用拓扑结构揭示逻辑数据的内在不变性，提升神经符号推理效率
The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data
- 将形式逻辑映射为拓扑结构，捕捉数据深层不变特性
- 提出数据的拓扑对偶概念，解决符号推理中的扩展瓶颈
- 适合研究神经符号系统可解释性的研究人员
AlphaGeometry 是神经符号推理的里程碑，但其符号推理引擎存在对数线性扩展瓶颈，随问题复杂度增加而效率下降。近期技术报告指出，当前领域专用语言与自然语言在输入表示上可能同构，二者互换是性能不变变换，暗示现有神经引导依赖表层编码而非结构理解。本文通过提出一种逻辑到拓扑的编码方法，揭示模型潜在空间在输入空间变换下的结构不变性。基于观测逻辑，利用可观测理论中的可证明性与拓扑间的对偶关系，构建输入空间的逻辑-拓扑编码器。提出‘数据集的拓扑对偶’概念，融合形式逻辑、拓扑学与神经处理，为神经符号人工智能提供一种原理性框架，实现对模型探索复杂发现路径的机制可解释性。
原文摘要 · Abstract (English)
AlphaGeometry represents a milestone in neuro-symbolic reasoning, yet its architecture faces a log-linear scaling bottleneck within its symbolic deduction engine that limits its efficiency as problem complexity increases. Recent technical reports suggest that current domain-specific languages may be isomorphic as input representations to natural language, interchanging them acts as a performance-invariant transformation, implying that current neural guidance relies on superficial encodings rather than structural understanding. This paper addresses this representation bottleneck by proposing a logic-to-topology encoding designed to reveal the structural invariants of a model's latent space under a transformation of its input space. By leveraging the Logic of Observation, we utilize the duality between provability in observable theories and topologies to propose a logic-to-topology encoder for the input space. We introduce the concept of the "topological dual of a dataset", a transformation that bridges formal logic, topology, and neural processing. This framework serves as a Rosetta Stone for neuro-symbolic AI, providing a principled pathway for the mechanistic interpretability of how models navigate complex discovery paths.
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