提出物理感知学习理论，解决集合论中学习不可判定的悖论
Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints
- 用物理可实现协议替代抽象集合论学习者
- 有限精度下连续EMX问题变为可计算的可数问题
- 适用于量子、非局域等物理模型，具可判定性
超越二分类，学习能力在某些集合论模型中可能逻辑脆弱：例如在某些ZFC模型下，[0,1]的所有有限子集类是可学习的，而在另一些模型下则不是。我们认为该悖论源于操作性缺失——标准定义隐含了无限精度、非物理数据访问和不可表示输出等非现实资源。为此提出物理感知学习（PL），将学习能力定义在显式物理协议族上。有限精度粗粒化使连续EMX退化为可数问题，通过精确的前推/后推变换保持原目标，使该例在显式(ε,δ)样本复杂度下可证明可学习。对于量子数据，可接受的学习者恰好是d个副本上的POVM，将样本量转化为副本复杂度，并导出类似赫尔斯特（Helstrom）的下界。对有限无信道及量子模型，PL可行性变为线性或半定规划，因此是可判定的。
原文摘要 · Abstract (English)
Beyond binary classification, learnability can become a logically fragile notion: in EMX, even the class of all finite subsets of $[0,1]$ is learnable in some models of ZFC and not in others. We argue the paradox is operational. The standard definitions quantify over arbitrary set-theoretic learners that implicitly assume non-operational resources (infinite precision, unphysical data access, and non-representable outputs). We introduce physics-aware learnability (PL), which defines the learnability relative to an explicit access model -- a family of admissible physical protocols. Finite-precision coarse-graining reduces continuum EMX to a countable problem, via an exact pushforward/pullback reduction that preserves the EMX objective, making the independence example provably learnable with explicit $(ε,δ)$ sample complexity. For quantum data, admissible learners are exactly POVMs on $d$ copies, turning sample size into copy complexity and yielding Helstrom(-type) lower bounds. For finite no-signaling and quantum models, PL feasibility becomes linear or semidefinite and is therefore decidable.
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