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 foldr and foldl is given, showing in particular that foldl s = foldr (flip_s) holds under conditions milder than usually advocated.

I am professor of Computer Science at the Informatics Department of University of Minho and researcher at HASLab/ INESC TEC. I am also a member of IFIP WG 2.1 (Algorithmic Languages and Calculi) and of the Formal Methods Europe (FME) Association. I serve on the editorial board of Springer journal Formal Aspects of Computing.

My research interests are focused on formal methods, algebra of programming (program calculation) and functional programming. I’ve published recently on relation algebra and its application to programming. Currently, I am developing a linear algebra of programming which I want to apply to the verification of complex software systems, including quantum programming.