Discrete chemical reaction networks formalize the interactions of molecular species in a well-mixed solution as stochastic events. Given their basic mathematical and physical role, the computational power of chemical reaction networks has been widely studied in the molecular programming and distributed computing communities. While for Turing-universal systems there is a universal measure of optimal information encoding based on Kolmogorov complexity, chemical reaction networks are not Turing universal unless error and unbounded molecular counts are permitted. Nonetheless, here we show that the optimal number of reactions to generate a specific count $x \in \mathbb{N}$ with probability $1$ is asymptotically equal to a ``space-aware'' version of the Kolmogorov complexity of $x$, defined as $\mathrm{\widetilde{K}s}(x) = \min_p\left\{\lvert p \rvert / \log \lvert p \rvert + \log(\texttt{space}(\mathcal{U}(p))) : \mathcal{U}(p) = x \right\}$, where $p$ is a program for universal Turing machine $\mathcal{U}$. This version of Kolmogorov complexity incorporates not just the length of the shortest program for generating $x$, but also the space usage of that program. Probability $1$ computation is captured by the standard notion of stable computation from distributed computing, but we limit our consideration to chemical reaction networks obeying a stronger constraint: they ``know when they are done'' in the sense that they produce a special species to indicate completion. As part of our results, we develop a module for encoding and unpacking any $b$ bits of information via $O(b/\log{b})$ reactions, which is information-theoretically optimal for incompressible information. Our work provides one answer to the question of how succinctly chemical self-organization can be encoded -- in the sense of generating precise molecular counts of species as the desired state.

翻译：离散化学反应网络将均匀混合溶液中分子物种间的相互作用形式化为随机事件。鉴于其基础性数学与物理角色，化学反应网络的计算能力已在分子编程与分布式计算领域得到广泛研究。虽然图灵通用系统存在基于柯尔莫哥洛夫复杂度的最优信息编码普适度量，但化学反应网络除非允许误差及无界分子计数，否则不具备图灵通用性。然而，本文证明：以概率1生成特定计数$x \in \mathbb{N}$所需的最优反应数目渐近等于$x$的“空间感知”版本柯尔莫哥洛夫复杂度，定义为$\mathrm{\widetilde{K}s}(x) = \min_p\left\{\lvert p \rvert / \log \lvert p \rvert + \log(\texttt{space}(\mathcal{U}(p))) : \mathcal{U}(p) = x \right\}$，其中$p$为通用图灵机$\mathcal{U}$的程序。该版本柯尔莫哥洛夫复杂度不仅包含生成$x$的最短程序长度，还包含该程序的空间使用量。概率1计算由分布式计算中的标准稳定计算概念刻画，但我们将讨论范围限制在满足更强约束的化学反应网络上：这些网络“知晓完成时刻”，即通过产生特殊物种指示完成。作为研究结果的一部分，我们开发了一种通过$O(b/\log{b})$个反应编码与解包任意$b$位信息的模块，这对不可压缩信息而言在信息论意义上达到最优。本研究为“化学自组织能以多简洁方式进行编码”——即以生成精确分子物种计数作为目标状态——这一核心问题提供了答案。