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Abstract
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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Humans are good at understanding subjective or vague statements which, however, are hard to express in classical logic. Fuzzy logic is an evolution of classical logic that can cope with vague terms by handling degrees of truth and not just the crisp values true and false. Logic is the formal basis of computing, enabling the formal design of systems supported by tools such as model checkers and theorem provers.This paper shows how a model checker such as Alloy can evolve to handle both classical and fuzzy logic, enabling the specification of high-level quantitative relational models in the fuzzy domain. In particular, the paper showcases how QAlloy-F (a conservative, general-purpose quantitative extension to standard Alloy) can be used to tackle fuzzy problems, namely in the context of validating the design of fuzzy controllers. The evaluation of QAlloy-F against examples taken from various classes of fuzzy case studies shows the approach to be feasible.

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.