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\chapter{\texorpdfstring{\Glsxtrshort{DSL}}{DSL} embedding techniques}%
\label{chp:dsl_embedding_techniques}%
An \gls{EDSL} is a language embedded in a host language created for a specific domain\todo{citation needed?}.
Properties such as referential transparency, minimal syntax, powerful type systems and rich data types make \gls{FP} languages excellent candidates for hosting \glspl{EDSL}.

There are two flavours of \gls{DSL} embedding: deep- and shallow embedding~\citep{boulton_experience_1992}.
Shallow embedding---also called tagless embedding---models language constructs as functions in the host language.
As a result, adding new language constructs---extra functions---is easy.
However, the interpretation of the language is embedded in these functions, making it troublesome to add semantics since it requires updating all existing language constructs.

In contrast to shallow embedding, deep embedding---also called tagged embedding---models terms in the language as data types.
Interpretations are functions over these data types.

Consequently, adding new semantics, i.e.\ novel functions, is straightforward.
It can be stated that the language constructs are embedded in the functions that form a semantics.
If one wants to add a language construct, all semantics functions must be revisited and revised to avoid ending up with partial functions.

This juxtaposition has been known for many years~\citep{reynolds_user-defined_1978} and discussed by many others~\citep{krishnamurthi_synthesizing_1998} but most famously dubbed the \emph{expression problem} by Wadler~\citep{wadler_expression_1998}:

\begin{quote}
	The \emph{expression problem} is a new name for an old problem.
	The goal is to define a data type by cases, where one can add new cases to the data type and new functions over the data type, without recompiling existing code, and while retaining static type safety (e.g., no casts).
\end{quote}

Terms in an \glspl{EDSL} can have multiple interpretations\footnote{Interpretations are also called backends or views}, i.e.\ a term in the \gls{DSL} is just an interface.
Commonly used intepretations are printing, compiling, simulating, optimising, verifying, proving the program\etc.
There are two main flavours of embedding \glspl{DSL}.
Deep embedding---also called shallow--- models terms in the language as data types, interpretations are functions over these terms.
Shallow embedding---also called tagless---models terms in the language as functions, interpretations are embedded in these functions.

Most importantly, the two flavours differ on two axes: extensibility of language constructs and extensibility of interpretations.
\todo{elaborate}

\Cref{sec:deep_embedding} shows the basics of deep embedding.
\Cref{sec:shallow_embedding} shows the basics of shallow embedding including tagless embedding.
\Cref{sec:compare_embedding} compares the embedding technique.

In the following sections the basics of both techniques are explained.
A simple language with integers, booleans and some arithmetic operators is used as a running example.

\section{Deep embedding}\label{sec:deep_embedding}
In a deeply embedded \gls{DSL}, the language terms are represented as data type{(s)} in the host language.
Therefore, interpretations of the terms are functions that operate on these data types.
\Cref{lst:exdeep} shows an implementation for the example \gls{DSL}.

\begin{lstHaskell}[label={lst:exdeep},caption={A deeply embedded expression \gls{DSL}.}]
data Value = I Int | B Bool
data Expr
	= Lit  Value
	| Plus Expr Expr
	| Eq   Expr Expr
  deriving Show
\end{lstHaskell}

Implementing a printer for the language is straightforward, we just define a function that transforms the term to a string.

\begin{lstHaskell}[caption={A printer for the deeply embedded expression \gls{DSL}.}]
print :: Expr -> String
print (Lit i)    = show i
print (Plus l r) = "(" ++ print l ++ "+" ++ print r ++ ")"
print (Eq l r)   = "(" ++ print l ++ "==" ++ print r ++ ")"
\end{lstHaskell}

Adding a construct---for example subtraction---reveals the Achilles' heel of deep embedding, namely that we need to revisit the original data type \emph{and} all the existing views.
I.e.\ we need to add \haskellinline{| Sub Expr Expr} to the \haskellinline{Expr} data type.
Furthermore, we need to add \haskellinline{print (Sub l r) = ...} to the \haskellinline{print} view in order to not end up with a partial function.
This limitation can be overcome by lifting the views to classes (See \cref{chp:classy_deep_embedding}).

Implementing an evaluator for the language is possible without touching any original code, we just add a function operating on the \haskellinline{Expr} data type.
To store variables, it has an extra environment argument.
Here another downside of basic deep embedding arises immediately, the expressions are not typed, and therefore there has to be some type checking in the evaluation code.
Luckily this problem can be overcome by switching from regular \glspl{ADT} to \glspl{GADT}, resulting in the following data type and evaluator.

\begin{lstHaskell}[caption={An evaluator for the deeply embedded expression \gls{DSL}.}]
eval :: Expr -> Value
eval (Lit i)    = i
eval (Plus l r) = case (eval l, eval r) of
	(Lit (I l), Lit (I r)) -> I (l+r))
	(l, r)       -> error ("Can't add " ++ show l ++ " to " ++ show r)
eval (Eq l r) = case (eval l, eval r) of
	(Lit (I l), Lit (I r)) -> B (l==r)
	(Lit (B l), Lit (B r)) -> B (l==r)
	(l, r)       -> error ("Can't compare " ++ show l ++ " to " ++ show r)
\end{lstHaskell}

\subsection{\texorpdfstring{\Glsxtrlongpl{GADT}}{Generalised algebraic data types}}
Deep embedding has the advantage that it is easy to build and views are easy to add.
On the downside, the expressions created with this language are not necessarily type-safe.
In the given language it is possible to create an expression such as \haskellinline{LitI 4 `Plus` LitB True} that adds a boolean to an integer.
Extending the \gls{ADT} is easy and convenient but extending the views accordingly is tedious since it has to be done individually for all views.

The first downside of this type of \gls{EDSL} can be overcome by using \glspl{GADT}~\citep{cheney_first-class_2003}.
\Cref{lst:exdeepgadt} shows the same language, but type-safe with a \gls{GADT}.
\glspl{GADT} are not supported in the current version of \gls{CLEAN} and therefore the syntax is hypothetical (See \todo{insert link to appendix}).
However, it has been shown that \glspl{GADT} can be simulated using bimaps or projection pairs~\citep[\citesection{2.2}]{cheney_lightweight_2002}.
Unfortunately the lack of extendability remains a problem.
If a language construct is added, no compile time guarantee can be given that all views support it.

\begin{lstHaskell}[label={lst:exdeepgadt},caption={A deeply embedded expression \gls{DSL} using \glspl{GADT}.}]
data Expr a where
    Lit  :: Show a => a -> Expr a
    Plus :: Num a  => Expr a -> Expr a -> Expr a
    Eq   :: Eq a   => Expr a -> Expr a -> Expr Bool

eval :: Expr a -> a
eval (Lit i)    = i
eval (Plus l r) = eval l + eval r
eval (Eq l r)   = eval l == eval r
\end{lstHaskell}

\section{Shallow embedding}\label{sec:shallow_embedding}
In a shallowly \gls{EDSL} all language constructs are expressed as functions in the host language.
An evaluator view for the example language then can be implemented as the code shown in \cref{lst:exshallow}.
Note that the internals of the language could have been hidden using a reader monad.

\begin{lstHaskell}[label={lst:exshallow}, caption={A minimal shallow \gls{EDSL}.}]
type Env   = String -> Int
type DSL a = Env -> a

lit :: a -> DSL a
lit x = \e->x

var :: String -> DSL Int
var i = \e->retrEnv e i

plus :: DSL Int -> DSL Int -> DSL Int
plus x y = \e->x e + y e

eq :: Eq a => DSL a -> DSL a -> DSL Bool
eq x y = \e->x e == y e
\end{lstHaskell}

One of the advantages of shallowly embedding a language in a host language is its extendability.
It is very easy to add functionality because the compile time checks of the host language guarantee whether or not the functionality is available when used.
For example, adding a new construct---such as subtraction---is done as follows:

\begin{lstHaskell}[label={lst:exshallowsubst},caption={Adding subtraction to the shallow \gls{EDSL}.}]
sub :: DSL Int -> DSL Int -> DSL Int
sub x y = \e->x e - y e
\end{lstHaskell}

Moreover, the language is type safe as it is directly typed in the host language, i.e.\ \haskellinline{lit True `plus` lit 4} is rejected.
Another advantage is the intimate link with the host language, allowing for a lot more linguistic reuse such as the support of implicit sharing \citep{krishnamurthi_linguistic_2001}.

The downside of this method is extending the language with views.
It is nearly impossible to add views to a shallowly embedded language.
The only way of achieving this is by reimplementing all functions so that they run all backends at the same time or to create a single interpretation that produces a fold function \citep{gibbons_folding_2014}.

\subsection{Tagless-final embedding}\label{ssec:tagless}
By lifting the functions representing the \gls{DSL} terms to type classes, interpretations can be added.
This technique is called tagless-final---or class-based shallow---embedding.
The interface for the \gls{DSL} looks as follows:

\begin{lstHaskell}[label={lst:extagless},caption={A minimal tagless-final \gls{EDSL}.}]
class DSL v where
	lit :: a -> v a
	var :: String -> v a
	plus :: v Int -> v Int -> v Int
	eq :: Eq a => v a -> v a -> v Bool
\end{lstHaskell}

An interpretation of this view is a data type that implements the type class.

\begin{lstHaskell}[label={lst:extaglessprint},caption={A pretty printer for the tagless-final \gls{EDSL}.}]
data Print a = P {runPrint :: String}
instance DSL Print where
	lit a = P (show a)
	var i = P i
	plus x y = P ("(" ++ runPrint x ++ "+" ++ runPrint y ++ ")"
	eq x y = P ("(" ++ runPrint x ++ "==" ++ runPrint y ++ ")"
\end{lstHaskell}

Adding a language construct---e.g.\ subtraction---is a easy as adding a type class and providing instances for interpretations.

\begin{lstHaskell}[label={lst:extaglesssubt},caption={Adding subtraction to the shallow \gls{EDSL}.}]
class Sub v where
	sub :: v Int -> v Int -> v Int

instance Sub Print where
	sub x y = P ("(" ++ runPrint x ++ "-" ++ runPrint y ++ ")"
\end{lstHaskell}

Adding an interpretation means adding a data type and providing instances for the language constructs.

\begin{lstHaskell}[label={lst:extaglesseval},caption={An evaluator interpretation of the minimal tagless-final \gls{EDSL}.}]
data Eval a = Eval {runEval :: Env -> a}

instance DSL v where
	lit a = Eval (\_->a)
	var i = Eval (\e->retrEnv e i)
	plus x y = Eval (\e->runEval x e + runEval y e)
	eq x y = Eval (\e->runEval x e == runEval y e)

instance Sub Eval where
	sub x y = Eval (\e->runEval x e - runEval y e)
\end{lstHaskell}

\section{Comparison}\label{sec:compare_embedding}
Both flavours have their strengths and weaknesses and both flavours can be improved in order to mitigate (some of the) downsides.

\begin{table}[ht]
	\begin{threeparttable}[b]
		\caption{Comparison of embedding techniques, adapted from \citet[\citesection{3.6}]{sun_compositional_2022}}%
		\label{tbl:dsl_comparison}
		\begin{tabular}{lllllll}
			\toprule
									& Shallow & Deep  & Hybrid          & Poly           & Comp. & Classy\\
			\midrule
			Transcoding free        & yes     & yes   & no              & yes            & yes            & yes\\
			Linguistic reuse        & yes     & no    & partly\tnote{1} & yes            & yes            & no?\\
			Extend constructs       & yes     & no    & partly\tnote{1} & yes            & yes            & yes\\
			Extend interpretations  & no      & yes   & yes             & yes            & yes            & yes\\
			Transformations         & no      & yes   & yes             & maybe\tnote{2} & maybe\tnote{2} & yes\\
			Modular dependencies    & no      & maybe & maybe           & yes            & yes            & yes\\
			Nested pattern matching & no      & yes   & yes             & no             & maybe          & maybe\tnote{3}\\
			Type safe               & yes     & maybe & no              & yes            & yes            & yes\\
			\bottomrule
		\end{tabular}
		\begin{tablenotes}
			\item [1] Only in the shallowly embedded part.
			\item [2] Transformations require some ingenuity and are sometimes awkward to write.
			\item [3] It requires some---safe---form of dynamic typing.
		\end{tablenotes}
	\end{threeparttable}
\end{table}

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