Related papers: Algebraic foundations for qualitative calculi and …
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
Qualitative reasoning involves expressing and deriving knowledge based on qualitative terms such as natural language expressions, rather than strict mathematical quantities. Well over 40 qualitative calculi have been proposed so far, mostly…
We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such…
Conjunctive queries select and are expected to return certain tuples from a relational database. We study the potentially easier problem of counting all selected tuples, rather than enumerating them. In particular, we are interested in the…
We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes…
In this paper, we prove that if $\mathcal{A}$ is a unital separable $C^*$-algebra, $\mathcal{M}$ is a von Neumann algebra which has the Kirchberg's quotient weak expectation property (QWEP), and $\phi:\, \mathcal{A}\rightarrow \mathcal{M}$…
Consider any representation $\phi$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $\Lambda(g)$. We…
In safety-critical applications, guaranteeing the satisfaction of constraints over continuous environments is crucial, e.g., an autonomous agent should never crash into obstacles or go off-road. Neural models struggle in the presence of…
Alpay Algebra is introduced as a universal, category-theoretic framework that unifies classical algebraic structures with modern needs in symbolic recursion and explainable AI. Starting from a minimal list of axioms, we model each algebra…
Relation algebra and its reducts provide us with a strong tool for reasoning about nondeterministic programs and their partial correctness. Demonic calculus, introduced to model the behaviour of a machine where the demon is in control of…
Rapid development in numerical modelling of materials and the complexity of new models increases quickly together with their computational demands. Despite the growing performance of modern computers and clusters, calibration of such models…
The empirical results suggest that the learnability of a neural network is directly related to its size. To mathematically prove this, we borrow a tool in topological algebra: Betti numbers to measure the topological geometric complexity of…
We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative…
A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of complete binary trees whose leaves are labeled by letters of an…
We investigate notions of complete representation by partial functions, where the operations in the signature include antidomain restriction and may include composition, intersection, update, preferential union, domain, antidomain, and set…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
Deep neural networks can empirically perform efficient hierarchical learning, in which the layers learn useful representations of the data. However, how they make use of the intermediate representations are not explained by recent theories…
A strict interpretation of connectionism mandates complex networks of simple components. The question here is, is this simplicity to be interpreted in absolute terms? I conjecture that absolute simplicity might not be an essential attribute…
Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First,…
A classical result in approximation theory states that for any continuous function \( \varphi: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \(…