Related papers: Borel Functors and Infinitary Interpretations

The class of abelian $p$-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory $T_p$ whose models are each bi-interpretable with the disjoint union of an…

Homomorphisms are defined between the multiplicative group of an etale algebra of dimension 4 and the multiplicative group of a canonically associated etale algebra of degree 6 over an arbitrary field. These homomorphisms are used to relate…

A function from Baire space to the natural numbers is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working…

In this paper, we show that there is a one-to-one correspondence between operator monotone functions on the nonnegative reals and finite Borel measures on the unit interval. This correspondence appears as an integral representation of…

To understand changes in physical systems and facilitate decisions, explaining how model predictions are made is crucial. We use model-based interpretability, where models of physical systems are constructed by composing basic constructs…

On a finite structure, the polymorphism invariant relations are exactly the primitively positively definable relations. On infinite structures, these two sets of relations are different in general. Infinitary primitively positively…

We describe a general construction of finiteness spaces which subsumes the interpretations of all positive connectors of linear logic. We then show how to apply this construction to prove the existence of least fixpoints for particular…

We define the notion of a $\lambda$-definable category, a generalisation of the notion of definable category from the model theory of modules. Let ${\cal C}$ be a $\lambda$-accessible additive category. We characterise the additive functors…

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second…

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized…

We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S,…

We give a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. We show that any isomorphism-invariant property of a finite structure can be computed by counting the number of…

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…

We regard explanations as a blending of the input sample and the model's output and offer a few definitions that capture various desired properties of the function that generates these explanations. We study the links between these…

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Goedel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones.…

What is it to interpret the outputs of an opaque machine learning model. One approach is to develop interpretable machine learning techniques. These techniques aim to show how machine learning models function by providing either model…

The role of automorphisms of infinite-dimensional Lie algebras in conformal field theory is examined. Two main types of applications are discussed; they are related to the enhancement and reduction of symmetry, respectively. The structures…

We consider a category whose morphisms are bordisms of $n$-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric group. We…

Let C and D be quasi-categories (a.k.a. infinity-categories). Suppose also that one has an assignment sending commutative diagrams of C to commutative diagrams of D which respects face maps, but not necessarily degeneracy maps. (This is…

Mechanistic interpretability aims to understand how neural networks generalize beyond their training data by reverse-engineering their internal structures. We introduce patterning as the dual problem: given a desired form of generalization,…