# Defunctionalization

In programming languages, **defunctionalization** refers to a compile-time transformation which eliminates higher-order functions, replacing them by a single first-order *apply* function. The technique was first described by John C. Reynolds in his 1972 paper, "Definitional Interpreters for Higher-Order Programming Languages". Reynolds' observation was that a given program contains only finitely many function abstractions, so that each can be assigned (and replaced by) a unique identifier. Every function application within the program is then replaced by a call to the *apply* function with the function identifier as the first argument. The *apply* function's only job is to dispatch on this first argument, and then perform the instructions denoted by the function identifier on the remaining arguments.

One complication to this basic idea is that function abstractions may reference free variables. In such situations, defunctionalization must be preceded by closure conversion (lambda lifting), so that any free variables of a function abstraction are passed as extra arguments to *apply*. In addition, if closures are supported as first-class values, it becomes necessary to represent these captured bindings by creating data structures.

Instead of having a single *apply* function dispatch on all function abstractions in a program, various kinds of control flow analysis (including simple distinctions based on arity or type signature) can be employed to determine which function(s) may be called at each function application site, and a specialized *apply* function may be referenced instead. Alternately, the target language may support indirect calls through function pointers, which may be more efficient and extensible than a dispatch-based approach.

Besides its use as a compilation technique for higher-order functional languages, defunctionalization has been studied (particularly by Olivier Danvy and collaborators) as a way of mechanically transforming interpreters into abstract machines. Defunctionalization is also related to the technique from object-oriented programming of representing functions by function objects (as an alternative to closures).

## Example

This is an example given by Olivier Danvy, translated to Haskell:

Given the Tree datatype:

```
data Tree a = Leaf a
| Node (Tree a) (Tree a)
```

We will defunctionalize the following program:

```
cons :: a -> [a] -> [a]
cons x xs = x : xs
o :: (b -> c) -> (a -> b) -> a -> c
o f g x = f (g x)
flatten :: Tree t -> [t]
flatten t = walk t []
walk :: Tree t -> [t] -> [t]
walk (Leaf x) = cons x
walk (Node t1 t2) = o (walk t1) (walk t2)
```

We defunctionalize by replacing all higher-order functions (`cons`

, `o`

, and `walk`

) with a value of the `Lam`

datatype, and instead of calling them directly, we introduce an `apply`

function that interprets the datatype:

```
data Lam a = LamCons a
| LamO (Lam a) (Lam a)
apply :: Lam a -> [a] -> [a]
apply (LamCons x) xs = x : xs
apply (LamO f1 f2) xs = apply f1 (apply f2 xs)
cons_def :: a -> Lam a
cons_def x = LamCons x
o_def :: Lam a -> Lam a -> Lam a
o_def f1 f2 = LamO f1 f2
flatten_def :: Tree t -> [t]
flatten_def t = apply (walk_def t) []
walk_def :: Tree t -> Lam t
walk_def (Leaf x) = cons_def x
walk_def (Node t1 t2) = o_def (walk_def t1) (walk_def t2)
```

## See also

## References

- Reynolds, John (August 1972). "Definitional Interpreters for Higher-Order Programming Languages" (PDF).
*Proceedings of the ACM Annual Conference*. Boston, Massachusetts. pp. 717–740. doi:10.1145/800194.805852. - Danvy, Olivier; Nielsen, Lasse R. (2001). "Defunctionalization at Work" (PDF).
*Proceedings of the ACM SIGPLAN Conference on Principles and Practice of Declarative Programming*. pp. 162–174. doi:10.1145/773184.773202. (More comprehensive version: Technical Report BRICS-RS-01-23) - Danvy, Olivier; Millikin, Kevin R. (June 2009). "Refunctionalization at Work".
*Science of Computer Programming*.**74**(8): 534–549. doi:10.1016/j.scico.2007.10.007. (Also available as Technical Report BRICS-RS-07-7)

## External links

- Defunctionalization (Programming Languages). Oxford University.