Anonymous (co)inductive types: A way for structured recursion to cohabit with modular abstraction

We investigate the interaction between structured recursion combinators and modularization in the style of Standard ML. When built-in structured recursion combinators are straightforwardly added to a language like SML’97 or OCaml, they cannot operate over values of abstractly specified types. Consequently, when a program is modularized in an abstract and fine-grained way, the structured recursion combinators can hardly ever be used. We explore various ways of solving this incompatibility, presenting the possible solutions in our modular programming language with a verbose notation for inductive and coinductive types and operations. We propose structured recursion programming constructs that are, in our opinion, best suited for action across abstract module boundaries: anonymous inductive and coinductive types separated from sum and product types, and the related operations. We discuss advantages and disadvantages of the proposed programming idioms and their expressibility in other languages and formal systems, such as OCaml, Charity, Functorial ML and PolyP. We describe their precise typing and semantics as parts of our programming language. Finally we discuss other possible uses of the introduced constructs, related to both polytypism and modular programming and inspired by the notions of views and 2-level types.