Towards a Formal Distributional Semantics: Simulating Logical Calculi with Tensors
This work addresses the challenge of integrating empirical distributional semantics with compositional formal semantics for researchers in computational linguistics and AI, representing an incremental step in this ongoing effort.
The paper tackles the problem of reconciling distributional and formal semantics by demonstrating how tensors and matrices can simulate predicate logic, including quantifier-free predicate calculus and propositional calculi, and suggests a variant for modeling quantifiers with few non-linear operations.
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.