  In their 2006 seminal paper in Distributed Computing, Angluin et al. present
a construction that, given any Presburger predicate as input, outputs a
leaderless population protocol that decides the predicate. The protocol for a
predicate of size $m$ (when expressed as a Boolean combination of threshold and
remainder predicates with coefficients in binary) runs in $\mathcal{O}(m \cdot
n^2 \log n)$ expected number of interactions, which is almost optimal in $n$.
However, the number of states of the protocol is exponential in $m$. Blondin et
al. described in STACS 2020 another construction that produces protocols with a
polynomial number of states, but exponential expected number of interactions.
We present a construction that produces protocols with $\mathcal{O}(m)$ states
that run in expected $\mathcal{O}(m^{7} \cdot n^2)$ interactions, optimal in
$n$, for all inputs of size $\Omega(m)$. For this we introduce population
computers, a carefully crafted generalization of population protocols easier to
program, and show that our computers for Presburger predicates can be
translated into fast and succinct population protocols.
