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Language Generation in the Limit
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Shows theoretically that an agent can generate unseen valid strings from an unknown language in the limit for every countable collection of candidate languages, in contrast to Gold-Angluin identification, and gives uniform sample bounds for finite collections.
Problem
Formalizing language generation from positive examples under adversarial enumeration, asking whether generating unseen valid strings is possible in the limit even though identifying the language is impossible (Gold-Angluin).
Approach
Generation only requires outputting unseen strings guaranteed to lie in the target language, not identifying it. The algorithm outputs elements from the highest-indexed critical language (minimal relative to earlier consistent candidates), using finite prefixes and membership queries, so the output stays inside the true language eventually; for finite collections, the closure of the sample is infi
Key results
- For every countable collection of candidate languages, an algorithm generates unseen strings from the true language in the limit, using only membership queries.
- For finite collections, there is a uniform sample size t(C) = m*+1 after which the algorithm can generate an infinite sequence of unseen strings from the true language, where m* is the maximum finite intersection size over sub-collections.
- The generation result extends to prompted generation with robust prompts.
Surprising findings
- Language generation in the limit is possible for every countable collection of languages, even though identification in the limit is impossible for the same collections.
- Uniform sample bounds exist for generation with finite collections, whereas no uniform bound exists for identification (even two-language collections).
Limitations
- Results are purely theoretical and do not yield a practical LLM training method.
- Algorithm generation is not guaranteed to be efficient; it assumes membership queries and unbounded computation.
- Prompted generation result requires robust prompts or extra subset-query power for non-trivial prompts.
Experimental setup
Builds on: Gold-Angluin language identification in the limit
Closest papers by meaning
- Generating in the Limit with Infinitely Many Hallucinations (NeurIPS 2026 · similarity 0.91)
- On Union-Closedness of Language Generation (NeurIPS 2025 · similarity 0.87)
- Representative Language Generation (ICML 2025 · similarity 0.85)
- Agnostic Language Identification and Generation (NeurIPS 2026 · similarity 0.84)
- A Theory of Time-Sensitive Language Generation: Sparse Hallucination Beats Mode Collapse (NeurIPS 2026 · similarity 0.84)
- Language Modeling by Language Models (NeurIPS 2025 · similarity 0.83)
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