Chemical reaction networks (CRNs) model systems where molecules interact according to a finite set of reactions such as $A + B \to C$, representing that if a molecule of $A$ and $B$ collide, they disappear and a molecule of $C$ is produced. CRNs can compute Boolean-valued predicates $\phi:\mathbb{N}^d \to \{0,1\}$ and integer-valued functions $f:\mathbb{N}^d \to \mathbb{N}$; for instance $X_1 + X_2 \to Y$ computes the function $\min(x_1,x_2)$. We study the computational power of execution bounded CRNs, in which only a finite number of reactions can occur from the initial configuration (e.g., ruling out reversible reactions such as $A \rightleftharpoons B$). The power and composability of such CRNs depend crucially on some other modeling choices that do not affect the computational power of CRNs with unbounded executions, namely whether an initial leader is present, and whether (for predicates) all species are required to "vote" for the Boolean output. If the CRN starts with an initial leader, and can allow only the leader to vote, then all semilinear predicates and functions can be stably computed in $O(n \log n)$ parallel time by execution bounded CRNs. However, if no initial leader is allowed, all species vote, and the CRN is "noncollapsing" (does not shrink from initially large to final $O(1)$ size configurations), then execution bounded CRNs are severely limited, able to compute only eventually constant predicates. A key tool is to characterize execution bounded CRNs as precisely those with a nonnegative linear potential function that is strictly decreased by every reaction, a result that may be of independent interest.

翻译：化学反应网络（CRN）模拟了分子按照有限反应集（例如 $A + B \to C$，表示一个 $A$ 分子和一个 $B$ 分子碰撞后消失并产生一个 $C$ 分子）进行相互作用的系统。CRN 可以计算布尔值谓词 $\phi:\mathbb{N}^d \to \{0,1\}$ 和整数值函数 $f:\mathbb{N}^d \to \mathbb{N}$；例如 $X_1 + X_2 \to Y$ 计算函数 $\min(x_1,x_2)$。我们研究执行有界 CRN 的计算能力，此类 CRN 从初始配置出发只能发生有限数量的反应（例如，排除了诸如 $A \rightleftharpoons B$ 的可逆反应）。这类 CRN 的能力和可组合性关键取决于其他一些建模选择，而这些选择不影响无界执行 CRN 的计算能力，即初始领导者是否存在，以及（对于谓词）是否要求所有物种都对布尔输出进行"投票"。如果 CRN 开始时存在初始领导者，并且只允许领导者投票，那么所有半线性谓词和函数都可以由执行有界 CRN 在 $O(n \log n)$ 并行时间内稳定计算。然而，如果不允许初始领导者，所有物种都参与投票，并且 CRN 是"非坍缩"的（不会从初始的大规模配置缩小到最终 $O(1)$ 规模的配置），那么执行有界 CRN 的能力受到严重限制，只能计算最终恒定的谓词。一个关键工具是将执行有界 CRN 精确地表征为那些具有非负线性势函数且每个反应都严格降低该势函数的 CRN，这一结果可能具有独立的研究价值。