Embedding Tarskian Semantics in Vector Spaces

We propose a new linear algebraic approach to the computation of Tarskian semantics in logic. We embed a finite model M in first-order logic with N entities in N-dimensional Euclidean space R^N by mapping entities of M to N dimensional one-hot vectors and k-ary relations to order-k adjacency tensors (multi-way arrays). Second given a logical formula F in prenex normal form, we compile F into a set SigmaF of algebraic formulas in multi-linear algebra with a nonlinear operation. In this compilation, existential quantifiers are compiled into a specific type of tensors, e.g., identity matrices in the case of quantifying two occurrences of a variable. It is shown that a systematic evaluation of SigmaF in R^N gives the truth value, 1(true) or 0(false), of F in M. Based on this framework, we also propose an unprecedented way of computing the least models defined by Datalog programs in linear spaces via matrix equations and empirically show its effectiveness compared to state-of-the-art approaches.