Language Generation in the Limit
language-generation-in-the-limit-ca60d3c7·2 events·first seen 19d agoAliases: Language Generation in the Limit
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Language Generation in the Limit with Bounded Memory: Characterization via Sperner's Theorem
This paper studies language generation in the limit under bounded memory constraints, extending classical learning theory to the generation setting. The authors characterize when memoryless generation is possible, derive minimax density bounds using Sperner's theorem and symmetric chain decompositions, and show that adaptively chosen memory outperforms sliding-window memory. They also revisit incremental identification in the limit, finding that exact identification fails for collections of three or more languages but an approximate relaxation is achievable for all finite collections.
Formal theory shows infinite trivial output is provably necessary for AI systems generating valuable mathematics
A new arXiv paper models AI-assisted formal mathematics generation as a nested language-generation-in-the-limit problem, using a proof checker as a membership oracle and an adversarial enumeration of the mathematical literature as the signal for 'valuable' content. The authors prove a sharp dichotomy: generators emitting only finitely many trivial (correct but worthless) statements achieve at most α/2 coverage of unseen valuable mathematics, while allowing an infinite (but asymptotically vanishing) stream of trivia raises the optimum to 1−α/2. The central result is that a perfect verifier cannot substitute for mathematical taste, and the flood of certified-but-trivial output from AI proof systems is a provable mathematical necessity, not an engineering failure. The work formalizes the gap between formal verifiability and mathematical value, which is increasingly the binding constraint as AI-proof-assistant systems scale.