The 2P-Set Conflict-Free Replicated Data Type (CRDT) supports two phases for each possible element: in the first phase an element can be added to the set and the subsequent additions are ignored; in the second phase an element can be removed after which it will stay removed forever regardless of subsequent additions and removals. We generalize the 2P-Set to support an infinite sequence of alternating additions and removals of the same element. In the presence of concurrent additions and removals on different replicas, all replicas will eventually converge to the longest sequence of alternating additions and removals that follows causal history. The idea of converging on the longest-causal sequence of opposite operations had already been suggested in the context of an undo-redo framework but the design was neither given a name nor fully developed. In this paper, we present the full design directly, using nothing more than the basic formulation of state-based CRDTs. We also show the connection between the set-based definition of 2P-Set and the counter-based definition of the $\infty$P-Set with simple reasoning. We then give detailed proofs of convergence. The underlying \textit{grow-only dictionary of grow-only counters} on which the $\infty$P-Set is built may be used to build other state-based CRDTs. In addition, this paper should be useful as a pedagogical example for designing state-based CRDTs, and might help raise the profile of CRDTs based on \textit{longest sequence wins}.

翻译：2P-Set无冲突复制数据类型（CRDT）为每个可能的元素支持两个阶段：第一阶段可将元素添加到集合中，后续添加操作将被忽略；第二阶段可移除元素，此后无论后续进行多少次添加或移除操作，该元素都将永久保持移除状态。我们将2P-Set泛化为支持同一元素的无限交替添加与移除序列。当不同副本上存在并发的添加和移除操作时，所有副本最终将收敛到遵循因果历史的最长交替添加与移除序列。这一基于最长因果对立操作序列的收敛思想已在撤销-重做框架中被提出，但该设计既未被命名也未被完整开发。本文直接给出完整设计，仅需使用基于状态的CRDT基本公式即可实现。我们还通过简单推理展示了2P-Set的基于集合的定义与$\infty$P-Set的基于计数器的定义之间的联系。随后给出了收敛性的详细证明。构建$\infty$P-Set所依赖的底层数据结构——仅增长计数器的仅增长字典，可用于构建其他基于状态的CRDT。此外，本文可作为设计基于状态的CRDT的教学范例，并有助于提升基于"最长序列获胜"策略的CRDT的知名度。