RevCRN：基于化学反应网络的可逆模拟计算

中文摘要 AI 辅助

自20世纪中期以来，使用图灵机计算实数和函数的可计算性一直是理论计算机科学的核心领域。20世纪后期，研究表明化学反应可作为基于化学反应网络（Chemical Reaction Network，CRN）模型进行计算的基础。近期使用确定性化学反应网络（Deterministic Chemical Reaction Networks，DCRNs）计算实数的进展，已确定了众多DCRN可计算实数的类别。与此同时，R. Landauer和C. H. Bennett在20世纪60年代至21世纪初的研究表明，可逆计算相比不可逆方法具有显著优势，尤其是在能效方面，这推动了对可逆计算的广泛研究。本研究探讨使用可逆化学反应网络（Reversible Chemical Reaction Networks，RevCRNs）计算实数的可计算性，主要贡献有两点：（1）建立CRN可计算实数类别间的关系，包括李雅普诺夫CRN（Lyapunov CRN，$\boldsymbol{\reals_{LCRN}}$）、实时CRN（Real-Time CRN，$\boldsymbol{\reals_{RTCRN}}$）、有理数（$\boldsymbol{\rationals}$）和RevCRNs（$\boldsymbol{\reals_{RevCRN}}$），关键结果为：（i）有理数集是$\boldsymbol{\reals_{RevCRN}}$的严格子集；（ii）正代数数集（$\boldsymbol{ALG}$）、$\boldsymbol{\reals_{LCRN}}$与单物种RevCRN可计算实数集（$\boldsymbol{\reals_{RevCRN}^{1s}}$）相等；（iii）$\boldsymbol{\reals_{RTCRN}}$与$\boldsymbol{\reals_{RevCRN}}$存在非空交集；（iv）细致平衡RevCRN可计算实数集（$\boldsymbol{\reals_{RevCRN}^{DetBal}}$）是$\boldsymbol{ALG}$的子集；（2）探索$\boldsymbol{\reals_{RevCRN}}$内部的层级结构。最后，研究未明确$\boldsymbol{\reals_{RevCRN}}$与$\boldsymbol{\reals_{RTCRN}}$的精确关系，同时推测RevCRN可计算实数存在通用层级结构。

英文摘要

The computability of real numbers and functions using Turing Machines has been a central area of theoretical computer science since the mid-20th century. In the late 20th century, it was shown that chemical reactions can serve as a basis for computation using the Chemical Reaction Network (CRN) model. Recent advances in computing real numbers using Deterministic Chemical Reaction Networks (DCRNs) have identified numerous classes of DCRN-computable real numbers. In parallel, the works of R. Landauer and C. H. Bennett, spanning the 1960s to the early 2000s, showed that reversible computing offers significant advantages over irreversible methods, particularly in energy efficiency, motivating extensive research on reversible computation. In this work, we investigate the computability of real numbers using Reversible Chemical Reaction Networks (RevCRNs). The paper has two primary contributions: (1) establishing relationships among CRN-computable real number classes including Lyapunov CRN ($\mathbb{R}_{LCRN}$), Real-Time CRN ($\mathbb{R}_{RTCRN}$), rational numbers ($\mathbb{Q}$), and RevCRNs ($\mathbb{R}_{RevCRN}$), with key results: (i) $\mathbb{Q}$ is a strict subset of $\mathbb{R}_{RevCRN}$; (ii) the set of positive algebraic numbers ($ALG$), $\mathbb{R}_{LCRN}$, and real numbers computable by 1-species RevCRN ($\mathbb{R}_{RevCRN}^{1s}$) are equal; (iii) $\mathbb{R}_{RTCRN}$ and $\mathbb{R}_{RevCRN}$ exhibit non-empty overlap; and (iv) the set of real numbers computable by detailed-balanced RevCRNs ($\mathbb{R}^{DetBal}_{RevCRN}$) is a subset of $ALG$; and (2) exploring the existence of a hierarchy within $\mathbb{R}_{RevCRN}$. Finally, we leave open the exact relationship between $\mathbb{R}_{RevCRN}$ and $\mathbb{R}_{RTCRN}$ while conjecturing a general hierarchy of RevCRN-computable reals.

发表机构

• Iowa State University（爱荷华州立大学）
• Ames National Laboratory（埃姆斯国家实验室）

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