AI 中文总结
该研究提出化学反应网络中基于比率编码的瞬时算术计算框架，构建了四种基本运算的反应网络模块，可瞬时计算截断幂级数、矩阵乘积等，还扩展到带符号实数运算。
AI 中文摘要
我们开发了一种利用化学反应网络进行瞬时算术计算的通用框架。数值由物种浓度的比率编码：扩展非负值$a\in[0,+\infty]$通过比率$a=a_1/a_0$由计算对$(A_0,A_1)$表示。对于这种比率编码，我们的主要理论结果是前馈组合性定理：若一组模块满足两个易于验证的结构条件，则任何可容许的有限前馈组合都能瞬时计算对应的算术表达式，即每个中间比率和输出比率在所有正时刻均正确。我们通过为四个基本运算——恒等、求逆、乘法和加法——构建反应网络模块，验证每个模块均满足这些结构条件，使该框架具体化。作为应用，我们表明利用反应网络可瞬时计算截断幂级数和矩阵乘积，其规模反映了基础计算中的算术运算数量。我们还将非负比率编码扩展到带符号量，将实数表示为两个非负比率编码值的差；这支持实值输入上的加法和乘法。我们讨论了瞬时计算的实际意义，包括其与其他资源约束的关系，如所需化学物种的数量，以及读取最终结果的成本，这需要除法运算，而对于带符号情况还需要整流减法。
英文摘要
We develop a general framework for instantaneous arithmetic computation using chemical reaction networks. Numerical values are encoded by ratios of species concentrations: an extended nonnegative value $a\in[0,+\infty]$ is represented by a computational pair $(A_0,A_1)$ through the ratio $a=a_1/a_0$. For this ratio encoding, our main theoretical result is a feedforward compositionality theorem: if a collection of modules satisfies two easily checked structural conditions, then any admissible finite feedforward composition computes the corresponding arithmetic expression instantaneously, meaning that every intermediate and output ratio is correct at every positive time. We make this framework concrete by constructing reaction network modules for four elementary operations---identification, inversion, multiplication, and addition---and verifying that each satisfies these structural conditions. As applications, we show that truncated power series and matrix products can be computed instantaneously using reaction networks whose size reflects the number of arithmetic operations in the underlying computation. We also extend the nonnegative ratio encoding to signed quantities by representing a real number as the difference of two nonnegative ratio-encoded values; this allows addition and multiplication over real-valued inputs. We discuss the practical implications of instantaneous computation, including its relationship to other resource constraints such as the number of chemical species required and the cost of reading out the final answer, which requires division and, in the signed case, rectified subtraction.
