Corecursion - Definition

Definition

Initial data types can be defined as being the least fixpoint (up to isomorphism) of some type equation; the isomorphism is then given by an initial algebra. Dually, final (or terminal) data types can be defined as being the greatest fixpoint of a type equation; the isomorphism is then given by a final coalgebra.

If the domain of discourse is the category of sets and total functions, then final data types may contain infinite, non-wellfounded values, whereas initial types do not. On the other hand, if the domain of discourse is the category of complete partial orders and continuous functions, which corresponds roughly to the Haskell programming language, then final types coincide with initial types, and the corresponding final coalgebra and initial algebra form an isomorphism.

Corecursion is then a technique for recursively defining functions whose range (codomain) is a final data type, dual to the way that ordinary recursion recursively defines functions whose domain is an initial data type.

The discussion below provides several examples in Haskell that distinguish corecursion. Roughly speaking, if one were to port these definitions to the category of sets, they would still be corecursive. This informal usage is consistent with existing textbooks about Haskell. Also note that the examples used in this article predate the attempts to define corecursion and explain what it is.