Continuous Modal Logical Neural Networks: Modal Reasoning via Stochastic Accessibility

2026-03-04

AI Summary

This work proposes a novel framework for differentiable reasoning with modal logics—such as temporal, epistemic, and deontic logics—in continuous spaces, overcoming the limitations of traditional discrete Kripke structures. The approach embeds modal logic into continuous manifolds by modeling modal operators via Neural Stochastic Differential Equations (Neural SDEs) and incorporates logical constraints directly into the training objective through Logic-Informed Neural Networks (LINNs), guiding the generation of solutions that satisfy prescribed logical properties. Key contributions include the first differentiable embedding of modal logic in continuous domains, the use of stochastic diffusion to prevent degeneracy in universal and existential modalities, structural correspondences between modal operators and entropic risk measures as well as classical modal axioms, and a substantial reduction in memory complexity. Experiments on multi-robot hallucination detection, geometric reconstruction of Lorenz attractors, and deontic-norm-guided safe dynamics learning demonstrate the method’s effectiveness and logical consistency.

Technology Category

• Machine Learning: Statistical Relational/Logic Learning
• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose Fluid Logic, a paradigm in which modal logical reasoning, temporal, epistemic, doxastic, deontic, is lifted from discrete Kripke structures to continuous manifolds via Neural Stochastic Differential Equations (Neural SDEs). Each type of modal operator is backed by a dedicated Neural SDE, and nested formulas compose these SDEs in a single differentiable graph. A key instantiation is Logic-Informed Neural Networks (LINNs): analogous to Physics-Informed Neural Networks (PINNs), LINNs embed modal logical formulas such as ($\Box$ bounded) and ($\Diamond$ visits\_lobe) directly into the training loss, guiding neural networks to produce solutions that are structurally consistent with prescribed logical properties, without requiring knowledge of the governing equations. The resulting framework, Continuous Modal Logical Neural Networks (CMLNNs), yields several key properties: (i) stochastic diffusion prevents quantifier collapse ($\Box$ and $\Diamond$ differ), unlike deterministic ODEs; (ii) modal operators are entropic risk measures, sound with respect to risk-based semantics with explicit Monte Carlo concentration guarantees; (iii)SDE-induced accessibility provides structural correspondence with classical modal axioms; (iv) parameterizing accessibility through dynamics reduces memory from quadratic in world count to linear in parameters. Three case studies demonstrate that Fluid Logic and LINNs can guide neural networks to produce consistent solutions across diverse domains: epistemic/doxastic logic (multi-robot hallucination detection), temporal logic (recovering the Lorenz attractor geometry from logical constraints alone), and deontic logic (learning safe confinement dynamics from a logical specification).

Problem

• modal logic
• neural networks
• continuous reasoning
• stochastic accessibility
• logical consistency

Innovation

• Neural Stochastic Differential Equations
• Modal Logic
• Logic-Informed Neural Networks
• Continuous Reasoning
• Stochastic Accessibility