偏序上上下文学习的可识别性与序维极限

中文摘要 AI 辅助

上下文学习通常被形式化为从函数示例中进行推理，而偏序结合了传递性、反对称性与不可比性，因此有限的提示可能无法确定所查询的比较关系。我们建立了偏序上上下文学习的理论，该理论将逻辑可识别性、提示教学成本、结构复杂性与形式坐标解码器类的精确容量分离开来。版本空间语义明确了背景知识以及开放世界与封闭世界假设。对于包含正、负比较的有限开放世界提示，我们证明了精确的完备三分法：在对正演示进行自反传递闭包处理后，查询要么被强制为真，要么因每个真完备都会产生循环或违反负演示而被强制为假，要么保持真正的歧义性。对于已知的n元论域，我们将开放世界教学数表征为覆盖数加阻断集击中数，证明其在所有n元偏序上的最大值为n(n-1)，且仅由反链达到，并确定阻断项是开放世界而非完全哈塞语义的精确成本。我们形式化了依赖提示的s坐标解码器，并利用经典的坐标-序等价性得到精确表示边界：维度至多为s是必要且充分条件，而宽度至多为s是便利的充分条件。

英文摘要

In-context learning is commonly formalized as inference from examples of a function. Partial orders instead combine transitivity, antisymmetry, and incomparability, so a finite prompt may not determine a queried comparison. We develop a theory of in-context learning on partial orders that separates logical identifiability, prompt teaching cost, structural complexity, and the exact capacity of a formal coordinate-decoder class. A version-space semantics makes background knowledge and open- versus closed-world assumptions explicit. For finite open-world prompts with positive and negative comparisons, we prove an exact completion trichotomy: after taking the reflexive transitive closure of the positive demonstrations, a query is forced true, forced false because every true completion creates a cycle or violates a negative demonstration, or remains genuinely ambiguous. For a known $n$-element universe, we characterize the open-world teaching number as the number of covers plus a blocker-set hitting number, prove that its maximum over all $n$-element posets is $n(n-1)$ and is uniquely attained by the antichain, and identify the blocker term as the exact cost of open-world rather than complete-Hasse semantics. We formalize prompt-dependent $s$-coordinate decoders and use the classical coordinate-order equivalence to obtain an exact representation boundary: dimension at most $s$ is necessary and sufficient, while width at most $s$ is a convenient sufficient condition.

发表机构

• Indian Statistical Institute（印度统计研究所）

机构由 AI 辅助整理，请以论文原文为准。