The paper studies the problem of language generation in the limit, where a learner must produce valid unseen elements from any exhaustive positive presentation of an unknown infinite language. It establishes that generation is possible precisely when each target language admits a finite positive witness such that all targets activated by any finite sample share an infinite common intersection. The authors introduce a separation-width hierarchy to measure the size of compatible witnesses, showing that every level of the hierarchy occurs and that countable families admit singleton witnesses while more complex families require unbounded finite witnesses. The results are formalized and verified in Lean, with the development available on GitHub.
arXiv:2607. 23361v1 Announce Type: cross Abstract: Language generation in the limit is an elegant model introduced by Kleinberg and Mullainathan [KM24] to formally study language generation by an algorithm that learns solely based on example strings.
arXiv:2608. 01320v1 Announce Type: cross Abstract: Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples.
arXiv:2606. 25777v1 Announce Type: cross Abstract: We initiate a resource-aware theory of \textit{language generation in the limit} under the minimal constraint of space efficiency.
arXiv:2601. 21237v2 Announce Type: replace-cross Abstract: Kleinberg and Mullainathan recently proposed a formal framework for studying the phenomenon of language generation, called language generation in the limit.
arXiv:2605. 11644v3 Announce Type: replace-cross Abstract: Positive data can show that two tuple occurrences share a successful sentence context without certifying that they are safely interchangeable.
arXiv:2609.36663v1 Announce Type: new Abstract: We present GenLimitLib, a source-aligned Lean 4 library for language generation in the limit. Introduced by Kleinberg and Mullainathan at NeurIPS 2024,...
arXiv:2606. 14688v1 Announce Type: cross Abstract: AI systems coupled to proof assistants now generate formal mathematics at scale, and the gap between what a checker can verify and what a mathematician would value has become the binding constraint.
arXiv:2606. 07623v1 Announce Type: new Abstract: This paper develops a model-theoretic framework for verifying context-conditioned language-model behavior by replacing benchmark labels with finite semantic certificates.
The paper reports on a large‑scale verified search experiment using a 30B language model on a laptop, evaluating three operator packages—schematic notebooks, named obstacles, and behavioural repulsion—in a factorial design across nine construction problems. Results show that the full composition of operators closes the seed‑to‑record gap more effectively than any single component, increases construction‑hash diversity, and that memory plus repulsion consistently avoids collapse. A frontier proposer achieves similar gains in far fewer samples, but the search ultimately stalls near a plateau where the reference family is adopted and optimized only when provided as code.
arXiv:2606. 19788v1 Announce Type: new Abstract: We present CombEval, a dynamic benchmark for evaluating combinatorial counting in large language models.
arXiv:2607. 11606v1 Announce Type: cross Abstract: We study how little extra information is needed to make adversarial language learning possible.
arXiv:2608. 15979v1 Announce Type: new Abstract: Large language models produce outputs presented as discoveries - new proofs, conjectures, or molecules.