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Arithmetic Without Algorithms: Language Models Solve Math with a Bag of Heuristics
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Uses causal circuit analysis and neuron-level heuristic classification to show that pretrained LLMs perform arithmetic via a "bag of heuristics" rather than a robust algorithm or pure memorization.
Problem
Whether LLMs solve arithmetic reasoning with generalizable algorithms or by memorizing input-output patterns, and what mechanism implements arithmetic answers inside pretrained models.
Approach
Identifies the arithmetic circuit via activation patching, decomposes circuit MLP neurons into key-value heuristics, classifies neurons into human-defined heuristic types (range, modulo, pattern, identical operands, multi-result), and shows ablating heuristic neurons by type or by prompt reduces accuracy; the unordered combination of heuristics explains most accuracy and emerges early in training.
Key results
- Llama3-8B circuit achieves 0.96 average faithfulness across the four operators, and only ~200 neurons per layer (~1.5%) are needed for high faithfulness.
- 91% of the 3,200 top neurons (200 per layer across 16 layers, per operator) are classified as arithmetic heuristics.
- Heuristic neurons from the final Pythia-6.9B checkpoint explain on average 79% of heuristic contribution at intermediate checkpoints, and ablating 25 heuristic neurons per layer drives near-zero accuracy at all training stages.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Ablate neurons classified as a specific heuristic (ablation) | Llama3-8B arithmetic circuit · Correct arithmetic prompts associated with the ablated heuristic | accuracy: average drop of 29% out of 95% average pre-ablation accuracy | Hurt |
| Ablate neurons classified as a specific heuristic (ablation) | Llama3-8B arithmetic circuit · Prompts associated with the heuristic versus prompts not associated with it | accuracy | Mixed |
| Ablate the highest-scoring neurons from heuristics associated with each prompt rather than an equal number from unassociated heuristics (ablation) | Arithmetic models · Arithmetic prompts | accuracy | Hurt |
| Ablate 25 prompt-associated heuristic neurons per layer (ablation) | Pythia-6.9B checkpoints · Arithmetic prompts across training stages | accuracy: near-zero accuracy | Hurt |
| Use neurons mutual with the final checkpoint as the heuristic set (ablation) | Pythia-6.9B intermediate checkpoints · Arithmetic prompts across training | share of total heuristics' contribution to accuracy: 79% | Helped |
| Ablate prompt-associated heuristic neurons instead of a similar number of random heuristic neurons (ablation) | Pythia-6.9B checkpoints · Correct arithmetic prompts, including earlier checkpoints | accuracy | Hurt |
| Use the arithmetic circuit rather than the full model | Llama3-8B · Arithmetic prompts across four operators | faithfulness: 0.96 | Helped |
| Keep only 200 high-effect neurons per layer | Llama3-8B middle- and late-layer MLP circuit · Arithmetic prompts | faithfulness: only 200 neurons (roughly 1.5%) per layer are needed to achieve high faithfulness | Helped |
| Keep the top 1% of MLP0 neurons in the circuit | Llama3-8B arithmetic circuit · Arithmetic prompts | faithfulness: 1% | Helped |
Surprising findings
- Ablating neurons of a heuristic type causes an average 29% accuracy drop out of 95% pre-ablation accuracy, so no single heuristic is sufficient.
- Incorrect prompts do not have fewer associated heuristic neurons than correct ones; failures are attributed to weaker total logit contribution instead.
Limitations
- Heuristic types are human-defined and may impose human biases on model abstractions.
- Analysis is limited to LLMs whose tokenizers merge multiple digits; single-digit tokenization could yield different conclusions.
- Heuristic classification threshold was chosen manually with estimated false classifications, with no ground-truth neuron-to-heuristic labels.
Experimental setup
Datasets: synthetic arithmetic prompts (operands in [0,300], four operators)
Benchmarks: synthetic arithmetic prompts
Metrics: faithfulness accuracy normalized logit knockout accuracy drop probe accuracy
Baselines: Fourier-feature account of addition (Zhou et al.) Nanda et al. modular addition circuit
Builds on: activation patching key-value memories logit lens TransformerLens
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