Orthologics for Cones

TL;DR

This paper introduces a logic extension for geometric structures like convex cones, combining lattice theory with convex optimization to enhance knowledge representation in data-driven and logical applications.

Contribution

It develops an extension of minimal orthologic with a partial modularity rule tailored for closed convex cones, integrating convexity and conicity into logical reasoning.

Findings

Extended orthologic with modularity rule for convex cones

Logic supports full orthonegation in geometric structures

Combines data structure efficiency with expressive logical capabilities

Abstract

In applications that use knowledge representation (KR) techniques, in particular those that combine data-driven and logic methods, the domain of objects is not an abstract unstructured domain, but it exhibits a dedicated, deep structure of geometric objects. One example is the class of convex sets used to model natural concepts in conceptual spaces, which also links via convex optimization techniques to machine learning. In this paper we study logics for such geometric structures. Using the machinery of lattice theory, we describe an extension of minimal orthologic with a partial modularity rule that holds for closed convex cones. This logic combines a feasible data structure (exploiting convexity/conicity) with sufficient expressivity, including full orthonegation (exploiting conicity).