Dense Language Generation Made Simple: Deterministic, Randomized, and Multi-Order Algorithms

Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples. In this model, an adversary chooses an unknown language from a countable family and enumerates its elements in an arbitrary order, while the generator must eventually output only elements of the language that have not yet appeared in the enumeration. Reliable generation is thus formalized through two eventual guarantees: validity and novelty relative to the observed data. To further quantify the breadth of the generator's outputs, Kleinberg and Wei (FOCS 2025, STOC 2026) introduced lower density as a measure of output coverage. Given an order representing the importance or relevance of possible outputs, lower density is the asymptotic lower bound, as $n$ grows, on the fraction of the first $n$ elements of the target language that the generator outputs before they appear in the data. Kleinberg and Wei showed that $1/2$ is the optimal lower-density guarantee for deterministic algorithms. We develop a simple and unified framework for obtaining optimal lower-density guarantees. We first give a deterministic algorithm that recovers the optimal guarantee of $1/2$ with a significantly simpler analysis than prior work. We then demonstrate the flexibility of our framework through two extensions. First, against an oblivious adversary, randomization raises the optimal guarantee to $1-1/e$. Second, for any finite collection of orders, the optimal deterministic and randomized guarantees can be achieved simultaneously with respect to every order, so accommodating multiple notions of importance or relevance entails no loss in the optimal guarantee.

What drives performance in molecular MPNNs? An operator-level factorial benchmark

Message-passing neural networks (MPNNs) are widely used for molecular property prediction, but their deployment as monolithic architectures makes it difficult to identify how specific message-passing operators affect performance. We present an operator-level factorial benchmark that decomposes 2D molecular MPNNs into the three families of message-seed initialization, node-edge fusion, and node update operators. The resulting 84 configurations are benchmarked on ten MoleculeNet datasets under a shared experimental setup and statistical analysis protocol. Across this controlled design, performance variation is associated primarily with message construction rather than update complexity. Message-seed initialization shows significant family-level effects for both regression and classification, node-edge fusion shows a significant family-level effect for regression with descriptive advantages for concatenation-based mixing, and the update family shows no statistically supported effect for either endpoint family. A representation probe into the Quinethazone molecule further demonstrates that concatenation-based mixing can better differentiate chemically distinct heteroatoms and withstand oversmoothing than Hadamard gating. Representative configurations selected separately for classification and regression recover competitive performance relative to established molecular graph neural network (GNN) baselines, ranking numerically best on eight of ten benchmark datasets. These empirical results are interpreted through concise mechanistic analyses of representative node-edge fusion and update operators. Our findings provide empirical design heuristics for molecular MPNNs by turning model design from a search over monolithic architectures into a targeted assessment of where and how chemical information enters the message-passing pipeline.