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

2025-07-21

AI Summary

This work investigates fundamental limits of language generation across three dimensions: (1) closure properties of nonuniformly generated language classes under finite union; (2) impact of noise corruption and sample sparsity on learnability; and (3) power of membership query feedback in enhancing generative capacity. Using formal language theory and computability-theoretic analysis, we establish that the class of nonuniformly generated languages is not closed under finite union—a first such result. We further show that even a single noisy string suffices to strictly separate noisy from noise-free generation capabilities. Finally, we prove that infinite membership queries enable strictly stronger generation than any feedback-free model. Collectively, these results provide precise characterizations—both in terms of learning power and closure properties—for three key generative paradigms: noisy, sparse-sample, and feedback-driven learning. The work thereby lays a rigorous theoretical foundation for robustness and interactivity in language learning.

Technology Category

• Natural Language Processing: Generation
• Machine Learning: Learning Theory
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Resolves union-closedness in language generation limits
• Characterizes noisy and non-noisy generation equivalences
• Determines power of feedback in generation models

Innovation

• Resolves union-closedness in language generation
• Equivalence of noisy and non-noisy generation models
• Infinite feedback queries enhance generation power