Using matrices to model symbolic relationship

We describe a way of learning matrix representations of objects and relationships. The goal of learning is to allow multiplication of matrices t o represent symbolic relationships between objects and symbolic relationships between relationships, which is the main novelty of the method. We demonstrate that this leads to ex- cellent generalization in two different domains: modular arithmetic and family relationships. We show that the same system can learn first-o rder propositions such as (2,5) ∈ +3 or (Christopher, Penelope) ∈ has wife, and higher-order propositions such as (3,+3) ∈ plus and (+3, −3) ∈ inverse or (has husband, has wife) ∈ higher oppsex. We further demonstrate that the system understands how higher-order propositions are related to first-order on es by showing that it can correctly answer questions about first-order propositions involving the relations +3 or has wife even though it has not been trained on any first-order example s involving these relations.