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Language Generation in the Limit: Noise, Loss, and Feedback

Yannan Bai, Debmalya Panigrahi, Ian Zhang

TL;DR

This work advances the theory of language generation in the limit by resolving the finite-union question via a counterexample that separates union-closedness from naive union-building intuitions. It then systematically analyzes natural extensions—lossy and noisy settings, and feedback-based generation—deriving precise equivalences and separations: lossiness and noise levels can dramatically alter generatability, infinite feedback strictly increases power, and finite feedback does not; moreover, any countable language family can be non-uniformly identified with feedback. The results offer a detailed map of how omissions, incorrect outputs, and interactive queries affect the learnability and generation of languages in this framework, tying together a spectrum of models through tight characterizations and isomorphism arguments. Overall, the paper clarifies the boundaries of what can be learned or generated in the limit under realistic perturbations and interactive constraints, with implications for formal models of language learning and verification. $t^\\star$, $K$, $S_t$, and other quantities are used to formalize these notions throughout the work.

Abstract

Kleinberg and Mullainathan (2024) recently proposed a formal framework called language generation in the limit and showed that given a sequence of example strings from an unknown target language drawn from any countable collection, an algorithm can correctly generate unseen strings from the target language within finite time. This notion was further refined by Li, Raman, and Tewari (2024), who defined stricter categories of non-uniform and uniform generation. They showed that a finite union of uniformly generatable collections is generatable in the limit, and asked if the same is true for non-uniform generation. We begin by resolving the question in the negative: we give a uniformly generatable collection and a non-uniformly generatable collection whose union is not generatable in the limit. We then use facets of this construction to further our understanding of several variants of language generation. The first two, generation with noise and without samples, were introduced by Raman and Raman (2025) and Li, Raman, and Tewari (2024) respectively. We show the equivalence of these models for uniform and non-uniform generation, and provide a characterization of non-uniform noisy generation. The former paper asked if there is any separation between noisy and non-noisy generation in the limit -- we show that such a separation exists even with a single noisy string. Finally, we study the framework of generation with feedback, introduced by Charikar and Pabbaraju (2025), where the algorithm is strengthened by allowing it to ask membership queries. We show finite queries add no power, but infinite queries yield a strictly more powerful model. In summary, the results in this paper resolve the union-closedness of language generation in the limit, and leverage those techniques (and others) to give precise characterizations for natural variants that incorporate noise, loss, and feedback.

Language Generation in the Limit: Noise, Loss, and Feedback

TL;DR

This work advances the theory of language generation in the limit by resolving the finite-union question via a counterexample that separates union-closedness from naive union-building intuitions. It then systematically analyzes natural extensions—lossy and noisy settings, and feedback-based generation—deriving precise equivalences and separations: lossiness and noise levels can dramatically alter generatability, infinite feedback strictly increases power, and finite feedback does not; moreover, any countable language family can be non-uniformly identified with feedback. The results offer a detailed map of how omissions, incorrect outputs, and interactive queries affect the learnability and generation of languages in this framework, tying together a spectrum of models through tight characterizations and isomorphism arguments. Overall, the paper clarifies the boundaries of what can be learned or generated in the limit under realistic perturbations and interactive constraints, with implications for formal models of language learning and verification. , , , and other quantities are used to formalize these notions throughout the work.

Abstract

Kleinberg and Mullainathan (2024) recently proposed a formal framework called language generation in the limit and showed that given a sequence of example strings from an unknown target language drawn from any countable collection, an algorithm can correctly generate unseen strings from the target language within finite time. This notion was further refined by Li, Raman, and Tewari (2024), who defined stricter categories of non-uniform and uniform generation. They showed that a finite union of uniformly generatable collections is generatable in the limit, and asked if the same is true for non-uniform generation. We begin by resolving the question in the negative: we give a uniformly generatable collection and a non-uniformly generatable collection whose union is not generatable in the limit. We then use facets of this construction to further our understanding of several variants of language generation. The first two, generation with noise and without samples, were introduced by Raman and Raman (2025) and Li, Raman, and Tewari (2024) respectively. We show the equivalence of these models for uniform and non-uniform generation, and provide a characterization of non-uniform noisy generation. The former paper asked if there is any separation between noisy and non-noisy generation in the limit -- we show that such a separation exists even with a single noisy string. Finally, we study the framework of generation with feedback, introduced by Charikar and Pabbaraju (2025), where the algorithm is strengthened by allowing it to ask membership queries. We show finite queries add no power, but infinite queries yield a strictly more powerful model. In summary, the results in this paper resolve the union-closedness of language generation in the limit, and leverage those techniques (and others) to give precise characterizations for natural variants that incorporate noise, loss, and feedback.

Paper Structure

This paper contains 18 sections, 29 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Our Results.
  3. Preliminaries.
  4. Generation in the Limit.
  5. Noisy Generation.
  6. Projection.
  7. Generation in the Limit is Not Closed under Finite Union.
  8. Lossy Generation.
  9. Generation Without Samples.
  10. Generation with Infinite Omissions.
  11. Generation with Finite Omissions.
  12. Generation with Noise.
  13. Identification and Generation with Feedback.
  14. Generation in the Limit with Feedback.
  15. Generation in the Limit with Finite Feedback.
  16. ...and 3 more sections

Key Result

Theorem 1.1

There exist collections $\mathcal{C}_1$ and $\mathcal{C}_2$ such that $\mathcal{C}_1$ is non-uniformly generatable (without samples) and $\mathcal{C}_2$ is uniformly generatable (without samples), but $\mathcal{C}_1 \cup \mathcal{C}_2$ is not generatable in the limit.

Figures (2)

  • Figure 1.1: A figurative representation of the relationship between the various models of language generation that we consider in this paper. The equivalence of (non)-uniform noisy and (non)-uniform generation without samples is given by \ref{['thm:equi_lossy_noisy']}. The separations between noisy generation in the limit, genereration with noise level $i$, generation in the limit, and generation in the limit with infinite feedback are given by \ref{['thm:noise_sensitivity', 'thm:fine_grained', 'thm:feedback_combined']}.
  • Figure 3.1: An example of possible values of $x$ and $z$ at the end of stage $2$ where $t_0 = 1$, $t_1 = 5$, and $t_2 = 8$. The red dots represent the incorrect values output by the algorithm at times $z_{t_0}$, $z_{t_1}$, and $z_{t_2}$.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1: Generator algorithm LRT25
  • Definition 2.2: Generation in the limit KM24
  • ...and 74 more