Entropy of Ukrainian

Proving Shannon-type entropy inequalities is a fundamental task in information theory that often requires constructing non-trivial linear combinations of known constraints, which is a combinatorial search problem that scales poorly with the number of random variables. We investigate whether small-scale large language models (0.6B--1.7B parameters), fine-tuned on atomic proof steps and combined with guided beam search, can automate this process. On a held-out test set of 60 inequalities spanning n=10 to 15 variables, our 0.6B fine-tuned model achieves an 85% proof success rate with tree search. GPT-5.5 solves 1.7% samples under zero-shot prompting while Psitip solves 33.3% samples. A systematic ablation study across training context length (4096 vs.\ 8192 tokens) and data distribution (n=9-skewed vs not skewed) reveals that a 4096-token not skewed training distribution yields the best performance, with extended context and skewed data providing no marginal benefit. We further identify two dominant failure modes -- format failures and step quality degradation -- and verify that the beam-scoring heuristic is essential via a controlled ablation (random scoring reduces success from 83% to 23%).

Tokenizer fertility the number of tokens per word imposes a hidden cost on non-English NLP. We measure fertility for ten foundation models across 25 European languages on parallel text, producing the first controlled tokenizer tax map for the continent. The tax spans 2.5x from English (1.2 tokens/word) to Greek/Maltese (~3.1), following a clear hierarchy: Romance (1.5-1.7), Germanic (1.7-1.9), Slavic (2.2-2.5), Uralic/Baltic (2.7-3.0). Ukrainian (2.7) pays 15-18% more than cognate Slavic languages, reflecting underrepresentation in pre-training data. Fertility rankings are domain-invariant across three text registers (rho > 0.97). A subword analysis reveals that high-fertility tokenizers fragment morphological boundaries rather than preserving them. Cross-lingual few-shot evaluation on four Slavic languages shows that few-shot effects are model-intrinsic, not language-dependent. We release all measurements as a public dataset.

Fine-tuned Large Language Models (LLMs) dominate in Ukrainian grammatical error correction (GEC), while API-accessed LLMs remain nearly untested on minimal-edit benchmarks. We evaluate 11 commercial LLMs from four providers and one open-source Ukrainian model on the UNLP 2023 GEC-only benchmark, comparing zero-shot, few-shot, minimal-edits, and LLM-assisted prompt optimization strategies. Our best configuration (Gemini 3.1-Pro) reaches F0.5=69.22, closing over 90% of the gap to fine-tuned SOTA (F0.5=73.14). For zero-shot prompts, only Claude models benefit from Ukrainian instructions. However, the best overall results for all models use Ukrainian minimal-edits prompts, whose language-specific rules require Ukrainian to express precisely. LLM-assisted prompt optimization on top of minimal-edits + few-shot achieves the highest score. Detailed minimal-edits instructions yield the largest gains for punctuation and case errors but cause the model to abandon several low-frequency categories. Delving into error analysis, we identify five recurring overcorrection patterns tied to Ukrainian-specific linguistic phenomena. Code, prompts, and outputs are publicly available.