发表机构
University of Crete; Institute of Computer Science, FORTH; Center for Quantum Science & Technologies, FORTH-QuTech（克里特大学; FORTH计算机研究所; FORTH-QuTech量子科学与技术中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出不确定性敏感性框架，定义成本变化与不确定性消减定理，引入参数化类 $P_U[K]$ 和 $NP_U[K]$，以解释现实世界可处理性，并支持SLA、智能体AI及量子加速。
AI 中文摘要
经典问题 $P \stackrel{?}{=} NP$ 是否是理解现实世界计算复杂性的正确问题？传统的最坏情况分析将复杂性仅表征为输入规模 $n$ 的函数。然而，那些在一般情况下成本呈指数级的任务，一旦已知特定的结构约束或部分输入，往往能以多项式甚至线性时间运行。例如，划分问题（Partition Problem）在其输入满足一个简单的结构性质时，可在线性时间内求解，随后是线性时间验证的常数时间核心步骤，尽管在一般情况下没有已知的多项式时间算法；相比之下，整数分解（Integer Factorization）则抵抗结构知识：没有已知的可检查输入性质能改进亚指数的数域筛（number-field-sieve）界限。经典最坏情况分析无法解释这种差异，因为它将整个难度谱系压缩为一个悲观的界限。本文正式发展了一个全面的参数化和不确定性感知的计算复杂性框架。我们引入了“对不确定性的敏感性”（Sensitivity to Uncertainty, SU）的概念，并形式化了在给定输入性质 $K$ 下的成本变化 $\u0394_P(n, K)$；我们定义了不确定性消失度量 $\u03A8(n, K)$，并证明了连接成本变化与输入不确定性的基本“不确定性消减定理”（Uncertainty Reduction Theorem）；我们引入了参数化类 $P_U[K]$ 和 $NP_U[K]$，作为更实用的、基于实例级别的基础，专注于可认证的结构，而非分布典型性，从而弥合了理论复杂性与现实世界可处理性之间的差距。我们进一步展示了相同的不确定性分析如何支持实践：它通过契约式设计（Design-by-Contract）实现确定性服务等级协议（SLA），为智能体AI中的工具使用和推理拆分提供信息，并解释了量子计算机何时实现指数级加速。
英文摘要
Is the classical question $P \stackrel{?}{=} NP$ the right one for understanding the complexity of real-world computation? Traditional worst-case analysis characterizes complexity solely as a function of input size $n$. Yet tasks whose cost is exponential in general often run in polynomial or even linear time once specific structural constraints or partial inputs are known. For instance, the Partition Problem is solvable in linear time, a constant-time core step following linear-time verification, when its inputs satisfy a simple structural property, although no polynomial-time algorithm is known for it in general; Integer Factorization, by contrast, resists structural knowledge: no checkable property of the input is known to improve on the sub-exponential number-field-sieve bound. Classical worst-case analysis cannot explain this difference, because it collapses a whole spectrum of difficulty into a single pessimistic bound. This paper formally develops a comprehensive parametric and uncertainty-aware framework for computational complexity. We introduce the notion of \emph{Sensitivity to Uncertainty} (SU) and formalize the cost variation $Δ_P(n, K)$ under a given input property $K$; we define the uncertainty-vanishing metric $Ψ(n, K)$ and prove a fundamental \emph{Uncertainty Reduction Theorem} bridging cost variation and input uncertainty; and we introduce the parametric classes $P_U[K]$ and $NP_U[K]$ as a more practical, instance-level foundation, focused on certifiable structure, rather than distributional typicality, that resolves the gap between theoretical complexity and real-world tractability. We further show how the same uncertainty analysis supports practice: it enables deterministic SLA via Design-by-Contract informs tool-use and reasoning splits in agentic AI, and provides an explanation of when quantum computers achieve exponential speedup.
Comments35 pages
