Related papers: Limit Complexities, Minimal Descriptions, and $n$-…

The famous G\"odel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but…

Let $K \subset {\mathbb R}^n$ be a compact definable set in an o-minimal structure over $\mathbb R$, e.g., a semi-algebraic or a subanalytic set. A definable family $\{ S_\delta|\> 0< \delta \in {\mathbb R} \}$ of compact subsets of $K$, is…

This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that…

The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's…

We fix a number field $K$ and study statistical properties of the ring $\mathcal{O}_K[\gamma]\cap K$ as $\gamma$ varies over algebraic numbers of a fixed degree $n\geq 2$. Given $k\geq 1$, we explicitly compute the density of $\gamma$ for…

Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic…

We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of "low description complexity". Specifically, we consider $m\times n$ random matrices…

Let M be ternary, homogeneous and simple. We prove that if M is finitely constrained, then it is supersimple with finite SU-rank and dependence is $k$-trivial for some $k < \omega$ and for finite sets of real elements. Now suppose that, in…

Consider a binary string $x$ of length $n$ whose Kolmogorov complexity is $\alpha n$ for some $\alpha<1$. We want to increase the complexity of $x$ by changing a small fraction of bits in $x$. This is always possible: Buhrman, Fortnow,…

Algorithmic statistics studies explanations of observed data that are good in the algorithmic sense: an explanation should be simple i.e. should have small Kolmogorov complexity and capture all the algorithmically discoverable regularities…

We consider word complexity and topological entropy for random substitution subshifts. In contrast to previous work, we do not assume that the underlying random substitution is compatible. We show that the subshift of a primitive random…

We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this…

We prove a precise formula for the minimal number K(n) such that every binary word of length $n$ can be divided into K(n) palindromes. Also we estimate the average number $\ol K(n)$ of palindromes composing a random binary word of the…

Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This…

Kolmogorov argued that the concept of information exists also in problems with no underlying stochastic model (as Shannon's information representation) for instance, the information contained in an algorithm or in the genome. He introduced…

The Kochen-Specker theorem states that exclusive and complete deterministic outcome assignments are impossible for certain sets of measurements, called Kochen-Specker (KS) sets. A straightforward consequence is that KS sets do not have…

An accurate assessment of a model's complexity is crucial for topics such as interpretation, generalization, and model selection. However, most existing complexity measures either rely on heuristic assumptions or are computationally…

Given a simple undirected graph, one can construct from it a $c$-step nilpotent Lie algebra for every $c \geq 2$ and over any field $K$, in particular also over the real and complex numbers. These Lie algebras form an important class of…

We give a construction of a large first-order definable family of subrings of finitely generated fields $K$ of any characteristic. We deduce that for any such $K$ there exists a first-order sentence $\varphi_K$ characterising $K$ in the…

This paper studies sequence prediction based on the monotone Kolmogorov complexity Km=-log m, i.e. based on universal deterministic/one-part MDL. m is extremely close to Solomonoff's universal prior M, the latter being an excellent…