Ontology Neural Networks for Topologically Conditioned Constraint Satisfaction

TL;DR

This work addresses the challenge of maintaining semantic coherence while satisfying heterogeneous constraints in neuro-symbolic reasoning by extending Ontology Neural Networks (ONNs) with topological conditioning and gradient stabilization. The authors introduce Forman-Ricci curvature-guided step sizing, Deep Delta Learning for stable rank-one constraint projections, and CMA-ES for derivative-free hyperparameter optimization, integrated via the LOGOS projection loop. Empirical results show seed-independent convergence and sub-quadratic scaling up to twenty-node graphs, achieving a mean final energy of $E_{final} \approx 1.15$ compared with baseline values around $11$ and a substantial reduction in variance. These findings demonstrate that topological structure can inform gradient-based optimization in constrained neuro-symbolic systems, enabling robust, scalable, and interpretable constraint satisfaction suitable for medium-scale planning and reasoning tasks.

Abstract

Neuro-symbolic reasoning systems face fundamental challenges in maintaining semantic coherence while satisfying physical and logical constraints. Building upon our previous work on Ontology Neural Networks, we present an enhanced framework that integrates topological conditioning with gradient stabilization mechanisms. The approach employs Forman-Ricci curvature to capture graph topology, Deep Delta Learning for stable rank-one perturbations during constraint projection, and Covariance Matrix Adaptation Evolution Strategy for parameter optimization. Experimental evaluation across multiple problem sizes demonstrates that the method achieves mean energy reduction to 1.15 compared to baseline values of 11.68, with 95 percent success rate in constraint satisfaction tasks. The framework exhibits seed-independent convergence and graceful scaling behavior up to twenty-node problems, suggesting that topological structure can inform gradient-based optimization without sacrificing interpretability or computational efficiency.

Figures (11)

• Figure 1: ONN System Overview: The pipeline begins with raw semantic graph $\mathcal{G}_{\text{raw}}$ containing $N$ nodes with 64-dimensional state vectors, passes through LOGOS (Logical Ontological Generator for Self-Adjustment) projection ($T_{\text{max}} = 10$ iterations, $\tau = 10^{-6}$ tolerance) for constraint satisfaction, and integrates CMA-ES ($\lambda_{\text{pop}} = 4 + \lfloor 3\ln d \rfloor$ population) for parameter optimization. The system balances data fidelity ($\lambda_{\text{data}}$), physical constraints ($\lambda_{\text{phys}}$), and logical consistency ($\lambda_{\text{logic}}$) through iterative refinement.
• Figure 2: State Vector Anatomy: Each node's 64-dimensional state $\mathbf{s}_v \in \mathbb{R}^{64}$ decomposes into Bound ($\mathbf{b}_v \in \mathbb{R}^{16}$, physical boundaries), Form ($\mathbf{f}_v \in \mathbb{R}^{32}$, structural/visual properties), and Intent ($\mathbf{i}_v \in \mathbb{R}^{16}$, functional purpose). Edge weights are computed via cosine similarity of connected node states.
• Figure 3: Data-Logic-Physics Trade-off Triangle: The optimization navigates a three-way tension between data fidelity, logical consistency, and physical feasibility. Moving nodes to satisfy one constraint type often violates others, requiring careful balancing via weighted loss components.
• Figure 4: LOGOS Inner Loop Mechanics: The iterative projection process cycles through (1) residual computation from constraint violations, (2) Delta operator update step, (3) projection onto constraint manifold, and (4) convergence check. The loop terminates when violations fall below tolerance or maximum iterations are reached.
• Figure 5: Meta-LOGOS Acceptance Gate Flow: The system monitors convergence metrics (gradient norm, energy change, constraint violations) and makes adaptive decisions: continue optimization, trigger branching for exploration, or reduce step size. This meta-level control prevents premature convergence and divergence.