KnowledgeBerg: Evaluating Systematic Knowledge Coverage and Compositional Reasoning in Large Language Models

Many real-world questions appear deceptively simple yet implicitly demand two capabilities: (i) systematic coverage of a bounded knowledge universe and (ii) compositional set-based reasoning over that universe, a phenomenon we term "the tip of the iceberg." We formalize this challenge through two orthogonal dimensions: knowledge width, the cardinality of the required universe, and reasoning depth, the number of compositional set operations. We introduce KNOWLEDGEBERG, a benchmark of 4,800 multiple-choice questions derived from 1,183 enumeration seeds spanning 10 domains and 17 languages, with universes grounded in authoritative sources to ensure reproducibility. Representative open-source LLMs demonstrate severe limitations, achieving only 5.26-36.88 F1 on universe enumeration and 16.00-44.19 accuracy on knowledge-grounded reasoning. Diagnostic analyses reveal three stages of failure: completeness, or missing knowledge; awareness, or failure to identify requirements; and application, or incorrect reasoning execution. This pattern persists across languages and model scales. Although testtime compute and retrieval augmentation yield measurable gains-up to 4.35 and 3.78 points, respectively-substantial gaps remain, exposing limitations in how current LLMs organize structured knowledge and execute compositional reasoning over bounded domains. The dataset is available at https://huggingface. co/datasets/2npc/KnowledgeBerg . When looking at all constellations, which type of theme appears most often? A: Biological theme B: Mythological theme C: Instruments D: Abstract concepts (Tip) Question (Iceberg) Knowledge and Reasoning To determine which category is most prevalent, we must carefully classify all 88 IAU constellations.