Dynamics Reveals Structure: Challenging the Linear Propagation Assumption
This addresses structural limitations in neural network reasoning, potentially impacting AI systems requiring logical coherence, but it is incremental as it builds on existing formalizations.
The paper investigates the Linear Propagation Assumption (LPA) in neural networks, showing that it fails for composition due to a bilinearity conflict with negation, which may explain issues like knowledge editing failures and the reversal curse.
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.