Related papers: Orthologics for Cones
Ontology Alignment (OA) is essential for enabling semantic interoperability across heterogeneous knowledge systems. While recent advances have focused on large language models (LLMs) for capturing contextual semantics, this work revisits…
We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary…
Tree convex sets refer to a collection of sets such that each set in the collection is a subtree of a tree whose nodes are the elements of these sets. They extend the concept of row convex sets each of which is an interval over a total…
The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…
In this paper we show that states, transitions and behavior of concurrent systems can often be modeled as sheaves over a suitable topological space. In this context, geometric logic can be used to describe which local properties (i.e.…
Many classes of images exhibit rotational symmetry. Convolutional neural networks are sometimes trained using data augmentation to exploit this, but they are still required to learn the rotation equivariance properties from the data.…
Reasoning over knowledge graphs (KGs) is a challenging task that requires a deep understanding of the complex relationships between entities and the underlying logic of their relations. Current approaches rely on learning geometries to…
Service robots need common-sense knowledge to help humans in everyday situations as it enables them to understand the context of their actions. However, approaches that use ontologies face a challenge because common-sense knowledge is often…
A convex optimization problem in conic form involves minimizing a linear functional over the intersection of a convex cone and an affine subspace. In some cases, it is possible to replace a conic formulation using a certain cone, with a…
The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the…
In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g.…
The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions…
Knowledge Graphs (KGs) are increasingly used to represent and explore complex, interconnected data across diverse domains. However, existing KG visualization systems remain limited because they fail to provide the context of user questions.…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
The integration of knowledge extracted from different models described by domain experts or from models generated by machine learning algorithms is strongly conditioned by the lack of an appropriated framework to specify and integrate…
Convex neural codes are subsets of the Boolean lattice that record the intersection patterns of convex sets in Euclidean space. Much work in recent years has focused on finding combinatorial criteria on codes that can be used to classify…
It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system…
The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by convex regions in this space. After…
Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing the image of the standard…
Let a $R$-body be a closed set, complement of union of open balls of radius $R$ in the Euclidean space. Properties generalizing similar ones for convex sets are proved for the family of $R$-bodies; properties for the family of sets…