The Geometry of Reasoning: Flowing Logics in Representation Space
This provides a new interpretability tool for analyzing LLMs' reasoning behavior, which is incremental as it builds on existing representation space studies.
The paper tackles the problem of understanding how large language models (LLMs) reason by proposing a geometric framework that models reasoning as flows in representation space, with empirical validation showing that logical statements control these flows' velocities.
We study how large language models (LLMs) ``think'' through their representation space. We propose a novel geometric framework that models an LLM's reasoning as flows -- embedding trajectories evolving where logic goes. We disentangle logical structure from semantics by employing the same natural deduction propositions with varied semantic carriers, allowing us to test whether LLMs internalize logic beyond surface form. This perspective connects reasoning with geometric quantities such as position, velocity, and curvature, enabling formal analysis in representation and concept spaces. Our theory establishes: (1) LLM reasoning corresponds to smooth flows in representation space, and (2) logical statements act as local controllers of these flows' velocities. Using learned representation proxies, we design controlled experiments to visualize and quantify reasoning flows, providing empirical validation of our theoretical framework. Our work serves as both a conceptual foundation and practical tools for studying reasoning phenomenon, offering a new lens for interpretability and formal analysis of LLMs' behavior.