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 to 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-order 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 ones 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 examples involving these relations.