Related papers: Relativity of Depth and Sophistication
Measure and integral are two closely related, but distinct objects of study. Nonetheless, they are both real-valued lattice valuations: order preserving real-valued functions $\phi$ on a lattice $L$ which are modular, i.e.,…
Inverse problems are characterized by their inherent non-uniqueness and sensitivity with respect to data perturbations. Their stable solution requires the application of regularization methods including variational and iterative…
Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the…
Comparing the top $k$ elements between two or more ranked results is a common task in many contexts and settings. A few measures have been proposed to compare top $k$ lists with attractive mathematical properties, but they face a number of…
Analogical reasoning is a powerful qualitative reasoning tool that enables humans to connect two situations, and to generalize their knowledge from familiar to novel situations. Cognitive Science research provides valuable insights into the…
The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events. The paper is divided into two parts. In…
Convex analysis is a modern branch of mathematics with many applications. As Large Language Models (LLMs) start to automate research-level math and sciences, it is important for LLMs to demonstrate the ability to understand and reason with…
Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to ambiguity and confusion in their meaning. In this paper we use information theory to provide abstract and concise…
Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in $\IR^d$. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional…
We consider a natural measure of relevance: the reduction in optimal prediction risk in the presence of side information. For any given loss function, this relevance measure captures the benefit of side information for performing inference…
We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes…
Issues concerning intelligent data analysis occurring in machine learning are investigated. A scheme for synthesizing correct supervised classification procedures is proposed. These procedures are focused on specifying partial order…
Many philosophers of physics maintain that a physical theory that exhibits (certain kinds of) symmetries is flawed, on the grounds that such theories posit "excess structure". In an influential paper, Dewar [2019, "Sophistication about…
The classical view of epistemic logic is that an agent knows all the logical consequences of their knowledge base. This assumption of logical omniscience is often unrealistic and makes reasoning computationally intractable. One approach to…
This paper primarily demonstrates a method to quantitatively assess the alignment between multi-step, structured reasoning in large language models and human preferences. We introduce the Alignment Score, a semantic-level metric that…
We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth $D(G)$ of a graph $G$ is equal to the minimum quantifier depth of a sentence defining $G$…
A general notion of information-related complexity applicable to both natural and man-made systems is proposed. The overall approach is to explicitly consider a rational agent performing a certain task with a quantifiable degree of success.…
In experimental applications of bounded-reasoning models, behavior is often summarized by distributions of "levels". We argue that such summaries conflate two conceptually distinct dimensions: a player's type, capturing beliefs about what…
The ability to abstract, count, and use System~2 reasoning are well-known manifestations of intelligence and understanding. In this paper, we argue, using the example of the ``Look and Say" puzzle, that although deep neural networks can…
Intuitively, the concept of similarity is the notion to measure an inexact matching between two entities of the same reference set. The notions of similarity and its close relative dissimilarity are widely used in many fields of Artificial…