Related papers: Dimension in team semantics
We investigate the expressive power of first-order quantifications in the context of monadic second-order logic over pictures. We show that k+1 set quantifier alternations allow to define a picture language that cannot be defined using k…
The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This…
Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic…
Formal definitions of quantities, quantity spaces, dimensions and dimension groups are introduced. Based on these concepts, a theoretical framework and a practical algorithm for dimensional analysis are developed, and examples of…
We introduce a notion of dimension for the solution set of a system of algebraic difference equations that measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but, as we…
Team semantics is the mathematical framework of modern logics of dependence and independence in which formulae are interpreted by sets of assignments (teams) instead of single assignments as in first-order logic. In order to deepen the…
Sets with atoms serve as an alternative to ZFC foundations for mathematics, where some infinite, though highly symmetric sets, behave in a finitistic way. Therefore, one can try to carry over analysis of the classical algorithms from finite…
Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{ \triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at most…
We construct a new kind of measures, called projection families, which generalize the classical notion of vector and operator-valued measures. The maximal class of reasonable functions admits an integral with respect to a projection family,…
The Ulam's metric is the minimal number of moves consisting in removal of one element from a permutation and its subsequent reinsertion in different place, to go between two given permutations. Thet elements that are not moved create…
A dimension group is a partially ordered countable group such that (1) every finite subset is contained in an ordered subgroup which is a finite direct power of Z and (2) the group has an order unit i.e. a positive element u such that every…
We define and study logics in the framework of probabilistic team semantics and over metafinite structures. Our work is paralleled by the recent development of novel axiomatizable and tractable logics in team semantics that are closed under…
We consider the problem of computing the measure of a regular set of infinite binary trees. While the general case remains unsolved, we show that the measure of a language can be computed when the set is given in one of the following three…
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…
It is well known that dependence logic captures the complexity class NP, and it has recently been shown that inclusion logic captures P on ordered models. These results demonstrate that team semantics offers interesting new possibilities…
One of the main issues in modern network science is the phenomenon of cascading failures of a small number of attacks. Here we define the dimension of a network to be the maximal number of functions or features of nodes of the network. It…
We translate notions and results of decomposition and dimension theories for module categories, into the lattice environment. In particular we translate dimension theory in module categories to complete modular upper-continuous lattices.
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of…
The rooted tree is an important data structure, and the subtree size, height, and depth are naturally defined attributes of every node. We consider the problem of the existence of a k-ary tree given a list of attribute sequences. We give…
We study fragments of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these…