aiqbench analysis
This is a theory preprint on learning formal languages from positive examples only. It argues that a contrastive scorer trained on self-generated negatives can provably rank in-language strings above perturbations of them, with polynomial sample complexity under a faithfulness condition. For ML readers, the useful pieces are the negative-sampling design that avoids a global normalizer and the Metropolis-Hastings generator built from the same scorer. The guarantees rest on assumptions about the operator and scorer, and the abstract reports no experiments, so the practical reach is still unclear.
A Reverse-Gold Theorem: Positive-Data Identification by Substrate-Bounded Contrastive Learners
arXiv:2610.28747v1 Announce Type: cross Abstract: Gold's theorem shows that no learner can identify an arbitrary language in a superfinite class from positive presentations alone. We prove a positive result of a different shape. Fix an innate \emph{recursive contrast operator} $R$ mapping any string to a finite set of structurally bounded perturbations of it, and a discriminative scorer of bounded capacity trained by InfoNCE-style contrastive estimation \citep{vandenoord2018,macollins2018} on self-generated perturbations. Under a faithfulness condition relating operator, scorer, and target, the trained scorer achieves \emph{PAC conditional identification} from positive data alone: with high probability it ranks all but an $\varepsilon$-fraction of positives above every out-of-language perturbation in their own orbit, with sample complexity polynomial in the capacity, the orbit size, the score bound, and $1/\varepsilon$. The negatives are constructed by the learner, not supplied by an oracle, and no global normalizing constant over strings is ever computed. The misranking bound follows from the geometry of the contrastive loss---per-example loss floors plus a definite loss increment under misranking---yielding an $\varepsilon^{-2}$ rate, together with an explicit approximation gap that quantifies the cost of in-language (``colliding'') perturbations and vanishes for orbit-pure operators. The ranking guarantee lifts to a global membership classifier under a margin condition, and the same operator composed with the learned scorer is a normalization-free Metropolis--Hastings generator. We instantiate the hypotheses for substitutable context-free languages \citep{clarkeyraud2007} and the subregular classes \citep{heinz2010,hayashikamigaito2025}, and state an open conjecture locating the construction in the mildly context-sensitive hierarchy.
Abstract:Gold's theorem shows that no learner can identify an arbitrary language in a superfinite class from positive presentations alone. We prove a positive result of a different shape. Fix an innate \emph{recursive contrast operator} $R$ mapping any string to a finite set of structurally bounded perturbations of it, and a discriminative scorer of bounded capacity trained by InfoNCE-style contrastive estimation \citep{vandenoord2018,macollins2018} on self-generated perturbations. Under a faithfulness condition relating operator, scorer, and target, the trained scorer achieves \emph{PAC conditional identification} from positive data alone: with high probability it ranks all but an $\varepsilon$-fraction of positives above every out-of-language perturbation in their own orbit, with sample complexity polynomial in the capacity, the orbit size, the score bound, and $1/\varepsilon$. The negatives are constructed by the learner, not supplied by an oracle, and no global normalizing constant over strings is ever computed. The misranking bound follows from the geometry of the contrastive loss---per-example loss floors plus a definite loss increment under misranking---yielding an $\varepsilon^{-2}$ rate, together with an explicit approximation gap that quantifies the cost of in-language (``colliding'') perturbations and vanishes for orbit-pure operators. The ranking guarantee lifts to a global membership classifier under a margin condition, and the same operator composed with the learned scorer is a normalization-free Metropolis--Hastings generator. We instantiate the hypotheses for substitutable context-free languages \citep{clarkeyraud2007} and the subregular classes \citep{heinz2010,hayashikamigaito2025}, and state an open conjecture locating the construction in the mildly context-sensitive hierarchy.
Machine Learning (stat.ML); Computation and Language (cs.CL); Machine Learning (cs.LG); Neurons and Cognition (q-bio.NC)
arXiv:2610.28747 [stat.ML]
(or arXiv:2610.28747v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.28747 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 341 words