We extend classical methods of computational complexity to the setting of distributed computing, where they prove even more effective in some respects than in their original context. Instead of a single computer, several networked computers communicate via synchronous message-passing to collectively solve some decision problem related to the network topology. Their running time is limited in two ways: 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 the computers, 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：我们将计算复杂性的经典方法拓展至分布式计算场景，并发现该方法在某些方面比原始语境更具效用。此处并非采用单台计算机，而是多台联网计算机通过同步消息传递进行协作，以解决与网络拓扑相关的判定问题。其运行时间受两方面限制：通信轮次被常数界约束，且每台计算机的计算步数由其本地输入规模及接收消息规模的多项式函数界定。通过让两个玩家交替为计算机分配证书，我们得到了多项式层次结构（进而包括复杂度类 $\mathbf{P}$ 与 $\mathbf{NP}$）的推广形式。随后我们将复杂性理论的关键成果延伸至该场景，特别是库克-列文定理（将布尔可满足性认定为 $\mathbf{NP}$ 完全问题）与费金定理（将 $\mathbf{NP}$ 刻画为可用存在二阶逻辑表达的问题）。当网络仅含单台计算机时，原始结论可作为特例重现。但更令人惊讶的是，在一般情形下分离复杂度类的任务反而变得更容易：我们可证明该层次结构是无限的，而关于单台计算机是否具有相同属性仍为悬而未决的公开难题（相比之下，若我们的层次结构坍塌，则将导致多项式层次结构的坍塌）。作为应用，我们提出量词交替可作为衡量分布式计算问题局部性的新方法。