相关论文: An effective version of definability in metric mod…

When quantitative models are used to support decision-making on complex and important topics, understanding a model's ``reasoning'' can increase trust in its predictions, expose hidden biases, or reduce vulnerability to adversarial attacks.…

One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…

The concept of complementability is extended from bounded operators to densely defined operators on Hilbert spaces. By introducing appropriate projections and decomposition techniques, a framework is developed for analyzing…

The usual definition of the set of constructible reals is $\Sigma ^1_2$. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be…

Let $\cal R$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture from [5]. The…

We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define…

Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and…

We prove that if $A$ is a computable Hopfian finitely presented structure, then $A$ has a computable $d$-$\Sigma_2$ Scott sentence if and only if the weak Whitehead problem for $A$ is decidable. We use this to infer that every hyperbolic…

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…

We prove that universal differentiability sets in Euclidean spaces possess distinctive structural properties. Namely, we show that any universal differentiability set contains a `kernel' in which the points of differentiability of each…

We consider the extension of two variable logic with quantifiers that state that the number of elements where a formula holds should belong to a given ultimately periodic set. We show that both satisfiability and finite satisfiability of…

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…

We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank $\omega_1^{CK}$, the computable infinitary theory is $\aleph_0$-categorical. Millar and Sacks asked whether this…

We extend the classical notion of solvability to a lambda-calculus equipped with pattern matching. We prove that solvability can be characterized by means of typability and inhabitation in an intersection type system P based on…

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various…

Motivated by the notion of strong computable type for sets in computable analysis, we define the notion of strong computable type for $G$-shifts, where $G$ is a finitely generated group with decidable word problem. A $G$-shift has strong…

We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory,…

Probabilistic programming is considered as a framework, in which basic components of cognitive architectures can be represented in unified and elegant fashion. At the same time, necessity of adopting some component of cognitive…

We show that arithmetic definability of henselian valuations is preserved by the tilting correspondence. Moreover, we show that if a perfectoid valuation is arithmetically definable, then no parameters are needed. We also investigate…