Towards a Formal Distributional Semantics: Simulating Logical Calculi with Tensors

TL;DR

This paper explores how tensors and matrices can be used to simulate predicate logic and propositional calculus within distributional semantics, aiming to bridge empirical and formal semantic models.

Contribution

It demonstrates how tensor calculus can model predicate logic and propositional connectives, proposing variants capable of handling quantifiers with minimal non-linear operations.

Findings

Tensor interpretations of logical connectives

Simulation of predicate calculus with tensors

Proposed tensor variants for quantifiers

Abstract

The development of compositional distributional models of semantics reconciling the empirical aspects of distributional semantics with the compositional aspects of formal semantics is a popular topic in the contemporary literature. This paper seeks to bring this reconciliation one step further by showing how the mathematical constructs commonly used in compositional distributional models, such as tensors and matrices, can be used to simulate different aspects of predicate logic. This paper discusses how the canonical isomorphism between tensors and multilinear maps can be exploited to simulate a full-blown quantifier-free predicate calculus using tensors. It provides tensor interpretations of the set of logical connectives required to model propositional calculi. It suggests a variant of these tensor calculi capable of modelling quantifiers, using few non-linear operations. It finally discusses the relation between these variants, and how this relation should constitute the subject of future work.