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:$lambda\; x\_1\; ldots\; x\_\{A\_i\}\; .\; lambda\; c\_1\; ldots\; c\_N\; .\; c\_i\; (x\_1\; c\_1\; ldots\; c\_N)\; ldots\; (x\_\{A\_i\}\; c\_1\; ldots\; c\_N)$

cott 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::

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

* [http://cl.cse.wustl.edu/archon/archon.pdf Directly Reflective Meta-Programming]