Orthologics for Cones
This work addresses a domain-specific need in knowledge representation for applications combining data-driven and logic methods, but it appears incremental as it builds on existing lattice theory and orthologic frameworks.
The paper tackles the problem of developing logics for geometric structures in knowledge representation, specifically for closed convex cones, by extending minimal orthologic with a partial modularity rule to combine feasible data structures with expressivity.
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).