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

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. 1 INTRODUCTION As the capabilities of language models (LMs) continue to grow, users apply them to increasingly challenging and diverse tasks, accompanied by evolving expectations (Zhao et al., 2024; Tamkin et al., 2024; Kwa et al., 2025) . Consequently, it becomes impractical to include every task of interest in a model's training prior to deployment. In this context, task-level generalization-the ability of a model to perform novel tasks at inference time-becomes highly crucial and valued. Prior work shows that LMs already exhibit this capability to a significant extent through in-context learning (Brown et al., 2020; Chen et al., 2022; Min et al., 2022a). The underlying mechanisms of this behavior are being actively investigated, with work on induction heads (Olsson et al., 2022) and function vectors (Hendel et al., 2023; Todd et al., 2024) offering substantial insights on pattern matching tasks (i.e., ) and mapping-style tasks (e.g., France: Paris, Australia: → Canberra). However, our understanding is still limited, especially regarding more complex generalization scenarios involving multi-step reasoning or newly-defined concepts in the task. In this work, we aim to enhance our understanding of how models handle novelty and unconventionality with one counterfactual task: off-by-one addition (i.e., 1+1=3, 2+2=5, 3+3=?). For humans, this task consists of two sequential steps: standard addition, followed by an unexpected increment of one to the sum. When a language model is prompted to perform this task with in-context learning, we anticipate two possible outcomes: (1) the model acquires the intended +1 operation and thus outputs 7, or (2) it adheres to fundamental arithmetic rules and outputs 6. We begin our study by evaluating six contemporary LMs on off-by-one addition. Our findings indicate that all evaluated models consistently demonstrate the first outcome, effectively leveraging in-context