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

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\hyphenation{op-tical net-works semi-conduc-tor}

% Macro for certain acronyms in small caps. Doesn't work with the
% default font, though (it contains no smallcaps it seems).
\def\acro#1{{\small{#1}}}
\def\VHDL{\acro{VHDL}}
\def\GHC{\acro{GHC}}
\def\CLaSH{{\small{C}}$\lambda$a{\small{SH}}}

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%include polycode.fmt
%include clash.fmt

\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
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\author{\IEEEauthorblockN{Christiaan P.R. Baaij, Matthijs Kooijman, Jan Kuper, Marco E.T. Gerards, Bert Molenkamp, Sabih H. Gerez}
\IEEEauthorblockA{University of Twente, Department of EEMCS\\
P.O. Box 217, 7500 AE, Enschede, The Netherlands\\
c.p.r.baaij@@utwente.nl, matthijs@@stdin.nl, j.kuper@@utwente.nl}}
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% Springfield, USA\\
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% \IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
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% Telephone: (800) 555--1212\\
% Fax: (888) 555--1212}}

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%Eldon Tyrell\IEEEauthorrefmark{4}}
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%Georgia Institute of Technology,
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% make the title area
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\begin{abstract}
%\boldmath
\CLaSH\ is a functional hardware description language that borrows both its 
syntax and semantics from the functional programming language Haskell. Circuit 
descriptions can be translated to synthesizable VHDL using the prototype 
\CLaSH\ compiler. As the circuit descriptions are made in plain Haskell, 
simulations can also be compiled by a Haskell compiler.

The use of polymorphism and higher-order functions allow a circuit designer to 
describe more abstract and general specifications than are possible in the 
traditional hardware description languages.
\end{abstract}
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\section{Introduction}
Hardware description languages has allowed the productivity of hardware 
engineers to keep pace with the development of chip technology. Standard 
Hardware description languages, like \VHDL~\cite{VHDL2008} and 
Verilog~\cite{Verilog}, allowed an engineer to describe circuits using a 
programming language. These standard languages are very good at describing 
detailed hardware properties such as timing behavior, but are generally 
cumbersome in expressing higher-level abstractions. In an attempt to raise the 
abstraction level of the descriptions, a great number of approaches based on 
functional languages has been proposed \cite{T-Ruby,Hydra,HML2,Hawk1,Lava,
ForSyDe1,Wired,reFLect}. The idea of using functional languages for hardware 
descriptions started in the early 1980s \cite{Cardelli1981, muFP,DAISY,FHDL}, 
a time which also saw the birth of the currently popular hardware description 
languages such as \VHDL. The merit of using a functional language to describe 
hardware comes from the fact that combinatorial circuits can be directly 
modeled as mathematical functions and that functional languages are very good 
at describing and composing mathematical functions.

In an attempt to decrease the amount of work involved with creating all the 
required tooling, such as parsers and type-checkers, many functional hardware 
description languages are embedded as a domain specific language inside the 
functional language Haskell \cite{Hydra,Hawk1,Lava,ForSyDe1,Wired}. This 
means that a developer is given a library of Haskell~\cite{Haskell} functions 
and types that together form the language primitives of the domain specific 
language. As a result of how the signals are modeled and abstracted, the 
functions used to describe a circuit also build a large domain-specific 
datatype (hidden from the designer) which can be further processed by an 
embedded compiler. This compiler actually runs in the same environment as the 
description; as a result compile-time and run-time become hard to define, as 
the embedded compiler is usually compiled by the same Haskell compiler as the 
circuit description itself.

The approach taken in this research is not to make another domain specific 
language embedded in Haskell, but to use (a subset of) the Haskell language 
itself for the purpose of describing hardware. By taking this approach, we can 
capture certain language constructs, such as Haskell's choice elements 
(if-constructs, case-constructs, pattern matching, etc.), which are not 
available in the functional hardware description languages that are embedded 
in Haskell as a domain specific languages. As far as the authors know, such 
extensive support for choice-elements is new in the domain of functional 
hardware description languages. As the hardware descriptions are plain Haskell 
functions, these descriptions can be compiled for simulation using an 
optimizing Haskell compiler such as the Glasgow Haskell Compiler (\GHC)~\cite{ghc}.

Where descriptions in a conventional hardware description language have an 
explicit clock for the purpose state and synchronicity, the clock is implied 
in this research. A developer describes the behavior of the hardware between 
clock cycles, as such, only synchronous systems can be described. Many 
functional hardware description model signals as a stream of all values over 
time; state is then modeled as a delay on this stream of values. The approach 
taken in this research is to make the current state of a circuit part of the 
input of the function and the updated state part of the output.

Like the standard hardware description languages, descriptions made in a 
functional hardware description language must eventually be converted into a 
netlist. This research also features a prototype translator called \CLaSH\ 
(pronounced: clash), which 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.

\section{Hardware description in Haskell}

  \subsection{Function application}
    The basic syntactic 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 output port can have a complex type (such as a tuple), so having just 
    a single output port does not pose any limitation. The arguments of a 
    function applications are assigned to a signal, 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,% aswell, 
    creating a hierarchical description of the hardware. This separation in 
    different components makes the resulting \VHDL\ output easier to read and 
    debug.

    As an example we can see the netlist of the |mac| function in
    \Cref{img:mac-comb}; the |mac| function applies both the |mul| and |add|
    function to calculate $a * b + c$:
    
    \begin{code}
    mac a b c = add (mul a b) c
    \end{code}
    
    \begin{figure}
    \centerline{\includegraphics{mac.svg}}
    \caption{Combinatorial Multiply-Accumulate}
    \label{img:mac-comb}
    \end{figure}
    
    The result of using a complex input type can be seen in 
    \cref{img:mac-comb-nocurry} where the |mac| function now uses a single
    input tuple for the |a|, |b|, and |c| arguments:
    
    \begin{code}
    mac (a, b, c) = add (mul a b) c
    \end{code}
    
    \begin{figure}
    \centerline{\includegraphics{mac-nocurry.svg}}
    \caption{Combinatorial Multiply-Accumulate (complex input)}
    \label{img:mac-comb-nocurry}
    \end{figure}

  \subsection{Choice}
    In Haskell, choice can be achieved by a large set of language constructs, 
    consisting of: \hs{case} constructs, \hs{if-then-else} constructs, 
    pattern matching, and guards. The easiest of these are the \hs{case} 
    constructs (\hs{if} expressions can be very directly translated to 
    \hs{case} expressions). A \hs{case} construct is translated to a 
    multiplexer, where the control value is linked to the selection port and 
    the  output of each case is linked to the corresponding input port on 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. 
    We can see two versions of a contrived example below, the first 
    using a \hs{case} construct and the other using a \hs{if-then-else} 
    constructs, in the code below. The example sums two values when they are 
    equal or non-equal (depending on the predicate given) and returns 0 
    otherwise. Both versions of the example roughly correspond to the same 
    netlist, which is depicted in \Cref{img:choice}.
    
    \begin{code}
    sumif pred a b = case pred of
      Eq ->   case a == b of
        True    -> a + b
        False   -> 0
      Neq ->  case a != b of
        True    -> a + b
        False   -> 0
    \end{code}

    \begin{code}
    sumif pred a b = 
      if pred == Eq then 
        if a == b then a + b else 0
      else 
        if a != b then a + b else 0
    \end{code}

    \begin{figure}
    \centerline{\includegraphics{choice-case.svg}}
    \caption{Choice - sumif}
    \label{img:choice}
    \end{figure}

    A slightly more complex (but very powerful) form of choice is pattern 
    matching. A function can be defined in multiple clauses, where each clause 
    specifies a pattern. When the arguments match the pattern, the 
    corresponding clause will be used. Expressions can also contain guards, 
    where the expression is only executed if the guard evaluates to true. Like 
    \hs{if-then-else} constructs, pattern matching and guards have a 
    (straightforward) translation to \hs{case} constructs and can as such be 
    mapped to multiplexers. A third version of the earlier example, using both 
    pattern matching and guards, can be seen below. The version using pattern 
    matching and guards also has roughly the same netlist representation 
    (\Cref{img:choice}) as the earlier two versions of the example.
    
    \begin{code}
    sumif Eq a b    | a == b = a + b
    sumif Neq a b   | a != b = a + b
    sumif _ _ _     = 0
    \end{code}

    % \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, as such this 
    section will 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 direct translation is defined within the \CLaSH\ 
    compiler; the term user-defined types should not require any further 
    elaboration. The translatable types are also inferable by the compiler, 
    meaning that a developer does not have to annotate every function with a 
    type signature.
  
    % 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 direct translation defined within the \CLaSH\
    compiler:
    \begin{xlist}
      \item[\bf{Bit}]
        This is the most basic type available. It can have two values:
        \hs{Low} and \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}
        and \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} construct, which requires a \hs{Bool} value for 
        the condition.
      \item[\bf{SizedWord}, \bf{SizedInt}]
        These are types to represent integers. A \hs{SizedWord} is unsigned,
        while a \hs{SizedInt} is signed. Both are parametrizable in their 
        size. 
        % , 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 fixed 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 the \hs{a} is the element 
        type, and \hs{n} is the length of the vector.
        % 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}]
        This is another type to describe integers, but unlike the previous
        two it has no specific bit-width, but an 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. The main purpose of the \hs{Index} type is to be 
        used as an index to a \hs{Vector}.

        % \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, {\small{GADT}}s, etc.) which are not standard Haskell. 
    As it is currently unclear how these advanced type constructs correspond 
    with hardware, they are for now unsupported by the \CLaSH\ compiler

    Only an algebraic datatype declaration actually introduces a
    completely new type. Type synonyms and renaming constructs only define new 
    names for existing types, where synonyms are completely interchangeable 
    and renaming constructs need explicit conversions. Therefore, these do not 
    need any particular translation, a synonym or renamed type will just use 
    the same representation as the original type. For algebraic types, we can 
    make the following distinctions: 

    \begin{xlist}
      \item[\bf{Single constructor}]
        Algebraic datatypes with a single constructor with one or more
        fields, are essentially a way to pack a few values together in a
        record-like structure. Haskell's built-in tuple types are also defined 
        as single constructor algebraic types  An example of a single 
        constructor type 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 a way to get an enumeration-like type
        containing alternatives. Note that Haskell's \hs{Bool} type is also 
        defined as an enumeration type, but we have a fixed translation for 
        that. An example of such an enum type is the type that represents the
        colors in 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 not
        currently supported.
    \end{xlist}

  \subsection{Polymorphism}
    A powerful construct in most functional 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{append} function, which appends an element to a vector:
    \begin{code}
    append :: [a|n] -> a -> [a|n + 1]
    \end{code}

    This type is parameterized by \hs{a}, which can contain any type at
    all. This means that \hs{append} can append an element to a vector,
    regardless of the type of the elements in the list (as long as the type of 
    the value to be added is of the same type as the values in the vector). 
    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.

    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 
    type classes, where a class definition provides the general interface of a 
    function, and class 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 developer can make use 
    of this ad-hoc polymorphism by adding a 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. 
    
    As an example we will take a look at type signature of the function 
    \hs{sum}, 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  
    we know that the addition (+) operator is defined for that type. 
    \CLaSH's built-in numerical types are also instances of the \hs{Num}
    class, so we can use the addition operator on \hs{SizedWords} as
    well as on \hs{SizedInts}.

    In \CLaSH, parametric 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 (as 
    they are 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 built-in type classes for its built-in function, such as: \hs{Num} 
    for numerical operations, \hs{Eq} for the equality operators, and
    \hs{Ord} for the comparison/order operators.

  \subsection{Higher-order functions \& values}
    Another powerful abstraction mechanism in functional languages, is
    the concept of \emph{higher-order functions}, or \emph{functions as
    a first class value}. 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:
    
    \begin{code}
    negVector xs = map not xs
    \end{code}

    The code above defines a function \hs{negVector}, which takes a vector of
    booleans, 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 vector elements, other than that it must be the same type as 
    the input type 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 readily 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:

    \begin{code}
    map ((+) 1) xs
    \end{code}

    Here, the expression \hs{(+) 1} is the partial application of the
    plus operator to the value \hs{1}, which is again a function that
    adds one to its 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:

    \begin{code}
    map (\x -> x + 1) xs
    \end{code}

    Finally, higher order arguments are not limited to just built-in
    functions, but any function defined in \CLaSH\ can have function
    arguments. This allows the hardware designer to use a powerful
    abstraction mechanism in his designs and have an optimal amount of
    code reuse.

    \comment{TODO: Describe ALU example (no code)}

  \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 or 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. In this sense the
    descriptions made in \CLaSH are the describing the combinatorial 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}:
    
    \begin{code}
    macS (State c) a b = (State c', outp)
      where
        outp  = mac a b c
        c'    = outp
    \end{code}
    
    \begin{figure}
    \centerline{\includegraphics{mac-state.svg}}
    \caption{Stateful Multiply-Accumulate}
    \label{img:mac-state}
    \end{figure}
    
    The \hs{State} keyword 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 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 constructs, as state values are just normal values. We can simulate 
    stateful descriptions using the recursive \hs{run} function:
    
    \begin{code}
    run f s (i:inps) = o : (run f s' inps)
      where
        (s', o) = f s i
    \end{code}
    
    The \hs{run} function maps a list of inputs over the function that a 
    developer wants to simulate, passing the state to each new iteration. Each
    value in the input list corresponds to exactly one cycle of the (implicit) 
    clock. The result of the simulation is a list of outputs for every clock
    cycle. As both the \hs{run} function and the hardware description are 
    plain hardware, the complete simulation can be compiled by an optimizing
    Haskell compiler.
    
\section{\CLaSH\ prototype}

foo\par bar

\section{Use cases}
As an example of a common hardware design where the use of higher-order
functions leads to a very natural description is a FIR filter, which is 
basically the dot-product of two vectors:

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

A 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 FIR 
filter is indeed equivalent to the equation of the dot-product, which is 
shown below:

\begin{equation}
\mathbf{x}\bullet\mathbf{y} = \sum\nolimits_{i = 0}^{n - 1} {x_i \cdot y_i } 
\end{equation}

We can easily and directly implement the equation for the dot-product
using higher-order functions:

\begin{code}
xs *+* ys = foldl1 (+) (zipWith (*) xs hs)
\end{code}

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]} $\equiv$ \hs{[3,8]}).

The \hs{foldl1} function takes a 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 from 
the vector. This continues until the end of the vector is reached. The 
result of the \hs{foldl1} function is the result of the last application.
As you can see, the \hs{zipWith (*)} function is just pairwise 
multiplication and the \hs{foldl1 (+)} function is just summation.

Returning to the actual FIR filter, we will slightly change the
equation belong to it, so as to make the translation to code more obvious.
What we will do is change the definition of the vector of input samples.
So, instead of having the input sample received at time
$t$ stored in $x_t$, $x_0$ now always stores the current 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}

Consider that the vector \hs{hs} contains the FIR coefficients and the 
vector \hs{xs} contains the current input sample in front and older 
samples behind. The function that shifts the input samples is shown below:

\begin{code}
x >> xs = x +> tail xs  
\end{code}

Where the \hs{tail} function returns all but the first element of a 
vector, and the concatenate operator ($\succ$) adds a new element to the 
left of a vector. The complete definition of the FIR filter then becomes:

\begin{code}
fir (State (xs,hs)) x = (State (x >> xs,hs), xs *+* hs)
\end{code}

The resulting netlist of a 4-taps FIR filter based on the above definition
is depicted in \Cref{img:4tapfir}.

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

\section{Related 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. Published work suggests that there is no direct 
simulation support for \acro{HML}, but that a description in \acro{HML} has to 
be translated to \VHDL\ and that the translated description can than be 
simulated in a \VHDL\ simulator. Also not all of the mentioned language 
features of \acro{HML} could be translated to hardware. The \CLaSH\ compiler 
on the other hand can correctly translate all of the language constructs 
mentioned in this paper to a netlist format.

Like this work, many functional hardware description languages have some sort 
of foundation in the functional programming language Haskell. 
Hawk~\cite{Hawk1} uses Haskell to describe system-level executable 
specifications used to model the behavior of superscalar microprocessors. Hawk 
specifications can be simulated, but there seems to be no support for 
automated circuit synthesis. 

The ForSyDe~\cite{ForSyDe2} system uses Haskell to specify abstract system 
models, which can (manually) be transformed into an implementation model using 
semantic preserving transformations. 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 simulation and  synthesis backends, 
though synthesis is restricted to the synchronous subset of the ForSyDe 
language. Unlike \CLaSH\ there is no support for the automated synthesis of description that contain polymorphism or higher-order functions.

Lava~\cite{Lava} is a hardware description language that focuses on the 
structural representation of hardware. Besides support for simulation and 
circuit synthesis, Lava descriptions can be interfaced with formal method 
tools for formal verification. Lava descriptions are actually circuit 
generators when viewed from a synthesis viewpoint, in that the language 
elements of Haskell, such as choice, can be used to guide the circuit 
generation. If a developer wants to insert a choice element inside an actual 
circuit he will have to specify this explicitly as a component. 

In this respect \CLaSH\ differs from Lava, in that all the choice elements, 
such as case-statements and pattern matching, are synthesized to choice 
elements in the eventual circuit. As such, richer control structures can both 
be specified and synthesized in \CLaSH\ compared to any of the languages 
mentioned in this section.

The merits of polymorphic typing, combined with higher-order functions, are 
now also recognized in the `main-stream' hardware description languages, 
exemplified by the new \VHDL-2008 standard~\cite{VHDL2008}. \VHDL-2008 has 
support to specify types as generics, thus allowing a developer to describe 
polymorphic components. Note that those types still require an explicit 
generic map, whereas type-inference and type-specialization are implicit in 
\CLaSH.

% 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.
% Note that \label must occur AFTER (or within) \caption.
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\section{Conclusion}
The conclusion goes here.




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% use section* for acknowledgement
\section*{Acknowledgment}


The authors would like to thank...





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% that's all folks
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