# Principled Understanding of Generalization for Generative Transformer Models in Arithmetic Reasoning Tasks

arXiv: https://arxiv.org/abs/2407.17963 · HTML: https://mlsanity.com/paper/2407.17963/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 15 of its 27 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 15 of 15)

- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [AI Alignment: A Comprehensive Survey](https://mlsanity.com/paper/2310.19852/) (2023)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Scaling Laws and Interpretability of Learning from Repeated Data](https://mlsanity.com/paper/2205.10487/) (2022)
- [Investigating the Limitations of Transformers with Simple Arithmetic Tasks](https://mlsanity.com/paper/2102.13019/) (2021)
- [Teaching Arithmetic to Small Transformers](https://mlsanity.com/paper/2307.03381/) (2023)
- [Transformers Can Do Arithmetic with the Right Embeddings](https://mlsanity.com/paper/2405.17399/) (2024)
- [Limitations of Language Models in Arithmetic and Symbolic Induction](https://mlsanity.com/paper/2208.05051/) (2022)
- [Length Generalization in Arithmetic Transformers](https://mlsanity.com/paper/2306.15400/) (2023)
- [Searching for Needles in a Haystack: On the Role of Incidental Bilingualism in PaLM's Translation Capability](https://mlsanity.com/paper/2305.10266/) (2023)
- [Location Attention for Extrapolation to Longer Sequences](https://mlsanity.com/paper/1911.03872/) (2019)
- [From GPT-4 to Gemini and Beyond: Assessing the Landscape of MLLMs on Generalizability, Trustworthiness and Causality through Four Modalities](https://mlsanity.com/paper/2401.15071/) (2024)
- [From Interpolation to Extrapolation: Complete Length Generalization for Arithmetic Transformers](https://mlsanity.com/paper/2310.11984/) (2023)

## Built on by (most cited first, 5 of 5)

- [The Two-Stage Decision-Sampling Hypothesis: Understanding the Emergence of Self-Reflection in RL-Trained LLMs](https://mlsanity.com/paper/2601.01580/) (2026)
- [Beyond the Training Distribution: Mapping Generalization Boundaries in Neural Program Synthesis](https://mlsanity.com/paper/2604.27551/) (2026)
- [When LLMs Stop Following Steps: A Diagnostic Study of Procedural Execution in Language Models](https://mlsanity.com/paper/2605.00817/) (2026)
- [When Tokenization is Secretly Output Supervision](https://mlsanity.com/paper/2609.01386/) (2026)
- [Distance generalization in transformers: why bother with positional encoding?](https://mlsanity.com/paper/2609.11913/) (2026)

## Abstract (arXiv)

Transformer-based models excel in various tasks but their generalization capabilities, especially in arithmetic reasoning, remain incompletely understood. Arithmetic tasks provide a controlled framework to explore these capabilities, yet performance anomalies persist, such as inconsistent effectiveness in multiplication and erratic generalization in modular addition (e.g., modulo 100 vs. 101). This paper develops a unified theoretical framework for understanding the generalization behaviors of transformers in arithmetic tasks, focusing on length generalization. Through detailed analysis of addition, multiplication, and modular operations, we reveal that translation invariance in addition aligns with relative positional encoding for robust generalization, while base mismatch in modular operations disrupts this alignment. Experiments across GPT-family models validate our framework, confirming its ability to predict generalization behaviors. Our work highlights the importance of task structure and training data distribution for achieving data-efficient and structure-aware training, providing a systematic approach to understanding of length generalization in transformers.