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.

翻译：在2006年《分布式计算》的 seminal 论文中，Angluin等人提出了一种构造方法：对于任意输入的Presburger谓词，该方法可生成一个无需领导者的判断该谓词的人口协议。当谓词的大小为$m$（表示为阈值谓词和余数谓词的布尔组合，且系数为二进制）时，该协议的期望交互次数为 $\mathcal{O}(m \cdot n^2 \log n)$ ，这使得其在$n$上几乎达到最优。然而，该协议的状态数随$m$呈指数增长。Blondin等人在STACS 2020上描述了另一种构造方法，该方法生成的协议具有多项式级的状态数，但期望交互次数呈指数增长。我们提出了一种新构造方法：对于所有规模为 $\Omega(m)$ 的输入，该方法可生成具有 $\mathcal{O}(m)$ 个状态且期望交互次数为 $\mathcal{O}(m^{7} \cdot n^2)$ 的协议（在$n$上达到最优）。为此，我们引入了“人口计算机”——一种对人口协议进行精心概括、易于编程的泛化模型，并证明针对Presburger谓词的此类计算机可转化为快速且简洁的人口协议。