Basic Reasoning with Tensor Product Representations

Abstract: In this paper we present the initial development of a general theory for mapping inference in predicate logic to computation over Tensor Product Representations (TPRs; Smolensky (1990), Smolensky & Legendre (2006)). After an initial brief synopsis of TPRs (Section 0), we begin with particular examples of inference with TPRs in the 'bAbI' question-answering task of Weston et al. (2015) (Section 1). We then present a simplification of the general analysis that suffices for the bAbI task (Section 2). Finally, we lay out the general treatment of inference over TPRs (Section 3). We also show the simplification in Section 2 derives the inference methods described in Lee et al. (2016); this shows how the simple methods of Lee et al. (2016) can be formally extended to more general reasoning tasks.