有界模拟复杂度
Bounded Analog Complexity
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中文总结 AI 辅助
研究针对当前模拟复杂度理论中状态变量无界的问题，提出有界模拟复杂度理论及有界替代编译方法，证明编译后系统挂钟时间特性，展示具体构造的细粒度有界时间复杂度，还证明了有界GPACs在指数运算下的性质及编译管道保留时间复杂度类。
中文摘要 AI 辅助
当前基于通用模拟计算机（GPAC）模型和多项式常微分方程的模拟复杂度理论允许无界状态变量，这对化学反应网络和其他实验室规模的模拟计算机来说在物理上不现实。我们开发了一种有界模拟复杂度理论，其中所有状态变量都保持在紧凑区间内，且物理时间（挂钟时间）是唯一发散的资源。主要技术贡献是有界替代编译，它将无界多项式常微分方程系统转换为有界系统，同时保留计算限制和时间精度保证。我们证明了通过算法编译后的系统，其挂钟时间在原始系统的弧长和物理时间上是多项式的。还展示了具体构造以证明细粒度有界时间复杂度。此外，有界GPACs在指数运算下封闭，完整的GPAC到CRN编译管道通过读出模块的低通滤波器分析保留时间复杂度类。
英文摘要
Current analog complexity theory, built on the General-Purpose Analog Computer (GPAC) model and polynomial ODEs, allows unbounded state variables -- an assumption that is physically unrealistic for chemical reaction networks and other laboratory-scale analog computers. We develop a bounded analog complexity theory in which all state variables remain in compact intervals and physical time (wall-clock time) is the only diverging resource. Our main technical contribution is bounded surrogate compilation, a compilation framework that transforms unbounded polynomial ODE systems into bounded ones while preserving computational limits and time-to-precision guarantees. We prove that if a system is compiled into a bounded system through our algorithm, the wall-clock time of the compiled system is polynomial in the arc length and physical time of the original system. We exhibit concrete constructions demonstrating fine-grained bounded time complexity -- a tunable polynomial-degree family, a Lambert-$W$-based system achieving $Θ(r\log r)$ time-to-precision (where $r$ is the desired precision parameter, in nats: $|x(t)-α|<e^{-r}$), and an iterated-logarithm tower realizing arbitrarily high complexity classes -- all for the task of computing the constant 1. We show that bounded GPACs are closed under exponentiation ($α^β$) with time complexity equal to the harder input, and that the full GPAC-to-CRN compilation pipeline preserves time complexity class via a low-pass filter analysis of readout modules.
