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How Do Transformers Learn Variable Binding in Symbolic Programs?
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Trains a small 37.8M-parameter GPT-2-style Transformer from scratch on synthetic Python-like variable-assignment programs and uses causal interventions and training checkpoints to show how it learns variable dereferencing.
Problem
How neural networks without built-in binding operations acquire variable binding (dereferencing queried variables through assignment chains) during training, and what mechanisms implement it.
Approach
Track the developmental trajectory of a from-scratch Transformer on a variable dereferencing task with distractor chains, and use counterfactual interchange interventions on the residual stream and attention heads to show the model uses the residual stream as addressable memory while layering a systematic mechanism on top of early line-position heuristics.
Key results
- Model reaches over 99.9% test accuracy after training, with three phases: chance-level (~12%), early-line heuristic phase (~56%), and systematic dereferencing (~99%).
- Causal patching identifies specialized heads (e.g., 6.5, 7.2, 8.3, 9.4, 11.2/11.3/11.7) routing variable values along reference chains.
- Interventions on numerical-constant and variable-name subspaces succeed at 92.17% and 87.08%, supporting addressable subspaces in the residual stream.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train a Transformer from scratch on synthetic variable-assignment programs | Transformer trained from scratch · Variable dereferencing test set | accuracy | Helped |
| Progress from early-line heuristics to systematic variable dereferencing | Transformer trained from scratch · Variable dereferencing test set | accuracy: 12% to 56%; 56% to 99% | Helped |
| Swap the numerical-constant residual stream subspace with its counterfactual | Transformer trained from scratch · Variable dereferencing test programs | patching success rate: 92.17% | Helped |
| Swap the variable-name residual stream subspace with its counterfactual | Transformer trained from scratch · Variable dereferencing test programs | patching success rate: 87.08% | Helped |
| Use the line-1 heuristic on programs with answers on line 1 | Transformer trained from scratch · Programs with answer on line 1 | accuracy: over 90% | Helped |
| Use the line-2 heuristic on programs with answers on line 2 | Transformer trained from scratch · Programs with answer on line 2 | accuracy: approximately 65% | Helped |
| Apply the line-1 heuristic to programs with lengths substantially different from training | Transformer trained from scratch · Programs with 2–25 lines, answer on line 1 | accuracy | Hurt |
| Use the systematic variable-binding mechanism on programs with more than four hops | Transformer trained from scratch · Programs with up to 13 hops, answer not on lines 1 or 2 | accuracy | Helped |
| Rely on the line-1 heuristic as hop count increases | Transformer trained from scratch · Programs with up to 13 hops, answer on line 1 | accuracy | Hurt |
Surprising findings
- The final solution builds on rather than replaces early line-position heuristics, which persist throughout training.
- Linear probes fail to decode full program state (8.90% state accuracy at best), suggesting no complete linearly decodable state representation.
Limitations
- Task uses a tiny synthetic vocabulary (26 variables, digits 0-9) and fixed 16-line program structure, limiting generality.
- Line-1 heuristic fails to generalize to program lengths far from 16 lines.
Experimental setup
Datasets: synthetic variable-assignment programs (500,000 generated)
Benchmarks: synthetic program test set (49,000 programs)
Metrics: accuracy patching success rate logit difference linear probe accuracy
Baselines: line-1 heuristic line-2 heuristic
Builds on: GPT-2 interchange intervention causal tracing UMAP
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