良定义但非预定的持久数学对象、嵌入观察者与 P = NP 的可证明性

中文摘要 AI 辅助

我们发展了一个关于“良定义性”与“预定”之间的基础性区分。一个持久的数学或计算对象可能在每次查询时返回唯一答案，并保留所有先前返回的答案，但其对尚未查询的输入的答案并不由一个完整的、与历史无关的扩展来表示。我们称这样的对象为“良定义但非预定的”。接下来，我们制定了一种基于观察者的语义。所有观察者都居住在一个共享的计算世界中，他们的交互修改一个共同的全局状态，并且他们可以访问一个已实现的历史。在共享历史、历史不可逆性、持久性和历史不可区分性的公理下，嵌入的观察者无法仅从已实现的历史中确定其兼容扩展是预定的还是逐步形成的。一个持续进化的多项式时间可计算预言机提供了一个具体例子，并在一种明确对历史敏感的语义下产生了确定性与非确定性之间的分离。然后，我们推导出一个关于 P = NP 可证明性的条件性结果。结论是一个观察者相对不可证明性定理，而不是经典 P ≠ NP 的无条件证明。

英文摘要

We develop a foundational distinction between \emph{well-definedness} and \emph{predetermination}. A persistent mathematical or computational object may return a unique answer whenever a query is made and preserve every previously returned answer, while its answers to as-yet unqueried inputs are not represented by one completed, history-independent extension. We call such an object \emph{well-defined but non-predetermined}. We next formulate an observer-based semantics. All observers inhabit one shared computational world, their interactions modify one common global state, and they have access to one realized history. Under axioms of shared history, historical irreversibility, persistence, and historical indistinguishability, embedded observers cannot establish from the realized history alone whether its compatible extension was predetermined or progressively formed. A persistently evolutionary polynomial time computable oracle supplies a concrete example and yields a deterministic--nondeterministic separation under an explicitly history-sensitive semantics. We then derive a conditional result concerning the provability of \(P=NP\). The conclusion is an observer-relative unprovability theorem, not an unconditional proof of classical \(P\neq NP\).

发表机构

• University of Fribourg（弗里堡大学）

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