Update introduction of the type section, and remove parts about built-in types

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

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\begin{document}
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% paper title
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\title{C$\lambda$aSH: Structural Descriptions \\ of Synchronous Hardware using Haskell}


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\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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\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 basic combinatorial circuits are equivalent 
to 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 language. As the hardware descriptions are plain Haskell 
functions, these descriptions can be compiled for simulation using using the 
optimizing Haskell compiler \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. The functions describe the behavior of the hardware between 
clock cycles, as such, only synchronous systems can be described. Many 
functional hardware description models 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: every function becomes a component, every function argument is an
    input port and the result value is of a function is an output port. This 
    output port can have a complex type (such as a tuple), so having just a 
    single output port does not create a limitation. Each function application 
    in turn becomes a component instantiation. Here, the result of each 
    argument expression is assigned to a signal, which is mapped to the 
    corresponding input port. 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}}
    \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}}
    \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 (and \hs{if} expressions, which can be very directly translated 
    to \hs{case} expressions). 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, 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.
    
    \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}

    Both versions of the example correspond to the same netlist, which is 
    depicted in \Cref{img:choice}.

    \begin{figure}
    \centerline{\includegraphics{choice-case}}
    \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. A 
    pattern match (with optional guards) can be to a conditional assignments 
    in \VHDL, where the conditions are an equality test of the argument and 
    one of the patterns (combined with the guard if was present). A third 
    version of the earlier example, using both pattern matching and guards, 
    can be seen below:
    
    \begin{code}
    sumif Eq a b    | a == b = a + b
    sumif Neq a b   | a != b = a + b
    sumif _ _ _     = 0
    \end{code}
    
    The version using pattern matching and guards has the same netlist 
    representation (\Cref{img:choice}) as the earlier two versions of the 
    example.

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

  \subsection{Types}
    Haskell is a strongly-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.
  
    % 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}
    \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 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{RangedWord}]
        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.
        A \hs{RangedWord} only has an upper bound, its lower bound is
        implicitly zero. The main purpose of the \hs{RangedWord} 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 renamings 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. These are not currently supported.

    Only an algebraic datatype declaration actually introduces a
    completely new type, for which we provide the \VHDL\ translation
    below. Type synonyms and renamings only define new names for
    existing types, where synonyms are completely interchangeable and
    renamings need explicit conversion. Therefore, these do not need
    any particular \VHDL\ translation, a synonym or renamed type will
    just use the same representation as the original type. The
    distinction between a renaming and a synonym does no longer matter
    in hardware and can be disregarded in the generated \VHDL. For algebraic 
    types, we can make the following distinction: 

    \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. An example of such a type is the following pair 
        of integers:

        \begin{code}
        data IntPair = IntPair Int Int
        \end{code}

        Haskell's builtin tuple types are also defined as single
        constructor algebraic types and are translated according to this
        rule by the \CLaSH\ compiler. 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. 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{Polymorphic functions}
    A powerful construct in most functional language is polymorphism.
    This means the arguments of a function (and consequentially, values
    within the function as well) do not need to have a fixed type.
    Haskell supports \emph{parametric polymorphism}, meaning a
    function's type can be parameterized with another type.

    As an example of a polymorphic function, consider the following
    \hs{append} function's type:
    
    \comment{TODO: Use vectors instead of lists?}

    \begin{code}
    append :: [a] -> a -> [a]
    \end{code}

    This type is parameterized by \hs{a}, which can contain any type at
    all. This means that append can append an element to a list,
    regardless of the type of the elements in the list (but the element
    added must match the elements in the list, since there is only one
    \hs{a}).

    This kind of polymorphism is extremely useful in hardware designs to
    make operations work on a vector without knowing exactly what elements
    are inside, routing signals without knowing exactly what kinds of
    signals these are, or working with a vector without knowing exactly
    how long it is. Polymorphism also plays an important role in most
    higher order functions, as we will see in the next section.

    The previous example showed unconstrained polymorphism \comment{(TODO: How 
    is this really called?)}: \hs{a} can have \emph{any} type. 
    Furthermore,Haskell supports limiting the types of a type parameter to 
    specific class of types. An example of such a type class is the 
    \hs{Num} class, which contains all of Haskell's numerical types.

    Now, take the addition operator, which has the following type:

    \begin{code}
    (+) :: Num a => a -> a -> a
    \end{code}

    This type is again parameterized by \hs{a}, but it can only contain
    types that are \emph{instances} of the \emph{type class} \hs{Num}.
    Our numerical built-in types are also instances of the \hs{Num}
    class, so we can use the addition operator on \hs{SizedWords} as
    well as on {SizedInts}.

    In \CLaSH, unconstrained polymorphism is completely supported. Any
    function defined can have any number of unconstrained type
    parameters. The \CLaSH\ compiler will infer the type of every such
    argument depending on how the function is applied. There is one
    exception to this: The top level function that is translated, can
    not have any polymorphic arguments (since it is never applied, so
    there is no way to find out the actual types for the type
    parameters).

    \CLaSH\ does not support user-defined type classes, but does use some
    of the builtin ones for its builtin functions (like \hs{Num} and
    \hs{Eq}).

  \subsection{Higher order}
    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. Let's clarify that with an example:
    
    \begin{code}
    notList xs = map not xs
    \end{code}

    This defines a function \hs{notList}, with a single list of booleans
    \hs{xs} as an argument, which simply negates all of the booleans in
    the list. To do this, it uses the function \hs{map}, which takes
    \emph{another function} as its first argument and applies that other
    function to each element in the list, returning again a list of the
    results.

    As you can see, the \hs{map} function is a higher order function,
    since it takes another function as an argument. Also note that
    \hs{map} is again a polymorphic function: It does not pose any
    constraints on the type of elements in the list passed, other than
    that it must be the same as the type of the argument the passed
    function accepts. The type of elements in the resulting list is of
    course equal to the return type of the function passed (which need
    not be the same as the type of elements in the input list). Both of
    these can be readily seen from the type of \hs{map}:

    \begin{code}
    map :: (a -> b) -> [a] -> [b]
    \end{code}
    
    As an example from a common hardware design, let's look at the
    equation of a FIR filter.

    \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 $t$.

    This is easily and directly implemented using higher order
    functions. 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. How \hs{xs} gets its value will be
    show in the next section about state.

    \begin{code}
    fir ... = foldl1 (+) (zipwith (*) xs hs)
    \end{code}

    Here, the \hs{zipwith} function is very similar to the \hs{map}
    function: It takes a function two lists and then applies the
    function to each of the elements of the two lists pairwise
    (\emph{e.g.}, \hs{zipwith (+) [1, 2] [3, 4]} becomes 
    \hs{[1 + 3, 2 + 4]}.

    The \hs{foldl1} function takes a function and a single list and applies the
    function to the first two elements of the list. It then applies to
    function to the result of the first application and the next element
    from the list. This continues until the end of the list 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.

    To make the correspondence between the code and the equation even
    more obvious, we turn the list of input samples in the equation
    around. So, instead of having the the input sample received at time
    $t$ 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 better suits
    the \hs{x} from the code):

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

    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 labmda expression allows one to introduce an anonymous function
    in any expression. Consider the following expression, which again
    adds one to every element of a list:

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

    Finally, higher order arguments are not limited to just builtin
    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 
    for a combinatorial circuit, where the output solely depends on the 
    inputs. When a circuit has state however, it can no longer be simply
    described by a pure function. Simply removing the purity property is not a 
    valid option, as the language would then lose many of it mathematical 
    properties. In an effort to include the concept of state in pure 
    functions, the current value of the state is made an argument of the  
    function; the updated state becomes part of the result.
    
    A simple example is the description of an accumulator circuit:
    \begin{code}
    acc :: Word -> State Word -> (State Word, Word)
    acc inp (State s) = (State s', outp)
      where
        outp  = s + inp
        s'    = outp
    \end{code}
    This approach makes the state of a function very explicit: which variables 
    are part of the state is completely determined by the type signature. This 
    approach to state is well suited to be used in combination with the 
    existing code and language features, such as all the choice constructs, as 
    state values are just normal values.
\section{\CLaSH\ prototype}

foo\par bar

\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}, and that the translation to \VHDL\ is only partial.

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. ForSyDe has 
several simulation and synthesis backends, though synthesis is restricted to 
the synchronous subset of the ForSyDe language.

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

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\section{Conclusion}
The conclusion goes here.




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\section*{Acknowledgment}


The authors would like to thank...





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