Logical Neural Networks

TL;DR

This paper introduces Logical Neural Networks, a framework combining neural learning with symbolic logic, enabling interpretable reasoning, logical inference, and handling inconsistent or incomplete knowledge.

Contribution

It presents a novel neural-symbolic model with interpretable formulas, omnidirectional inference, and logical reasoning capabilities, including theorem proving.

Findings

Supports classical first-order logic reasoning

Achieves resilience to inconsistent knowledge

Handles incomplete knowledge with probabilistic semantics

Abstract

We propose a novel framework seamlessly providing key properties of both neural nets (learning) and symbolic logic (knowledge and reasoning). Every neuron has a meaning as a component of a formula in a weighted real-valued logic, yielding a highly intepretable disentangled representation. Inference is omnidirectional rather than focused on predefined target variables, and corresponds to logical reasoning, including classical first-order logic theorem proving as a special case. The model is end-to-end differentiable, and learning minimizes a novel loss function capturing logical contradiction, yielding resilience to inconsistent knowledge. It also enables the open-world assumption by maintaining bounds on truth values which can have probabilistic semantics, yielding resilience to incomplete knowledge.