On the Topological Foundation of Learning and Memory

TL;DR

This paper introduces a topological framework for cognition, modeling stable structures and dynamic flows in learning and memory through algebraic topology, unifying various cognitive theories.

Contribution

It presents a novel algebraic topology-based foundation for cognition, linking structure, flow, and uncertainty principles to explain learning and memory processes.

Findings

Homological Parity Principle distinguishes stable and dynamic cognitive components.

Framework unifies semantic and episodic memory within a topological model.

Recovers and generalizes existing theories like Free Energy Principle and Integrated Information.

Abstract

We propose a formal foundation for cognition rooted in algebraic topology, built on a Homological Parity Principle. This posits that even-dimensional homology represents stable Structure/Context (e.g., generative models), while odd-dimensional homology represents dynamic Flow/Content (e.g., sensory/memory data). Cognition is governed by the Context-Content Uncertainty Principle (CCUP), a dynamical cycle aligning these parities. This framework distinguishes two modes: Inference (waking), where the scaffold predicts the flow (a Context-before-Content process); and Learning (sleep), an inverted Structure-before-Specificity process where memory traces sculpt the scaffold. This parity interpretation unifies cognitive functions like semantic and episodic memory and provides a structural generalization of existing theories, recasting Friston's Free Energy Principle and Tonini's Integrated Information in topological terms.