# On Language Generation in the Limit with Bounded Memory

arXiv: https://arxiv.org/abs/2605.30324 · HTML: https://mlsanity.com/paper/2605.30324/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 37 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 21 of 21)

- [Language Generation in the Limit](https://mlsanity.com/paper/2404.06757/) (2024)
- [Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning](https://mlsanity.com/paper/1602.05161/) (2016)
- [Generation through the lens of learning theory](https://mlsanity.com/paper/2410.13714/) (2024)
- [Memory-Sample Tradeoffs for Linear Regression with Small Error](https://mlsanity.com/paper/1904.08544/) (2019)
- [Density Measures for Language Generation](https://mlsanity.com/paper/2504.14370/) (2025)
- [Memory Bounds for the Experts Problem](https://mlsanity.com/paper/2204.09837/) (2022)
- [Generation from Noisy Examples](https://mlsanity.com/paper/2501.04179/) (2025)
- [Representative Language Generation](https://mlsanity.com/paper/2505.21819/) (2025)
- [On Union-Closedness of Language Generation](https://mlsanity.com/paper/2506.18642/) (2025)
- [Language Generation in the Limit: Noise, Loss, and Feedback](https://mlsanity.com/paper/2507.15319/) (2025)
- [Near Optimal Memory-Regret Tradeoff for Online Learning](https://mlsanity.com/paper/2303.01673/) (2023)
- [(Im)possibility of Automated Hallucination Detection in Large Language Models](https://mlsanity.com/paper/2504.17004/) (2025)
- [On Characterizations for Language Generation: Interplay of Hallucinations, Breadth, and Stability](https://mlsanity.com/paper/2412.18530/) (2024)
- [Language Generation and Identification From Partial Enumeration: Tight Density Bounds and Topological Characterizations](https://mlsanity.com/paper/2511.05295/) (2025)
- [Pareto-optimal Non-uniform Language Generation](https://mlsanity.com/paper/2510.02795/) (2025)
- [Language Generation with Infinite Contamination](https://mlsanity.com/paper/2511.07417/) (2025)
- [A Characterization of List Language Identification in the Limit](https://mlsanity.com/paper/2511.04103/) (2025)
- [Language Generation: Complexity Barriers and Implications for Learning](https://mlsanity.com/paper/2511.05759/) (2025)
- [Safe Language Generation in the Limit](https://mlsanity.com/paper/2601.08648/) (2026)
- [Bounded Memory Active Learning through Enriched Queries](https://mlsanity.com/paper/2102.05047/) (2021)
- [Learning Languages in the Limit from Positive Information with Finitely Many Memory Changes](https://mlsanity.com/paper/2010.04782/) (2020)

## Built on by (most cited first, 3 of 3)

- [Flood and Harvest: The Provable Necessity of Trivia for Generating Valuable Mathematics via the Lens of Language Generation in the Limit](https://mlsanity.com/paper/2606.14688/) (2026)
- [Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarchy](https://mlsanity.com/paper/2609.10525/) (2026)
- [GenLimitLib: A Formal Library for Language Generation in the Limit and AI-Assisted Mathematical Research](https://mlsanity.com/paper/2609.36663/) (2026)

## Abstract (arXiv)

We study language generation in the limit under bounded memory. In this task, a learner observes examples from an unknown target language one at a time and must eventually output only new valid examples. Prior work assumes access to the entire history, a strong assumption since realistic algorithms retain limited past information. Classical work in learning theory shows memory constraints dramatically alter learnability; we extend this to language generation. First, we study memoryless generators. Under a mild enumeration restriction, every countable collection of infinite languages remains generable without memory. Without this restriction, we exactly characterize when memoryless generation is possible. For finite collections, we characterize the optimal minimax density achievable by memoryless generators -- the best density guaranteed against any collection of a given size. This combinatorial bound relies on Sperner's theorem and symmetric chain decompositions. We further show that a sliding window of the last W examples does not improve this worst-case density, whereas allowing it to store b adaptively chosen past examples improves the achievable density for every b ≥ 1. Finally, we revisit identification in the limit, where the learner must converge to a single correct hypothesis for the target language. We focus on its incremental variant, where the learner remembers only its previous guess. Here, although exact identification fails on a collection of just three languages, a mild relaxation requiring convergence to an ``approximate'' version of the target is achievable for every finite collection. These results show bounded memory affects these tasks differently: generation remains achievable for every countable collection, while density and identification are confined to finite collections, with guarantees weakening as the collection grows.