Related papers: Topological arguments for Kolmogorov complexity

We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the well-known 2log n upper bound on the Kolmogorov complexity of initial segments of r.e.\ sets is optimal and characterize…

This paper is a contribution to understanding what properties should a topological algebra on a Stone space satisfy to be profinite. We reformulate and simplify proofs for some known properties using syntactic congruences. We also clarify…

We examine Gromov's method of selecting a point "heavily covered" by simplices formed by a given finite point sets, in order to understand the dependence of the heavily covered point on parameters. We have no continuous dependence, but…

Classical versions of Kolmogorov complexity are incomputable. Nevertheless, in 1975 Solovay showed that there are computable functions $f > K+O(1)$ such that for infinitely many strings $\sigma$, $f(\sigma)=K(\sigma)+O(1)$, where $K$…

Multilayer networks preserve full information about the different interactions among the constituents of a complex system, and have recently proven quite useful in modelling transportation networks, social circles, and the human brain. A…

In contrast with the notion of complexity, a set $A$ is called anti-complex if the Kolmogorov complexity of the initial segments of $A$ chosen by a recursive function is always bounded by the identity function. We show that, as for…

Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory -- in the discrete setting of bits and Turing machines. Over real numbers, on the other hand, the…

Chaitin's incompleteness theorem states that sufficiently rich formal systems cannot prove lower bounds on Kolmogorov complexity. In this paper we extend this theorem by showing theories that prove the Kolmogorov complexity of a large (but…

Kolmogorov complexity and algorithmic probability are defined only up to an additive resp. multiplicative constant, since their actual values depend on the choice of the universal reference computer. In this paper, we analyze a natural…

Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always…

We prove (a weak version of) Arnold's Chord Conjecture using Gromov's ``classical'' idea in to produce holomorphic disks with boundary on a Lagrangian submanifold.

We prove a criterion for $k-$formality of arrangements, using a complex constructed from vector spaces introduced in \cite{bt}. As an application, we give a simple description of $k-$formality of graphic arrangements: Let $G$ be a connected…

We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial $t(n)\geq (1+\varepsilon)n, \varepsilon>0$, the following are equivalent: - One-way functions exists (which in turn is equivalent to…

In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…

Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…

We show how the framework of crossed simplicial groups may be used to provide a classification of topological field theories on open cobordism categories defined by reductions of the structure group to a planar Lie group. Such theories are…

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let $X$ be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $d$. Informally, the theorem states that if $X$…

We extend Deligne's original argument showing that locally coherent topoi have enough points, clarified using collage diagrams. We show that our refinement of Deligne's technique can be adapted to recover every existing result of this kind,…

Coherence simplices are generic topological correlation-function defects supported by a hierarchy of coherence functions. We classify coherence simplices based on their topology and discuss their structure and dynamics, together with their…

In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an…