Abstract
The R/G approach to the development of interfering programs was initiated by the pioneering work of Cliff Jones (1981) on a relational basis. R/G has been the subject of much research since then, most of it deviating from the original relational set-up. This paper looks at such early work from a historical perspective and shows how it can be approached and extended using state-of-the-art relational algebra. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

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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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