Related papers: Describing Groups
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of $\mathcal{L}_{\omega_1 \omega}$)…
The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real…
Hard instances of natural computational problems are often elusive. In this note we present an example of a natural decision problem, the word problem for a certain finitely presented group, whose hard instances are easy to find. More…
We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted…
We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$…
The word problem is an old and central problem in (computational) group theory. It is well-known that the word problem is undecidable in general, but decidable for specific types of presentations. Consistent polycyclic presentations are an…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous…
In this paper, we describe an algorithm that efficiently collect relations in class groups of number fields defined by a small defining polynomial. This conditional improvement consists in testing directly the smoothness of principal ideals…
Improving the explainability of the results from machine learning methods has become an important research goal. Here, we study the problem of making clusters more interpretable by extending a recent approach of [Davidson et al., NeurIPS…
We answer two questions on the complexities of decision problems of groups, each related to a classical result. First, C. Miller characterized the complexity of the isomorphism problem for finitely presented groups in 1971. We do the same…
For any limit ordinal $\lambda$, we construct a linear order $L_\lambda$ whose Scott complexity is $\Sigma_{\lambda+1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there…
We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning,…
We study formal languages which are capable of fully expressing quantitative probabilistic reasoning and do-calculus reasoning for causal effects, from a computational complexity perspective. We focus on satisfiability problems whose…
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…
We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…