Related papers: Unifying Functional Interpretations: Past and Futu…

Fuzzy implication functions are a key area of study in fuzzy logic, extending the classical logical conditional to handle truth degrees in the interval $[0,1]$. While existing literature often focuses on a limited number of families, in the…

This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory. Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated…

The main subjects of this text are: (1) Generalization of concepts and operations, like distance and size, to situations where they are not definable in the usual way. (2) A pragmatic theory of handling contradictions using reliability of…

G\"odel logic with the projection operator Delta (G_Delta) is an important many-valued as well as intermediate logic. In contrast to classical logic, the validity and the satisfiability problems of G_Delta are not directly dual to each…

The paper discusses the role of interpretations, understood as multifunctions that preserve and reflect logical consequence, as refinement witnesses in the general setting of pi-institutions. This leads to a smooth generalization of the…

The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus…

Functional Distributional Semantics is a framework that aims to learn, from text, semantic representations which can be interpreted in terms of truth. Here we make two contributions to this framework. The first is to show how a type of…

To build intelligent machine learning systems, there are two broad approaches. One approach is to build inherently interpretable models, as endeavored by the growing field of causal representation learning. The other approach is to build…

Ordinary and transfinite recursion and induction and ZF set theory are used to construct from a fully interpreted object language and from an extra formula a new language. It is fully interpreted under a suitably defined interpretation.…

There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…

During the last twenty years or so a wide range of realizability interpretations of classical analysis have been developed. In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with…

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition…

Interpretability aims to explain the behavior of deep neural networks. Despite rapid growth, there is mounting concern that much of this work has not translated into practical impact, raising questions about its relevance and utility. This…

We define two versions of compositions of matrix-valued rational functions of appropriate sizes and whenever analytic at infinity, offer a set of formulas for the corresponding state-space realization, in terms of the realizations of the…

The investigations reported in this paper center on the process of dynamic uncertainty assessment during interpretation tasks in real domain. In particular, we are interested here in the nature of the control structure of computer programs…

This paper will develop a single framework for unifying, simplifying and extending our prior results about axiom systems that retain a partial knowledge of their own consistency, via an axiomatic declaration of self-consistency. Its perhaps…

The present book gives a systematic overview of function theory and the theory of Stieltjes integral. In particular, we give a detailed account of the theory of functions of bounded variation and of the theory of regulated functions (=…

In this review, we examine the problem of designing interpretable and explainable machine learning models. Interpretability and explainability lie at the core of many machine learning and statistical applications in medicine, economics,…

Mechanistic interpretability (MI) is an emerging sub-field of interpretability that seeks to understand a neural network model by reverse-engineering its internal computations. Recently, MI has garnered significant attention for…

The need for revolution in modern physics is a well known and often broached subject, however, the precision and success of current models narrows the possible changes to such a great degree that there appears to be no major change…