Mogensen–Scott encoding

In computer science, Scott encoding is a way to embed inductive datatypes in the lambda calculus. Mogensen–Scott encoding extends and slightly modifies this to an embedding of all terms of the untyped lambda calculus.

Definition

Let D be a datatype with N constructors, $\{C_i\}_{i=1}^N$, such that constructor C_i has arity A_i.

Church encoding

For comparison, the Church encoding of constructor C_i of D is

$\lambda x_1 \ldots x_{A_i} . \lambda c_1 \ldots c_N . c_i (x_1 c_1 \ldots c_N) \ldots (x_{A_i} c_1 \ldots c_N)$

Scott encoding

The Scott encoding of constructor C_i of D is

$\lambda x_1 \ldots x_{A_i} . \lambda c_1 \ldots c_N . c_i x_1 \ldots x_{A_i}$

Mogensen–Scott encoding

Mogensen extends Scott encoding to all untyped lambda terms:

$\begin{matrix} [x] & = & \lambda a b c . a x \\ \ [M N] & = & \lambda a b c . b [M] [N] \\ \ [\lambda x . M] & = & \lambda a b c . c (\lambda x . [M]) \\ \end{matrix}$

Comparison to the Church encoding

The Scott and Church encodings coincide on enumerated datatypes such as the boolean datatype.

Church encoded data and operations on them are typable in system F, but Scott encoded data and operations are not obviously typable in system F. Universal as well as recursive types appear to be required, and since strong normalization does not hold for recursively typed lambda calculus, termination of programs manipulating Scott-encoded data cannot be established by determining well-typedness of such programs.

References

• Directly Reflective Meta-Programming
• Torben Mogensen (1992). Efficient Self-Interpretation in Lambda Calculus. Journal of Functional Programming.