Modal Logical Neural Networks

TL;DR

Modal Logical Neural Networks (MLNNs) integrate deep learning with modal logic semantics, enabling differentiable reasoning about necessity and possibility, and can learn relational structures from data.

Contribution

The paper introduces MLNNs, a novel neurosymbolic framework that combines modal logic with neural networks, allowing flexible, differentiable reasoning and relational structure learning.

Findings

MLNNs improve logical consistency in various tasks.

They can learn relational structures from data.

Demonstrated effectiveness in four case studies.

Abstract

We propose Modal Logical Neural Networks (MLNNs), a neurosymbolic framework that integrates deep learning with the formal semantics of modal logic, enabling reasoning about necessity and possibility. Drawing on Kripke semantics, we introduce specialized neurons for the modal operators $\Box$ and $\Diamond$ that operate over a set of possible worlds, enabling the framework to act as a differentiable ``logical guardrail.'' The architecture is highly flexible: the accessibility relation between worlds can either be fixed by the user to enforce known rules or, as an inductive feature, be parameterized by a neural network. This allows the model to optionally learn the relational structure of a logical system from data while simultaneously performing deductive reasoning within that structure. This versatile construction is designed for flexibility. The entire framework is differentiable from end to end, with learning driven by minimizing a logical contradiction loss. This not only makes the system resilient to inconsistent knowledge but also enables it to learn nonlinear relationships that can help define the logic of a problem space. We illustrate MLNNs on four case studies: grammatical guardrailing, multi-agent epistemic trust, detecting constructive deception in natural language negotiation, and combinatorial constraint satisfaction in Sudoku. These experiments demonstrate how enforcing or learning accessibility can increase logical consistency and interpretability without changing the underlying task architecture.