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arxiv: 2605.30324 · v1 · pith:R6W6DF6Wnew · submitted 2026-05-28 · 💻 cs.DS · cs.AI· cs.CL· cs.LG· stat.ML

On Language Generation in the Limit with Bounded Memory

Jon Kleinberg , Anay Mehrotra , Amin Saberi , Grigoris Velegkas This is my paper

Pith reviewed 2026-06-28 23:54 UTC · model grok-4.3

classification 💻 cs.DS cs.AIcs.CLcs.LGstat.ML
keywords language generationbounded memoryidentification in the limitminimax densitySperner's theoremenumeration restrictioninfinite languages
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The pith

Every countable collection of infinite languages can be generated without memory under a mild enumeration restriction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how bounded memory affects language generation in the limit. It establishes that under a mild enumeration restriction, memoryless generators can handle every countable collection of infinite languages. Without the restriction, it provides an exact characterization of when memoryless generation is possible. For finite collections, it determines the optimal minimax density using combinatorial arguments from Sperner's theorem. Adaptive storage of past examples improves density, unlike fixed windows, and approximate identification works for finite collections while exact does not.

Core claim

Under a mild enumeration restriction, every countable collection of infinite languages remains generable without memory. Without this restriction, memoryless generation is possible only for collections meeting specific conditions. For finite collections, the optimal minimax density achievable by memoryless generators is characterized via Sperner's theorem and symmetric chain decompositions. A sliding window of fixed size does not improve the worst-case density, but storing b adaptively chosen past examples does improve it for every b at least 1. For incremental identification in the limit, exact identification fails on collections of three languages, but a mild relaxation to approximate vers

What carries the argument

The mild enumeration restriction on the collection of languages that permits memoryless generation for any countable collection of infinite languages.

If this is right

  • Memoryless generation works for all countable collections under the restriction.
  • Optimal minimax density for finite collections is characterized via Sperner's theorem and symmetric chain decompositions.
  • Storing b adaptively chosen past examples improves density for every b >= 1.
  • Approximate identification succeeds for all finite collections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Generation may be more resilient to memory limits than identification for infinite collections.
  • The contrast between tasks points to the need for task-specific memory strategies in learning algorithms.
  • Collections violating the enumeration restriction likely require some memory for generation.

Load-bearing premise

The collection of languages must satisfy a mild enumeration restriction for the result that every countable collection is generable without memory to hold.

What would settle it

A countable collection of infinite languages violating the mild enumeration restriction for which no memoryless generator exists, or one satisfying it that still cannot be generated memorylessly.

read the original abstract

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 $b \geq 1$. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies language generation in the limit under bounded memory. It proves that under a mild enumeration restriction every countable collection of infinite languages admits a memoryless generator; without the restriction it gives an exact characterization of memoryless generation. For finite collections it derives the optimal minimax density via Sperner's theorem and symmetric chain decompositions. It shows that a sliding window of fixed width W yields no improvement over the memoryless bound while adaptive storage of b past examples improves the density for every b≥1. For incremental identification in the limit, exact identification fails already on collections of size three, but an approximate variant succeeds for every finite collection.

Significance. If the derivations hold, the work extends classical memory-constrained learnability results to generation tasks and demonstrates that generation remains feasible for countable collections while identification and density guarantees are limited to finite collections. The exact characterizations, the parameter-free Sperner-based minimax densities, and the separation between sliding-window and adaptive memory models are concrete strengths that supply falsifiable combinatorial predictions.

major comments (2)
  1. [§3] §3 (memoryless generation under the mild enumeration restriction): the central positive result for countable collections is stated to require the restriction, yet no counter-example collection violating the restriction is exhibited to show necessity; without such an example the precise scope of the 'mild' condition remains load-bearing for the claim that the result covers 'every countable collection'.
  2. [§5.1] §5.1 (adaptive storage of b examples): the statement that density improves for every b≥1 is load-bearing for the memory-model comparison, but the quantitative improvement (or the new minimax density as a function of b) is not stated explicitly, making it impossible to verify whether the improvement is strict or merely non-decreasing.
minor comments (3)
  1. [identification section] The definition of 'approximate' identification (the relaxation used for the positive finite-collection result) should be stated formally with the same precision as the exact-identification definition, preferably in a dedicated subsection.
  2. Notation for the density function (e.g., the mapping from collection size to minimax density) is introduced without a single consolidated table or equation; a summary table would improve readability.
  3. [§5] The abstract claims the sliding-window result follows from the memoryless case, but the manuscript should add a one-sentence pointer to the precise lemma that shows no additional information is gained from the window.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments. We address each major comment below and will make the indicated revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (memoryless generation under the mild enumeration restriction): the central positive result for countable collections is stated to require the restriction, yet no counter-example collection violating the restriction is exhibited to show necessity; without such an example the precise scope of the 'mild' condition remains load-bearing for the claim that the result covers 'every countable collection'.

    Authors: The manuscript already supplies an exact characterization of memoryless generation in the unrestricted case. This characterization directly identifies the collections for which memoryless generation is impossible and thereby establishes necessity of the enumeration restriction for the 'every countable collection' statement. To make the necessity concrete and ease verification, we will add an explicit counterexample collection that violates the mild enumeration restriction in the revised manuscript. revision: yes

  2. Referee: [§5.1] §5.1 (adaptive storage of b examples): the statement that density improves for every b≥1 is load-bearing for the memory-model comparison, but the quantitative improvement (or the new minimax density as a function of b) is not stated explicitly, making it impossible to verify whether the improvement is strict or merely non-decreasing.

    Authors: The proof in §5.1 constructs, for each b≥1, an adaptive strategy whose worst-case density strictly exceeds the memoryless minimax density obtained from Sperner's theorem. While the explicit functional dependence on b is not written out, the argument shows the improvement is strict rather than merely non-decreasing. We will state the improved minimax density explicitly as a function of b in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript presents positive results on memoryless generation under an enumeration restriction, an exact characterization for the unrestricted case, Sperner-based minimax densities for finite collections, and comparisons of bounded-memory variants, all grounded in external combinatorial facts (Sperner's theorem, symmetric chain decompositions) rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equation or claim in the provided abstract or structure reduces by construction to its own inputs; the derivation remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the results rest on standard set-theoretic and combinatorial background plus the stated enumeration restriction.

axioms (1)
  • domain assumption Target languages are infinite
    Invoked for the memoryless generation claim on countable collections.

pith-pipeline@v0.9.1-grok · 5844 in / 1141 out tokens · 27905 ms · 2026-06-28T23:54:43.347075+00:00 · methodology

discussion (0)

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Works this paper leans on

14 extracted references · 11 canonical work pages · 1 internal anchor

  1. [1]

    08648(cit

    arXiv: 2601.08648 [cs.CL].URL: https://arxiv.org/abs/2601. 08648(cit. on p. 10). [ABCK25] Marcelo Arenas, Pablo Barceló, Luis Cofré, and Alexander Kozachinskiy.Language Generation: Complexity Barriers and Implications for Learning

    work page internal anchor Pith review arXiv

  2. [2]

    Finding Patterns Common to a Set of Strings (Extended Abstract)

    arXiv: 2511.05759 [cs.CL].URL: https: //arxiv.org/abs/2511.05759(cit. on p. 10). [Ang79] Dana Angluin. “Finding Patterns Common to a Set of Strings (Extended Abstract)”. In:Proceed- ings of the Eleventh Annual ACM Symposium on Theory of Computing. STOC ’79. Atlanta, Georgia, USA: Association for Computing Machinery, 1979, pp. 130–141.ISBN: 9781450374385.D...

    work page doi:10.1145/800135.804406.url:https://doi.org/10.1145/800135.804406(cit 1979

  3. [3]

    Pareto-optimal Non-uniform Language Generation

    Proceedings of Machine Learning Research. PMLR, 2025, pp. 854–887.URL: https://proceedings.mlr.press/v291/charikar25a.html (cit. on pp. 1, 2, 9, 10, 12, 33). [CP26] Moses Charikar and Chirag Pabbaraju. “Pareto-optimal Non-uniform Language Generation”. In:Proceedings of the 37th International Conference on Algorithmic Learning Theory. ALT

    2025

  4. [4]

    A Characterization of List Language Identification in the Limit

    URL:https://arxiv.org/abs/2510.02795(cit. on p. 10). [CPT25] Moses Charikar, Chirag Pabbaraju, and Ambuj Tewari. “A Characterization of List Language Identification in the Limit”. In:arXiv preprint arXiv:2511.04103(2025) (cit. on p. 10). [Eng97] Konrad Engel.Sperner theory. Vol

    work page arXiv 2025

  5. [5]

    Space-Efficient Language Generation in the Limit

    Cambridge University Press, 1997 (cit. on p. 21). [FPP+26] Nicolas Flammarion, Chirag Pabbaraju, Hristo Papazov, Miltiadis Stouras, and Ola Svensson. “Space-Efficient Language Generation in the Limit”. In:Proceedings of the 39th Annual Conference on Learning Theory. Ed. by Steve Hanneke and Tor Lattimore. Proceedings of Machine Learning Research. To appea...

    work page doi:10.1016/s0019-9958(67)91165-5(cit 1997

  6. [6]

    On Union-Closedness of Language Generation

    Proceedings of Machine Learning Research. PMLR, 2021, pp. 2358–2387 (cit. on p. 1). [HKMV25] Steve Hanneke, Amin Karbasi, Anay Mehrotra, and Grigoris Velegkas. “On Union-Closedness of Language Generation”. In:Advances in Neural Information Processing Systems. Vol

    2021

  7. [7]

    Incremental Learning with Temporary Memory

    URL:https://openreview.net/forum?id=6h7HLx1kbH(cit. on p. 10). [JLMZ10] Sanjay Jain, Steffen Lange, Samuel E. Moelius, and Sandra Zilles. “Incremental Learning with Temporary Memory”. In:Theoretical Computer Science411.31–33 (2010), pp. 2757–2772.DOI: 10.1016/j.tcs.2010.03.029(cit. on pp. 10, 11). [KM24] Jon Kleinberg and Sendhil Mullainathan. “Language G...

    work page doi:10.1016/j.tcs.2010.03.029(cit 2010

  8. [8]

    Advances in Neural Information Processing Systems , volume =

    2024.URL: https://arxiv.org/abs/2404.06757 (cit. on pp. 1, 2, 4, 6, 10, 12, 13, 35). [KMSV25] Amin Karbasi, Omar Montasser, John Sous, and Grigoris Velegkas. “(Im)possibility of Auto- mated Hallucination Detection in Large Language Models”. In:Second Conference on Language Modeling. 2025.URL:https://openreview.net/forum?id=e5jWdZIX0Q(cit. on p. 10). [KMV2...

    work page arXiv 2024

  9. [9]

    Iterative Learning from Positive Data and Counters

    2026.URL: https://arxiv. org/abs/2412.18530(cit. on pp. 1, 2, 9, 10, 12, 33). [Köt14] Timo Kötzing. “Iterative Learning from Positive Data and Counters”. In:Theoretical Computer Science519 (2014), pp. 155–169.DOI:10.1016/j.tcs.2013.09.023(cit. on p. 11). 37 [KS21] Timo Kötzing and Karen Seidel. “Learning Languages in the Limit from Positive Information wi...

    work page doi:10.1016/j.tcs.2013.09.023(cit 2026

  10. [10]

    2023 , volume =

    arXiv: 2511.07417 [stat.ML].URL: https://arxiv.org/abs/ 2511.07417(cit. on pp. 2, 10). [PR23] Binghui Peng and Aviad Rubinstein. “Near Optimal Memory-Regret Tradeoff for Online Learning”. In:Proceedings of the 64th Annual IEEE Symposium on Foundations of Computer Science (FOCS). 2023, pp. 1171–1194.DOI:10.1109/FOCS57990.2023.00069(cit. on p. 1). [PRR25] C...

    work page doi:10.1109/focs57990.2023.00069(cit 2023

  11. [11]

    Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning

    Proceedings of Machine Learning Research. PMLR, 2025, pp. 48518–48541.URL: https://proceedings.mlr.press/v267/peale25a.html (cit. on p. 10). [Raz16] Ran Raz. “Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning”. In:Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS). 2016, pp. 266–275.DOI:1...

    work page doi:10.1109/focs.2016.36(cit 2025

  12. [12]

    Generation from Noisy Examples

    Proceedings of Machine Learning Research. PMLR, 2025, pp. 4740–4776.URL: https://proceedings.mlr.press/v291/raman25a.html (cit. on pp. 10, 12). 38 [RR25] Ananth Raman and Vinod Raman. “Generation from Noisy Examples”. In:Proceedings of the 42nd International Conference on Machine Learning(July 13–19, 2025). Ed. by Aarti Singh, Maryam Fazel, Daniel Hsu, Si...

    2025

  13. [13]

    Memory-Sample Tradeoffs for Linear Regression with Small Error

    Proceedings of Machine Learning Research. PMLR, 2025, pp. 51079–51093. URL:https://proceedings.mlr.press/v267/raman25a.html(cit. on p. 10). [SSV19] Vatsal Sharan, Aaron Sidford, and Gregory Valiant. “Memory-Sample Tradeoffs for Linear Regression with Small Error”. In:Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. 2019, pp. 890...

    work page doi:10.1145/3313276.3316403(cit 2025

  14. [14]

    Hardness of approximation in p via short cycle removal: cycle detection, distance oracles, and beyond , booktitle =

    Proceedings of Machine Learning Research. PMLR, 2016, pp. 1490–1516 (cit. on p. 1). [SWXZ22] Vaidehi Srinivas, David P . Woodruff, Ziyu Xu, and Samson Zhou. “Memory Bounds for the Experts Problem”. In:Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. 2022, pp. 1158–1171.DOI:10.1145/3519935.3520069(cit. on p. 1). 39 A Further Resu...

    work page doi:10.1145/3519935.3520069(cit 2016