Title: Embedding Tarskian Semantics in Vector Spaces,
Abstract: 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 Sigma_F 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 Sigma_F 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.