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\begin{document}

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\section*{defunctionalization}

\hypertarget{defunctionalization}{}\section*{{Defunctionalization}}\label{defunctionalization}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{program_transformation}{Program transformation}\dotfill \pageref*{program_transformation} \linebreak
\noindent\hyperlink{connection_to_adjoint_functor_theorems}{Connection to adjoint functor theorems}\dotfill \pageref*{connection_to_adjoint_functor_theorems} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Defunctionalization} is a method of converting a [[functional programming|functional program]] into one that involves no higher order functions nor lambda expressions. It is a whole program transformation.

\hypertarget{program_transformation}{}\subsection*{{Program transformation}}\label{program_transformation}

The idea is it to consider a type $Defun(A,B)$ for every pair of types $A,B$ as a proxy for the true [[function type]] $A\to B$. This type $Defun(A,B)$ is a [[coproduct]]

\begin{displaymath}
Defun(A,B)=\coprod_{x_1:C_1,\dots,x_n:C_n\, \vdash \,\lambda y.\,t\,  :\,  A\to B}C_1\times \dots \times C_n
\end{displaymath}
over all [[lambda abstractions]] $\lambda y.\,t$ that appear syntactically in the program. Notice that lambda abstractions may have free variables $x_1:C_1,\dots,x_n:C_n$, because they may appear deep inside the program. An inhabitant of the type $Defun(A,B)$ is sometimes called a \emph{closure}, because it is a syntactic expression together with an environment: a valuation for its free variables.

The type $Defun(A,B)$ has a canonical evaluation function $eval:Defun(A,B)\times A\to B$ roughly given by

\begin{displaymath}
eval(((x_1:C_1\dots x_n:C_n\vdash \lambda y.\,t:A\to B),(x_1,\dots x_n)),y)=t
\end{displaymath}
and so it behaves like a function space.

Since $A$, $B$ and the $C_i$’s can also be defunctionalized, this is a method to transform a whole functional program into one with no lambda expressions nor any true function types.

The coproduct type $Defun(A,B)$ is typically not a true function space in the sense of [[cartesian closed categories]] for two reasons.

\begin{itemize}%
\item it does not admit arbitrary lambda abstractions -- just the lambda abstractions that happen to be included in the coproduct.


\item it does not equate lambda abstractions that are semantically equal but syntactically different.



\end{itemize}
(Thus it violates both existence and uniqueness in the universal property.)

\hypertarget{connection_to_adjoint_functor_theorems}{}\subsection*{{Connection to adjoint functor theorems}}\label{connection_to_adjoint_functor_theorems}

Recall that a cartesian [[internal hom]] $B^A$, if it exists, is a [[representable functor|representation]] of $Hom(-\times A,B)$. As such, in many concrete situations, we can deduce its existence from an [[adjoint functor theorem]]. Recall that the [[solution set condition]] in this instance requires a set $I$ and an $I$-indexed family of objects $C_i$ together with morphisms

\begin{displaymath}
f_i:C_i\times A\to B
\end{displaymath}
such that every morphism $f:C\times A\to B$ factors as

\begin{displaymath}
C\times A \stackrel{g\times A}\to C_i\times A\stackrel{f_i}\to B
\end{displaymath}
for some $i\in I$ and $g\colon C\to C_i$. Then, as is usually observed, the coproduct

\begin{displaymath}
\coprod_{i\in I}C_i
\end{displaymath}
is a weak internal hom, from which the internal hom is recovered by taking a [[coequalizer]]. This coproduct is strongly reminiscent of the coproduct type $Defun(A,B)$ used in defunctionalization.

\hypertarget{example}{}\subsection*{{Example}}\label{example}

Consider the [[factorial]] function in [[continuation-passing style]], written in [[Haskell]]:

\begin{verbatim}factcps :: Integer -> (Integer -> Integer) -> Integer
factcps x k = if x == 0 then k 1 else factcps (x-1) (\y -> k (x * y))

fact :: int -> int
fact x = factcps x (\x -> x)\end{verbatim}
The idea is that {\colorbox[rgb]{1.00,0.93,1.00}{\tt factcps\char32x\char32k}} calculates the factorial of {\colorbox[rgb]{1.00,0.93,1.00}{\tt x}} but does not return the result, instead, it calls the ``continuation'' {\colorbox[rgb]{1.00,0.93,1.00}{\tt k}} with the result. Since {\colorbox[rgb]{1.00,0.93,1.00}{\tt fact\char32\char53}} calls {\colorbox[rgb]{1.00,0.93,1.00}{\tt factcps\char32\char53}} with the identity function, it will eventually return {\colorbox[rgb]{1.00,0.93,1.00}{\tt \char49\char50\char48}}.

If this is the whole program, then there are only two [[lambda abstractions]] we need to deal with, both of type {\colorbox[rgb]{1.00,0.93,1.00}{\tt Integer\char32\char45\char62\char32Integer}}. The first one {\colorbox[rgb]{1.00,0.93,1.00}{\tt ~y\char32\char45\char62\char32k\char32\char40x\char32\char42\char32y\char41}} has free variables {\colorbox[rgb]{1.00,0.93,1.00}{\tt x}} and {\colorbox[rgb]{1.00,0.93,1.00}{\tt k}}; we'll call it {\colorbox[rgb]{1.00,0.93,1.00}{\tt Multiply}}. The second {\colorbox[rgb]{1.00,0.93,1.00}{\tt ~x\char32\char45\char62\char32x}} has no free variables; we'll call it {\colorbox[rgb]{1.00,0.93,1.00}{\tt Identity}}.

To defunctionalize this we define consider a type {\colorbox[rgb]{1.00,0.93,1.00}{\tt DefunIntInt}}, which is $Defun(Integer,Integer)$ above:

\begin{verbatim}data DefunIntInt = Multiply Integer DefunIntInt | Identity
eval :: DefunIntInt -> Integer -> Integer 
eval (Multiply x k) y = eval k (x * y)
eval (Identity) x = x\end{verbatim}
And now we can write the program without [[lambda abstractions]]:

\begin{verbatim}factcps :: Integer -> DefunIntInt -> Integer
factcps x k = if x == 0 then eval k 1 else factcps (x-1) (Multiply x k)

fact :: Integer -> Integer
fact x = factcps x Identity\end{verbatim}
In fact, in this example, the type {\colorbox[rgb]{1.00,0.93,1.00}{\tt DefunIntInt}} is isomorphic to the type of integer lists, and squinting slightly it can be seen as a call stack.

\hypertarget{references}{}\subsection*{{References}}\label{references}

Defunctionalization was probably introduced by John Reynolds in

\begin{itemize}%
\item [[John C. Reynolds]], Definitional Interpreters for Higher-Order Programming Languages. In Proceedings of the Annual ACM Conference. 1972. \href{https://homepages.inf.ed.ac.uk/wadler/papers/papers-we-love/reynolds-definitional-interpreters-1998.pdf}{pdf}

\end{itemize}
There and elsewhere a common theme is to first convert programs to [[continuation-passing style]], which makes the control stack explicit, and then to defunctionalize, which makes the control stack look like a stack; the net result is a kind-of abstract machine.



\end{document}