Tokenization Constraints in LLMs: A Study of Symbolic and Arithmetic Reasoning Limits

TL;DR

This paper investigates how tokenization schemes, especially subword methods like BPE, limit the symbolic and arithmetic reasoning capabilities of language models, highlighting the importance of token granularity for effective reasoning.

Contribution

It introduces the concept of Token Awareness to formalize the impact of token structure on reasoning and demonstrates how atomically-aligned token formats improve symbolic generalization in LLMs.

Findings

Token structure significantly affects reasoning performance.

Atomically-aligned token formats enable better symbolic generalization.

Small models can outperform larger ones with proper tokenization.

Abstract

Tokenization is the first - and often underappreciated - layer of computation in language models. While Chain-of-Thought (CoT) prompting enables transformer models to approximate recurrent computation by externalizing intermediate steps, we show that the success of such reasoning is fundamentally bounded by the structure of tokenized inputs. This work presents a theoretical and empirical investigation into how tokenization schemes, particularly subword-based methods like byte-pair encoding (BPE), impede symbolic computation by merging or obscuring atomic reasoning units. We introduce the notion of Token Awareness to formalize how poor token granularity disrupts logical alignment and prevents models from generalizing symbolic procedures. Through systematic evaluation on arithmetic and symbolic tasks, we demonstrate that token structure dramatically affect reasoning performance, causing failure even with CoT, while atomically-aligned formats unlock strong generalization, allowing small models (e.g., GPT-4o-mini) to outperform larger systems (e.g., o1) in structured reasoning. Our findings reveal that symbolic reasoning ability in LLMs is not purely architectural, but deeply conditioned on token-level representations.