Embedding Tarskian Semantics in Vector Spaces

TL;DR

This paper introduces a linear algebraic method to compute Tarskian semantics by embedding finite models into Euclidean space, enabling algebraic evaluation of logical formulas and least models with promising empirical results.

Contribution

It presents a novel approach to logic semantics using tensor algebra and linear algebra, including a new method for computing least models of Datalog programs.

Findings

Systematic evaluation of algebraic formulas yields truth values in R^N.

Effective computation of least models via matrix equations.

Empirical results outperform existing methods.

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.