Continuous Modal Logical Neural Networks: Modal Reasoning via Stochastic Accessibility

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).

💡 Research Summary

The paper introduces Continuous Modal Logical Neural Networks (CMLNNs), a novel framework that lifts modal logical reasoning from discrete Kripke structures to continuous manifolds by representing each modal operator (□, ♢, epistemic K, doxastic B, deontic O, etc.) as a dedicated Neural Stochastic Differential Equation (Neural SDE). In this “Fluid Logic” paradigm, worlds are points in ℝᵈ and accessibility is defined probabilistically: a world w′ is accessible from w under operator i if w′ lies within the distribution of sample paths generated by the SDE associated with i.

The authors define the semantics of □ and ♢ using entropic risk measures. For a given atomic formula φ, each Monte‑Carlo sampled path ωₙ yields a per‑path soft‑minimum (gₙ) and soft‑maximum (hₙ) of the robustness scores Lφ and Uφ evaluated along a time grid. The global □ and ♢ operators are then computed as
L□ᵢ φ(w) = –τ₍ω₎ log