We consider three classification systems for distributed decision tasks: With unbounded computation and certificates, defined by Balliu, D'Angelo, Fraigniaud, and Olivetti [JCSS'18], and with (two flavors of) polynomially bounded local computation and certificates, defined in recent works by Aldema Tshuva and Oshman [OPODIS'23], and by Reiter [PODC'24]. The latter two differ in the way they evaluate the polynomial bounds: the former considers polynomials with respect to the size of the graph, while the latter refers to being polynomial in the size of each node's local neighborhood. We start by revisiting decision without certificates. For this scenario, we show that the latter two definitions coincide: roughly, a node cannot know the graph size, and thus can only use a running time dependent on its neighborhood. We then consider decision with certificates. With existential certificates ($Σ_1$-type classes), a larger running time defines strictly larger classes of languages: when it grows from being polynomial in each node's view, through polynomial in the graph's size, and to unbounded, the derived classes strictly contain each other. With universal certificates ($Π_1$-type classes), on the other hand, we prove a surprising incomparability result: having running time bounded by the graph size sometimes allows us to decide languages undecidable even with unbounded certificates. We complement these results with other containment and separation results, which together portray a surprisingly complex lattice of strict containment relations between the classes at the base of the three classification systems.

翻译：我们考虑分布式决策任务的三种分类系统：由Balliu、D'Angelo、Fraigniaud和Olivetti [JCSS'18]定义的无界计算与证书系统，以及近期Aldema Tshuva与Oshman [OPODIS'23]及Reiter [PODC'24]工作中定义的（两种变体的）多项式有界局部计算与证书系统。后两者的区别在于多项式有界性的评估方式：前者考虑相对于图规模的多项式，而后者则指每个节点局部邻域规模的多项式。我们从无证书决策场景的再审视开始。针对该场景，我们证明后两种定义是等价的：粗略而言，节点无法获知图的规模，因此仅能使用依赖于其邻域的运行时间。随后我们考虑有证书决策。对于存在性证书（Σ₁型类），更长的运行时间定义出严格更大的语言类：当运行时间从每个节点视角的多项式增长为图规模的多项式，再至无界时，所导出的类之间构成严格包含关系。而对于全称性证书（Π₁型类），我们则证明了令人惊奇的不可比性结果：受图规模约束的运行时间有时能判定即使在无界证书下也不可判定的语言。我们通过其他包含与分离结果补充这些发现，共同描绘出三个分类系统基础类之间令人惊讶的复杂严格包含关系格。