We extend classical methods of computational complexity to the setting of distributed computing, where they are sometimes more effective than in their original context. Our focus is on distributed decision in the LOCAL model, where multiple networked computers communicate via synchronous message-passing to collectively answer a question about their network topology. Rather unusually, we impose two orthogonal constraints on the running time of this model: the number of communication rounds is bounded by a constant, and the number of computation steps of each computer is polynomially bounded by the size of its local input and the messages it receives. By letting two players take turns assigning certificates to all computers in the network, we obtain a generalization of the polynomial hierarchy (and hence of the complexity classes $\mathbf{P}$ and $\mathbf{NP}$). We then extend some key results of complexity theory to this setting, in particular the Cook-Levin theorem (which identifies Boolean satisfiability as a complete problem for $\mathbf{NP}$), and Fagin's theorem (which characterizes $\mathbf{NP}$ as the problems expressible in existential second-order logic). The original results can be recovered as the special case where the network consists of a single computer. But perhaps more surprisingly, the task of separating complexity classes becomes easier in the general case: we can show that our hierarchy is infinite, while it remains notoriously open whether the same is true in the case of a single computer. (By contrast, a collapse of our hierarchy would have implied a collapse of the polynomial hierarchy.) As an application, we propose quantifier alternation as a new approach to measuring the locality of problems in distributed computing.

翻译：我们将经典的计算复杂性方法推广到分布式计算环境中，在此环境中这些方法有时比原始场景更有效。我们重点关注LOCAL模型中的分布式决策问题：多个联网计算机通过同步消息传递进行通信，共同回答关于其网络拓扑结构的问题。不同寻常的是，我们对该模型的运行时间施加了两个正交约束：通信轮数有常数上限，且每台计算机的计算步骤数受其局部输入和接收消息大小的多项式界限约束。通过让两个玩家轮流为网络中的每台计算机分配证书，我们获得了多项式层级（进而推广了复杂度类$\mathbf{P}$和$\mathbf{NP}$）的泛化形式。随后我们将复杂性理论的一些关键结果推广到该场景，特别是库克-列文定理（该定理将布尔可满足性确定为$\mathbf{NP}$的完备问题）和费金定理（该定理将$\mathbf{NP}$刻画为可用存在二阶逻辑表达的问题）。当网络仅包含单台计算机时，原始结果可作为特例恢复。但或许更令人惊讶的是，在一般情况下分离复杂度类的任务反而变得更容易：我们可以证明我们的层级是无穷的，而单台计算机情形下该性质是否成立仍是著名的未决问题。（相比之下，如果我们的层级发生坍塌，将蕴含多项式层级的坍塌。）作为应用，我们提出量词交替作为衡量分布式计算问题局部性的新方法。