Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarchy — Paper

Abstract

-cross Abstract: Language generation in the limit asks for valid unseen elements from every exhaustive positive presentation of an unknown infinite language. We characterize this task for arbitrary families over a countable universe. Generation is possible exactly when each target can be assigned a finite positive witness so that the targets activated by any finite sample have an infinite common intersection. The necessary direction follows from a universal normalization: a search through unconfirmed histories converts any successful generator into one depending only on the observed set. We then ask how large compatible witnesses must be. Positive separation width records the smallest uniform size bound, with two further levels for unbounded finite witnesses and the absence of any compatible finite-witness assignment. Every level occurs. Countable families admit singleton witnesses, explicit families realize every finite width, and a union of two families with infinite common cores requires unbounded finite witnesses. Finally, countable-support and finite-profile obstructions explain why local combinatorial data cannot determine generation in the limit. The characterization and full width hierarchy are checked in Lean, including the simplified normalization and a direct diagonal capture lemma. The accompanying Lean development is maintained at https://github.com/xiaoyulics/language-generation-characterization

Authors 4

Xiaoyu Li, Andi Han, Jiaojiao Jiang, Junbin Gao

Specification

Official page: arxiv.org/abs/2609.10525
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
Arxiv announce type: replace
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
arXiv id: 2609.10525
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
Categories: cs.FL, cs.LG
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
PDF: arxiv.org/pdf/2609.10525
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
Primary category: cs.FL
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh

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T19

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7 h ago

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Authors: Xiaoyu Li, Andi Han, Jiaojiao Jiang

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Value	Valid from → to	Status	Source	Confidence	Extractor
https://arxiv.org/pdf/2609.10525	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Claims are temporal and append-only: a new observation closes the previous claim (valid_to) instead of overwriting it. Conflicting claims from different sources are kept side by side and flagged — never averaged. Methodology →

11 Sept 2026
New paperPaperCharacterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarchy
New paper: Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarchy
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Source	Document	Type	Tier	Last observed	Snapshots
arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.LG	feed	T1· Official	5 h ago	1

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