推翻了生成与符号AI中确定性与泛化范围的普适权衡假说
No Universal Hyperbola: A Formal Disproof of the Epistemic Trade-Off Between Certainty and Scope in Symbolic and Generative AI
- 用编码理论和算法信息论证明该权衡关系在标准定义下自相矛盾
- 构造反例证伪了基于普通柯尔莫哥洛夫复杂度的版本
- 连熵重构版也无法恢复普遍性，适用于逻辑与AI哲学研究者
针对近期提出的符号与生成式AI中认知确定性与作用范围之间存在普适双曲线权衡的猜想，本文通过严格的逻辑数学分析予以否定。确定性定义为输入空间上最坏情况下的正确概率，作用范围定义为输入集与输出集的柯尔莫哥洛夫复杂度之和。利用编码理论与算法信息论的基本结论，首先证明当使用前缀（自界定、前缀无歧义）柯尔莫哥洛夫复杂度时，该猜想导致内部矛盾；其次，当使用普通柯尔莫哥洛夫复杂度时，可通过构造反例直接证伪。由此确立主定理：在所提定义下，不存在普遍成立的“确定性-作用范围”双曲线边界。进一步表明，后续以香农联合熵替代柯尔莫哥洛夫复杂度并重新定义认知确定性的修正版本，亦无法恢复其普遍性。
原文摘要 · Abstract (English)
In direct response to requests for a logico-mathematical test of the conjecture, we formally disprove a recently conjectured artificial intelligence trade-off between epistemic certainty and scope in its published universal hyperbolic product form, as introduced in Philosophy and Technology. Certainty is defined as the worst-case correctness probability over the input space, and scope as the sum of the Kolmogorov complexities of the input and output sets. Using standard facts from coding theory and algorithmic information theory, we show, first, that when the conjecture is instantiated with prefix (self-delimiting, prefix-free) Kolmogorov complexity, it leads to an internal inconsistency, and second, that when it is instantiated with plain Kolmogorov complexity, it is refuted by a constructive counterexample. These results establish a main theorem: contrary to the conjecture's claim, no universal "certainty-scope" hyperbola holds as a general bound under the published definitions. We further show that a subsequent "entropy-based" revision, replacing the Kolmogorov scope with Shannon joint entropy and redefining the epistemic certainty level accordingly, cannot restore universality either.
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