Programming Metamorphic Algorithms: An Experiment in Type-Driven Algorithm Design

In dependently typed programming, proofs of basic, structural properties can be embedded implicitly into programs and do not need to be written explicitly. Besides saving the effort of writing separate proofs, a most distinguishing and fascinating aspect of dependently typed programming is that it makes the idea of interactive type-driven development much more powerful, where expressive type information becomes useful hints that help the programmer to complete a program. There have not been many attempts at exploiting the full potential of the idea, though. As a departure from the usual properties dealt with in dependently typed programming, and as a demonstration that the idea of interactive type-driven development has more potential to be discovered, we conduct an experiment in "type-driven algorithm design": we develop algorithms from their specifications encoded in sophisticated types, to see how useful the hints provided by a type-aware interactive development environment can be. The algorithmic problem we choose is metamorphisms, whose definitional behaviour is consuming a data structure to compute an intermediate value and then producing a codata structure from that value, but there are other ways to compute metamorphisms. We develop Gibbons's streaming algorithm and Nakano's jigsaw model in the interactive development environment provided by the dependently typed language Agda, turning intuitive ideas about these algorithms into formal conditions and programs that are correct by construction.