Language Models Use Trigonometry to Do Addition (2502.00873v1)

Abstract: Mathematical reasoning is an increasingly important indicator of LLM capabilities, yet we lack understanding of how LLMs process even simple mathematical tasks. To address this, we reverse engineer how three mid-sized LLMs compute addition. We first discover that numbers are represented in these LLMs as a generalized helix, which is strongly causally implicated for the tasks of addition and subtraction, and is also causally relevant for integer division, multiplication, and modular arithmetic. We then propose that LLMs compute addition by manipulating this generalized helix using the "Clock" algorithm: to solve $a+b$, the helices for $a$ and $b$ are manipulated to produce the $a+b$ answer helix which is then read out to model logits. We model influential MLP outputs, attention head outputs, and even individual neuron preactivations with these helices and verify our understanding with causal interventions. By demonstrating that LLMs represent numbers on a helix and manipulate this helix to perform addition, we present the first representation-level explanation of an LLM's mathematical capability.

• The paper finds mid-sized language models encode numbers on a generalized helix and likely use a "Clock" algorithm, not linear methods, to perform addition between 0 and 99.
• Causal analysis through activation patching supports that these helical structures and the "Clock" algorithm are central mechanisms for the addition task.
• Understanding these internal mechanisms could lead to improving LLM precision and reliability for mathematical reasoning and suggests similar structures may exist for other operations.

LLMs Use Trigonometry to Do Addition

The paper "LLMs Use Trigonometry to Do Addition" provides a detailed examination of the mechanisms by which LLMs execute mathematical tasks, particularly the operation of addition. Through a rigorous analysis, the authors investigate how mid-sized LLMs like GPT-J, Pythia-6.9B, and Llama3.1-8B undertake the mathematical function of addition within the range of numbers from 0 to 99. By reverse-engineering these models, the paper offers a mechanistic explanation at a representation level, emphasizing that numbers in LLMs are manifested on a generalized helix and computation is possibly executed via a method coined the "Clock" algorithm.

The paper's foundational hypothesis is that LLMs encode numbers not linearly but as helices, a discovery the authors validate through various methods including causal interventions. The research identifies that LLMs process addition by rotating these helices akin to clock arms, which juxtaposes traditional mathematical operations undertaken in a linear manner. The research then deep dives into model architecture specifics, examining multi-layer perceptrons (MLPs), attention heads, and individual neurons’ roles in this computation.

A significant takeaway from the research is the introduction of the "Clock" algorithm, which is posited to help LLMs compute the addition of two numbers by manipulating their helical representations such that they produce a resultant helical form for the sum when processed through the model's neuron paths. This algorithm, as described, enables an array of units within the model to enact modular arithmetic operations, highlighting the potential efficiency of LLM methodologies beyond rote linear calculations.

One of the most compelling elements of this paper is its systematic causal analysis through activation patching, which supports the argument that these helical structures and the Clock algorithm are indeed central to the addition task. This methodology offers a significant advancement in understanding LLM internal mechanics, supporting reproducibility and transparency within AI research. The interventions show that helices not only embody numeric representations but actively facilitate computational processes, a finding that may bear implications for improving model robustness.

Furthermore, the authors explore related work, discussing contrasts with other mathematical reasoning approaches in LLMs, such as those described by the "Pizza" algorithm for angular mathematics. The discussion extends to other AI computational heuristics, situating the Clock algorithm within a broader scientific dialogue about machine learning interpretability.

For practical applications, the findings highlight a potential avenue to enhance LLM precision and reliability by understanding and manipulating the mechanisms these models use for mathematical reasoning. This could affect numerous high-stakes applications where arithmetic efficiency and accuracy are paramount, fostering the deployment of LLMs in fields necessitating exactitude like financial modeling and scientific computation.

Looking forward, the authors speculate about the extensibility of the Clock algorithm and the helical representation across other mathematical operations, hinting at the possibility of uncovering similarly structured solutions for more complex calculations. There is an implicit suggestion that a deeper understanding of these algorithms could lead to designing even more capable and efficient LLMs.

In sum, this paper provides an intricate, data-driven perspective on the mathematical capabilities of LLMs, with the identification of the Clock algorithm as a cornerstone of numerical addition being a significant contribution to the field of machine learning interpretability. This approach and its findings are poised to influence both theoretical exploration and the practical deployment of AI in contexts requiring nuanced mathematical reasoning.