  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.
