Understanding as No-Arbitrage: Bounded Dutch Books as a Definition and Training Objective for Language Models

תקציר מקורי באנגליתarXiv:2609.39341v1 Announce Type: cross Abstract: Does a language model merely predict tokens, or does it understand what it says? We make this question measurable by defining "understanding" through the lens of no-arbitrage. A model understands a vocabulary to a certain degree if a computationally bounded trader cannot extract guaranteed profit by betting against the model's probabilities on logically related claims (a "Dutch book"). We establish three theoretical results: first, because full logical coherence is computationally intractable, understanding is inherently graded, not absolute. Second, we prove that the exact optimum of standard next-token prediction is inherently incoherent across different question formats; the flaw lies in the training objective, not the architecture. Thir