Related papers: Predicativity through transfinite reflection
We consider affine representable algebras, that is, finitely generated algebras over a field that can be embedded into some matrix algebra over a commutative algebra. We show that this algebra can in fact be chosen to be a polynomial…
In this paper we explore the representation property over sets. This property generalizes constructibility, however is weak enough to enable us to prove that the class of theories $T$ whose models are representable is exactly the class of…
We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in…
Explainability is needed to establish confidence in machine learning results. Some explainable methods take a post hoc approach to explain the weights of machine learning models, others highlight areas of the input contributing to…
We study whether a Large Language Model can learn the deterministic sequence of trees generated by the iterated prime factorization of the natural numbers. Each integer is mapped into a rooted planar tree and the resulting sequence $…
Nowadays, neural network (NN) and deep learning (DL) techniques are widely adopted in many applications, including recommender systems. Given the sparse and stochastic nature of collaborative filtering (CF) data, recent works have…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
The Arithmetic Fourier Transform is a numerical formulation for computing Fourier series and Taylor series coefficients. It competes with the Fast Fourier Transform in terms of speed and efficiency, requiring only addition operations and…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
Deeply-learned planning methods are often based on learning representations that are optimized for unrelated tasks. For example, they might be trained on reconstructing the environment. These representations are then combined with predictor…
Discrete mathematics is the foundation of computer science. It focuses on concepts and reasoning methods that are studied using math notations. It has long been argued that discrete math is better taught with programming, which takes…
Recent foundation models for tabular data achieve strong task-specific performance via in-context learning. Nevertheless, they focus on direct prediction by encapsulating both representation learning and task-specific inference inside a…
A wide variety of model explanation approaches have been proposed in recent years, all guided by very different rationales and heuristics. In this paper, we take a new route and cast interpretability as a statistical inference problem. We…
A new characterization of provably recursive functions of first-order arithmetic is described. Its main feature is using only terms consisting of 0, the successor S and variables in the quantifier rules, namely, universal elimination and…
Numerous approaches have been recently proposed for learning fair representations that mitigate unfair outcomes in prediction tasks. A key motivation for these methods is that the representations can be used by third parties with unknown…
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.
The advent of deep machine learning platforms such as Tensorflow and Pytorch, developed in expressive high-level languages such as Python, have allowed more expressive representations of deep neural network architectures. We argue that such…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
The task of event detection and classification is central to most information retrieval applications. We show that a Transformer based architecture can effectively model event extraction as a sequence labeling task. We propose a combination…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…