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（CRNs）模拟分子根据有限反应集相互作用的系统，例如 $A + B \to C$ 表示若 $A$ 和 $B$ 分子碰撞，它们消失并生成一个 $C$ 分子。CRNs 可计算布尔值谓词 $\phi:\mathbb{N}^d \to \{0,1\}$ 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 $\min(x_1,x_2)$。我们研究有界执行CRNs的计算能力，此类网络中从初始构型出发只能发生有限次反应（例如排除 $A \rightleftharpoons B$ 这类可逆反应）。此类CRNs的能力与可组合性关键取决于其他建模选择——这些选择对无界执行CRNs的计算能力无影响——即是否存在初始领导者，以及（对于谓词）是否要求所有物种为布尔输出“投票”。若CRN启动时存在初始领导者且仅允许领导者投票，则所有半线性谓词和函数均能被有界执行CRNs在 $O(n \log n)$ 并行时间内稳定计算。然而，若不允许初始领导者、所有物种均需投票且CRN为“非坍缩型”（不会从初始大尺度构型收缩为最终 $O(1)$ 规模构型），则有界执行CRNs严重受限，仅能计算最终常值谓词。关键工具是将有界执行CRNs精确刻画为具有非负线性势函数且每一次反应均严格降低该势函数的网络——该结果可能具有独立研究价值。