Embedding Tarskian Semantics in Vector Spaces
This work addresses the challenge of efficiently computing logical semantics for researchers in computational logic and AI, offering a novel linear algebraic approach that is incremental in applying existing mathematical concepts to logic.
The authors tackled the problem of computing Tarskian semantics in logic by embedding finite models and logical formulas into vector spaces, resulting in a method that systematically evaluates formulas to determine truth values (1 or 0) and shows effectiveness in computing least models for Datalog programs compared to state-of-the-art approaches.
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.