Stabilizing Logical Composition in Differentiable Logic Gate Networks

Differentiable logic gate networks (DLGNs) can be discretized into efficient hard logic circuits after training, but their intermediate relaxed Boolean features can be sensitive to input perturbations. This property is particularly relevant in DLGNs, where intermediate features serve as operands for subsequent logic gates and therefore directly participate in downstream logical composition. In this paper, we study convolutional DLGNs from the perspective of Boolean feature stability. We show that the bilinear interactions in relaxed logic gates induce feature-dependent local Jacobians, such that the local sensitivity of a gate varies with the values of its Boolean operands. Motivated by this analysis, we propose feature-level Jacobian regularization to stabilize selected intermediate relaxed Boolean features. We further introduce Binarization-Aware Jacobian regularization (BAJ), which assigns larger penalties to features near the binarization boundary, where perturbations are more likely to alter their Boolean interpretation. Experiments across image classification benchmarks show that the proposed regularizers improve classification performance and reduce the finite-perturbation sensitivity of intermediate Boolean features. Downstream analyses further show that feature stabilization affects subsequent logical computation. These results suggest that stabilizing intermediate Boolean operands is an effective approach to improving the stability of logical composition in convolutional DLGNs.