How Do Transformers Learn Variable Binding in Symbolic Programs?

TL;DR

This paper demonstrates how Transformer models can learn to perform variable binding and dereferencing in symbolic programs through training, developing a systematic mechanism that mimics symbolic reasoning without explicit architectural features.

Contribution

It reveals the developmental stages and mechanisms by which Transformers acquire variable binding capabilities, including the use of residual streams as addressable memory.

Findings

Transformers develop a systematic dereferencing mechanism during training.

Attention heads learn to route information across token positions.

The model can dynamically track variable bindings across layers.

Abstract

Variable binding -- the ability to associate variables with values -- is fundamental to symbolic computation and cognition. Although classical architectures typically implement variable binding via addressable memory, it is not well understood how modern neural networks lacking built-in binding operations may acquire this capacity. We investigate this by training a Transformer to dereference queried variables in symbolic programs where variables are assigned either numerical constants or other variables. Each program requires following chains of variable assignments up to four steps deep to find the queried value, and also contains irrelevant chains of assignments acting as distractors. Our analysis reveals a developmental trajectory with three distinct phases during training: (1) random prediction of numerical constants, (2) a shallow heuristic prioritizing early variable assignments, and (3) the emergence of a systematic mechanism for dereferencing assignment chains. Using causal interventions, we find that the model learns to exploit the residual stream as an addressable memory space, with specialized attention heads routing information across token positions. This mechanism allows the model to dynamically track variable bindings across layers, resulting in accurate dereferencing. Our results show how Transformer models can learn to implement systematic variable binding without explicit architectural support, bridging connectionist and symbolic approaches. To facilitate reproducible research, we developed Variable Scope, an interactive web platform for exploring our findings at https://variablescope.org