# Function Induction and Task Generalization: An Interpretability Study with Off-by-One Addition

arXiv: https://arxiv.org/abs/2507.09875 · HTML: https://mlsanity.com/paper/2507.09875/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 65 references that are in the map.
- First posted on arXiv 14 Jul 2025.

## Builds on (most cited first, 30 of 49)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [WildChat: 1M ChatGPT Interaction Logs in the Wild](https://mlsanity.com/paper/2405.01470/) (2024)
- [MetaICL: Learning to Learn In Context](https://mlsanity.com/paper/2110.15943/) (2021)
- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Larger language models do in-context learning differently](https://mlsanity.com/paper/2303.03846/) (2023)
- [In-Context Learning Creates Task Vectors](https://mlsanity.com/paper/2310.15916/) (2023)
- [Function Vectors in Large Language Models](https://mlsanity.com/paper/2310.15213/) (2023)
- [Reasoning or Reciting? Exploring the Capabilities and Limitations of Language Models Through Counterfactual Tasks](https://mlsanity.com/paper/2307.02477/) (2023)
- [Analyzing the Structure of Attention in a Transformer Language Model](https://mlsanity.com/paper/1906.04284/) (2019)
- [Retrieval Head Mechanistically Explains Long-Context Factuality](https://mlsanity.com/paper/2404.15574/) (2024)
- [When Attention Sink Emerges in Language Models: An Empirical View](https://mlsanity.com/paper/2410.10781/) (2024)
- [How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model](https://mlsanity.com/paper/2305.00586/) (2023)
- [Do Large Language Models Latently Perform Multi-Hop Reasoning?](https://mlsanity.com/paper/2402.16837/) (2024)
- [Meta-learning via Language Model In-context Tuning](https://mlsanity.com/paper/2110.07814/) (2021)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Fine-Tuning Enhances Existing Mechanisms: A Case Study on Entity Tracking](https://mlsanity.com/paper/2402.14811/) (2024)
- [Why do LLMs attend to the first token?](https://mlsanity.com/paper/2504.02732/) (2025)
- [Ground-Truth Labels Matter: A Deeper Look into Input-Label Demonstrations](https://mlsanity.com/paper/2205.12685/) (2022)

## Built on by (most cited first, 2 of 2)

- [Punctuation and Predicates in Language Models](https://mlsanity.com/paper/2508.14067/) (2025)
- [Test, then Route: How Language Models Execute In-Context Conditional Rules Across Models and Languages](https://mlsanity.com/paper/2608.04183/) (2026)

## Abstract (arXiv)

Large language models demonstrate the intriguing ability to perform unseen tasks via in-context learning. However, it remains unclear what mechanisms inside the model drive such task-level generalization. In this work, we approach this question through the lens of off-by-one addition (i.e., 1+1=3, 2+2=5, 3+3=?), a two-step, counterfactual task with an unexpected +1 function as a second step. Leveraging circuit-style interpretability techniques such as path patching, we analyze the models' internal computations behind their performance and present three key findings. First, we identify a mechanism that explains the model's generalization from standard addition to off-by-one addition. It resembles the induction head mechanism described in prior work, yet operates at a higher level of abstraction; we therefore term it "function induction" in this work. Second, we show that the induction of the +1 function is governed by multiple attention heads in parallel, each of which emits a distinct piece of the +1 function. Finally, we find that this function induction mechanism is reused in a broader range of tasks, including synthetic tasks such as shifted multiple-choice QA and algorithmic tasks such as base-8 addition. Overall, our findings offer deeper insights into how reusable and composable structures within language models enable task-level generalization.