Bounded Analog Complexity

2026-07-13

AI Summary

This work addresses a key limitation in existing analog complexity theory, which assumes unbounded state variables and thus fails to capture the physical constraints inherent in systems such as chemical reaction networks (CRNs). To overcome this, the paper introduces a bounded analog complexity framework that compiles unbounded polynomial ordinary differential equation systems into equivalent bounded counterparts whose state variables remain confined to compact intervals, with physical time as the sole divergent resource. Built upon the General Purpose Analog Computer (GPAC) model and integrating low-pass filtering with a CRN compilation pipeline, the approach yields concrete systems of tunable complexity while preserving time–accuracy guarantees. Key contributions include proving closure of bounded GPACs under exponentiation, constructing a Lambert-W system achieving Θ(r log r) time–accuracy performance, and realizing iterated logarithmic towers of arbitrary height, all while ensuring the compiled system’s physical time scales polynomially with the arc length and time of the original system.

Abstract

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.

Problem

• analog complexity
• bounded state variables
• physical realizability
• chemical reaction networks
• GPAC

Innovation

• bounded analog complexity
• bounded surrogate compilation
• polynomial ODEs
• time-to-precision
• GPAC-to-CRN compilation