+ \subsection{Polymorphic functions}
+ A powerful construct in most functional language is polymorphism.
+ This means the arguments of a function (and consequentially, values
+ within the function as well) do not need to have a fixed type.
+ Haskell supports \emph{parametric polymorphism}, meaning a
+ function's type can be parameterized with another type.
+
+ As an example of a polymorphic function, consider the following
+ \hs{append} function's type:
+
+ TODO: Use vectors instead of lists?
+
+ \begin{code}
+ append :: [a] -> a -> [a]
+ \end{code}
+
+ This type is parameterized by \hs{a}, which can contain any type at
+ all. This means that append can append an element to a list,
+ regardless of the type of the elements in the list (but the element
+ added must match the elements in the list, since there is only one
+ \hs{a}).
+
+ This kind of polymorphism is extremely useful in hardware designs to
+ make operations work on a vector without knowing exactly what elements
+ are inside, routing signals without knowing exactly what kinds of
+ signals these are, or working with a vector without knowing exactly
+ how long it is. Polymorphism also plays an important role in most
+ higher order functions, as we will see in the next section.
+
+ The previous example showed unconstrained polymorphism (TODO: How is
+ this really called?): \hs{a} can have \emph{any} type. Furthermore,
+ Haskell supports limiting the types of a type parameter to specific
+ class of types. An example of such a type class is the \hs{Num}
+ class, which contains all of Haskell's numerical types.
+
+ Now, take the addition operator, which has the following type:
+
+ \begin{code}
+ (+) :: Num a => a -> a -> a
+ \end{code}
+
+ This type is again parameterized by \hs{a}, but it can only contain
+ types that are \emph{instances} of the \emph{type class} \hs{Num}.
+ Our numerical built-in types are also instances of the \hs{Num}
+ class, so we can use the addition operator on \hs{SizedWords} as
+ well as on {SizedInts}.
+
+ In \CLaSH, unconstrained polymorphism is completely supported. Any
+ function defined can have any number of unconstrained type
+ parameters. The \CLaSH compiler will infer the type of every such
+ argument depending on how the function is applied. There is one
+ exception to this: The top level function that is translated, can
+ not have any polymorphic arguments (since it is never applied, so
+ there is no way to find out the actual types for the type
+ parameters).
+
+ \CLaSH does not support user-defined type classes, but does use some
+ of the builtin ones for its builtin functions (like \hs{Num} and
+ \hs{Eq}).
+
+ \subsection{State}
+ A very important concept in hardware it the concept of state. In a
+ stateful design, the outputs depend on the history of the inputs, or the
+ state. State is usually stored in registers, which retain their value
+ during a clock cycle. As we want to describe more than simple
+ combinatorial designs, \CLaSH\ needs an abstraction mechanism for state.
+
+ An important property in Haskell, and in most other functional languages,
+ is \emph{purity}. A function is said to be \emph{pure} if it satisfies two
+ conditions:
+ \begin{inparaenum}
+ \item given the same arguments twice, it should return the same value in
+ both cases, and
+ \item when the function is called, it should not have observable
+ side-effects.
+ \end{inparaenum}
+ This purity property is important for functional languages, since it
+ enables all kinds of mathematical reasoning that could not be guaranteed
+ correct for impure functions. Pure functions are as such a perfect match
+ for a combinatorial circuit, where the output solely depends on the
+ inputs. When a circuit has state however, it can no longer be simply
+ described by a pure function. Simply removing the purity property is not a
+ valid option, as the language would then lose many of it mathematical
+ properties. In an effort to include the concept of state in pure
+ functions, the current value of the state is made an argument of the
+ function; the updated state becomes part of the result.
+
+ A simple example is the description of an accumulator circuit:
+ \begin{code}
+ acc :: Word -> State Word -> (State Word, Word)
+ acc inp (State s) = (State s', outp)
+ where
+ outp = s + inp
+ s' = outp
+ \end{code}
+ This approach makes the state of a function very explicit: which variables
+ are part of the state is completely determined by the type signature. This
+ approach to state is well suited to be used in combination with the
+ existing code and language features, such as all the choice constructs, as
+ state values are just normal values.