Modal Logic Neural Networks

Antonin Sulc; Noor Naddour

cs.LGDec 3, 2025

Modal Logic Neural Networks

Authors: Antonin Sulc, Noor Naddour

Organizations: Lawrence Berkeley National Laboratory, Berkeley, CA, USA · The University of Queensland, Brisbane, QLD, Australia

Abstract

Neural Networks are indispensable to natural sciences and society. Their impact extends from applications in public health to workforce productivity. Here, we introduce Modal Logic Neural Networks (MLNNs) -- an end-to-end differentiable logical neural network realisation of modal logic which evaluates a learnable truth function across possible-world semantics. This neural architecture handles para-consistency and inconsistency via a learnable world accessibility relation and valuation function. Because the modality is fixed by which frame axioms the relation satisfies rather than by the operator, one differentiable engine covers the epistemic, doxastic, deontic and temporal readings, with applications from verification of reactive and distributed systems to legal discourse and microeconomic utility models. In this paper, we introduce a model of differentiable Kripke semantics, and establish their soundness, convergence, and structural guarantees. We show four applications, in which the learned relation reads as a trust matrix, an operating-regime embedding with safety bounds, a temporal precedence order, and a recovered constraint graph.

Figures & tables

Panel of Figure 1 — Figure 1: Modal Logical Neural Networks. Left: three worlds, each with its own truth value $V_{w}(\phi)$ , joined by learned accessibility weights $A_{\theta}\in[0,1]$ saying how much one world’s verdict bears on another’s. The weights are the parameters. One modal neuron firing at $w_{1}$ (Eq. 2 ) gives necessity $\Box\phi(w_{1}){=}\operatorname{smin}_{\tau}[(1{-}A_{\theta}){+}V]{\approx}0.45$ (“how true in every world $w_{1}$ sees”) and possibility $\Diamond\phi(w_{1}){=}\operatorname{smax}_{\tau}[A_{\theta}{+}V{-}1]{\approx}0.90$ (“in some ”). They bracket $V_{w_{1}}{=}0.9$ , so the bound is consistent and $L_{\text{contra}}{=}0$ ; had they crossed, that gap is the training signal. Right: the encoder $f_{\theta}$ turns worlds into points $h_{w}$ .

Figure 2: Turbofan Wear: three analyses of one relation trained only on the deontic contradiction. (a) the embedding $f_{\theta}$ (t-SNE) coloured by RUL: worlds organise along a wear manifold. (b) the same embedding, same axes, coloured by operating regime. (c) the deontic bounds against RUL. Both $V_{\phi}$ and $A_{\theta}$ read sensors only; no RUL is read at inference. Panels are seed $45$ , quoted statistics over $3$ seeds.

Panel of Figure 3 — Figure 3: The object the modal layer produces ( llama3 ): the in-trust $\overline{A_{\theta}[\cdot,j]}$ others accord each agent. (a) Subtle, per round — the saboteur drains to $0.018$ from round $4$ , honest agents hold near $0.98$ , the decoy, penalised only for tone, settles at $0.59$ ; detection evaluates the top- $3$ mean, not this curve. (b) a saboteur backing the budget from round $9$ (shaded): the default persistent summary stays collapsed at $0.009$ , a forgiving $W{=}4$ window lets trust return: one knob, same pipeline. (c) training: the gap reaches $90\%$ of its final $0.89$ by epoch $142$ .

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Differentiable Logics are deployed in neuro-symbolic learning tasks as a way of embedding logical constraints in the training objective of neural networks. A differentiable logic consists of a syntax to write logical properties and a semantics to interpret them as real-valued functions to be folded in the loss function. A defining trade-off of the field is that between logical properties of the connectives, and analytic concerns for the semantics, with both aspects being relevant in applications. At one extreme we find fuzzy logics, that have well-established algebraic and proof-theoretic foundations, and at the other ad-hoc differentiable logics like Fischer's DL2, conceived for deep learning applications. However, no satisfactory foundation has emerged yet. We propose a resolution to this long-standing tension via a novel logic, Quantitative Linear Logic (QLL), with foundational ambitions. Our design is driven by naturality -- the idea that, since logical constraints are translated to losses, the semantics of the connectives should be pertinent operations used in ML practice (that is, sum and log-sum-exp) on additive quantities (like logits). We then judge the result on two aspects: logical adequacy -- that they satisfy most of the standard logical laws of Linear Logic; and empirical effectiveness -- test-time performance (as measured by adversarial attacks) is well-correlated to the actual verification of the logical constraints (as measured by off-the-shelf neural network verifiers), which makes QLL stand out among SoTA techniques.

Abstract

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