Dynamics Reveals Structure: Challenging the Linear Propagation Assumption

TL;DR

This paper investigates the geometric and algebraic limitations of the Linear Propagation Assumption in neural networks, revealing fundamental obstructions that impact logical reasoning and knowledge editing capabilities.

Contribution

It formalizes the LPA using relation algebra, proves key limitations for composition, and links these to structural issues in neural network reasoning.

Findings

Negation and converse require tensor factorization for direction-agnostic propagation.

Composition reduces to conjunction, which must be bilinear on linear features.

Bilinearity conflicts with negation, causing feature map collapse.

Abstract

Neural networks adapt through first-order parameter updates, yet it remains unclear whether such updates preserve logical coherence. We investigate the geometric limits of the Linear Propagation Assumption (LPA), the premise that local updates coherently propagate to logical consequences. To formalize this, we adopt relation algebra and study three core operations on relations: negation flips truth values, converse swaps argument order, and composition chains relations. For negation and converse, we prove that guaranteeing direction-agnostic first-order propagation necessitates a tensor factorization separating entity-pair context from relation content. However, for composition, we identify a fundamental obstruction. We show that composition reduces to conjunction, and prove that any conjunction well-defined on linear features must be bilinear. Since bilinearity is incompatible with negation, this forces the feature map to collapse. These results suggest that failures in knowledge editing, the reversal curse, and multi-hop reasoning may stem from common structural limitations inherent to the LPA.