Bruno Bauwens

Linear List-Approximation for Short Programs (or the Power of a Few Random Bits)

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(log2 n)-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(log2 n) 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.

Short Lists with Short Programs in Short Time

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|).

Conditional Probabilities and van Lambalgen’s Theorem Revisited

The definition of conditional probability in the case of continuous distributions (for almost all conditions) was an important step in the development of mathematical theory of probabilities. Can we define this notion in algorithmic probability theory for individual random conditions? Can we define randomness with respect to the conditional probability distributions? Can van Lambalgen’s theorem (relating randomness of a pair and its elements) be generalized to conditional probabilities? We discuss the developments in this direction. We present almost no new results trying to put known results into perspective and explain their proofs in a more intuitive way. We assume that the reader is familiar with basic notions of measure theory and algorithmic randomness (see, e.g., Shen et al. ??2013 or Shen ??2015 for a short introduction).

Notes on Sum-Tests and Independence Tests

AbstractWe 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

Sophistication vs Logical Depth

\Sigma ^{0}_{1}

Conditional Measure and the Violation of Van Lambalgen's Theorem for Martin-Löf Randomness

-sum-test is bounded, but unbounded

An Additivity Theorem for Plain Kolmogorov Complexity

\Pi ^{0}_{1}

Complexity of complexity and maximal plain versus prefix-free kolmogorov complexity

-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

LZW-Kernel: fast kernel utilizing variable length code blocks from LZW compressors for protein sequence classification

\Pi ^{0}_{1}

Relating and Contrasting Plain and Prefix Kolmogorov Complexity

-sum-tests exist, whereas in the binary case of independence tests we prove that they do not exist.