Tensor logic

Tensor logic treats logical symbols, predicates, and composition operators as vectors, matrices, and higher-order tensors so that symbolic structures and inference can be represented, manipulated and learned via multilinear algebra and gradient-based optimization in AI systems. It builds on ideas such as Tensor Product Representations (Smolensky) and low-rank tensor factorization to bind role–filler pairs, perform unification via tensor contraction, and implement logical connectives as parameterized multilinear maps; this gives a continuous, compositional substrate that bridges symbolic reasoning and subsymbolic ML (Machine Learning) models. Practically, tensor logic appears in neurosymbolic architectures, differentiable theorem proving, knowledge-graph embeddings (e.g., RESCAL-style tensor factorization), and tensorized neural networks where high-order interactions encode relational structure. The approach is theoretically linked to linear/monoidal categorial views of logic (where ⊗ expresses composition) and to tensor networks from physics; its main engineering trade-offs are expressivity versus memory and compute, which are commonly addressed with low-rank decompositions, tensor-train formats, or learned multilinear operators that approximate high-order tensors.

First used circa 1990 (Smolensky’s tensor product representation lineage); gained wider traction in the 2000s–2010s with tensor-factorization methods for relational learning and again in the 2010s–2020s as neurosymbolic and differentiable reasoning approaches in AI/ML (Machine Learning) matured.