Related papers: Unifying Functional Interpretations: Past and Futu…
Goedel's functional "Dialectica" interpretation can be used to extract functional programs from non-constructive proofs in arithmetic by employing two sorts of higher-order witnessing terms: positive realisers and negative counterexamples.…
Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods,…
G\"odel's Dialectica interpretation is a fundamental tool for the extraction of computational content from proofs, and plays a central role in today's proof mining program. In the past decades, it has also been studied from the perspective…
This paper summarises the current state-of-the art in the study of compositionality in distributional semantics, and major challenges for this area. We single out generalised quantifiers and intensional semantics as areas on which to focus…
The purpose of the present paper is to address multiple aspects of the Fuglede question dealing (Fourier spectra vs geometry) with a variety of $L^2$ contexts where we make precise the interplay between the three sides of the question: (i)…
The rise of foundation models has transformed machine learning research, prompting efforts to uncover their inner workings and develop more efficient and reliable applications for better control. While significant progress has been made in…
This is a brief survey of recent results by the authors devoted to one of the most important operators of integral geometry. Basic facts about the analytic family of cosine transforms on the unit sphere and the corresponding Funk transform…
Differentiable logics (DL) have recently been proposed as a method of training neural networks to satisfy logical specifications. A DL consists of a syntax in which specifications are stated and an interpretation function that translates…
The question in the title is ambiguous. At least the understanding of words essentially different and function theory should be clarified. We discuss approaches to do that. We also present a new framework for analytic function theories…
The problem of giving a computational meaning to classical reasoning lies at the heart of logic. This article surveys three famous solutions to this problem - the epsilon calculus, modified realizability and the dialectica interpretation -…
Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are (1) to consider extensions of…
This book is based on notes compiled over the many years I have been teaching the course "Applied Functional Analysis" in the first year of the Master programme at Delft University of Technology, for students with previous exposure to the…
This essay provides a historical, philosophical, and critical overview of the development of the Transactional Interpretation of Quantum Mechanics (TI). It is separated into two parts. Part I presents the history and development of TI from…
The purpose of this article is to study the role of G\"odel's functional interpretation in the extraction of programs from proofs in well quasi-order theory. The main focus is on the interpretation of Nash-Williams' famous minimal bad…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…
In [Hitzler and Wendt 2002, 2005], a new methodology has been proposed which allows to derive uniform characterizations of different declarative semantics for logic programs with negation. One result from this work is that the well-founded…
Large language models (LLMs) have achieved remarkable capabilities across diverse tasks, yet their internal decision-making processes remain largely opaque. Mechanistic interpretability (i.e., the systematic study of how neural networks…
Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a…
The original derivation of Power Functional Theory, Schmidt and Brader, JCP 138, 214101 (2013), is reworked in some detail with a view to clarifying and simplifying the logic and making explicit the various functional dependencies. We note…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…