The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data

TL;DR

This paper introduces a logic-to-topology encoding that reveals structural invariants in neural models' latent spaces, aiming to improve neuro-symbolic reasoning efficiency and interpretability.

Contribution

It proposes the topological dual of a dataset using the Logic of Observation to connect formal logic, topology, and neural processing.

Findings

Introduces the concept of the topological dual of a dataset.

Provides a framework for interpretability of neuro-symbolic models.

Bridges logic, topology, and neural processing for better understanding.

Abstract

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.