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Experience in teaching functional programming (FP) on a relational basis has led the author to focus on a graphical style of expression and reasoning in which a geometric construct shines: the (semi) commutative square. In the classroom this is termed the magic square (MS), since virtually everything that we do in logic, FP, database modeling, formal semantics and so on fits in some MS geometry. The sides of each magic square are binary relations and the square itself is a comparison of two paths, each involving two sides. MSs compose and have a number of useful properties. Among several examples given in the paper ranging over different application domains, free-theorem MSs are shown to be particularly elegant and productive. Helped by a little bit of Galois connections, a generic, induction-free theory for ${\mathsf{foldr}}$ and $\mathsf{foldl}$ is given, showing in particular that ${\mathsf{foldl} \, {{s}}{}\mathrel{=}\mathsf{foldr}{({flip} \unicode{x005F}{s})}{}}$ holds under conditions milder than usually advocated.

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There are now multiple proposals for Conflict-free Replicated Data Types (CRDTs) in SQL databases aimed at distributed systems. Some, such as ElectricSQL, provide only relational tables as convergent replicated maps, but this omits semantics that would be useful for merging updates. Others, such as Pg\_crdt, provide access to a rich library of encapsulated column types. However, this puts merge and query processing outside the scope of the query optimizer and restricts the ability of an administrator to influence access paths with materialization and indexes. Our proposal, CRDV, overcomes this challenge by using two layers implemented as SQL views: The first provides a replicated relational table from an update history, while the second implements varied and rich types on top of the replicated table. This allows the definition of merge semantics, or even entire new data types, in SQL itself, and enables global optimization of user queries together with merge operations. Therefore, it naturally extends the scope of query optimization and local transactions to operations on replicated data, can be used to reproduce the functionality of common CRDTs with simple SQL idioms, and results in better performance than alternatives.

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Developers of data-intensive georeplicated applications face a difficult decision when selecting a database system. As captured by the CAP theorem, CP systems such as Spanner provide strong consistency that greatly simplifies application development. AP systems such as AntidoteDB providing Transactional Causal Consistency (TCC), ensure availability in face of network partitions and isolate performance from wide-area round-trip times, but avoid lost-update anomalies only when values can be merged. Ideally, an application should be able to adapt to current data and network conditions by selecting which transactional consistency to use for each transaction. In this paper, we test the hypothesis that a georeplicated database system can be built at its core providing only TCC, hence, being AP, but allow an application to execute some transactions under Snapshot Isolation (SI), hence CP. Our main result is showing that this can be achieved even when all the interaction happens through the TCC database system, without additional communication channels between the participants. A preliminary experimental evaluation with a proof-of-concept implementation using AntidoteDB shows that this approach is feasible.