Update text on fir filter use-case

[matthijs/master-project/dsd-paper.git] / cλash.lhs

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% Macro for certain acronyms in small caps. Doesn't work with the
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\def\acro#1{{\small{#1}}}
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\def\acrotiny#1{{\scriptsize{#1}}}
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\def\CLaSH{{\small{C}}$\lambda$a{\small{SH}}}
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\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{C$\lambda$aSH: Structural Descriptions \\ of Synchronous Hardware using Haskell}


% author names and affiliations
% use a multiple column layout for up to three different
% affiliations
\author{\IEEEauthorblockN{Christiaan P.R. Baaij, Matthijs Kooijman, Jan Kuper, Marco E.T. Gerards}%, Bert Molenkamp, Sabih H. Gerez}
\IEEEauthorblockA{%Computer Architecture for Embedded Systems (CAES)\\ 
Department of EEMCS, University of Twente\\
P.O. Box 217, 7500 AE, Enschede, The Netherlands\\
c.p.r.baaij@@utwente.nl, matthijs@@stdin.nl, j.kuper@@utwente.nl}
\thanks{Supported through the FP7 project: S(o)OS (248465)}
}
% \and
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% Springfield, USA\\
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% \IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
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%Homer Simpson\IEEEauthorrefmark{2},
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%Montgomery Scott\IEEEauthorrefmark{3} and
%Eldon Tyrell\IEEEauthorrefmark{4}}
%\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\
%Georgia Institute of Technology,
%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
%\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\
%Email: homer@thesimpsons.com}
%\IEEEauthorblockA{\IEEEauthorrefmark{3}Starfleet Academy, San Francisco, California 96678-2391\\
%Telephone: (800) 555--1212, Fax: (888) 555--1212}
%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}




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% make the title area
\maketitle

\begin{abstract}
%\boldmath
\CLaSH\ is a functional hardware description language that borrows both its 
syntax and semantics from the functional programming language Haskell. 
Polymorphism and higher-order functions provide a level of abstraction and 
generality that allow a circuit designer to describe circuits in a more 
natural way than possible in a traditional hardware description language.

Circuit descriptions can be translated to synthesizable VHDL using the 
prototype \CLaSH\ compiler. As the circuit descriptions, simulation code, and 
test input are also valid Haskell, complete simulations can be compiled as an 
executable binary by a Haskell compiler allowing high-speed simulation and 
analysis.

% \CLaSH\ supports stateful descriptions by explicitly making the current 
% state an argument of the function, and the updated state part of the result. 
% This makes \CLaSH\ descriptions in essence the combinational parts of a 
% mealy machine.
\end{abstract}
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% math in the abstract anyway.

% no keywords




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\section{Introduction}
Hardware description languages (\acrop{HDL}) have not allowed the productivity 
of hardware engineers to keep pace with the development of chip technology. 
While traditional \acrop{HDL}, like \VHDL~\cite{VHDL2008} and 
Verilog~\cite{Verilog}, are very good at describing detailed hardware 
properties such as timing behavior, they are generally cumbersome in 
expressing the higher-level abstractions needed for today's large and complex 
circuit designs. In an attempt to raise the abstraction level of the 
descriptions, a great number of approaches based on functional languages has 
been proposed \cite{Cardelli1981,muFP,DAISY,T-Ruby,HML2,Hydra,Hawk1,Lava,
Wired,ForSyDe1,reFLect}. The idea of using functional languages for hardware 
descriptions started in the early 1980s \cite{Cardelli1981,muFP,DAISY}, a 
time which also saw the birth of the currently popular \acrop{HDL}, such as 
\VHDL. Functional languages are especially well suited to describe hardware 
because combinational circuits can be directly modeled as mathematical 
functions and functional languages are very good at describing and composing 
these functions.

In an attempt to ease the prototyping process of the language, such as 
creating all the required tooling, like parsers and type-checkers, many 
functional \acrop{HDL} \cite{Hydra,Hawk1,Lava,Wired} are embedded as a domain 
specific language (\acro{DSL}) within the functional language Haskell 
\cite{Haskell}. This means that a developer is given a library of Haskell 
functions and types that together form the language primitives of the 
\acro{DSL}. The primitive functions used to describe a circuit do not actually 
process any signals, they instead compose a large domain-specific graph 
(which is usually hidden from the designer). This graph is then further 
processed by an embedded circuit compiler which can perform for example 
simulation or synthesis. As Haskell's choice elements (\hs{case}-expressions, 
pattern-matching etc.) are evaluated at the time the domain-specific graph is 
being build, they are no longer visible to the embedded compiler that 
processes the datatype. Consequently, it is impossible to capture Haskell's 
choice elements within a circuit description when taking the embedded language 
approach. This does not mean that circuits specified in an embedded language 
can not contain choice, just that choice elements only exists as functions, 
e.g. a multiplexer function, and not as language elements.

The approach taken in this research is to use (a subset of) the Haskell 
language \emph{itself} for the purpose of describing hardware. By taking this 
approach, this research \emph{can} capture certain language constructs, like 
all of Haskell's choice elements, within circuit descriptions. The more 
advanced features of Haskel, such as polymorphic typing and higher-order 
function, are also supported.

% supporting polymorphism, higher-order functions and such an extensive array 
% of choice-elements, combined with a very concise way of specifying circuits 
% is new in the domain of (functional) \acrop{HDL}. 
% As the hardware descriptions are plain Haskell 
% functions, these descriptions can be compiled to an executable binary
% for simulation using an optimizing Haskell compiler such as the Glasgow
% Haskell Compiler (\GHC)~\cite{ghc}.

Where descriptions in a conventional \acro{HDL} have an explicit clock for the 
purposes state and synchronicity, the clock is implicit for the descriptions and research presented in this paper. A circuit designer describes the behavior of the hardware between clock cycles. Many functional \acrop{HDL} model signals as a stream of all values over time; state is then modeled as a delay on this stream of values. Descriptions presented in this research make the current state an additional input and the updated state a part of their output. This abstraction of state and time limits the descriptions to synchronous hardware, there is however room within the language to eventually add a different abstraction mechanism that will allow for the modeling of asynchronous systems.

Like the traditional \acrop{HDL}, descriptions made in a functional \acro{HDL} 
must eventually be converted into a netlist. This research also features a 
prototype translator, which has the same name as the language: 
\CLaSH\footnote{\CLaSHtiny: \acrotiny{CAES} Language for Synchronous Hardware} 
(pronounced: clash). This compiler converts the Haskell code to equivalently 
behaving synthesizable \VHDL\ code, ready to be converted to an actual netlist 
format by an (optimizing) \VHDL\ synthesis tool.

To the best knowledge of the authors, \CLaSH\ is the only (functional) 
\acro{HDL} that allows circuit specification to be written in a very concise 
way and at the same time support such advanced features as polymorphic typing, 
user-defined higher-order functions and pattern matching.

\section{Hardware description in Haskell}
The following section describes the basic language elements of \CLaSH\ and the 
extensiveness of the support of these elements within the \CLaSH\ compiler. In 
various subsections, the relation between the language elements and their 
eventual netlist representation is also highlighted. 

  \subsection{Function application}
    Two basic elements of a functional program are functions and function 
    application. These have a single obvious translation to a netlist format: 
    \begin{inparaenum}
      \item every function is translated to a component, 
      \item every function argument is translated to an input port,
      \item the result value of a function is translated to an output port, 
            and
      \item function applications are translated to component instantiations.
    \end{inparaenum} 
    The result value can have a composite type (such as a tuple), so having 
    just a single result value does not pose any limitation. The actual 
    arguments of a function application are assigned to signals, which are 
    then mapped to the corresponding input ports of the component. The output 
    port of the function is also mapped to a signal, which is used as the 
    result of the application itself. Since every top level function generates 
    its own component, the hierarchy of function calls is reflected in the 
    final netlist. %, creating a hierarchical description of the hardware. 
    % The separation in different components makes it easier for a developer 
    % to understand and possibly hand-optimize the resulting \VHDL\ output of 
    % the \CLaSH\ compiler.

    The short example (\ref{code:mac}) seen below gives a demonstration of 
    the conciseness that can be achieved with \CLaSH\ when compared with 
    other (more traditional) \acrop{HDL}. The example is a combinational 
    multiply-accumulate circuit that works for \emph{any} word length (this 
    type of polymorphism will be further elaborated in 
    \Cref{sec:polymorhpism}). The corresponding netlist is depicted in 
    \Cref{img:mac-comb}.
    
    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    mac a b c = add (mul a b) c
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:mac}
      \end{example}
    \end{minipage}
    
    \begin{figure}
    \centerline{\includegraphics{mac.svg}}
    \caption{Combinational Multiply-Accumulate}
    \label{img:mac-comb}
    \vspace{-1.5em}
    \end{figure}
    
    The use of a composite result value is demonstrated in the next example 
    (\ref{code:mac-composite}), where the multiply-accumulate circuit not only 
    returns the accumulation result, but also the intermediate multiplication 
    result. Its corresponding netlist can be seen in 
    \Cref{img:mac-comb-composite}.
    
    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    mac a b c = (z, add z c)
      where
        z = mul a b
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:mac-composite}
      \end{example}
    \end{minipage}
    \vspace{-1.5em}
    
    \begin{figure}
    \vspace{1em}
    \centerline{\includegraphics{mac-nocurry.svg}}
    \caption{Combinational Multiply-Accumulate (composite output)}
    \label{img:mac-comb-composite}
    \vspace{-1.5em}
    \end{figure}

  \subsection{Choice}
    In Haskell, choice can be achieved by a large set of syntactic elements, 
    consisting of: \hs{case} expressions, \hs{if-then-else} expressions, 
    pattern matching, and guards. The most general of these are the \hs{case} 
    expressions (\hs{if} expressions can be directly translated to 
    \hs{case} expressions). When transforming a \CLaSH\ description to a   
    netlist, a \hs{case} expression is translated to a multiplexer. The 
    control value of the \hs{case} expression is fed into a number of 
    comparators and their combined output forms the selection port of the 
    multiplexer. The result of each alternative in the \hs{case} expression is 
    linked to the corresponding input port of the multiplexer.
    % A \hs{case} expression can in turn simply be translated to a conditional 
    % assignment in \VHDL, where the conditions use equality comparisons 
    % against the constructors in the \hs{case} expressions. 
    
    % Two versions of a contrived example are displayed below, the first  
    % (\ref{lst:code3}) using a \hs{case} expression and the second 
    % (\ref{lst:code4}) using an \hs{if-then-else} expression. Both examples 
    % sum two values when they are equal or non-equal (depending on the given 
    % predicate, the \hs{pred} variable) and return 0 otherwise. 
    
    An code example (\ref{code:counter1}) that uses a \hs{case} expression and 
    \hs{if-then-else} expressions is shown below. The function counts up or 
    down depending on the \hs{direction} variable, and has a \hs{wrap} 
    variable that determines both the upper bound and wrap-around point of the 
    counter. The \hs{direction} variable is of the following, user-defined, 
    enumeration datatype:
    
    \begin{code}
    data Direction = Up | Down
    \end{code}

    The naive netlist corresponding to this example is depicted in 
    \Cref{img:counter}. Note that the \hs{direction} variable is only
    compared to \hs{Up}, as an inequality immediately implies that 
    \hs{direction} is \hs{Down}.

    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}    
    counter direction wrap x = case direction of
        Up    -> if   x < wrap  then 
                      x + 1     else 
                      0
        Down  -> if   x > 0   then 
                      x - 1   else 
                      wrap
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:counter1}
      \end{example}
    \end{minipage}
    
    % \hspace{-1.7em}
    % \begin{minipage}{0.93\linewidth}
    % \begin{code}
    % sumif pred a b = 
    %   if pred == Equal then 
    %     if a == b then a + b else 0
    %   else 
    %     if a != b then a + b else 0
    % \end{code}
    % \end{minipage}
    % \begin{minipage}{0.07\linewidth}
    %   \begin{example}
    %   \label{lst:code4}
    %   \end{example}
    % \end{minipage}

    % \begin{figure}
    % \vspace{1em}
    % \centerline{\includegraphics{choice-case.svg}}
    % \caption{Choice - sumif}
    % \label{img:choice}
    % \vspace{-1.5em}
    % \end{figure}

    \begin{figure}
    \centerline{\includegraphics{counter.svg}}
    \caption{Counter netlist}
    \label{img:counter}
    \vspace{-2em}
    \end{figure}

    A user-friendly and also very powerful form of choice that is not found in 
    the traditional hardware description languages is pattern matching. A 
    function can be defined in multiple clauses, where each clause corresponds 
    to a pattern. When an argument matches a pattern, the corresponding clause 
    will be used. Expressions can also contain guards, where the expression is 
    only executed if the guard evaluates to true, and continues with the next 
    clause if the guard evaluates to false. Like \hs{if-then-else} 
    expressions, pattern matching and guards have a (straightforward) 
    translation to \hs{case} expressions and can as such be mapped to 
    multiplexers. A second version (\ref{code:counter2}) of the earlier 
    example, now using both pattern matching and guards, can be seen below. 
    The guard is the expression that follows the vertical bar (\hs{|}) and 
    precedes the assignment operator (\hs{=}). The \hs{otherwise} guards 
    always evaluate to \hs{true}.
    
    The version using pattern matching and guards corresponds to the same 
    naive netlist representation (\Cref{img:counter}) as the earlier example.
    
    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    counter Up    wrap x  | x < wrap    = x + 1
                          | otherwise   = 0
    counter Down  wrap x  | x > 0       = x - 1
                          | otherwise   = wrap
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:counter2}
      \end{example}
    \end{minipage}

    % \begin{figure}
    % \centerline{\includegraphics{choice-ifthenelse}}
    % \caption{Choice - \emph{if-then-else}}
    % \label{img:choice}
    % \end{figure}

  \subsection{Types}
    Haskell is a statically-typed language, meaning that the type of a 
    variable or function is determined at compile-time. Not all of Haskell's 
    typing constructs have a clear translation to hardware, this section will 
    therefore only deal with the types that do have a clear correspondence 
    to hardware. The translatable types are divided into two categories: 
    \emph{built-in} types and \emph{user-defined} types. Built-in types are 
    those types for which a fixed translation is defined within the \CLaSH\ 
    compiler. The \CLaSH\ compiler has generic translation rules to
    translate the user-defined types, which are described later on.

    The \CLaSH\ compiler is able to infer unspecified (polymorphic) types,
    meaning that a developer does not have to annotate every function with a 
    type signature. % (even if it is good practice to do so).
    Given that the top-level entity of a circuit design is annotated with 
    concrete/monomorphic types, the \CLaSH\ compiler can specialize 
    polymorphic functions to functions with concrete types.
  
    % Translation of two most basic functional concepts has been
    % discussed: function application and choice. Before looking further
    % into less obvious concepts like higher-order expressions and
    % polymorphism, the possible types that can be used in hardware
    % descriptions will be discussed.
    % 
    % Some way is needed to translate every value used to its hardware
    % equivalents. In particular, this means a hardware equivalent for
    % every \emph{type} used in a hardware description is needed.
    % 
    % The following types are \emph{built-in}, meaning that their hardware
    % translation is fixed into the \CLaSH\ compiler. A designer can also
    % define his own types, which will be translated into hardware types
    % using translation rules that are discussed later on.

  \subsubsection{Built-in types}
    The following types have fixed translations defined within the \CLaSH\
    compiler:
    \begin{xlist}
      \item[\bf{Bit}]
        the most basic type available. It can have two values:
        \hs{Low} or \hs{High}. 
        % It is mapped directly onto the \texttt{std\_logic} \VHDL\ type. 
      \item[\bf{Bool}]
        this is a basic logic type. It can have two values: \hs{True}
        or \hs{False}. 
        % It is translated to \texttt{std\_logic} exactly like the \hs{Bit} 
        % type (where a value of \hs{True} corresponds to a value of 
        % \hs{High}). 
        Supporting the Bool type is required in order to support the
        \hs{if-then-else} expression, which requires a \hs{Bool} value for 
        the condition.
      \item[\bf{Signed}, \bf{Unsigned}]
        these are types to represent integers and both are parametrizable in 
        their size. The overflow behavior of the numeric operators defined for 
        these types is \emph{wrap-around}.
        % , so you can define an unsigned word of 32 bits wide as follows:

        % \begin{code}
        % type Word32 = SizedWord D32
        % \end{code}

        % Here, a type synonym \hs{Word32} is defined that is equal to the
        % \hs{SizedWord} type constructor applied to the type \hs{D32}. 
        % \hs{D32} is the \emph{type level representation} of the decimal 
        % number 32, making the \hs{Word32} type a 32-bit unsigned word. These 
        % types are translated to the \VHDL\ \texttt{unsigned} and 
        % \texttt{signed} respectively.
      \item[\bf{Vector}]
        this is a vector type that can contain elements of any other type and
        has a static length. The \hs{Vector} type constructor takes two type 
        arguments: the length of the vector and the type of the elements 
        contained in it. The short-hand notation used for the vector type in  
        the rest of paper is: \hs{[a|n]}, where \hs{a} is the element 
        type, and \hs{n} is the length of the vector. Note that this is
        a notation used in this paper only, vectors are slightly more
        verbose in real \CLaSH\ descriptions.
        % The state type of an 8 element register bank would then for example 
        % be:

        % \begin{code}
        % type RegisterState = Vector D8 Word32
        % \end{code}

        % Here, a type synonym \hs{RegisterState} is defined that is equal to
        % the \hs{Vector} type constructor applied to the types \hs{D8} (The 
        % type level representation of the decimal number 8) and \hs{Word32} 
        % (The 32 bit word type as defined above). In other words, the 
        % \hs{RegisterState} type is a vector of 8 32-bit words. A fixed size 
        % vector is translated to a \VHDL\ array type.
      \item[\bf{Index}]
        the main purpose of the \hs{Index} type is to be used as an index into 
        a \hs{Vector}, and has no specified bit-size, but a specified upper 
        bound. This means that its range is not limited to powers of two, but 
        can be any number. An \hs{Index} only has an upper bound, its lower 
        bound is implicitly zero. If a value of this type exceeds either 
        bounds, an error will be thrown at \emph{simulation}-time. 

        % \comment{TODO: Perhaps remove this example?} To define an index for 
        % the 8 element vector above, we would do:

        % \begin{code}
        % type RegisterIndex = RangedWord D7
        % \end{code}

        % Here, a type synonym \hs{RegisterIndex} is defined that is equal to
        % the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
        % other words, this defines an unsigned word with values from
        % 0 to 7 (inclusive). This word can be be used to index the
        % 8 element vector \hs{RegisterState} above. This type is translated 
        % to the \texttt{unsigned} \VHDL type.
    \end{xlist}

  \subsubsection{User-defined types}
    % There are three ways to define new types in Haskell: algebraic
    % data-types with the \hs{data} keyword, type synonyms with the \hs{type}
    % keyword and datatype renaming constructs with the \hs{newtype} keyword. 
    % \GHC\ offers a few more advanced ways to introduce types (type families,
    % existential typing, {\acro{GADT}}s, etc.) which are not standard 
    % Haskell. As it is currently unclear how these advanced type constructs 
    % correspond to hardware, they are for now unsupported by the \CLaSH\ 
    % compiler.
    A completely new type is introduced by an algebraic datatype declaration 
    which is defined using the \hs{data} keyword. Type synonyms can be 
    introduced using the \hs{type} keyword.
    % Only an algebraic datatype declaration actually introduces a
    % completely new type. Type synonyms and type renaming only define new 
    % names for existing types, where synonyms are completely interchangeable 
    % and a type renaming requires an explicit conversion. 
    Type synonyms do not need any particular translation, as a synonym  will 
    just use the same representation as the original type. 
    
    Algebraic datatypes can be categorized as follows:
    \begin{xlist}
      \item[\bf{Single constructor}]
        Algebraic datatypes with a single constructor with one or more
        fields allow values to be packed together in a record-like structure. 
        Haskell's built-in tuple types are also defined as single constructor 
        algebraic types (using a bit of syntactic sugar). An example of a 
        single constructor type with multiple fields is the following pair of 
        integers:
        \begin{code}
        data IntPair = IntPair Int Int
        \end{code}
        % These types are translated to \VHDL\ record types, with one field 
        % for every field in the constructor.
      \item[\bf{No fields}]
        Algebraic datatypes with multiple constructors, but without any
        fields are essentially enumeration types. Note that Haskell's 
        \hs{Bool} type is also defined as an enumeration type, but that there 
        is a fixed translation for that type within the \CLaSH\ compiler. An 
        example of an enumeration type definition is the definition for a 
        traffic light:
        \begin{code}
        data TrafficLight = Red | Orange | Green
        \end{code}
        % These types are translated to \VHDL\ enumerations, with one 
        % value for each constructor. This allows references to these 
        % constructors to be translated to the corresponding enumeration 
        % value.
      \item[\bf{Multiple constructors with fields}]
        Algebraic datatypes with multiple constructors, where at least
        one of these constructors has one or more fields are currently not 
        supported. Additional research is required to allow for the overlap of
        the fields belonging to the different constructors.
    \end{xlist}

  \subsection{Polymorphism}\label{sec:polymorhpism}
    A powerful feature of most (functional) programming languages is 
    polymorphism, it allows a function to handle values of different data 
    types in a uniform way. Haskell supports \emph{parametric 
    polymorphism}~\cite{polymorphism}, meaning functions can be written 
    without mention of any specific type and can be used transparently with 
    any number of new types.

    As an example of a parametric polymorphic function, consider the type of 
    the following \hs{first} function, which returns the first element of a 
    tuple:\footnote{The \hs{::} operator is used to annotate a function
    with its type.}
    
    \begin{code}
    first :: (a,b) -> a
    \end{code}

    This type is parameterized in both \hs{a} and \hs{b}, which can both 
    represent any type at all (as long as that type is supported by the 
    \CLaSH\ compiler). This means that \hs{first} works for any tuple, 
    regardless of what elements it contains. This kind of polymorphism is 
    extremely useful in hardware designs, for example when routing signals 
    without knowing their exact type, or specifying vector operations that 
    work on vectors of any length and element type. Polymorphism also plays an 
    important role in most higher order functions, as will be shown in the 
    next section.

    Another type of polymorphism is \emph{ad-hoc 
    polymorphism}~\cite{polymorphism}, which refers to polymorphic 
    functions which can be applied to arguments of different types, but which 
    behave differently depending on the type of the argument to which they are 
    applied. In Haskell, ad-hoc polymorphism is achieved through the use of 
    \emph{type classes}, where a class definition provides the general 
    interface of a function, and class \emph{instances} define the 
    functionality for the specific types. An example of such a type class is 
    the \hs{Num} class, which contains all of Haskell's numerical operations. 
    A designer can make use of this ad-hoc polymorphism by adding a 
    \emph{constraint} to a parametrically polymorphic type variable. Such a 
    constraint indicates that the type variable can only be instantiated to a 
    type whose members supports the overloaded functions associated with the 
    type class. 
    
    An example of a type signature that includes such a constraint if the 
    signature of the \hs{sum} function, which sums the values in a vector:
    \begin{code}
    sum :: Num a => [a|n] -> 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}, so that  
    the compiler knows that the addition (+) operator is defined for that 
    type.
    
    A place where class constraints also play a role is in the size and range 
    parameters of the \hs{Vector} and numeric types. The reason being that 
    these parameters have to be limited to types that can represent 
    \emph{natural} numbers. The complete type of for example the \hs{Vector} 
    type is:
    \begin{code}
    Natural n => Vector n a
    \end{code}    
    
    % \CLaSH's built-in numerical types are also instances of the \hs{Num} 
    % class. 
    % so we can use the addition operator (and thus the \hs{sum}
    % function) with \hs{Signed} as well as with \hs{Unsigned}.

    \CLaSH\ supports both parametric polymorphism and ad-hoc polymorphism. Any 
    function defined can have any number of unconstrained type parameters. A
    circuit designer can also specify his own type classes and corresponding 
    instances. The \CLaSH\ compiler will infer the type of every polymorphic 
    argument depending on how the function is applied. There is however one 
    constraint: the top level function that is being translated can not have 
    any polymorphic arguments. The arguments of the top-level can not be 
    polymorphic as the function is never applied and consequently there is no 
    way to determine the actual types for the type parameters. 
    
    With regard to the built-in types, it should be noted that members of 
    some of the standard Haskell type classes are supported as built-in 
    functions. These include: the numerial operators of \hs{Num}, the equality 
    operators of \hs{Eq}, and the comparison/order operators of \hs{Ord}.

  \subsection{Higher-order functions \& values}
    Another powerful abstraction mechanism in functional languages, is
    the concept of \emph{functions as a first class value}, also called 
    \emph{higher-order functions}. This allows a function to be treated as a
    value and be passed around, even as the argument of another
    function. The following example should clarify this concept:
    
    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    %format not = "\mathit{not}"
    \begin{code}
    negateVector xs = map not xs
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:negatevector}
      \end{example}
    \end{minipage}

    The code above defines the \hs{negateVector} function, which takes a 
    vector of booleans, \hs{xs}, and returns a vector where all the values are 
    negated. It achieves this by calling the \hs{map} function, and passing it 
    \emph{another function}, boolean negation, and the vector of booleans, 
    \hs{xs}. The \hs{map} function applies the negation function to all the 
    elements in the vector.

    The \hs{map} function is called a higher-order function, since it takes 
    another function as an argument. Also note that \hs{map} is again a 
    parametric polymorphic function: it does not pose any constraints on the 
    type of the input vector, other than that its elements must have the same 
    type as the first argument of the function passed to \hs{map}. The element 
    type of the resulting vector is equal to the return type of the function 
    passed, which need not necessarily be the same as the element type of the 
    input vector. All of these characteristics can be inferred from the type 
    signature belonging to \hs{map}:

    \begin{code}
    map :: (a -> b) -> [a|n] -> [b|n]
    \end{code}

    So far, only functions have been used as higher-order values. In
    Haskell, there are two more ways to obtain a function-typed value:
    partial application and lambda abstraction. Partial application
    means that a function that takes multiple arguments can be applied
    to a single argument, and the result will again be a function (but
    that takes one argument less). As an example, consider the following
    expression, that adds one to every element of a vector:

    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    map (add 1) xs
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:partialapplication}
      \end{example}
    \end{minipage}

    Here, the expression \hs{(add 1)} is the partial application of the
    addition function to the value \hs{1}, which is again a function that
    adds one to its (next) argument. A lambda expression allows one to 
    introduce an anonymous function in any expression. Consider the following 
    expression, which again adds one to every element of a vector:

    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    map (\x -> x + 1) xs
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:lambdaexpression}
      \end{example}
    \end{minipage}

    Finally, not only built-in functions can have higher order arguments (such 
    as the \hs{map} function), but any function defined in \CLaSH\ may have 
    functions as arguments. This allows the circuit designer to use a 
    powerful amount of code reuse. The only exception is again the top-level 
    function: if a function-typed argument is not applied with an actual 
    function, no hardware can be generated.    

    An example of a common circuit where higher-order functions and partial 
    application lead to a very concise and natural description is a crossbar.
    The code (\ref{code:crossbar}) for this example can be seen below:
    
    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    crossbar inputs selects = map (mux inputs) selects
      where
        mux inp x = (inp ! x)
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:crossbar}
      \end{example}
    \end{minipage}

    The crossbar is polymorphic in the width of the input (defined by the 
    length of \hs{inputs}), the width of the output (defined by the length of
    \hs{selects}), and the signal type (defined by the element type of 
    \hs{inputs}). The type-checker can also automatically infer that 
    \hs{selects} is a vector of \hs{Index} values due to the use of the vector
    indexing operator (\hs{!}).

  \subsection{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 \CLaSH\ has to be able to describe more than 
    simple combinational designs, there is a need for an abstraction mechanism 
    for state.

    An important property in Haskell, and in many 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 that the function has no 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 combinational circuits, 
    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 \CLaSH\ deals with the concept of state in pure functions by making 
    the current state an additional argument of the function, and the 
    updated state part of result. In this sense the descriptions made in 
    \CLaSH\ are the combinational parts of a mealy machine.
    
    A simple example is adding an accumulator register to the earlier 
    multiply-accumulate circuit, of which the resulting netlist can be seen in 
    \Cref{img:mac-state}:
    
    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    macS (State c) a b = (State c', c')
      where
        c' = mac a b c
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:macstate}
      \end{example}
    \end{minipage}
    
    Note that the \hs{macS} function returns both the new state and the value
    of the output port. The \hs{State} wrapper indicates which arguments are 
    part of the current state, and what part of the output is part of the 
    updated state. This aspect will also be reflected in the type signature of 
    the function. Abstracting the state of a circuit in this way makes it 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 elements, as state values are just normal values. Stateful   
    descriptions are simulated using the recursive \hs{run} function:
    
    \hspace{-1.7em}
    \begin{minipage}{0.93\linewidth}
    \begin{code}
    run f s (i : inps) = o : (run f s' inps)
      where
        (s', o) = f s i
    \end{code}
    \end{minipage}
    \begin{minipage}{0.07\linewidth}
      \begin{example}
      \label{code:run}
      \end{example}
    \end{minipage}
    
    The \hs{(:)} operator is the list concatenation operator, where the 
    left-hand side is the head of a list and the right-hand side is the 
    remainder of the list. The \hs{run} function applies the function the 
    developer wants to simulate, \hs{f}, to the current state, \hs{s}, and the 
    first input value, \hs{i}. The result is the first output value, \hs{o}, 
    and the updated state \hs{s'}. The next iteration of the \hs{run} function 
    is then called with the updated state, \hs{s'}, and the rest of the 
    inputs, \hs{inps}. For the time being, and in the context of this paper, 
    it is assumed that there is one input per clock cycle. Also note how the 
    order of the input, output, and state in the \hs{run} function corresponds 
    with the order of the input, output and state of the \hs{macS} function 
    described earlier.

    \begin{figure}
    \centerline{\includegraphics{mac-state.svg}}
    \caption{Stateful Multiply-Accumulate}
    \label{img:mac-state}
    \vspace{-1.5em}
    \end{figure}
    
    As the \hs{run} function, the hardware description, and the test 
    inputs are also valid Haskell, the complete simulation can be compiled to 
    an executable binary by an optimizing Haskell compiler, or executed in an 
    Haskell interpreter. Both simulation paths require less effort from a 
    circuit designer than first translating the description to \VHDL\ and then 
    running a \VHDL\ simulation; it is also very likely that both simulation 
    paths are much faster.
    
\section{The \CLaSH\ compiler}
An important aspect in this research is the creation of the prototype 
compiler, which allows us to translate descriptions made in the \CLaSH\ 
language as described in the previous section to synthesizable \VHDL. 
% , allowing a designer to actually run a \CLaSH\ design on an \acro{FPGA}.

The Glasgow Haskell Compiler (\GHC)~\cite{ghc} is an open-source Haskell 
compiler that also provides a high level API to most of its internals. The 
availability of this high-level API obviated the need to design many of the 
tedious parts of the prototype compiler, such as the parser, semantics 
checker, and especially the type-checker. These parts together form the 
front-end of the prototype compiler pipeline, as seen in
\Cref{img:compilerpipeline}.

\begin{figure}
\vspace{1em}
\centerline{\includegraphics{compilerpipeline.svg}}
\caption{\CLaSHtiny\ compiler pipeline}
\label{img:compilerpipeline}
\vspace{-1.5em}
\end{figure}

The output of the \GHC\ front-end consists of the translation of the original 
Haskell description to \emph{Core}~\cite{Sulzmann2007}, which is a smaller, 
typed, functional language. This \emph{Core} language is relatively easy to 
process compared to the larger Haskell language. A description in \emph{Core} 
can still contain elements which have no direct translation to hardware, such 
as polymorphic types and function-valued arguments. Such a description needs 
to be transformed to a \emph{normal form}, which only contains elements that 
have a direct translation. The second stage of the compiler, the 
\emph{normalization} phase, exhaustively applies a set of 
\emph{meaning-preserving} transformations on the \emph{Core} description until 
this description is in a \emph{normal form}. This set of transformations 
includes transformations typically found in reduction systems and lambda 
calculus~\cite{lambdacalculus}, such as $\beta$-reduction and 
$\eta$-expansion. It also includes self-defined transformations that are 
responsible for the reduction of higher-order functions to `regular' 
first-order functions, and specializing polymorphic types to concrete types.

The final step in the compiler pipeline is the translation to a \VHDL\ 
\emph{netlist}, which is a straightforward process due to resemblance of a 
normalized description and a set of concurrent signal assignments. The 
end-product of the \CLaSH\ compiler is called a \VHDL\ \emph{netlist} as the 
result resembles an actual netlist description, and the fact that it is \VHDL\ 
is only an implementation detail; the output could for example also be in 
Verilog.

\section{Use cases}
\label{sec:usecases}
\subsection{FIR Filter}
As an example of a common hardware design where the relation between functional languages and mathematical functions, combined with the use of higher-order functions leads to a very natural description is a \acro{FIR} filter:

\begin{equation}
y_t  = \sum\nolimits_{i = 0}^{n - 1} {x_{t - i}  \cdot h_i } 
\end{equation}

A \acro{FIR} filter multiplies fixed constants ($h$) with the current 
and a few previous input samples ($x$). Each of these multiplications
are summed, to produce the result at time $t$. The equation of a \acro{FIR} 
filter is equivalent to the equation of the dot-product, which is shown below:

\begin{equation}
\mathbf{a}\bullet\mathbf{b} = \sum\nolimits_{i = 0}^{n - 1} {a_i \cdot b_i } 
\end{equation}

The equation for the dot-product is easily and directly implemented using 
higher-order functions:

\hspace{-1.7em}
\begin{minipage}{0.93\linewidth}
\begin{code}
as *+* bs = fold (+) (zipWith (*) as bs)
\end{code}
\end{minipage}
\begin{minipage}{0.07\linewidth}
  \begin{example}
  \label{code:dotproduct}
  \end{example}
\end{minipage}

The \hs{zipWith} function is very similar to the \hs{map} function seen 
earlier: It takes a function, two vectors, and then applies the function to 
each of the elements in the two vectors pairwise (\emph{e.g.}, \hs{zipWith (*) 
[1, 2] [3, 4]} becomes \hs{[1 * 3, 2 * 4]}).

The \hs{fold} function takes a binary function, a single vector, and applies 
the function to the first two elements of the vector. It then applies the
function to the result of the first application and the next element in the 
vector. This continues until the end of the vector is reached. The result of 
the \hs{fold} function is the result of the last application. It is obvious 
that the \hs{zipWith (*)} function is pairwise multiplication and that the 
\hs{fold (+)} function is summation.
% Returning to the actual \acro{FIR} filter, we will slightly change the 
% equation describing it, so as to make the translation to code more obvious and 
% concise. What we do is change the definition of the vector of input samples 
% and delay the computation by one sample. Instead of having the input sample 
% received at time $t$ stored in $x_t$, $x_0$ now always stores the newest 
% sample, and $x_i$ stores the $ith$ previous sample. This changes the equation 
% to the following (note that this is completely equivalent to the original 
% equation, just with a different definition of $x$ that will better suit the 
% transformation to code):
% 
% \begin{equation}
% y_t  = \sum\nolimits_{i = 0}^{n - 1} {x_i  \cdot h_i } 
% \end{equation}
The complete definition of the \acro{FIR} filter in \CLaSH\ is:

\hspace{-1.7em}
\begin{minipage}{0.93\linewidth}
\begin{code}
fir (State (xs,hs)) x = 
  (State (shiftInto x xs,hs), (x +> xs) *+* hs)
\end{code}
\end{minipage}
\begin{minipage}{0.07\linewidth}
  \begin{example}
  \label{code:fir}
  \end{example}
\end{minipage}

Where the vector \hs{xs} contains the previous input samples, the vector 
\hs{hs} contains the \acro{FIR} coefficients, and \hs{x} is the current input 
sample. The concatenate operator (\hs{+>}) creates a new vector by placing the 
current sample (\hs{x}) in front of the previous samples vector (\hs{xs}). The 
code for the \hs{shiftInto} function, that adds the new input sample (\hs{x}) 
to the list of previous input samples (\hs{xs}) and removes the oldest sample, 
is shown below:

\hspace{-1.7em}
\begin{minipage}{0.93\linewidth}
\begin{code}
shiftInto x xs = x +> init xs  
\end{code}
\end{minipage}
\begin{minipage}{0.07\linewidth}
  \begin{example}
  \label{code:shiftinto}
  \end{example}
\end{minipage}

Where the \hs{init} function returns all but the last element of a vector. 
The resulting netlist of a 4-taps \acro{FIR} filter, created by specializing 
the vectors of the \acro{FIR} code to a length of 4, is depicted in 
\Cref{img:4tapfir}.

\begin{figure}
\centerline{\includegraphics{4tapfir.svg}}
\caption{4-taps \acrotiny{FIR} Filter}
\label{img:4tapfir}
\vspace{-1.5em}
\end{figure}

\subsection{Higher-order CPU}
The following simple \acro{CPU} is an example of user-defined higher-order
functions and pattern matching. The \acro{CPU} consists of four function 
units, of which three have a fixed function and one can perform certain less
common operations. 

The \acro{CPU} contains a number of data sources, represented by the 
horizontal wires in \Cref{img:highordcpu}. These data sources offer the 
previous output of every function unit, along with the single data input of 
the \acro{CPU} and two fixed initialization values.

Each of the function units has both its operands connected to all data
sources, and can be programmed to select any data source for either
operand. In addition, the leftmost function unit has an additional
opcode input to select the operation it performs. The previous output of the 
rightmost function unit is the output of the entire \acro{CPU}.

The code of the function unit (\ref{code:functionunit}), which arranges the operand selection for the function unit, is shown below. Note that the actual operation that takes place inside the function unit is supplied as the (higher-order) argument \hs{op}, which is a function that takes two arguments.

\hspace{-1.7em}
\begin{minipage}{0.93\linewidth}
\begin{code}
fu op inputs (addr1, addr2) = regIn
  where
    in1     = inputs!addr1
    in2     = inputs!addr2
    regIn   = op in1 in2
\end{code}
\end{minipage}
\begin{minipage}{0.07\linewidth}
  \begin{example}
  \label{code:functionunit}
  \end{example}
\end{minipage}

The \hs{multiop} function (\ref{code:multiop}) defines the operation that takes place in the leftmost function unit. It is essentially a simple three operation \acro{ALU} that makes good use of pattern matching and guards in its description. The \hs{shift} function used here shifts its first operand by the number of bits indicated in the second operand, the \hs{xor} function produces
the bitwise xor of its operands.

\hspace{-1.7em}
\begin{minipage}{0.93\linewidth}
\begin{code}
data Opcode = Shift | Xor | Equal

multiop :: Opcode -> Word -> Word -> Word
multiop Shift   a b                 = shift a b
multiop Xor     a b                 = xor a b
multiop Equal   a b   | a == b      = 1
                      | otherwise   = 0
\end{code}
\end{minipage}
\begin{minipage}{0.07\linewidth}
  \begin{example}
  \label{code:multiop}
  \end{example}
\end{minipage}

The \acro{CPU} function (\ref{code:cpu}) ties everything together. It applies 
the function unit (\hs{fu}) to several operations, to create a different 
function unit each time. The first application is interesting, as it does not 
just pass a function to \hs{fu}, but a partial application of \hs{multiop}. 
This demonstrates how one function unit can effectively get extra inputs 
compared to the others.

The vector \hs{inputs} is the set of data sources, which is passed to
each function unit as a set of possible operants. The \acro{CPU} also receives 
a vector of address pairs, which are used by each function unit to select 
their operand. 
% The application of the function units to the \hs{inputs} and
% \hs{addrs} arguments seems quite repetitive and could be rewritten to use
% a combination of the \hs{map} and \hs{zipwith} functions instead.
% However, the prototype compiler does not currently support working with 
% lists of functions, so a more explicit version of the code is given instead.

\hspace{-1.7em}
\begin{minipage}{0.93\linewidth}
\begin{code}
type CpuState = State [Word | 4]

cpu :: CpuState -> Word -> [(Index 6, Index 6) | 4] 
       -> Opcode -> (CpuState, Word)
cpu (State s) input addrs opc = (State s', out)
  where
    s'    =   [ fu (multiop opc)  inputs (addrs!0)
              , fu add            inputs (addrs!1)
              , fu sub            inputs (addrs!2)
              , fu mul            inputs (addrs!3)
              ]
    inputs    =   0 +> (1 +> (input +> s))
    out       =   last s
\end{code}
\end{minipage}
\begin{minipage}{0.07\linewidth}
  \begin{example}
  \label{code:cpu}
  \end{example}
\end{minipage}

While this is still a simple example, it could form the basis of an actual 
design, in which the same techniques can be reused.

\section{Related work}
This section describes the features of existing (functional) hardware 
description languages and highlights the advantages that this research has 
over existing work.

% Many functional hardware description languages have been developed over the 
% years. Early work includes such languages as $\mu$\acro{FP}~\cite{muFP}, an 
% extension of Backus' \acro{FP} language to synchronous streams, designed 
% particularly for describing and reasoning about regular circuits. The 
% Ruby~\cite{Ruby} language uses relations, instead of functions, to describe 
% circuits, and has a particular focus on layout. 

\acro{HML}~\cite{HML2} is a hardware modeling language based on the strict 
functional language \acro{ML}, and has support for polymorphic types and 
higher-order functions. There is no direct simulation support for \acro{HML}, 
so a description in \acro{HML} has to be translated to \VHDL\ and that the 
translated description can then be simulated in a \VHDL\ simulator. Certain 
aspects of HML, such as higher-order functions are however not supported by 
the \VHDL\ translator~\cite{HML3}. The \CLaSH\ compiler on the other hand can 
correctly translate all of the language constructs mentioned in this paper.

\begin{figure}
\centerline{\includegraphics{highordcpu.svg}}
\caption{CPU with higher-order Function Units}
\label{img:highordcpu}
\vspace{-1.5em}
\end{figure}

Like the research presented in this paper, many functional hardware 
description languages have some sort of foundation in the functional 
programming language Haskell. Hawk~\cite{Hawk1} is a hardware modeling 
language embedded in Haskell and has sequential environments that make it 
easier to specify stateful computation. Hawk specifications can be simulated; 
to the best knowledge of the authors there is however no support for automated 
circuit synthesis. 

The ForSyDe~\cite{ForSyDe2} system uses Haskell to specify abstract system 
models. A designer can model systems using heterogeneous models of 
computation, which include continuous time, synchronous and untimed models of 
computation. Using so-called domain interfaces a designer can simulate 
electronic systems which have both analog as digital parts. ForSyDe has 
several backends including simulation and automated synthesis, though 
automated synthesis is restricted to the synchronous model of computation. 
Though ForSyDe offers higher-order functions and polymorphism, ForSyDe's 
choice elements are limited to \hs{if} and \hs{case} expressions. ForSyDe's 
explicit conversions, where function have to be wrapped in processes and 
processes have to be wrapped in systems, combined with the explicit 
instantiations of components, also makes ForSyDe more verbose than \CLaSH.

Lava~\cite{Lava,kansaslava} is a hardware description language, embedded in 
Haskell, and focuses on the structural representation of hardware. Like 
\CLaSH, Lava has support for polymorphic types and higher-order functions. 
Besides support for simulation and circuit synthesis, Lava descriptions can be 
interfaced with formal method tools for formal verification. As discussed in 
the introduction, taking the embedded language approach does not allow for 
Haskell's choice elements to be captured within the circuit descriptions. In 
this respect \CLaSH\ differs from Lava, in that all of Haskell's choice 
elements, such as \hs{case}-expressions and pattern matching, are synthesized 
to choice elements in the eventual circuit. Consequently, descriptions 
containing rich control structures can be specified in a more user-friendly 
way in \CLaSH\ than possible within Lava, and are hence less error-prone.

Bluespec~\cite{Bluespec} is a high-level synthesis language that features 
guarded atomic transactions and allows for the automated derivation of control 
structures based on these atomic transactions. Bluespec, like \CLaSH, supports 
polymorphic typing and function-valued arguments. Bluespec's syntax and 
language features \emph{had} their basis in Haskell. However, in order to 
appeal to the users of the traditional \acrop{HDL}, Bluespec has adapted 
imperative features and a syntax that resembles Verilog. As a result, Bluespec 
is (unnecessarily) verbose when compared to \CLaSH.

The merits of polymorphic typing and function-valued arguments are now also 
recognized in the traditional \acrop{HDL}, exemplified by the new \VHDL-2008 
standard~\cite{VHDL2008}. \VHDL-2008 support for generics has been extended to 
types and subprograms, allowing a designer to describe components with 
polymorphic ports and function-valued arguments. Note that the types and 
subprograms still require an explicit generic map, whereas types can be 
automatically inferred, and function-values can be automatically propagated 
by the \CLaSH\ compiler. There are also no (generally available) \VHDL\ 
synthesis tools that currently support the \VHDL-2008 standard.

% Wired~\cite{Wired},, T-Ruby~\cite{T-Ruby}, Hydra~\cite{Hydra}. 
% 
% A functional language designed specifically for hardware design is 
% $re{\mathit{FL}}^{ect}$~\cite{reFLect}, which draws experience from earlier 
% language called \acro{FL}~\cite{FL} to la

% An example of a floating figure using the graphicx package.
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%\label{fig_sim}
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\section{Conclusion}
This research demonstrates once more that functional languages are well suited 
for hardware descriptions: function applications provide an elegant notation 
for component instantiation. While circuit descriptions made in \CLaSH\ are 
very concise when compared to other (traditional) \acrop{HDL}, their intended 
functionality remains clear. Where \CLaSH\ goes beyond the existing 
(functional) hardware descriptions languages is the inclusion of advanced 
choice elements, such as pattern matching and guards, that are well suited to 
describe the conditional assignments in control-oriented circuits. Besides 
being able to translate these basic constructs to synthesizable \VHDL, the 
prototype compiler can also correctly translate descriptions that contain both 
polymorphic types and user-defined higher-order functions.

% Where recent functional hardware description languages have mostly opted to 
% embed themselves in an existing functional language, this research features 
% a `true' compiler. As a result there is a clear distinction between 
% compile-time and run-time, which allows a myriad of choice constructs to be 
% part of the actual circuit description; a feature the embedded hardware 
% description languages do not offer.

Besides simple circuits such as variants of both the \acro{FIR} filter and 
the higher-order \acro{CPU} shown in \Cref{sec:usecases}, the \CLaSH\ compiler 
has also been able to translate non-trivial functional descriptions such as a 
streaming reduction circuit~\cite{reductioncircuit} for floating point 
numbers.

\section{Future Work}
The choice of describing state explicitly as extra arguments and results can 
be seen as a mixed blessing. Even though the description that use state are 
usually very clear, one finds that distributing and collecting substate can 
become tedious and even error-prone. Removing the required boilerplate for 
distribution and collection, or finding a more suitable abstraction mechanism 
for state would make \CLaSH\ easier to use.

The transformations in normalization phase of the prototype compiler are 
developed in an ad-hoc manner, which makes the existence of many desirable 
properties unclear. Such properties include whether the complete set of 
transformations will always lead to a normal form or if the normalization 
process always terminates. Though extensive use of the compiler suggests that 
these properties usually hold, they have not been formally proven. A 
systematic approach to defining the set of transformations allows one to proof 
that the earlier mentioned properties do indeed exist.

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