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The Foundations of Tokenization: Statistical and Computational Concerns

Juan Luis Gastaldi, John Terilla, Luca Malagutti, Brian DuSell, Tim Vieira, Ryan Cotterell

TL;DR

The Foundations of Tokenization: Statistical and Computational Concerns develops a formal theory of tokenization by modeling encoders and decoders as stochastic maps between character strings and token sequences. It introduces a Fundamental Principle: a tokenizer preserves estimator consistency exactly when $\kappa\tau p^*=p^*$, enabling precise characterizations of when tokenization is statistically sound. The work analyzes statistical issues like inconsistency and ambiguity arising from noninjective encoders/decoders and addresses computational aspects such as finiteness and sequentiality via multiplicativity and bounded variation, linking practical tokenizers (e.g., BPE, WordPiece) to finite-state representations. Overall, the framework provides rigorous conditions for robust tokenization in neural language models and offers insights for theory-driven improvements and reliable evaluation.

Abstract

Tokenization - the practice of converting strings of characters from an alphabet into sequences of tokens over a vocabulary - is a critical step in the NLP pipeline. The use of token representations is widely credited with increased model performance but is also the source of many undesirable behaviors, such as spurious ambiguity or inconsistency. Despite its recognized importance as a standard representation method in NLP, the theoretical underpinnings of tokenization are not yet fully understood. In particular, the impact of tokenization on language model estimation has been investigated primarily through empirical means. The present paper contributes to addressing this theoretical gap by proposing a unified formal framework for representing and analyzing tokenizer models. Based on the category of stochastic maps, this framework enables us to establish general conditions for a principled use of tokenizers and, most importantly, the necessary and sufficient conditions for a tokenizer model to preserve the consistency of statistical estimators. In addition, we discuss statistical and computational concerns crucial for designing and implementing tokenizer models, such as inconsistency, ambiguity, finiteness, and sequentiality. The framework and results advanced in this paper contribute to building robust theoretical foundations for representations in neural language modeling that can inform future theoretical and empirical research.

The Foundations of Tokenization: Statistical and Computational Concerns

TL;DR

The Foundations of Tokenization: Statistical and Computational Concerns develops a formal theory of tokenization by modeling encoders and decoders as stochastic maps between character strings and token sequences. It introduces a Fundamental Principle: a tokenizer preserves estimator consistency exactly when , enabling precise characterizations of when tokenization is statistically sound. The work analyzes statistical issues like inconsistency and ambiguity arising from noninjective encoders/decoders and addresses computational aspects such as finiteness and sequentiality via multiplicativity and bounded variation, linking practical tokenizers (e.g., BPE, WordPiece) to finite-state representations. Overall, the framework provides rigorous conditions for robust tokenization in neural language models and offers insights for theory-driven improvements and reliable evaluation.

Abstract

Tokenization - the practice of converting strings of characters from an alphabet into sequences of tokens over a vocabulary - is a critical step in the NLP pipeline. The use of token representations is widely credited with increased model performance but is also the source of many undesirable behaviors, such as spurious ambiguity or inconsistency. Despite its recognized importance as a standard representation method in NLP, the theoretical underpinnings of tokenization are not yet fully understood. In particular, the impact of tokenization on language model estimation has been investigated primarily through empirical means. The present paper contributes to addressing this theoretical gap by proposing a unified formal framework for representing and analyzing tokenizer models. Based on the category of stochastic maps, this framework enables us to establish general conditions for a principled use of tokenizers and, most importantly, the necessary and sufficient conditions for a tokenizer model to preserve the consistency of statistical estimators. In addition, we discuss statistical and computational concerns crucial for designing and implementing tokenizer models, such as inconsistency, ambiguity, finiteness, and sequentiality. The framework and results advanced in this paper contribute to building robust theoretical foundations for representations in neural language modeling that can inform future theoretical and empirical research.
Paper Structure (17 sections, 17 theorems, 15 equations, 1 figure)

This paper contains 17 sections, 17 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Formal Languages, Estimators, and Stochastic Maps
  4. Notation and Terminology
  5. A Formal Framework for Tokenization
  6. Statistical Concerns: Inconsistency and Ambiguity
  7. Noninjective $\boldsymbol{\tau}$ and Inconsistency.
  8. Noninjective $\boldsymbol{\kappa}$ and Ambiguity.
  9. Computational Concerns: Finiteness and Sequentiality
  10. Multiplicativity and Finiteness.
  11. Bounded Variation and Sequentiality.
  12. Conclusion
  13. Acknowledgements
  14. Appendix
  15. Proofs
  16. ...and 2 more sections

Key Result

Lemma 3.1

Let $\{p_n\}$ be a sequence of probability distributions over a countable set $X$ that converges pointwise to a probability distribution $p$. Then $\lim_{n\to \infty}\sum_{x\in X} |p_n\xspace(x)-p\xspace(x)| = 0$.

Figures (1)

  • Figure 1: Example of an inconsistent tokenizer

Theorems & Definitions (40)

  • Definition 3.1
  • Lemma 3.1
  • Corollary 3.0.1
  • Lemma 3.2
  • Theorem 3.1: Fundamental Principle of Tokenization
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.1
  • Proposition 3.2
  • Definition 4.1
  • ...and 30 more