The Church-Scott representation of inductive and coinductive data

Data in the lambda calculus is usually represented using the "Church encoding", which gives closed terms for the constructors and which naturally allows to define functions by iteration. An additional nice feature is that in system F (polymorphically typed lambda calculus) one can define types for this data and the iteration scheme is well-typed. A problem is that primitive recursion is not directly available: it can be coded in terms of iteration at the cost of inefficiency (e.g. a predecessor with linear run-time). The much less well-known Scott encoding has case distinction as a primitive. The terms are not typable in system F and there is no iteration scheme, but there is a constant time destructor (e.g. predecessor). We present a unification of the Church and Scott definition of data types, the Church-Scott encoding, which has primitive recursion as basic. For the numerals, these are also known as ‘Parigot numerals’. We show how these can be typed in the polymorphic lambda calculus extended with recursive types. We show that this also works for the dual case, co-inductive data types with primitive co-recursion. We single out the major schemes for defining functions over data-types: the iteration scheme, the case scheme and the primitive recursion scheme, and we show how they are tightly connected to the Church, the Scott and the Church-Scott representation. We also single out the duals of these schemes: co-iteration, co-case and primitive co-recursion as schemes for defining function to a co-data-type. A major advantage is that all these are encodings in pure untyped λ-calculus and that strong normalization is guaranteed, because the terms are typable in the polymorphic lambda calculus extended with recursive types. 1998 ACM Subject Classification F.1.1 Models of Computation, D.1.1 Applicative (Functional) Programming, D.3.3 Language Constructs and Features, F.4.1 Mathematical Logic, F.3.3 Studies of Program Constructs