\chapter[chap:description]{Hardware description}
  This chapter will provide an overview of the hardware description language
  that was created and the issues that have arisen in the process. It will
  focus on the issues of the language, not the implementation. The prototype
  implementation will be discussed in \in{chapter}[chap:prototype].

  \todo{Shortshort introduction to Cλash (Bit, Word, and, not, etc.)}

  When translating Haskell to hardware, we need to make choices about what kind
  of hardware to generate for which Haskell constructs. When faced with
  choices, we've tried to stick with the most obvious choice wherever
  possible. In a lot of cases, when you look at a hardware description it is
  comletely clear what hardware is described. We want our translator to
  generate exactly that hardware whenever possible, to minimize the amount of
  surprise for people working with it.
   
  In this chapter we try to describe how we interpret a Haskell program from a
  hardware perspective. We provide a description of each Haskell language
  element that needs translation, to provide a clear picture of what is
  supported and how.

  \section[sec:description:application]{Function application}
  \todo{Sidenote: Inputs vs arguments}
  The basic syntactic element of a functional program are functions and
  function application. These have a single obvious \small{VHDL} translation: Each
  function becomes a hardware component, where each argument is an input port
  and the result value is the (single) output port. This output port can have
  a complex type (such as a tuple), so having just a single output port does
  not pose a limitation.

  Each function application in turn becomes 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.

  \in{Example}[ex:And3] shows a simple program using only function
  application and the corresponding architecture.

\startbuffer[And3]
-- | A simple function that returns 
--   conjunction of three bits
and3 :: Bit -> Bit -> Bit -> Bit
and3 a b c = and (and a b) c
\stopbuffer

\todo{Mirror this image vertically}
  \startuseMPgraphic{And3}
    save a, b, c, anda, andb, out;

    % I/O ports
    newCircle.a(btex $a$ etex) "framed(false)";
    newCircle.b(btex $b$ etex) "framed(false)";
    newCircle.c(btex $c$ etex) "framed(false)";
    newCircle.out(btex $out$ etex) "framed(false)";

    % Components
    newCircle.anda(btex $and$ etex);
    newCircle.andb(btex $and$ etex);

    a.c    = origin;
    b.c    = a.c + (0cm, 1cm);
    c.c    = b.c + (0cm, 1cm);
    anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
    andb.c = midpoint(b.c, c.c) + (4cm, 0cm);

    out.c   = andb.c + (2cm, 0cm);

    % Draw objects and lines
    drawObj(a, b, c, anda, andb, out);

    ncarc(a)(anda) "arcangle(-10)";
    ncarc(b)(anda);
    ncarc(anda)(andb);
    ncarc(c)(andb);
    ncline(andb)(out);
  \stopuseMPgraphic

  \placeexample[here][ex:And3]{Simple three input and gate.}
    \startcombination[2*1]
      {\typebufferhs{And3}}{Haskell description using function applications.}
      {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
    \stopcombination

  \note{This section should also mention hierarchy, top level functions and
  subcircuits}

  \section{Choice}
    Although describing components and connections allows us to describe a lot of
    hardware designs already, there is an obvious thing missing: Choice. We
    need some way to be able to choose between values based on another value.
    In Haskell, choice is achieved by \hs{case} expressions, \hs{if} expressions or
    pattern matching.

    An obvious way to add choice to our language without having to recognize
    any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
    function. This function would take three arguments: The condition, the
    value to return when the condition is true and the value to return when
    the condition is false.

    This \hs{if} function would then essentially describe a multiplexer and
    allows us to describe any architecture that uses multiplexers. \fxnote{Are
    there other mechanisms of choice in hardware?}

    However, to be able to describe our hardware in a more convenient way, we
    also want to translate Haskell's choice mechanisms. The easiest of these
    are of course case expressions (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, where the conditions use
    equality comparisons against the constructors in the \hs{case} expressions.

    \todo{Assignment vs multiplexers}

    In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
    scrutinizing a boolean value. The corresponding architecture has a
    comparator to determine which of the constructors is on the \hs{in}
    input. There is a multiplexer to select the output signal. The two options
    for the output signals are just constants, but these could have been more
    complex expressions (in which case also both of them would be working in
    parallel, regardless of which output would be chosen eventually).
    
    If we would translate a Boolean to a bit value, we could of course remove
    the comparator and directly feed 'in' into the multiplex (or even use an
    inverter instead of a multiplexer).  However, we will try to make a
    general translation, which works for all possible \hs{case} expressions.
    Optimizations such as these are left for the \VHDL synthesizer, which
    handles them very well.

    \todo{Be more explicit about >2 alternatives}

    \startbuffer[CaseInv]
    inv :: Bool -> Bool
    inv True  = False
    inv False = True
    \stopbuffer

    \startuseMPgraphic{CaseInv}
      save in, truecmp, falseout, trueout, out, cmp, mux;

      % I/O ports
      newCircle.in(btex $in$ etex) "framed(false)";
      newCircle.out(btex $out$ etex) "framed(false)";
      % Constants
      newBox.truecmp(btex $True$ etex) "framed(false)";
      newBox.trueout(btex $True$ etex) "framed(false)";
      newBox.falseout(btex $False$ etex) "framed(false)";

      % Components
      newCircle.cmp(btex $==$ etex);
      newMux.mux;

      in.c       = origin;
      cmp.c      = in.c + (3cm, 0cm);
      truecmp.c  = cmp.c + (-1cm, 1cm);
      mux.sel    = cmp.e + (1cm, -1cm);
      falseout.c = mux.inpa - (2cm, 0cm);
      trueout.c  = mux.inpb - (2cm, 0cm);
      out.c      = mux.out + (2cm, 0cm);

      % Draw objects and lines
      drawObj(in, out, truecmp, trueout, falseout, cmp, mux);

      ncline(in)(cmp);
      ncarc(truecmp)(cmp);
      nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
      ncline(falseout)(mux) "posB(inpa)";
      ncline(trueout)(mux) "posB(inpb)";
      ncline(mux)(out) "posA(out)";
    \stopuseMPgraphic

    \placeexample[here][ex:CaseInv]{Simple inverter.}
      \startcombination[2*1]
        {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
        {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
      \stopcombination

    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.

    \startbuffer[PatternInv]
    inv :: Bool -> Bool
    inv True = False
    inv False = True
    \stopbuffer

    \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.
    Describes the same architecture as \in{example}[ex:CaseInv].}
        {\typebufferhs{CaseInv}}

    The architecture described by \in{example}[ex:PatternInv] is of course the
    same one as the one in \in{example}[ex:CaseInv]. The general interpretation
    of pattern matching is also similar to that of \hs{case} expressions: Generate
    hardware for each of the clauses (like each of the clauses of a \hs{case}
    expression) and connect them to the function output through (a number of
    nested) multiplexers. These multiplexers are driven by comparators and
    other logic, that check each pattern in turn.

  \section{Types}
    We've seen the two most basic functional concepts translated to hardware:
    Function application and choice. Before we look further into less obvious
    concepts like higher order expressions and polymorphism, we will have a
    look at the types of the values we use in our descriptions.

    When working with values in our descriptions, we'll also need to provide
    some way to translate the values used to hardware equivalents. In
    particular, this means having to come up with a hardware equivalent for
    every \emph{type} used in our program.

    Since most functional languages have a lot of standard types that are
    hard to translate (integers without a fixed size, lists without a static
    length, etc.), we will start out by defining a number of \quote{builtin}
    types ourselves. These types are builtin in the sense that our compiler
    will have a fixed VHDL type for these. User defined types, on the other
    hand, will have their hardware type derived directly from their Haskell
    declaration automatically, according to the rules we sketch here.

    \todo{Introduce Haskell type syntax (type constructors, type application,
    :: operator?)}

    \subsection{Builtin types}
      The language currently supports the following builtin types. Of these,
      only the \hs{Bool} type is supported by Haskell out of the box (the
      others are defined by the Cλash package, so they are user-defined types
      from Haskell's point of view).

      \startdesc{\hs{Bit}}
        This is the most basic type available. It is mapped directly onto
        the \type{std_logic} \small{VHDL} type. Mapping this to the
        \type{bit} type might make more sense (since the Haskell version
        only has two values), but using \type{std_logic} is more standard
        (and allowed for some experimentation with don't care values)

        \todo{Sidenote bit vs stdlogic}
      \stopdesc
      \startdesc{\hs{Bool}}
        This is the only builtin Haskell type supported and is translated
        exactly like the Bit type (where a value of \hs{True} corresponds to a
        value of \hs{High}). Supporting the Bool type is particularly
        useful to support \hs{if ... then ... else ...} expressions, which
        always have a \hs{Bool} value for the condition.

        A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
      \stopdesc
      \startdesc{\hs{SizedWord}, \hs{SizedInt}}
        These are types to represent integers. A \hs{SizedWord} is unsigned,
        while a \hs{SizedInt} is signed. These types are parameterized by a
        length type, so you can define an unsigned word of 32 bits wide as
        follows:
        
        \starthaskell
          type Word32 = SizedWord D32
        \stophaskell

        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 \small{VHDL} \type{unsigned} and
        \type{signed} respectively.
        \todo{Sidenote on dependent typing?}
      \stopdesc
      \startdesc{\hs{Vector}}
        This is a vector type, that can contain elements of any other type and
        has a fixed length. It has two type parameters: Its
        length and the type of the elements contained in it. By putting the
        length parameter in the type, the length of a vector can be determined
        at compile time, instead of only at runtime for conventional lists.

        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:

        \starthaskell
        type RegisterState = Vector D8 Word32
        \stophaskell

        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 \small{VHDL} array type.
      \stopdesc
      \startdesc{\hs{RangedWord}}
        This is another type to describe integers, but unlike the previous
        two it has no specific bitwidth, 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. There is a lot of added implementation complexity
        when adding a lower bound and having just an upper bound was enough
        for the primary purpose of this type: Typesafely indexing vectors.

        To define an index for the 8 element vector above, we would do:

        \starthaskell
        type Register = RangedWord D7
        \stophaskell

        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 \type{unsigned} \small{VHDL} type.
      \stopdesc
      \fxnote{There should be a reference to Christiaan's work here.}

    \subsection{User-defined types}
      There are three ways to define new types in Haskell: Algebraic
      datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
      keyword and type renamings with the \hs{newtype} keyword. This
      explicitly excludes more advanced type creation from \GHC extensions
      such as type families, existential typing, \small{GADT}s, etc.

      The first of these actually introduces a new type, for which we provide
      the \VHDL translation below. The latter two only define new names for
      existing types (where synonyms are completely interchangeable and
      renamings need explicit conversion). Therefore, these don't need any
      particular \VHDL translation, a synonym or renamed type will just use
      the same representation as the equivalent type.

      For algebraic types, we can make the following distinction:

      \startdesc{Product types}
        A product type is an algebraic datatype with a single constructor with
        two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
        is essentially a way to pack a few values together in a record-like
        structure. In fact, the builtin tuple types are just algebraic product
        types (and are thus supported in exactly the same way).

        The "product" in its name refers to the collection of values belonging
        to this type. The collection for a product type is the cartesian
        product of the collections for the types of its fields.

        These types are translated to \VHDL, record types, with one field for
        every field in the constructor. This translation applies to all single
        constructor algebraic datatypes, including those with no fields (unit
        types) and just one field (which are technically not a product).
      \stopdesc
      \startdesc{Enumerated types}
        An enumerated type is an algebraic datatype with multiple constructors, but
        none of them have fields. This is essentially a way to get an
        enum-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.
      \stopdesc
      \startdesc{Sum types}
        A sum type is an algebraic datatype with multiple constructors, where
        the constructors have one or more fields. Technically, a type with
        more than one field per constructor is a sum of products type, but
        for our purposes this distinction does not really make a difference,
        so we'll leave it out.

        Sum types are currently not supported by the prototype, since there is
        no obvious \VHDL alternative. They can easily be emulated, however, as
        we will see from an example:

        \starthaskell
        data Sum = A Bit Word | B Word 
        \stophaskell

        An obvious way to translate this would be to create an enumeration to
        distinguish the constructors and then create a big record that
        contains all the fields of all the constructors. This is the same
        translation that would result from the following enumeration and
        product type (using a tuple for clarity):

        \starthaskell
        data SumC = A | B
        type Sum = (SumC, Bit, Word, Word)
        \stophaskell

        Here, the \hs{SumC} type effectively signals which of the latter three
        fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
        last one if \hs{B}), all the other ones have no useful value.

        An obvious problem with this naive approach is the space usage: The
        example above generates a fairly big \VHDL type. However, we can be
        sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
        at the same time, so this is a waste of space.

        Obviously, we could do some duplication detection here to reuse a
        particular field for another constructor, but this would only
        partially solve the problem. If I would, for example, have an array of
        8 bits and a 8 bit unsiged word, these are different types and could
        not be shared. However, in the final hardware, both of these types
        would simply be 8 bit connections, so we have a 100\% size increase by
        not sharing these.
      \stopdesc

      Another interesting case is that of recursive types. In Haskell, an
      algebraic datatype can be recursive: Any of its field types can be (or
      contain) the type being defined. The most well-known recursive type is
      probably the list type, which is defined is:
      
      \starthaskell
      data List t = Empty | Cons t (List t)
      \stophaskell

      Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
      \hs{:}, but this would make the definition harder to read.  This
      immediately shows the problem with recursive types: What hardware type
      to allocate here? 
      
      If we would use the naive approach for sum types we described above, we
      would create a record where the first field is an enumeration to
      distinguish \hs{Empty} from \hs{Cons}. Furthermore, we would add two
      more fields: One with the (\VHDL equivalent of) type \hs{t} (assuming we
      actually know what type this is at compile time, this should not be a
      problem) and a second one with type \hs{List t}.  The latter one is of
      course a problem: This is exactly the type we were trying to translate
      in the first place. 
      
      Our \VHDL type will thus become infinitely deep. In other words, there
      is no way to statically determine how long (deep) the list will be
      (it could even be infinite).
      
      In general, we can say we can never properly translate recursive types:
      All recursive types have a potentially infinite value (even though in
      practice they will have a bounded value, there is no way for the
      compiler to determine an upper bound on its size).

  \subsection{Partial application}
  Now we've seen how to to translate application, choice and types, we will
  get to a more complex concept: Partial applications. A \emph{partial
  application} is any application whose (return) type is (again) a function
  type.

  From this, we can see that the translation rules for full application do not
  apply to a partial application. \in{Example}[ex:Quadruple] shows an example
  use of partial application and the corresponding architecture.

\startbuffer[Quadruple]
-- | Multiply the input word by four.
quadruple :: Word -> Word
quadruple n = mul (mul n)
  where
    mul = (*) 2
\stopbuffer

  \startuseMPgraphic{Quadruple}
    save in, two, mula, mulb, out;

    % I/O ports
    newCircle.in(btex $n$ etex) "framed(false)";
    newCircle.two(btex $2$ etex) "framed(false)";
    newCircle.out(btex $out$ etex) "framed(false)";

    % Components
    newCircle.mula(btex $\times$ etex);
    newCircle.mulb(btex $\times$ etex);

    two.c    = origin;
    in.c     = two.c + (0cm, 1cm);
    mula.c  = in.c + (2cm, 0cm);
    mulb.c  = mula.c + (2cm, 0cm);
    out.c   = mulb.c + (2cm, 0cm);

    % Draw objects and lines
    drawObj(in, two, mula, mulb, out);

    nccurve(two)(mula) "angleA(0)", "angleB(45)";
    nccurve(two)(mulb) "angleA(0)", "angleB(45)";
    ncline(in)(mula);
    ncline(mula)(mulb);
    ncline(mulb)(out);
  \stopuseMPgraphic

  \placeexample[here][ex:Quadruple]{Simple three port and.}
    \startcombination[2*1]
      {\typebufferhs{Quadruple}}{Haskell description using function applications.}
      {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
    \stopcombination

  Here, the definition of mul is a partial function application: It applies
  the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
  resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
  expression is again a function, we can't generate hardware for it directly.
  This is because the hardware to generate for \hs{mul} depends completely on
  where and how it is used. In this example, it is even used twice.

  However, it is clear that the above hardware description actually describes
  valid hardware. In general, we can see that any partial applied function
  must eventually become completely applied, at which point we can generate
  hardware for it using the rules for function application given in
  \in{section}[sec:description:application]. It might mean that a partial
  application is passed around quite a bit (even beyond function boundaries),
  but eventually, the partial application will become completely applied.
  \todo{Provide a step-by-step example of how this works}

  \section{Costless specialization}
    Each (complete) function application in our description generates a
    component instantiation, or a specific piece of hardware in the final
    design. It is interesting to note that each application of a function
    generates a \emph{separate} piece of hardware. In the final design, none
    of the hardware is shared between applications, even when the applied
    function is the same (of course, if a particular value, such as the result
    of a function application, is used twice, it is not calculated twice).

    This is distinctly different from normal program compilation: Two separate
    calls to the same function share the same machine code. Having more
    machine code has implications for speed (due to less efficient caching)
    and memory usage. For normal compilation, it is therefore important to
    keep the amount of functions limited and maximize the code sharing.

    When generating hardware, this is hardly an issue. Having more \quote{code
    sharing} does reduce the amount of \small{VHDL} output (Since different
    component instantiations still share the same component), but after
    synthesis, the amount of hardware generated is not affected.

    In particular, if we would duplicate all functions so that there is a
    separate function for every application in the program (\eg, each function
    is then only applied exactly once), there would be no increase in hardware
    size whatsoever.
    
    Because of this, a common optimization technique called
    \emph{specialization} can be applied to hardware generation without any
    performance or area cost (unlike for software). 
   
   \fxnote{Perhaps these next three sections are a bit too
   implementation-oriented?}

    \subsection{Specialization}
      \defref{specialization}
      Given some function that has a \emph{domain} $D$ (\eg, the set of all
      possible arguments that could be applied), we create a specialized
      function with exactly the same behaviour, but with a domain $D' \subset
      D$. This subset can be chosen in all sorts of ways. Any subset is valid
      for the general definition of specialization, but in practice only some
      of them provide useful optimization opportunities.

      Common subsets include limiting a polymorphic argument to a single type
      (\ie, removing polymorphism) or limiting an argument to just a single
      value (\ie, cross-function constant propagation, effectively removing
      the argument). 
      
      Since we limit the argument domain of the specialized function, its
      definition can often be optimized further (since now more types or even
      values of arguments are already known). By replacing any application of
      the function that falls within the reduced domain by an application of
      the specialized version, the code gets faster (but the code also gets
      bigger, since we now have two versions instead of one). If we apply
      this technique often enough, we can often replace all applications of a
      function by specialized versions, allowing the original function to be
      removed (in some cases, this can even give a net reduction of the code
      compared to the non-specialized version).

      Specialization is useful for our hardware descriptions for functions
      that contain arguments that cannot be translated to hardware directly
      (polymorphic or higher order arguments, for example). If we can create
      specialized functions that remove the argument, or make it translatable,
      we can use specialization to make the original, untranslatable, function
      obsolete.

  \section{Higher order values}
    What holds for partial application, can be easily generalized to any
    higher order expression. This includes partial applications, plain
    variables (e.g., a binder referring to a top level function), lambda
    expressions and more complex expressions with a function type (a \hs{case}
    expression returning lambda's, for example).

    Each of these values cannot be directly represented in hardware (just like
    partial applications). Also, to make them representable, they need to be
    applied: function variables and partial applications will then eventually
    become complete applications, applied lambda expressions disappear by
    applying β-reduction, etc.

    So any higher order value will be \quote{pushed down} towards its
    application just like partial applications. Whenever a function boundary
    needs to be crossed, the called function can be specialized.
    
    \fxnote{This section needs improvement and an example}

  \section{Polymorphism}
    In Haskell, values can be \emph{polymorphic}: They can have multiple types. For
    example, the function \hs{fst :: (a, b) -> a} is an example of a
    polymorphic function: It works for tuples with any two element types. Haskell
    typeclasses allow a function to work on a specific set of types, but the
    general idea is the same. The opposite of this is a \emph{monomorphic}
    value, which has a single, fixed, type.

%    A type class is a collection of types for which some operations are
%    defined. It is thus possible for a value to be polymorphic while having
%    any number of \emph{class constraints}: The value is not defined for
%    every type, but only for types in the type class. An example of this is
%    the \hs{even :: (Integral a) => a -> Bool} function, which can map any
%    value of a type that is member of the \hs{Integral} type class 

    When generating hardware, polymorphism can't be easily translated. How
    many wires will you lay down for a value that could have any type? When
    type classes are involved, what hardware components will you lay down for
    a class method (whose behaviour depends on the type of its arguments)?
    Note that the language currently does not allow user-defined typeclasses,
    but does support partly some of the builtin typeclasses (like \hs{Num}).

    Fortunately, we can again use the principle of specialization: Since every
    function application generates a separate piece of hardware, we can know
    the types of all arguments exactly. Provided that we don't use existential
    typing, all of the polymorphic types in a function must depend on the
    types of the arguments (In other words, the only way to introduce a type
    variable is in a lambda abstraction).

    If a function is monomorphic, all values inside it are monomorphic as
    well, so any function that is applied within the function can only be
    applied to monomorphic values. The applied functions can then be
    specialized to work just for these specific types, removing the
    polymorphism from the applied functions as well.

    Our top level function must not have a polymorphic type (otherwise we
    wouldn't know the hardware interface to our top level function).

    By induction, this means that all functions that are (indirectly) called
    by our top level function (meaning all functions that are translated in
    the final hardware) become monomorphic.

  \section{State}
    A very important concept in hardware designs is \emph{state}. In a
    stateless (or, \emph{combinatoric}) design, every output is  directly and solely dependent on the
    inputs. In a stateful design, the outputs can depend on the history of
    inputs, or the \emph{state}. State is usually stored in \emph{registers},
    which retain their value during a clockcycle, and are typically updated at
    the start of every clockcycle. Since the updating of the state is tightly
    coupled (synchronized) to the clock signal, these state updates are often
    called \emph{synchronous} behaviour.

    \todo{Sidenote? Registers can contain any (complex) type}
  
    To make our hardware description language useful to describe more than
    simple combinatoric designs, we'll need to be able to describe state in
    some way.

    \subsection{Approaches to state}
      In Haskell, functions are always pure (except when using unsafe
      functions with the \hs{IO} monad, which is not supported by Cλash). This
      means that the output of a function solely depends on its inputs. If you
      evaluate a given function with given inputs, it will always provide the
      same output.

      \todo{Define pure}

      This is a perfect match for a combinatoric circuit, where the output
      also soley depends on the inputs. However, when state is involved, this
      no longer holds. Since we're in charge of our own language (or at least
      let's pretend we aren't using Haskell and we are), we could remove this
      purity constraint and allow a function to return different values
      depending on the cycle in which it is evaluated (or rather, the current
      state). However, this means that all kinds of interesting properties of
      our functional language get lost, and all kinds of transformations and
      optimizations might no longer be meaning preserving.

      Provided that we want to keep the function pure, the current state has
      to be present in the function's arguments in some way. There seem to be
      two obvious ways to do this: Adding the current state as an argument, or
      including the full history of each argument.

      \subsubsection{Stream arguments and results}
        Including the entire history of each input (\eg, the value of that
        input for each previous clockcycle) is an obvious way to make outputs
        depend on all previous input. This is easily done by making every
        input a list instead of a single value, containing all previous values
        as well as the current value.

        An obvious downside of this solution is that on each cycle, all the
        previous cycles must be resimulated to obtain the current state. To do
        this, it might be needed to have a recursive helper function as well,
        wich might be hard to be properly analyzed by the compiler.

        A slight variation on this approach is one taken by some of the other
        functional \small{HDL}s in the field: \todo{References to Lava,
        ForSyDe, ...} Make functions operate on complete streams. This means
        that a function is no longer called on every cycle, but just once. It
        takes stream as inputs instead of values, where each stream contains
        all the values for every clockcycle since system start. This is easily
        modeled using an (infinite) list, with one element for each clock
        cycle. Since the function is only evaluated once, its output must also
        be a stream. Note that, since we are working with infinite lists and
        still want to be able to simulate the system cycle-by-cycle, this
        relies heavily on the lazy semantics of Haskell.

        Since our inputs and outputs are streams, all other (intermediate)
        values must be streams. All of our primitive operators (\eg, addition,
        substraction, bitwise operations, etc.) must operate on streams as
        well (note that changing a single-element operation to a stream
        operation can done with \hs{map}, \hs{zipwith}, etc.).

        Note that the concept of \emph{state} is no more than having some way
        to communicate a value from one cycle to the next. By introducing a
        \hs{delay} function, we can do exactly that: Delay (each value in) a
        stream so that we can "look into" the past. This \hs{delay} function
        simply outputs a stream where each value is the same as the input
        value, but shifted one cycle. This causes a \quote{gap} at the
        beginning of the stream: What is the value of the delay output in the
        first cycle? For this, the \hs{delay} function has a second input
        (which is a value, not a stream!).

        \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
        style.

\startbuffer[DelayAcc]
acc :: Stream Word -> Stream Word
acc in = out
  where
    out = (delay out 0) + in
\stopbuffer

\startuseMPgraphic{DelayAcc}
  save in, out, add, reg;

  % I/O ports
  newCircle.in(btex $in$ etex) "framed(false)";
  newCircle.out(btex $out$ etex) "framed(false)";

  % Components
  newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
  newCircle.add(btex + etex);
  
  in.c    = origin;
  add.c   = in.c + (2cm, 0cm);
  out.c   = add.c + (2cm, 0cm);
  reg.c   = add.c + (0cm, 2cm);

  % Draw objects and lines
  drawObj(in, out, add, reg);

  nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
  nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
  ncline(in)(add);
  ncline(add)(out);
\stopuseMPgraphic


        \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
          \startcombination[2*1]
            {\typebufferhs{DelayAcc}}{Haskell description using streams.}
            {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
          \stopcombination


        This notation can be confusing (especially due to the loop in the
        definition of out), but is essentially easy to interpret. There is a
        single call to delay, resulting in a circuit with a single register,
        whose input is connected to \hs{out} (which is the output of the
        adder), and it's output is the expression \hs{delay out 0} (which is
        connected to one of the adder inputs).

        This notation has a number of downsides, amongst which are limited
        readability and ambiguity in the interpretation. \note{Reference
        Christiaan, who has done further investigation}
        
      \subsubsection{Explicit state arguments and results}
        A more explicit way to model state, is to simply add an extra argument
        containing the current state value. This allows an output to depend on
        both the inputs as well as the current state while keeping the
        function pure (letting the result depend only on the arguments), since
        the current state is now an argument.

        In Haskell, this would look like
        \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{Note that this example is not in the final
  Cλash syntax}. \todo{Referencing notfinalsyntax from Introduction.tex doesn't
  work}

\startbuffer[ExplicitAcc]
-- input -> current state -> (new state, output)
acc :: Word -> Word -> (Word, Word)
acc in s = (s', out)
  where
    out = s + in
    s'  = out
\stopbuffer

        \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
          \startcombination[2*1]
            {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
            % Picture is identical to the one we had just now.
            {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
          \stopcombination

        This approach makes a function's state very explicit, which state
        variables are used by a function can be completely determined from its
        type signature (as opposed to the stream approach, where a function
        looks the same from the outside, regardless of what state variables it
        uses (or whether it's stateful at all).

        This approach is the one chosen for Cλash and will be examined more
        closely below.

    \subsection{Explicit state specification}
      We've seen the concept of explicit state in a simple example below, but
      what are the implications of this approach?

      \subsubsection{Substates}
        Since a function's state is reflected directly in its type signature,
        if a function calls other stateful functions (\eg, has subcircuits), it
        has to somehow know the current state for these called functions. The
        only way to do this, is to put these \emph{substates} inside the
        caller's state. This means that a function's state is the sum of the
        states of all functions it calls, and its own state.

        This also means that the type of a function (at least the "state"
        part) is dependent on its own implementation and of the functions it
        calls.

        This is the major downside of this approach: The separation between
        interface and implementation is limited. However, since Cλash is not
        very suitable for separate compilation (see
        \in{section}[sec:prototype:separate]) this is not a big problem in
        practice.

        Additionally, when using a type synonym for the state type
        of each function, we can still provide explicit type signatures
        while keeping the state specification for a function near its
        definition only.
        \todo{Example}
    
      \subsubsection{Which arguments and results are stateful?}
        \fxnote{This section should get some examples}
        We need some way to know which arguments should become input ports and
        which argument(s?) should become the current state (\eg, be bound to
        the register outputs). This does not hold just for the top
        level function, but also for any subfunction. Or could we perhaps
        deduce the statefulness of subfunctions by analyzing the flow of data
        in the calling functions?

        To explore this matter, the following observeration is interesting: We
        get completely correct behaviour when we put all state registers in
        the top level entity (or even outside of it). All of the state
        arguments and results on subfunctions are treated as normal input and
        output ports. Effectively, a stateful function results in a stateless
        hardware component that has one of its input ports connected to the
        output of a register and one of its output ports connected to the
        input of the same register.

        \todo{Example}

        Of course, even though the hardware described like this has the
        correct behaviour, unless the layout tool does smart optimizations,
        there will be a lot of extra wire in the design (since registers will
        not be close to the component that uses them). Also, when working with
        the generated \small{VHDL} code, there will be a lot of extra ports
        just to pass on state values, which can get quite confusing.

        To fix this, we can simply \quote{push} the registers down into the
        subcircuits. When we see a register that is connected directly to a
        subcircuit, we remove the corresponding input and output port and put
        the register inside the subcircuit instead. This is slightly less
        trivial when looking at the Haskell code instead of the resulting
        circuit, but the idea is still the same.

        \todo{Example}

        However, when applying this technique, we might push registers down
        too far. When you intend to store a result of a stateless subfunction
        in the caller's state and pass the current value of that state
        variable to that same function, the register might get pushed down too
        far. It is impossible to distinguish this case from similar code where
        the called function is in fact stateful. From this we can conclude
        that we have to either:

        \todo{Example of wrong downpushing}

        \startitemize
        \item accept that the generated hardware might not be exactly what we
        intended, in some specific cases. In most cases, the hardware will be
        what we intended.
        \item explicitely annotate state arguments and results in the input
        description.
        \stopitemize

        The first option causes (non-obvious) exceptions in the language
        intepretation. Also, automatically determining where registers should
        end up is easier to implement correctly with explicit annotations, so
        for these reasons we will look at how this annotations could work.

        \todo{Sidenote: One or more state arguments?}

    \subsection{Explicit state annotation}
      To make our stateful descriptions unambigious and easier to translate,
      we need some way for the developer to describe which arguments and
      results are intended to become stateful.
    
      Roughly, we have two ways to achieve this:
      \startitemize[KR]
        \item Use some kind of annotation method or syntactic construction in
        the language to indicate exactly which argument and (part of the)
        result is stateful. This means that the annotation lives
        \quote{outside} of the function, it is completely invisible when
        looking at the function body.
        \item Use some kind of annotation on the type level, \ie give stateful
        arguments and stateful (parts of) results a different type. This has the
        potential to make this annotation visible inside the function as well,
        such that when looking at a value inside the function body you can
        tell if it's stateful by looking at its type. This could possibly make
        the translation process a lot easier, since less analysis of the
        program flow might be required.
      \stopitemize

      From these approaches, the type level \quote{annotations} have been
      implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
      the possible ways this could have been implemented.

  \todo{Note about conditions on state variables and checking them}

  \section[sec:recursion]{Recursion}
  An import concept in functional languages is recursion. In it's most basic
  form, recursion is a definition that is defined in terms of itself. A
  recursive function is thus a function that uses itself in its body. This
  usually requires multiple evaluations of this function, with changing
  arguments, until eventually an evaluation of the function no longer requires
  itself.

  Recursion in a hardware description is a bit of a funny thing. Usually,
  recursion is associated with a lot of nondeterminism, stack overflows, but
  also flexibility and expressive power.

  Given the notion that each function application will translate to a
  component instantiation, we are presented with a problem. A recursive
  function would translate to a component that contains itself. Or, more
  precisely, that contains an instance of itself. This instance would again
  contain an instance of itself, and again, into infinity. This is obviously a
  problem for generating hardware.

  This is expected for functions that describe infinite recursion. In that
  case, we can't generate hardware that shows correct behaviour in a single
  cycle (at best, we could generate hardware that needs an infinite number of
  cycles to complete).
  
  However, most recursive hardware descriptions will describe finite
  recursion. This is because the recursive call is done conditionally. There
  is usually a \hs{case} expression where at least one alternative does not contain
  the recursive call, which we call the "base case". If, for each call to the
  recursive function, we would be able to detect at compile time which
  alternative applies, we would be able to remove the \hs{case} expression and
  leave only the base case when it applies. This will ensure that expanding
  the recursive functions will terminate after a bounded number of expansions.

  This does imply the extra requirement that the base case is detectable at
  compile time. In particular, this means that the decision between the base
  case and the recursive case must not depend on runtime data.

  \subsection{List recursion}
  The most common deciding factor in recursion is the length of a list that is
  passed in as an argument. Since we represent lists as vectors that encode
  the length in the vector type, it seems easy to determine the base case. We
  can simply look at the argument type for this. However, it turns out that
  this is rather non-trivial to write down in Haskell already, not even
  looking at translation. As an example, we would like to write down something
  like this:
 
  \starthaskell
    sum :: Vector n Word -> Word
    sum xs = case null xs of
      True -> 0
      False -> head xs + sum (tail xs)
  \stophaskell

  However, the Haskell typechecker will now use the following reasoning (element
  type of the vector is left out). Below, we write conditions on type
  variables before the \hs{=>} operator. This is not completely valid Haskell
  syntax, but serves to illustrate the typechecker reasoning. Also note that a
  vector can never have a negative length, so \hs{Vector n} implicitly means
  \hs{(n >= 0) => Vector n}.

  \todo{This typechecker disregards the type signature}
  \startitemize
  \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
  \item This means that xs must have the type \hs{(n > 0) => Vector n}
  \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
  \item sum is called with the result of tail as an argument, which has the
  type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
  => Vector n}, which is the same as just \hs{Vector n}).
  \item This means that sum must have the type \hs{Vector n -> a}
  \item This is a contradiction between the type deduced from the body of sum
  (the input vector must be non-empty) and the use of sum (the input vector
  could have any length).
  \stopitemize

  As you can see, using a simple \hs{case} expression  at value level causes
  the type checker to always typecheck both alternatives, which can't be done!
  This is a fundamental problem, that would seem perfectly suited for a type
  class.  Considering that we need to switch between to implementations of the
  sum function, based on the type of the argument, this sounds like the
  perfect problem to solve with a type class. However, this approach has its
  own problems (not the least of them that you need to define a new typeclass
  for every recursive function you want to define).

  Another approach tried involved using GADTs to be able to do pattern
  matching on empty / non empty lists. While this worked partially, it also
  created problems with more complex expressions.

  \note{This should reference Christiaan}

  Evaluating all possible (and non-possible) ways to add recursion to our
  descriptions, it seems better to leave out list recursion alltogether. This
  allows us to focus on other interesting areas instead. By including
  (builtin) support for a number of higher order functions like map and fold,
  we can still express most of the things we would use list recursion for.
  
  \todo{Expand on this decision a bit}
 
  \subsection{General recursion}
  Of course there are other forms of recursion, that do not depend on the
  length (and thus type) of a list. For example, simple recursion using a
  counter could be expressed, but only translated to hardware for a fixed
  number of iterations. Also, this would require extensive support for compile
  time simplification (constant propagation) and compile time evaluation
  (evaluation of constant comparisons), to ensure non-termination. Even then, it
  is hard to really guarantee termination, since the user (or GHC desugarer)
  might use some obscure notation that results in a corner case of the
  simplifier that is not caught and thus non-termination.

  Due to these complications and limited time available, we leave other forms
  of recursion as future work as well.

% vim: set sw=2 sts=2 expandtab:
