# Executing Arithmetic: Fine-Tuning Large Language Models as Turing Machines

arXiv: https://arxiv.org/abs/2410.07896 · HTML: https://mlsanity.com/paper/2410.07896/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 22 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 20 of 20)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [Large Language Models Can Self-Improve](https://mlsanity.com/paper/2210.11610/) (2022)
- [Chameleon: Plug-and-Play Compositional Reasoning with Large Language Models](https://mlsanity.com/paper/2304.09842/) (2023)
- [ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving](https://mlsanity.com/paper/2309.17452/) (2023)
- [Reasoning or Reciting? Exploring the Capabilities and Limitations of Language Models Through Counterfactual Tasks](https://mlsanity.com/paper/2307.02477/) (2023)
- [ToolkenGPT: Augmenting Frozen Language Models with Massive Tools via Tool Embeddings](https://mlsanity.com/paper/2305.11554/) (2023)
- [Solving Challenging Math Word Problems Using GPT-4 Code Interpreter with Code-based Self-Verification](https://mlsanity.com/paper/2308.07921/) (2023)
- [MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning](https://mlsanity.com/paper/2310.03731/) (2023)
- [Solving Math Word Problems by Combining Language Models With Symbolic Solvers](https://mlsanity.com/paper/2304.09102/) (2023)
- [Teaching Arithmetic to Small Transformers](https://mlsanity.com/paper/2307.03381/) (2023)
- [Transformers Can Achieve Length Generalization But Not Robustly](https://mlsanity.com/paper/2402.09371/) (2024)
- [Case-Based or Rule-Based: How Do Transformers Do the Math?](https://mlsanity.com/paper/2402.17709/) (2024)

## Built on by (most cited first, 3 of 3)

- [A Survey on Mathematical Reasoning and Optimization with Large Language Models](https://mlsanity.com/paper/2503.17726/) (2025)
- [Unraveling Arithmetic in Large Language Models: The Role of Algebraic Structures](https://mlsanity.com/paper/2411.16260/) (2024)
- [Computational Reasoning of Large Language Models](https://mlsanity.com/paper/2504.20771/) (2025)

## Abstract (arXiv)

Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing and reasoning tasks. However, their performance in the foundational domain of arithmetic remains unsatisfactory. When dealing with arithmetic tasks, LLMs often memorize specific examples rather than learning the underlying computational logic, limiting their ability to generalize to new problems. In this paper, we propose a Composable Arithmetic Execution Framework (CAEF) that enables LLMs to learn to execute step-by-step computations by emulating Turing Machines, thereby gaining a genuine understanding of computational logic. Moreover, the proposed framework is highly scalable, allowing composing learned operators to significantly reduce the difficulty of learning complex operators. In our evaluation, CAEF achieves nearly 100% accuracy across seven common mathematical operations on the LLaMA 3.1-8B model, effectively supporting computations involving operands with up to 100 digits, a level where GPT-4o falls short noticeably in some settings.