Programming from Galois connections

Problem statements often resort to superlatives such as in e.g. “… the smallest such number”, “… the best approximation”, “… the longest such list” which lead to specifications made of two parts: one defining a broad class of solutions (the easy part) and the other requesting one particular such solution, optimal in some sense (the hard part).This article introduces a binary relational combinator which mirrors this linguistic structure and exploits its potential for calculating programs by optimization. This applies in particular to specifications written in the form of Galois connections, in which one of the adjoints delivers the optimal solution.The framework encompasses re-factoring of results previously developed by Bird and de Moor for greedy and dynamic programming, in a way which makes them less technically involved and therefore easier to understand and play with.