Kleinberg and Mullainathan recently proposed a formal framework for studying the phenomenon of language generation, called language generation in the limit. In this model, an adversary gives an enumeration of example strings from an unknown target language, and the algorithm is tasked with correctly generating unseen strings from the target language within finite time. Refined notions of non-uniform and uniform generation were later introduced by Li, Raman, and Tewari (2025), and a noisy model was introduced by Raman and Raman (2025), which allows the adversary to insert extraneous strings. A natural question in the noisy model is to quantify the effect of noise, by studying the impact of each additional extraneous string. We show two complementary results in this setting. We first show that for both uniform and non-uniform generation, a single noisy string strictly reduces the set of collections that can be generated, thus answering an open question in Raman and Raman (2025). Then, we show for both uniform and non-uniform generation that generation with a single noisy string is equivalent to generation with any finite amount of noise, sharply contrasting with the strict hierarchy for noisy generation in the limit shown by Bai, Panigrahi, and Zhang (2026). Finally, we leverage our previous results to provide the first known characterization for non-uniform noise-dependent generatability.

翻译：Kleinberg与Mullainathan近期提出了一个研究语言生成现象的形式化框架，称为极限语言生成。在该模型中，对手会枚举来自未知目标语言的示例字符串，而算法的任务是在有限时间内正确生成目标语言中未出现过的字符串。Li、Raman与Tewari（2025）随后提出了非均匀生成与均匀生成的细化概念，Raman与Raman（2025）则引入了噪声模型，允许对手插入无关字符串。噪声模型中一个自然的问题是通过研究每个额外无关字符串的影响来量化噪声效应。我们在此设定下展示了两项互补的结果。首先证明对于均匀与非均匀生成，单个噪声字符串都会严格缩减可生成的集合，从而回答了Raman与Raman（2025）中提出的开放性问题。随后证明对于均匀与非均匀生成，带单个噪声字符串的生成等价于任意有限噪声量的生成，这与Bai、Panigrahi及Zhang（2026）所揭示的极限噪声生成的严格层级结构形成鲜明对比。最后，我们利用前述结果为非均匀噪声依赖的可生成性提供了首个已知的特征刻画。