatics Online.

A c-short program for a string x is a description of x of length at most C(x) + c, where C(x) is the Kolmogorov complexity of x. We show that there exists a randomized algorithm that constructs a list of n elements that contains a O(log n)-short program for x. We also show a polynomial-time randomized construction that achieves the same list size for O(log2n)-short programs. These results beat the lower bounds shown by Bauwens et al. [1] for deterministic constructions of such lists. We also prove tight lower bounds for the main parameters of our result. The constructions use only O(log n) (O(log2n) for the polynomial-time result) random bits. Thus using only few random bits it is possible to do tasks that cannot be done by any deterministic algorithm regardless of its running time.

Symmetry of information states that C(x)+C(y|x)=C(x,y)+O(log(C(x))). In [Chernov, Shen, Vereshchagin, and Vovk, 2008] an online variant of Kolmogorov complexity is introduced and we show that a similar relation does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring x_1 x_2...x_n be the length of a shortest program that computes x_2 on input x_1, computes x_4 on input x_1 x_2 x_3, etc; and similar for odd complexity. We show that for all n there exists an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence x_2 x_1 x_4 x_3..., decreases the sum of odd and even complexity to C(x). Our result is related to the problem of inferrence of causality in timeseries.

Abstract Péter Gács showed (Gács 1974) 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., $C\left( {C\left( x \right)|x} \right) \ge {\rm{log}}n - {\rm{log}}^{\left( 2 \right)} n - O\left( 1 \right)$ (Here ${\rm{log}}^{\left( 2 \right)} i = {\rm{loglog}}i$.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound ${\rm{log}}n - O\left( 1 \right)$ We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that $|x| - C\left( x \right) \le c$ and $|x| + K\left( {|x|} \right) - K\left( x \right) \ge {\rm{log}}^{\left( 2 \right)} |x| - c{\rm{log}}^{\left( 3 \right)} |x|$ In fact, the results of Solovay and Gács are closely related. Using the result above, we provide a short proof for 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 $n - C\left( x \right) \le c$ and $n + K\left( n \right) - K\left( x \right) \ge K\left( {K\left( n \right)|n} \right) - 3K\left( {K\left( {K\left( n \right)|n} \right)|n} \right) - c.$ We also prove a close upper bound $K\left( {K\left( n \right)|n} \right) + O\left( 1 \right)$ Finally, we provide a direct game proof for Joseph Miller’s generalization (Miller 2006) of the same Solovay’s theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that $|x| + K\left( {|x|} \right) - K\left( x \right) \ge {\rm{log}}^{\left( 2 \right)} |x| - O\left( {{\rm{log}}^{\left( 3 \right)} |x|} \right).$

Given a machine U, a c-short program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a O(log|x|)-short program for x. We also show that there exist computable functions that map every x to a list of size O(|x|2) containing a O(1)-short program for x and this is essentially optimal because we prove that such a list must have size Ω(|x|2). Finally we show that for some machines, computable lists containing a shortest program must have length Ω(2|x|).

We study statistical sum-tests and independence tests, in particular for computably enumerable semimeasures on a discrete domain. Among other things, we prove that for universal semimeasures every $\Sigma ^{0}_{1}$ -sum-test is bounded, but unbounded $\Pi ^{0}_{1}$ -sum-tests exist, and we study to what extent the latter can be universal. For universal semimeasures, in the unary case of sum-test we leave open whether universal $\Pi ^{0}_{1}$ -sum-tests exist, whereas in the binary case of independence tests we prove that they do not exist.