On Language Generation in the Limit with Bounded Memory

Problem
This paper addresses the gap in understanding language generation under bounded memory constraints, a scenario that has not been thoroughly explored in existing literature. Prior research typically assumes that learners have access to the entire history of examples, which is unrealistic for practical algorithms that operate with limited memory. The authors aim to extend classical learning theory insights to the domain of language generation, particularly focusing on the implications of memory constraints on learnability and generation capabilities. This work is presented as a preprint and has not yet undergone peer review.

Method
The authors investigate two primary scenarios: memoryless generators and generators with bounded memory. For memoryless generators, they establish conditions under which infinite languages can be generated without memory, introducing a mild enumeration restriction. They characterize the optimal minimax density achievable by memoryless generators for finite collections, leveraging combinatorial principles such as Sperner’s theorem and symmetric chain decompositions. The study also examines the impact of a sliding window of the last ( W ) examples, concluding that it does not enhance worst-case density. However, allowing the storage of ( b ) adaptively chosen past examples improves density for any ( b \geq 1 ). Additionally, the paper revisits the concept of identification in the limit, particularly in its incremental form, where the learner retains only its previous guess. The authors demonstrate that while exact identification fails for three languages, a relaxed criterion for approximate convergence is achievable for any finite collection.

Results
The findings indicate that memoryless generation is feasible for every countable collection of languages under specific conditions, while the achievable density for finite collections is characterized by a minimax bound. The authors report that the worst-case density does not improve with a sliding window but does improve with adaptive memory storage. In terms of identification, the results show that exact identification is not possible for three languages, but approximate identification can be achieved for any finite collection. These results highlight the nuanced effects of memory constraints on language generation and identification tasks.

Limitations
The authors acknowledge that their results are confined to finite collections for density and identification tasks, with guarantees diminishing as the collection size increases. They do not address the implications of varying memory sizes beyond the specified bounds or the potential impact of different types of memory architectures. Additionally, the practical applicability of their theoretical findings in real-world language generation systems remains to be explored.

Why it matters
This work has significant implications for the design of language generation systems that operate under memory constraints, which is a common scenario in real-world applications. By elucidating the effects of bounded memory on generation and identification tasks, the findings can inform the development of more efficient algorithms that can operate with limited historical context. This research could pave the way for advancements in areas such as natural language processing, where memory efficiency is crucial for scalability and performance.

Authors: Jon Kleinberg, Anay Mehrotra, Amin Saberi, Grigoris Velegkas
Source: arXiv:2605.30324
URL: https://arxiv.org/abs/2605.30324v1