An \gls{EDSL} is a language embedded in a host language. \glspl{EDSL} can have
one or more backends or views. Commonly used views are pretty printing,
compiling, simulating, verifying and proving the program.  There are several
techniques available for creating \glspl{EDSL}. Each of them have their own
advantages and disadvantages in terms of extendability, typedness and view
support. In the following subsections each of the main techniques are briefly
explained.

\section{Deep embedding}
A deep \gls{EDSL} is a language represented as an \gls{ADT}. Views are
functions that transform something to the datatype or the other way around. As
an example we have the simple arithmetic \gls{EDSL} shown in
Listing~\ref{lst:exdeep}.

\begin{lstlisting}[label={lst:exdeep},%
	caption={A minimal deep \gls{EDSL}}]
:: DSL
	= LitI  Int
	| LitB  Bool
	| Var   String
	| Plus  DSL DSL
	| Minus DSL DSL
	| And   DSL DSL
	| Eq    DSL
\end{lstlisting}

Deep embedding has the advantage that it is very simple to build and the views
are easy to add. To the downside, the expressions created with this language
are not type-safe. In the given language it is possible to create an expression
such as \CI{Plus (LitI 4) (LitB True)} that adds a boolean to an integer.
Evermore so, extending the \gls{ADT} is easy and convenient but extending the
views accordingly is tedious and has to be done individually for all views.

The first downside of this type of \gls{EDSL} can be overcome by using
\glspl{GADT}~\cite{cheney_first-class_2003}. Listing~\ref{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. However, it has been shown that \glspl{GADT} can be simulated
using bimaps or projection pairs~\cite{cheney_lightweight_2002}. Unfortunately
the lack of extendability remains a problem. If a language construct is added,
no compile time guarantee is given that all views support it.

\begin{lstlisting}[label={lst:exdeepgadt},%
	caption={A minimal deep \gls{EDSL} using \glspl{GADT}}]
:: DSL a
	=     LitI  Int                   -> DSL Int
	|     LitB  Bool                  -> DSL Bool
	| E.e: Var   String                -> DSL e
	|     Plus  (DSL Int) (DSL Int)   -> DSL Int
	|     Minus (DSL Int) (DSL Int)   -> DSL Int
	|     And   (DSL Bool) (DSL Bool) -> DSL Bool
	| E.e: Eq (DSL e) (DSL e)          -> DSL Bool &  == e
\end{lstlisting}

\section{Shallow embedding}
In a shallowly \gls{EDSL} all language constructs are expressed as functions in
the host language. An evaluator view for our example language then looks
something like the code shown in Listing~\ref{lst:exshallow}. Note that much of
the internals of the language can be hidden using monads.

\begin{lstlisting}[label={lst:exshallow},%
	caption={A minimal shallow \gls{EDSL}}]
:: Env   = ...            // Some environment
:: DSL a = DSL (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 :: (DSL a) (DSL a) -> DSL Bool | == a
Eq x y = \e -> x e + y e
\end{lstlisting}

The advantage of shallowly embedding a language in a host language is its
extendability. It is very easy to add functionality and compile time checks
of the host language guarantee whether or not the functionality is available
when used. Moreover, the language is type safe as it is directly typed in the
host language.

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 decorating the datatype for the \gls{EDSL} with all the
information for all the views. This will mean that every component will have to
implement all views rendering it slow for multiple views and complex to
implement.

\section{Class based shallow embedding}
The third type of embedding is called class-based shallow embedding and has the
advantages of both shallow and deep embedding. In class-based shallow embedding
the language constructs are defined as type classes. This language is shown
with the new method in Listing~\ref{lst:exclassshallow}.

This type of embedding inherits the easiness of adding views from shallow
embedding. A view is just a different data type implementing one or more of the
type classes as shown in the aforementioned Listing where an evaluator and a
pretty printer are implemented.

Just as with \glspl{GADT}, type safety is guaranteed in deep embedding. Type
constraints are enforced through phantom types. One can add as many phantom
types as necessary. Lastly, extensions can be added easily, just as in
shallow embedding. When an extension is made in an existing class, all views
must be updated accordingly to prevent possible runtime errors. When an
extension is added in a new class, this problem does not arise and views can
choose to implement only parts of the collection of classes.

In contrast to deep embedding, it is very well possible to have multiple views
applied on the same expression. This is also shown in the following listing.

\begin{lstlisting}[label={lst:exclassshallow},%
	caption={A minimal class based shallow \gls{EDSL}}]
:: Env   = ...            // Some environment
:: Evaluator a = Evaluator (Env -> a)
:: PrettyPrinter a = PP String

class intArith where
	lit   :: t -> v t             | toString t
	add   :: (v t) (v t) -> (v t) | + t
	minus :: (v t) (v t) -> (v t) | - t

class boolArith where
	and :: (v Bool) (v Bool) -> (v Bool)
	eq  :: (v t)    (v t)    -> (v Bool) | == t

instance intArith Evaluator where
	lit x = \e->x
	add x y = ...

instance intArith PrettyPrinter where
	lit x = toString x
	add x y = x +++ "+" +++ y
	...

...

Start :: (PP String, Bool)
Start = (print e0, eval e0)
where
	e0 :: a Bool | intArith, boolArith a
	e0 = eq (lit 42) (lit 21 +. lit 21)

	print (PP p) = p
	eval (Evaluator e) env = e env
\end{lstlisting}
