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Case-Based or Rule-Based: How Do Transformers Do the Math?
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Uses leave-square-out data interventions on five math tasks to show fine-tuned transformers rely on case-based rather than rule-based reasoning, then proposes Rule-Following Fine-Tuning (RFFT) that makes GPT-3.5 generalize from 1-5 digit to 12-digit addition.
Problem
LLMs fail at simple math such as addition on inputs longer than training data; it is debated whether transformers learn underlying rules or recall similar training cases.
Approach
Distinguish case-based from rule-based reasoning via data interventions that remove a square region of training examples and test whether accuracy holes appear; then teach rule-following by putting explicit rules (a Python program) in the input and fine-tuning the model to quote and execute each rule step by step.
Key results
- RFFT on GPT-3.5-turbo trained on 100 samples of 1-5 digit addition keeps over 95% accuracy on 12-digit addition, over 40% higher than scratchpad.
- Llama-2-7B with RFFT maintains 91.1% accuracy on 9-digit sums, while scratchpad achieves under 40% on similar lengths.
- Leave-square-out on GPT-2 produces clear accuracy holes in test squares for addition, modular addition, base addition, and linear regression despite 100% random-split accuracy.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Provide explicit addition rules and train the model to recite and follow them step by step instead of using direct-answer or scratchpad fine-tuning | GPT-3.5-turbo-1106 · Addition, up to 12 digits | accuracy: over 95% | Helped |
| Provide explicit addition rules and train the model to follow them step by step rather than use scratchpad fine-tuning | Llama-2-7B · Addition, 6- to 9-digit sums | accuracy: 91.1%; less than 40% | Helped |
| Use scratchpad fine-tuning instead of direct-answer training (ablation) | GPT-2 and GPT-2-medium · Addition, Leave-Square-Out with square side length l_k≥20 | test accuracy | Helped |
| Use scratchpad fine-tuning instead of direct-answer training (ablation) | GPT-2 and GPT-2-medium · Addition, Leave-Square-Out with square side length l_k=10 | test accuracy | Hurt |
| Use scratchpad instead of direct-answer fine-tuning (ablation) | GPT-3.5-turbo-1106 · Addition, including 15-digit addition | accuracy: 46.2% on 15-digit addition with direct answer | Hurt |
| Train on a leave-square-out split rather than a random split (ablation) | GPT-2 and GPT-2-medium · Five math tasks: addition, modular addition, base addition, linear regression, and chicken & rabbit | test accuracy: 100%; above 95% | Hurt |
| Increase the left-out test-square side length (ablation) | GPT-2 and GPT-2-medium · Addition, Leave-Square-Out | test accuracy | Hurt |
| Increase addition training range and scale the left-out square accordingly (ablation) | GPT-2 · Addition, Leave-Square-Out over ranges [0,100), [0,200), and [0,500) | test accuracy | No clear effect |
| Replace base-9 digits 0–8 with letters A–I (ablation) | Fine-tuned Llama-2-7B · Base-9 addition, Leave-Square-Out | test accuracy | Hurt |
| Provide explicit rules in-context rather than direct-answer or scratchpad examples (ablation) | GPT-3.5-turbo-0125 · Familiar base-10 addition | accuracy | Hurt |
| Provide explicit rules in-context rather than direct-answer or scratchpad examples (ablation) | GPT-3.5-turbo-0125 · Unfamiliar base-9 addition | accuracy | Helped |
| Train on a random split instead of leaving out a test square (ablation) | Llama-2-7B trained from scratch · Base-9 addition | test accuracy: 30.2% to 92% | Helped |
Surprising findings
- Scratchpad fine-tuning of GPT-3.5 underperformed direct answer fine-tuning, contrary to expectation.
- Fine-tuned pretrained Llama-2-7B on base-9 addition with letter digits (A-I) showed a hole, suggesting pretraining biases rather than systematic rule learning.
Limitations
- RFFT is task-specific and requires carefully designed input-output sequences.
- Study focuses on fine-tuning for specific tasks rather than pretrained model reasoning mechanisms or prompting techniques like CoT.
- Errors persist at longer lengths due to execution mistakes (e.g., pop) and limited long-context ability.
Experimental setup
Datasets: synthetic addition modular addition base-9 addition linear regression chicken & rabbit problems last-letter concatenation
Benchmarks: synthetic addition test sets (1-9 and 6-15 digits) leave-square-out test squares on five math tasks
Metrics: accuracy accuracy over test squares accuracy gap train vs test
Baselines: direct answer fine-tuning scratchpad fine-tuning
Builds on: scratchpad fine-tuning grokking chain-of-thought
Closest papers by meaning
- Grokking of Implicit Reasoning in Transformers: A Mechanistic Journey to the Edge of Generalization (NeurIPS 2024 · similarity 0.86)
- Learning Theorem Rationale for Improving the Mathematical Reasoning Capability of Large Language Models (AAAI 2025 · similarity 0.86)
- Advancing Mathematical Reasoning in Language Models: The Impact of Problem-Solving Data, Data Synthesis Methods, and Training Stages (ICLR 2025 · similarity 0.86)
- When can transformers reason with abstract symbols? (ICLR 2024 · similarity 0.86)
- OMEGA: Can LLMs Reason Outside the Box in Math? Evaluating Exploratory, Compositional, and Transformative Generalization (NeurIPS 2025 · similarity 0.86)
- RMath: A Logic Reasoning-Focused Datasets Toward Mathematical Multistep Reasoning Tasks (AAAI 2025 · similarity 0.85)
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