We extend the notion of distributed decision in the framework of distributed network computing, inspired by both the polynomial hierarchy for Turing machines and recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree (MST) can be certified with $O(\log n)$-bit certificates in n-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires $\mathrm{\Omega }({\log }^{2}n)$-bit certificates if only the prover can act. The improvement can even be exponential for some simple graph properties. For instance, it is known that certifying the existence of a nontrivial automorphism requires $\mathrm{\Omega }(n^2)$ bits if only the prover can act. We show that there is a protocol with two interactions between the prover and the disprover that certifies nontrivial automorphism with $O(\log n)$-bit certificates. These results are achieved by defining and analyzing a local hierarchy of decision which generalizes the classical notions of proof-labeling schemes and locally checkable proofs.

我们在图灵机的多项式层次结构和所谓的分布式图自动机的最新结果的启发下，在分布式网络计算的框架中扩展了分布式决策的概念。我们表明，通过使用基于证明者和反对者之间的交互的分布式决策机制，可以极大地减少分发给节点以证明给定网络属性的证书的大小。例如，我们证明最小生成树（MST）可以通过$Ø（\mathrm{日志}ñ）$n节点图中的位证书，证明方和证明方之间只有一种交互，而众所周知，证明MST需要$\mathrm{\Omega }（{\mathrm{日志}}^2ñ）$位证明书（如果只有证明人可以采取行动）。对于某些简单的图形属性，改进甚至可以是指数级的。例如，众所周知，证明非平凡自同构的存在需要$\mathrm{\Omega }（{ñ}^2）$如果只有证明者可以采取行动，那么就位。我们证明，在证明者和反对者之间存在一种具有两个相互作用的协议，可以证明非平凡自同构具有$Ø（\mathrm{日志}ñ）$位证书。这些结果是通过定义和分析本地决策层次结构来实现的，该决策层次结构概括了证明标签方案和本地可检查证据的经典概念。