Related papers: Describing Groups
We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by $ R\le S \iff \ex f \, \forall x, y \, [xRy…
We study the complexity of locally checkable labeling (LCL) problems on $\mathbb{Z}^n$ from the point of view of descriptive set theory, computability theory, and factors of i.i.d. Our results separate various complexity classes that were…
We define a notion of complexity, which quantifies the nonlinearity of the computation of a neural network, as well as a complementary measure of the effective dimension of feature representations. We investigate these observables both for…
We study the problem of finding a subgroup of a given order in a finite group, where the group is represented by its Cayley table. We analyze the complexity of the problem in the special case of abelian groups and present an optimal…
Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincar\'e…
This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of…
Despite the growing popularity of explainable and interpretable machine learning, there is still surprisingly limited work on inherently interpretable clustering methods. Recently, there has been a surge of interest in explaining the…
We introduce the notion of a powerfully solvable group. These are powerful groups possessing an abelian series of a special kind. These groups include in particular the class of powerfully nilpotent groups. We will also see that for a…
In this manuscript, we derive the principle of conservation of computational complexity. We measure computational complexity as the number of binary computations (decisions) required to solve a problem. Every problem then defines a unique…
The theory of computational complexity focuses on functions and, hence, studies programs whose interactive behavior is reduced to a simple question/answer pattern. We propose a broader theory whose ultimate goal is expressing and analyzing…
We survey recent results concerning the complexity of regular languages represented by their minimal deterministic finite automata. In addition to the quotient complexity of the language -- which is the number of its (left) quotients, and…
Looking at the automata defined over a group alphabet as a nearring, we see that they are a highly complicated structure. As with ring theory, one method to deal with complexity is to look at semisimplicity modulo radical structures. We…
In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and…
We study the state complexity of regular operations in the class of ideal languages. A language L over an alphabet Sigma is a right (left) ideal if it satisfies L = L Sigma* (L = Sigma* L). It is a two-sided ideal if L = Sigma* L Sigma *,…
Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in…
We study first-order model checking, by which we refer to the problem of deciding whether or not a given first-order sentence is satisfied by a given finite structure. In particular, we aim to understand on which sets of sentences this…
We discuss the connection between computational social choice (comsoc) and computational complexity. We stress the work so far on, and urge continued focus on, two less-recognized aspects of this connection. Firstly, this is very much a…
Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important…
We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using…
We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing…