Mogensen–Scott encoding

In computer science, Scott encoding is a way to represent (recursive) data types in the lambda calculus. Church encoding performs a similar function. The data and operators form a mathematical structure which is embedded in the lambda calculus.

Whereas Church encoding starts with representations of the basic data types, and builds up from it, Scott encoding starts from the simplest method to compose algebraic data types.

Mogensen–Scott encoding extends and slightly modifies Scott encoding by applying the encoding to Metaprogramming. This encoding allows the representation of lambda calculus terms, as data, to be operated on by a meta program.

History
Scott encoding appears first in a set of unpublished lecture notes by Dana Scott whose first citation occurs in the book Combinatorial Logic, Volume II. Michel Parigot gave a logical interpretation of and strongly normalizing recursor for Scott-encoded numerals, referring to them as the "Stack type" representation of numbers. Torben Mogensen later extended Scott encoding for the encoding of Lambda terms as data.

Discussion
Lambda calculus allows data to be stored as parameters to a function that does not yet have all the parameters required for application. For example,


 * $$ ((\lambda x_1 \ldots x_n.\lambda c.c\ x_1 \ldots x_n)\ v_1 \ldots v_n)\ f $$

May be thought of as a record or struct where the fields $$ x_1 \ldots x_n $$ have been initialized with the values $$ v_1 \ldots v_n $$. These values may then be accessed by applying the term to a function f. This reduces to,


 * $$ f\ v_1 \ldots v_n $$

c may represent a constructor for an algebraic data type in functional languages such as Haskell. Now suppose there are N constructors, each with $$A_i$$ arguments;
 * $$\begin{array}{c|c|c}

\text{Constructor} & \text{Given arguments} & \text{Result} \\ \hline ((\lambda x_1 \ldots x_{A_1}.\lambda c_1 \ldots c_N.c_1\ x_1 \ldots x_{A_1})\ v_1 \ldots v_{A_1}) & f_1 \ldots f_N & f_1\ v_1 \ldots v_{A_1} \\ ((\lambda x_1 \ldots x_{A_2}.\lambda c_1 \ldots c_N.c_2\ x_1 \ldots x_{A_2})\ v_1 \ldots v_{A_2}) & f_1 \ldots f_N & f_2\ v_1 \ldots v_{A_2} \\ \vdots & \vdots & \vdots \\

((\lambda x_1 \ldots x_{A_N}.\lambda c_1 \ldots c_N.c_N\ x_1 \ldots x_{A_N})\ v_1 \ldots v_{A_N}) & f_1 \ldots f_N & f_N\ v_1 \ldots v_{A_N} \end{array}$$

Each constructor selects a different function from the function parameters $$ f_1 \ldots f_N $$. This provides branching in the process flow, based on the constructor. Each constructor may have a different arity (number of parameters). If the constructors have no parameters then the set of constructors acts like an enum; a type with a fixed number of values. If the constructors have parameters, recursive data structures may be constructed.

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

Scott encoding
The Scott encoding of constructor $$c_i$$ of the data type 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 encode any untyped lambda term as data. This allows a lambda term to be represented as data, within a Lambda calculus meta program. The meta function mse converts a lambda term into the corresponding data representation of the lambda term;
 * $$\begin{align}

\operatorname{mse}[x] & = \lambda a, b, c.a\ x \\ \operatorname{mse}[M\ N] & = \lambda a, b, c.b\ \operatorname{mse}[M]\ \operatorname{mse}[N] \\ \operatorname{mse}[\lambda x. M] & = \lambda a, b, c.c\ (\lambda x.\operatorname{mse}[M]) \\ \end{align}$$

The "lambda term" is represented as a tagged union with three cases:
 * Constructor a - a variable (arity 1, not recursive)
 * Constructor b - function application (arity 2, recursive in both arguments),
 * Constructor c - lambda-abstraction (arity 1, recursive).

For example,


 * $$ \begin{array}{l}

\operatorname{mse}[\lambda x.f\ (x\ x)]\\ \lambda a, b, c.c\ (\lambda x.\operatorname{mse}[f\ (x\ x)])\\ \lambda a, b, c.c\ (\lambda x.\lambda a, b, c.b\ \operatorname{mse}[f]\ \operatorname{mse}[x\ x])\\ \lambda a, b, c.c\ (\lambda x.\lambda a, b, c.b\ (\lambda a, b, c.a\ f)\ \operatorname{mse}[x\ x])\\ \lambda a, b, c.c\ (\lambda x.\lambda a, b, c.b\ (\lambda a, b, c.a\ f)\ (\lambda a, b, c.b\ \operatorname{mse}[x]\ \operatorname{mse}[x]))\\ \lambda a, b, c.c\ (\lambda x.\lambda a, b, c.b\ (\lambda a, b, c.a\ f)\ (\lambda a, b, c.b\ (\lambda a, b, c.a\ x)\ (\lambda a, b, c.a\ x))) \end{array} $$

Comparison to the Church encoding
The Scott encoding coincides with the Church encoding for booleans. Church encoding of pairs may be generalized to arbitrary data types by encoding $$c_i$$ of D above as


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

compare this to the Mogensen Scott encoding,
 * $$\lambda x_1 \ldots x_{A_i} . \lambda c_1 \ldots c_N . c_i x_1 \ldots x_{A_i}$$

With this generalization, the Scott and Church encodings coincide on all enumerated datatypes (such as the boolean datatype) because each constructor is a constant (no parameters).

Concerning the practicality of using either the Church or Scott encoding for programming, there is a symmetric trade-off: Church-encoded numerals support a constant-time addition operation and have no better than a linear-time predecessor operation; Scott-encoded numerals support a constant-time predecessor operation and have no better than a linear-time addition operation.

Type definitions
Church-encoded data and operations on them are typable in system F, as are Scott-encoded data and operations. However, the encoding is significantly more complicated.

The type of the Scott encoding of the natural numbers is the positive recursive type:


 * $$\mu X. \forall R. R \to (X \to R) \to R$$

Full recursive types are not part of System F, but positive recursive types are expressible in System F via the encoding:


 * $$\mu X. G[X] = \forall X. ((G[X] \to X) \to X)$$

Combining these two facts yields the System F type of the Scott encoding:


 * $$\forall X. (((\forall R. R \to (X \to R) \to R) \to X) \to X)$$

This can be contrasted with the type of the Church encoding:


 * $$\forall X. X \to (X \to X) \to X$$

The Church encoding is a second-order type, but the Scott encoding is fourth-order!