Artificial Neural networks (ANNs) are often treated as black-box models, making explainability a central challenge in deep learning. Many engineering methods have been proposed to approximately explain the ANN from various perspectives, such as feature attribution and visualization. However, it remains a long-standing open question whether the complex inference logic of an ANN can be explained exhaustively and concisely as sparse symbolic patterns. This raises a deeper inquiry: does the emergence of symbolic patterns reflect a natural law rather than chance? Here, we show that across a broad class of ANNs trained on diverse tasks, their inference logic can indeed be reformulated as sparse symbolic interactions. We further prove that two common mathematical criteria, which are implicitly required across tasks, lead to the emergence of such sparse symbolic interactions. Empirical evidence confirms that the two criteria hold for the majority of input samples in diverse models. Furthermore, the faithfulness of these interactions is also demonstrated by their strong sample-to-sample and model-to-model transferability, as well as their ability to explain the overall generalization power of ANNs. Our theoretical analysis and extensive experiments provide a solid foundation for symbolic explanations of ANNs, and offer novel insights into the ANN's generalization power. Our findings also highlight the potential of communicative learning, a paradigm in which the inference logic of an ANN can be directly inspected and tuned at the level of symbolic patterns, thus complementing traditional end-to-end learning paradigm. Finally, the observed emergence of symbolic patterns in ANNs suggests that similar symbolic representations may also emerge in other types of black-box systems under certain conditions, because our proof does not depend on any specific ANN architecture.
Artificial neural networks are often regarded as powerful yet opaque black boxes. Here, we demonstrate that learning in deep neural networks generates local symmetries known in graph theory as fibrations and coverings. We prove that covering symmetries are stable attractors of stochastic gradient descent. Consistent with this theory, we report the emergence of covering symmetries across major network architectures, including multilayer, convolutional, recurrent, and transformer networks. Exploiting these symmetries enables drastic model compression - reducing networks to 17% of their original size without sacrificing performance. Furthermore, controlled breaking of covering symmetry overcomes the loss of plasticity, achieving state-of-the-art performance in continual learning. The theoretical results provide a new foundation for AI systems based on symmetries that convert black boxes into interpretable colored graphs and enable more efficient inference and lifelong learning.
Osvaldo M Velarde, Lucas C Parra, Alireza Hashemi +1
A central goal of explainable AI is to express large language model (LLM) decision logic symbolically and ground it in internal mechanisms. Existing rule-extraction methods usually learn ungrounded symbolic surrogates, while mechanistic interpretability links behavior to neurons but often requires hand-crafted hypotheses and costly interventions. We introduce MechaRule, a pipeline that grounds rule extraction in LLM circuits by localizing sparse agonist activations whose ablation disrupts rule-related behavior. MechaRule rests on two findings. First, in a fixed baseline/flip regime, sparse agonist effects can exhibit overtopping: a few high-effect activations remain detectable within larger groups, dominate weaker ones, and flip many of the same examples. In such regimes, adaptive group testing with confidence-guided conservative pruning requires O(k log(N/k) + k) interventions over N candidates when k << N are agonists. Second, agonists are localized more reliably on data splits aligned with close-to-faithful rule behavior; spectral splits provide a rule-free fallback, whereas unfaithful splits degrade localization. Empirically, on arithmetic and jailbreaking, MechaRule recalls 97.0% of highest-effect agonists in matched brute-force validations at only 2.14% of exhaustive-ablation cost on average. Ablating the localized agonists eliminates 97.6--100.0% of eligible correct arithmetic answers and jailbreaks, and can correct arithmetic errors or induce jailbreaks by up to 72.8% and 32.5%.
Francesco Sovrano, Gabriele Dominici, Marc Langheinrich
Compositional generalization remains a foundational weakness of modern neural networks, limiting their robustness and applicability in domains requiring out-of-distribution reasoning. A central, yet unverified, assumption in neuro-symbolic AI is that compositional reasoning will emerge as a byproduct of successful symbol grounding. This work presents the first systematic empirical analysis to challenge this assumption by disentangling the contributions of grounding and reasoning. To operationalize this investigation, we introduce the Iterative Logic Tensor Network (iLTN), a fully differentiable architecture designed for multi-step deduction. Using a formal taxonomy of generalization -- probing for novel entities, unseen relations, and complex rule compositions -- we demonstrate that a model trained solely on a grounding objective fails to generalize. In contrast, our full iLTN, trained jointly on perceptual grounding and multi-step reasoning, achieves high zero-shot accuracy across all tasks. Our findings provide conclusive evidence that symbol grounding, while necessary, is insufficient for generalization, establishing that reasoning is not an emergent property but a distinct capability that requires an explicit learning objective.