Quantitative Linear Logic for Neuro-Symbolic Learning and Verification

2026-05-13

AI Summary

This work addresses the challenge of neural-symbolic learning in achieving differentiable semantics while preserving the formal properties of logical connectives. The paper proposes Quantitative Linear Logic (QLL), the first differentiable logic framework grounded in the principle of “naturality,” which directly maps logical constraints to additive and log-sum-exp operations commonly used in machine learning. By defining semantics in logit space, QLL seamlessly integrates neural network training with formal verification, maintaining both logical rigor and compatibility with gradient-based optimization. Experimental results demonstrate that QLL exhibits robustness under adversarial attacks that closely aligns with its formal verification guarantees, significantly outperforming existing neural-symbolic approaches.

Abstract

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.

Problem

• Neuro-Symbolic Learning
• Differentiable Logic
• Logical Constraints
• Linear Logic
• Quantitative Semantics

Innovation

• Quantitative Linear Logic
• Neuro-Symbolic Learning
• Differentiable Logic
• Logical Verification
• Natural Semantics