State-Based $\infty$P-Set Conflict-Free Replicated Data Type

***** This design is a duplicate of a Causal Length Set (see notes in the comments). We leave nonetheless the original paper here because the proofs are referred to in another submission.***** 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}.