Actually add bibtex entries for added references.

[matthijs/master-project/report.git] / Chapters / Prototype.tex

\chapter[chap:prototype]{Prototype}
  An important step in this research is the creation of a prototype compiler.
  Having this prototype allows us to apply the ideas from the previous chapter
  to actual hardware descriptions and evaluate their usefulness. Having a
  prototype also helps to find new techniques and test possible
  interpretations.

  Obviously the prototype was not created after all research
  ideas were formed, but its implementation has been interleaved with the
  research itself. Also, the prototype described here is the final version, it
  has gone through a number of design iterations which we will not completely
  describe here.

  \section[sec:prototype:input]{Input language}
    When implementing this prototype, the first question to ask is:
    Which (functional) language will be used to describe our hardware?
    (Note that this does not concern the \emph{implementation language}
    of the compiler, just the language \emph{translated by} the
    compiler).

    Initially, we have two choices:

    \startitemize
      \item Create a new functional language from scratch. This has the
      advantage of having a language that contains exactly those elements that
      are convenient for describing hardware and can contain special
      constructs that allows our hardware descriptions to be more powerful or
      concise.
      \item Use an existing language and create a new back-end for it. This has
      the advantage that existing tools can be reused, which will speed up
      development.
    \stopitemize


    \placeintermezzo{}{
      \startframedtext[width=8cm,background=box,frame=no]
      \startalignment[center]
        {\tfa No \small{EDSL} or Template Haskell}
      \stopalignment
      \blank[medium]

      Note that in this consideration, embedded domain-specific
      languages (\small{EDSL}) and Template Haskell (\small{TH})
      approaches have not been included. As we have seen in
      \in{section}[sec:context:fhdls], these approaches have all kinds
      of limitations on the description language that we would like to
      avoid.
      \stopframedtext
    }
    Considering that we required a prototype which should be working quickly,
    and that implementing parsers, semantic checkers and especially
    type-checkers is not exactly the Core of this research (but it is lots and
    lots of work!), using an existing language is the obvious choice. This
    also has the advantage that a large set of language features is available
    to experiment with and it is easy to find which features apply well and
    which do not. Another import advantage of using an existing language, is
    that simulation of the code becomes trivial. Since there are existing
    compilers and interpreters that can run the hardware description directly,
    it can be simulated without also having to write an interpreter for the
    new language.
    
    A possible second prototype could use a custom language with just the useful
    features (and possibly extra features that are specific to
    the domain of hardware description as well).

    The second choice to be made is which of the many existing languages to use. As
    mentioned before, the chosen language is Haskell.  This choice has not been the
    result of a thorough comparison of languages, for the simple reason that
    the requirements on the language were completely unclear at the start of
    this research. The fact that Haskell is a language with a broad spectrum
    of features, that it is commonly used in research projects and that the
    primary compiler, \GHC, provides a high level API to its internals, made
    Haskell an obvious choice.

  \section[sec:prototype:output]{Output format}
    The second important question is: what will be our output format?
    This output format should at least allow for programming the
    hardware design into a field-programmable gate array (\small{FPGA}).
    The choice of output format is thus limited by what hardware
    synthesis and programming tools can process.

    Looking at other tools in the industry, the Electronic Design Interchange
    Format (\small{EDIF}) is commonly used for storing intermediate
    \emph{netlists} (lists of components and connections between these
    components) and is commonly the target for \small{VHDL} and Verilog
    compilers.

    However, \small{EDIF} is not completely tool-independent. It specifies a
    meta-format, but the hardware components that can be used vary between
    various tool and hardware vendors, as well as the interpretation of the
    \small{EDIF} standard. \cite[li89]
   
    This means that when working with \small{EDIF}, our prototype would become
    technology dependent (\eg\ only work with \small{FPGA}s of a specific
    vendor, or even only with specific chips). This limits the applicability
    of our prototype. Also, the tools we would like to use for verifying,
    simulating and draw pretty pictures of our output (like Precision, or
    QuestaSim) are designed for \small{VHDL} or Verilog input.

    For these reasons, we will not use \small{EDIF}, but \small{VHDL} as our
    output language.  We choose \VHDL\ over Verilog simply because we are
    familiar with \small{VHDL} already. The differences between \small{VHDL}
    and Verilog are on the higher level, while we will be using \small{VHDL}
    mainly to write low level, netlist-like descriptions anyway.

    An added advantage of using VHDL is that we can profit from existing
    optimizations in VHDL synthesizers. A lot of optimizations are done on the
    VHDL level by existing tools. These tools have been under
    development for years, so it would not be reasonable to assume we
    could achieve a similar amount of optimization in our prototype (nor
    should it be a goal, considering this is just a prototype).

    \placeintermezzo{}{
      \startframedtext[width=8cm,background=box,frame=no]
      \startalignment[center]
        {\tfa Translation vs. compilation vs. synthesis}
      \stopalignment
      \blank[medium]
        In this thesis the words \emph{translation}, \emph{compilation} and
        sometimes \emph{synthesis} will be used interchangeably to refer to the
        process of translating the hardware description from the Haskell
        language to the \VHDL\ language.

        Similarly, the prototype created is referred to as both the
        \emph{translator} as well as the \emph{compiler}.

        The final part of this process is usually referred to as \emph{\VHDL\
        generation}.
      \stopframedtext
    }

    Note that we will be using \small{VHDL} as our output language, but will
    not use its full expressive power. Our output will be limited to using
    simple, structural descriptions, without any complex behavioral
    descriptions like arbitrary sequential statements (which might not
    be supported by all tools). This ensures that any tool that works
    with \VHDL\ will understand our output (most tools do not support
    synthesis of more complex \VHDL).  This also leaves open the option
    to switch to \small{EDIF} in the future, with minimal changes to the
    prototype.

  \section{Simulation and synthesis}
    As mentioned above, by using the Haskell language, we get simulation of
    our hardware descriptions almost for free. The only thing that is needed
    is to provide a Haskell implementation of all built-in functions that can
    be used by the Haskell interpreter to simulate them.

    The main topic of this thesis is therefore the path from the Haskell
    hardware descriptions to \small{FPGA} synthesis, focusing of course on the
    \VHDL\ generation. Since the \VHDL\ generation process preserves the meaning
    of the Haskell description exactly, any simulation done in Haskell
    \emph{should} produce identical results as the synthesized hardware.

  \section[sec:prototype:design]{Prototype design}
    As suggested above, we will use the Glasgow Haskell Compiler (\small{GHC}) to
    implement our prototype compiler. To understand the design of the
    compiler, we will first dive into the \small{GHC} compiler a bit. Its
    compilation consists of the following steps (slightly simplified):

    \startuseMPgraphic{ghc-pipeline}
      % Create objects
      save inp, front, desugar, simpl, back, out;
      newEmptyBox.inp(0,0);
      newBox.front(btex Frontend etex);
      newBox.desugar(btex Desugarer etex);
      newBox.simpl(btex Simplifier etex);
      newBox.back(btex Backend etex);
      newEmptyBox.out(0,0);

      % Space the boxes evenly
      inp.c - front.c = front.c - desugar.c = desugar.c - simpl.c 
        = simpl.c - back.c = back.c - out.c = (0, 1.5cm);
      out.c = origin;

      % Draw lines between the boxes. We make these lines "deferred" and give
      % them a name, so we can use ObjLabel to draw a label beside them.
      ncline.inp(inp)(front) "name(haskell)";
      ncline.front(front)(desugar) "name(ast)";
      ncline.desugar(desugar)(simpl) "name(core)";
      ncline.simpl(simpl)(back) "name(simplcore)";
      ncline.back(back)(out) "name(native)";
      ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)";
      ObjLabel.front(btex Haskell AST etex) "labpathname(ast)", "labdir(rt)";
      ObjLabel.desugar(btex Core etex) "labpathname(core)", "labdir(rt)";
      ObjLabel.simpl(btex Simplified core etex) "labpathname(simplcore)", "labdir(rt)";
      ObjLabel.back(btex Native code etex) "labpathname(native)", "labdir(rt)";

      % Draw the objects (and deferred labels)
      drawObj (inp, front, desugar, simpl, back, out);
    \stopuseMPgraphic
    \placefigure[right]{GHC compiler pipeline}{\startboxed \useMPgraphic{ghc-pipeline}\stopboxed}

    \startdesc{Frontend}
      This step takes the Haskell source files and parses them into an
      abstract syntax tree (\small{AST}). This \small{AST} can express the
      complete Haskell language and is thus a very complex one (in contrast
      with the Core \small{AST}, later on). All identifiers in this
      \small{AST} are resolved by the renamer and all types are checked by the
      type-checker.
    \stopdesc
    \startdesc{Desugaring}
      This steps takes the full \small{AST} and translates it to the
      \emph{Core} language. Core is a very small functional language with lazy
      semantics, that can still express everything Haskell can express. Its
      simpleness makes Core very suitable for further simplification and
      translation. Core is the language we will be working with as well.
    \stopdesc
    \startdesc{Simplification}
      Through a number of simplification steps (such as inlining, common
      sub-expression elimination, etc.) the Core program is simplified to make
      it faster or easier to process further.
    \stopdesc
    \startdesc{Backend}
      This step takes the simplified Core program and generates an actual
      runnable program for it. This is a big and complicated step we will not
      discuss it any further, since it is not required for our prototype.
    \stopdesc

    In this process, there are a number of places where we can start our work.
    Assuming that we do not want to deal with (or modify) parsing, type-checking
    and other front end business and that native code is not really a useful
    format anymore, we are left with the choice between the full Haskell
    \small{AST}, or the smaller (simplified) Core representation.

    The advantage of taking the full \small{AST} is that the exact structure
    of the source program is preserved. We can see exactly what the hardware
    description looks like and which syntax constructs were used. However,
    the full \small{AST} is a very complicated data-structure. If we are to
    handle everything it offers, we will quickly get a big compiler.

    Using the Core representation gives us a much more compact data-structure
    (a Core expression only uses 9 constructors). Note that this does not mean
    that the Core representation itself is smaller, on the contrary.
    Since the Core language has less constructs, most Core expressions
    are larger than the equivalent versions in Haskell.

    However, the fact that the Core language is so much smaller, means it is a
    lot easier to analyze and translate it into something else. For the same
    reason, \small{GHC} runs its simplifications and optimizations on the Core
    representation as well \cite[jones96].

    We will use the normal Core representation, not the simplified Core. Even
    though the simplified Core version is an equivalent, but simpler
    definition, some problems were encountered with it in practice. The
    simplifier restructures some (stateful) functions in a way the normalizer
    and the \VHDL\ generation cannot handle, leading to uncompilable programs
    (whereas the non-simplified version more closely resembles the original
    program, allowing the original to be written in a way that can be
    handled). This problem is further discussed in
    \in{section}[sec:normalization:stateproblems].
    
    \startuseMPgraphic{clash-pipeline}
      % Create objects
      save inp, front, norm, vhdl, out;
      newEmptyBox.inp(0,0);
      newBox.front(btex \small{GHC} front-end etex);
      newBox.norm(btex Normalization etex);
      newBox.vhdl(btex \small{VHDL} generation etex);
      newEmptyBox.out(0,0);

      % Space the boxes evenly
      inp.c - front.c = front.c - norm.c = norm.c - vhdl.c 
        = vhdl.c - out.c = (0, 1.5cm);
      out.c = origin;

      % Draw lines between the boxes. We make these lines "deferred" and give
      % them a name, so we can use ObjLabel to draw a label beside them.
      ncline.inp(inp)(front) "name(haskell)";
      ncline.front(front)(norm) "name(core)";
      ncline.norm(norm)(vhdl) "name(normal)";
      ncline.vhdl(vhdl)(out) "name(vhdl)";
      ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)";
      ObjLabel.front(btex Core etex) "labpathname(core)", "labdir(rt)";
      ObjLabel.norm(btex Normalized core etex) "labpathname(normal)", "labdir(rt)";
      ObjLabel.vhdl(btex \small{VHDL} description etex) "labpathname(vhdl)", "labdir(rt)";

      % Draw the objects (and deferred labels)
      drawObj (inp, front, norm, vhdl, out);
    \stopuseMPgraphic
    \placefigure[right]{Cλash compiler pipeline}{\startboxed \useMPgraphic{clash-pipeline}\stopboxed}

    The final prototype roughly consists of three steps:

    \page[no] % suppress page break here.
    \startdesc{Frontend}
      This is exactly the front-end from the \small{GHC} pipeline, that
      translates Haskell sources to a typed Core representation.
    \stopdesc
    \startdesc{Normalization}
      This is a step that transforms the Core representation into a normal
      form. This normal form is still expressed in the Core language, but has
      to adhere to an additional set of constraints. This normal form is less
      expressive than the full Core language (e.g., it can have limited 
      higher-order expressions, has a specific structure, etc.), but is
      also very close to directly describing hardware.
    \stopdesc
    \startdesc{\small{VHDL} generation}
      The last step takes the normal formed Core representation and generates
      \small{VHDL} for it. Since the normal form has a specific, hardware-like
      structure, this final step is very straightforward.
    \stopdesc
    
    The most interesting step in this process is the normalization step. That
    is where more complicated functional constructs, which have no direct
    hardware interpretation, are removed and translated into hardware
    constructs. This step is described in a lot of detail at
    \in{chapter}[chap:normalization].

    
    \defref{entry function}Translation of a hardware description always
    starts at a single function, which is referred to as the \emph{entry
    function}. \VHDL\ is generated for this function first, followed by
    any functions used by the entry functions (recursively).
    
  \section[sec:prototype:core]{The Core language}
    \defreftxt{Core}{the Core language}
    Most of the prototype deals with handling the program in the Core
    language. In this section we will show what this language looks like and
    how it works.

    The Core language is a functional language that describes
    \emph{expressions}. Every identifier used in Core is called a
    \emph{binder}, since it is bound to a value somewhere. On the highest
    level, a Core program is a collection of functions, each of which bind a
    binder (the function name) to an expression (the function value, which has
    a function type).

    The Core language itself does not prescribe any program structure
    (like modules, declarations, imports, etc.), only expression
    structure. In the \small{GHC} compiler, the Haskell module structure
    is used for the resulting Core code as well. Since this is not so
    relevant for understanding the Core language or the Normalization
    process, we will only look at the Core expression language here.

    Each Core expression consists of one of these possible expressions.

    \startdesc{Variable reference}
      \defref{variable reference}
      \startlambda
      bndr :: T
      \stoplambda
      This is a reference to a binder. It is written down as the
      name of the binder that is being referred to along with its type. The
      binder name should of course be bound in a containing scope
      (including top level scope, so a reference to a top level function
      is also a variable reference). Additionally, constructors from
      algebraic data-types also become variable references.

      In our examples, binders will commonly consist of a single
      characters, but they can have any length.

      The value of this expression is the value bound to the given
      binder.

      Each binder also carries around its type (explicitly shown above), but
      this is usually not shown in the Core expressions. Only when the type is
      relevant (when a new binder is introduced, for example) will it be
      shown. In other cases, the binder is either not relevant, or easily
      derived from the context of the expression. \todo{Ref sidenote on type
      annotations}
    \stopdesc

    \startdesc{Literal}
      \defref{literal}
      \startlambda
      10
      \stoplambda
      This is a literal. Only primitive types are supported, like
      chars, strings, integers and doubles. The types of these literals are the
      \quote{primitive}, unboxed versions, like \lam{Char\#} and \lam{Word\#}, not the
      normal Haskell versions (but there are built-in conversion
      functions). Without going into detail about these types, note that
      a few conversion functions exist to convert these to the normal
      (boxed) Haskell equivalents.
    \stopdesc

    \startdesc{Application}
      \defref{application}
      \startlambda
      func arg
      \stoplambda
      This is function application. Each application consists of two
      parts: the function part and the argument part. Applications are used
      for normal function \quote{calls}, but also for applying type
      abstractions and data constructors.

      In Core, there is no distinction between an operator and a
      function. This means that, for example the addition of two numbers
      looks like the following in Core:
      
      \startlambda
      (+) 1 2
      \stoplambda

      Where the function \quote{\lam{(+)}} is applied to the numbers 1
      and 2. However, to increase readability, an application of an
      operator like \lam{(+)} is sometimes written infix. In this case,
      the parenthesis are also left out, just like in Haskell. In other
      words, the following means exactly the same as the addition above:

      \startlambda
      1 + 2
      \stoplambda

      The value of an application is the value of the function part, with the
      first argument binder bound to the argument part.
    \stopdesc

    \startdesc{Lambda abstraction}
      \defref{lambda abstraction}
      \startlambda
      λbndr.body
      \stoplambda
      This is the basic lambda abstraction, as it occurs in lambda calculus.
      It consists of a binder part and a body part.  A lambda abstraction
      creates a function, that can be applied to an argument. The binder is
      usually a value binder, but it can also be a \emph{type binder} (or
      \emph{type variable}). The latter case introduces a new polymorphic
      variable, which can be used in types later on. See
      \in{section}[sec:prototype:coretypes] for details.
     
      The body of a lambda abstraction extends all the way to the end of
      the expression, or the closing bracket surrounding the lambda. In
      other words, the lambda abstraction \quote{operator} has the
      lowest priority of all.

      The value of an application is the value of the body part, with the
      binder bound to the value the entire lambda abstraction is applied to.
    \stopdesc

    \startdesc{Non-recursive let expression}
      \defref{let expression}
      \startlambda
      let bndr = value in body
      \stoplambda
      A let expression allows you to bind a binder to some value, while
      evaluating to some other value (for which that binder is in scope). This
      allows for sharing of sub-expressions (you can use a binder twice) and
      explicit \quote{naming} of arbitrary expressions. A binder is not
      in scope in the value bound it is bound to, so it is not possible
      to make recursive definitions with a non-recursive let expression
      (see the recursive form below).

      Even though this let expression is an extension on the basic lambda
      calculus, it is easily translated to a lambda abstraction. The let
      expression above would then become:

      \startlambda
      (λbndr.body) value
      \stoplambda

      This notion might be useful for verifying certain properties on
      transformations, since a lot of verification work has been done on
      lambda calculus already.

      The value of a let expression is the value of the body part, with the
      binder bound to the value. 
    \stopdesc

    \startdesc{Recursive let expression}
      \startlambda
      letrec
        bndr1 = value1
        \vdots
        bndrn = valuen
      in 
        body
      \stoplambda
      This is the recursive version of the let expression. In \small{GHC}'s
      Core implementation, non-recursive and recursive lets are not so
      distinct as we present them here, but this provides a clearer overview.
      
      The main difference with the normal let expression is that it can
      contain multiple bindings (or even none) and each of the binders
      is in scope in each of the values, in addition to the body. This
      allows for self-recursive or mutually recursive definitions.

      It is also possible to express a recursive let expression using
      normal lambda calculus, if we use the \emph{least fixed-point
      operator}, \lam{Y} (but the details are too complicated to help
      clarify the let expression, so this will not be explored further).
    \stopdesc

    \placeintermezzo{}{
      \startframedtext[width=8cm,background=box,frame=no]
      \startalignment[center]
        {\tfa Weak head normal form (\small{WHNF})}
      \stopalignment
      \blank[medium]
        An expression is in weak head normal form if it is either an
        constructor application or lambda abstraction. \cite[jones87]

        Without going into detail about the differences with head
        normal form and normal form, note that evaluating the scrutinee
        of a case expression to normal form (evaluating any function
        applications, variable references and case expressions) is
        sufficient to decide which case alternatives should be chosen.
        \todo{ref?}
      \stopframedtext

    }

    \startdesc{Case expression}
      \defref{case expression}
      \startlambda
        case scrutinee of bndr
          DEFAULT -> defaultbody
          C0 bndr0,0 ... bndr0,m -> body0
          \vdots
          Cn bndrn,0 ... bndrn,m -> bodyn
      \stoplambda

      A case expression is the only way in Core to choose between values. All
      \hs{if} expressions and pattern matchings from the original Haskell
      program have been translated to case expressions by the desugarer. 
      
      A case expression evaluates its scrutinee, which should have an
      algebraic datatype, into weak head normal form (\small{WHNF}) and
      (optionally) binds it to \lam{bndr}. If bndr is wild, \refdef{wild
      binders} it is left out.  Every alternative lists a single constructor
      (\lam{C0 ... Cn}). Based on the actual constructor of the scrutinee, the
      corresponding alternative is chosen. The binders in the chosen
      alternative (\lam{bndr0,0 ....  bndr0,m} are bound to the actual
      arguments to the constructor in the scrutinee.

      This is best illustrated with an example. Assume
      there is an algebraic datatype declared as follows\footnote{This
      datatype is not supported by the current Cλash implementation, but
      serves well to illustrate the case expression}:

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

      This is an algebraic datatype with two constructors, each getting
      a single argument. A case expression scrutinizing this datatype
      could look like the following:

      \startlambda
        case s of
          A word -> High
          B bit -> bit
      \stoplambda

      What this expression does is check the constructor of the
      scrutinee \lam{s}. If it is \lam{A}, it always evaluates to
      \lam{High}. If the constructor is \lam{B}, the binder \lam{bit} is
      bound to the argument passed to \lam{B} and the case expression
      evaluates to this bit.
      
      If none of the alternatives match, the \lam{DEFAULT} alternative
      is chosen. A case expression must always be exhaustive, \ie\ it
      must cover all possible constructors that the scrutinee can have
      (if all of them are covered explicitly, the \lam{DEFAULT}
      alternative can be left out).
      
      Since we can only match the top level constructor, there can be no overlap
      in the alternatives and thus order of alternatives is not relevant (though
      the \lam{DEFAULT} alternative must appear first for implementation
      efficiency).
      
      To support strictness, the scrutinee is always evaluated into
      \small{WHNF}, even when there is only a \lam{DEFAULT} alternative. This
      allows application of the strict function \lam{f} to the argument \lam{a}
      to be written like:

      \startlambda
      f (case a of arg DEFAULT -> arg)
      \stoplambda

      According to the \GHC\ documentation, this is the only use for the extra
      binder to which the scrutinee is bound.  When not using strictness
      annotations (which is rather pointless in hardware descriptions),
      \small{GHC} seems to never generate any code making use of this binder.
      In fact, \GHC\ has never been observed to generate code using this
      binder, even when strictness was involved.  Nonetheless, the prototype
      handles this binder as expected.

      Note that these case expressions are less powerful than the full Haskell
      case expressions. In particular, they do not support complex patterns like
      in Haskell. Only the constructor of an expression can be matched,
      complex patterns are implemented using multiple nested case expressions.

      Case expressions are also used for unpacking of algebraic data-types, even
      when there is only a single constructor. For examples, to add the elements
      of a tuple, the following Core is generated:

      \startlambda
      sum = λtuple.case tuple of
        (,) a b -> a + b
      \stoplambda
    
      Here, there is only a single alternative (but no \lam{DEFAULT}
      alternative, since the single alternative is already exhaustive). When
      its body is evaluated, the arguments to the tuple constructor \lam{(,)}
      (\eg, the elements of the tuple) are bound to \lam{a} and \lam{b}.
    \stopdesc

    \startdesc{Cast expression}
      \defref{cast expression}
      \startlambda
      body ▶ targettype
      \stoplambda
      A cast expression allows you to change the type of an expression to an
      equivalent type. Note that this is not meant to do any actual work, like
      conversion of data from one format to another, or force a complete type
      change. Instead, it is meant to change between different representations
      of the same type, \eg\ switch between types that are provably equal (but
      look different).
      
      In our hardware descriptions, we typically see casts to change between a
      Haskell newtype and its contained type, since those are effectively
      different types (so a cast is needed) with the same representation (but
      no work is done by the cast).

      More complex are types that are proven to be equal by the type-checker,
      but look different at first glance. To ensure that, once the type-checker
      has proven equality, this information sticks around, explicit casts are
      added. In our notation we only write the target type, but in reality a
      cast expressions carries around a \emph{coercion}, which can be seen as a
      proof of equality. \todo{Example}

      The value of a cast is the value of its body, unchanged. The type of this
      value is equal to the target type, not the type of its body.
    \stopdesc

    \startdesc{Note}
      The Core language in \small{GHC} allows adding \emph{notes}, which serve
      as hints to the inliner or add custom (string) annotations to a Core
      expression. These should not be generated normally, so these are not
      handled in any way in the prototype.
    \stopdesc

    \startdesc{Type}
      \defref{type expression}
      \startlambda
      @T
      \stoplambda
      It is possibly to use a Core type as a Core expression. To prevent
      confusion between types and values, the \lam{@} sign is used to
      explicitly mark a type that is used in a Core expression.
      
      For the actual types supported by Core, see
      \in{section}[sec:prototype:coretypes]. This \quote{lifting} of a
      type into the value domain is done to allow for type abstractions
      and applications to be handled as normal lambda abstractions and
      applications above. This means that a type expression in Core can
      only ever occur in the argument position of an application, and
      only if the type of the function that is applied to expects a type
      as the first argument. This happens in applications of all
      polymorphic functions. Consider the \lam{fst} function:

      \startlambda
      fst :: \forall t1. \forall t2. (t1, t2) ->t1 
      fst = λt1.λt2.λ(tup :: (t1, t2)). case tup of (,) a b -> a

      fstint :: (Int, Int) -> Int
      fstint = λa.λb.fst @Int @Int a b
      \stoplambda
          
      The type of \lam{fst} has two universally quantified type variables. When
      \lam{fst} is applied in \lam{fstint}, it is first applied to two types.
      (which are substituted for \lam{t1} and \lam{t2} in the type of \lam{fst}, so
      the actual type of arguments and result of \lam{fst} can be found:
      \lam{fst @Int @Int :: (Int, Int) -> Int}).
    \stopdesc

    \subsection[sec:prototype:coretypes]{Core type system}
      Whereas the expression syntax of Core is very simple, its type system is
      a bit more complicated. It turns out it is harder to \quote{desugar}
      Haskell's complex type system into something more simple. Most of the
      type system is thus very similar to that of Haskell.

      We will slightly limit our view on Core's type system, since the more
      complicated parts of it are only meant to support Haskell's (or rather,
      \GHC's) type extensions, such as existential types, \small{GADT}s, type
      families and other non-standard Haskell stuff which we do not (plan to)
      support.

      \placeintermezzo{}{
        \defref{id function}
        \startframedtext[width=8cm,background=box,frame=no]
        \startalignment[center]
          {\tfa The \hs{id} function}
        \stopalignment
        \blank[medium]
          A function that is probably present in every functional language, is
          the \emph{identity} function. This is the function that takes a
          single argument and simply returns it unmodified. In Haskell this
          function is called \hs{id} and can take an argument of any type
          (\ie, it is polymorphic).

          The \hs{id} function will be used in the examples every now and
          then.
        \stopframedtext
      }
      In Core, every expression is typed. The translation to Core happens
      after the type-checker, so types in Core are always correct as well
      (though you could of course construct invalidly typed expressions
      through the \GHC\ API).

      Any type in Core is one of the following:

      \startdesc{A type variable}
        \startlambda
        t
        \stoplambda

        This is a reference to a type defined elsewhere. This can either be a
        polymorphic type (like the latter two \lam{t}'s in \lam{id :: \forall t.
        t -> t}), or a type constructor (like \lam{Bool} in \lam{not :: Bool ->
        Bool}). Like in Haskell, polymorphic type variables always
        start with a lowercase letter, while type constructors always start
        with an uppercase letter.

        \todo{How to define (new) type constructors?}

        A special case of a type constructor is the \emph{function type
        constructor}, \lam{->}. This is a type constructor taking two arguments
        (using application below). The function type constructor is commonly
        written inline, so we write \lam{a -> b} when we really mean \lam{-> a
        b}, the function type constructor applied to \lam{a} and \lam{b}.

        Polymorphic type variables can only be defined by a lambda
        abstraction, see the forall type below.
      \stopdesc

      \startdesc{A type application}
        \startlambda
          Maybe Int
        \stoplambda

        This applies some type to another type. This is particularly used to
        apply type variables (type constructors) to their arguments.

        As mentioned above, applications of some type constructors have
        special notation. In particular, these are applications of the
        \emph{function type constructor} and \emph{tuple type constructors}:
        \startlambda
          foo :: t1 -> t2 
          foo' :: -> t1 t2 
          bar :: (t1, t2, t3)
          bar' :: (,,) t1 t2 t3
        \stoplambda
      \stopdesc

      \startdesc{The forall type}
        \startlambda
          id :: \forall t. t -> t
        \stoplambda
        The forall type introduces polymorphism. It is the only way to
        introduce new type variables, which are completely unconstrained (Any
        possible type can be assigned to it). Constraints can be added later
        using predicate types, see below.

        A forall type is always (and only) introduced by a type lambda
        expression. For example, the Core translation of the
        id function is:
        \startlambda
          id = λt.λ(x :: t).x
        \stoplambda

        Here, the type of the binder \lam{x} is \lam{t}, referring to the
        binder in the topmost lambda.

        When using a value with a forall type, the actual type
        used must be applied first. For example Haskell expression \hs{id
        True} (the function \hs{id} applied to the data-constructor \hs{True})
        translates to the following Core:

        \startlambda
        id @Bool True
        \stoplambda

        Here, id is first applied to the type to work with. Note that the type
        then changes from \lam{id :: \forall t. t -> t} to \lam{id @Bool ::
        Bool -> Bool}. Note that the type variable \lam{a} has been
        substituted with the actual type.

        In Haskell, forall types are usually not explicitly specified (The use
        of a lowercase type variable implicitly introduces a forall type for
        that variable). In fact, in standard Haskell there is no way to
        explicitly specify forall types. Through a language extension, the
        \hs{forall} keyword is available, but still optional for normal forall
        types (it is needed for \emph{existentially quantified types}, which
        Cλash does not support).
      \stopdesc

      \startdesc{Predicate type}
        \startlambda
          show :: \forall t. Show t ⇒ t → String
        \stoplambda
       
        \todo{Sidenote: type classes?}

        A predicate type introduces a constraint on a type variable introduced
        by a forall type (or type lambda). In the example above, the type
        variable \lam{t} can only contain types that are an \emph{instance} of
        the \emph{type class} \lam{Show}.

        There are other sorts of predicate types, used for the type families
        extension, which we will not discuss here.

        A predicate type is introduced by a lambda abstraction. Unlike with
        the forall type, this is a value lambda abstraction, that must be
        applied to a value. We call this value a \emph{dictionary}.

        Without going into the implementation details, a dictionary can be
        seen as a lookup table all the methods for a given (single) type class
        instance. This means that all the dictionaries for the same type class
        look the same (\eg\ contain methods with the same names). However,
        dictionaries for different instances of the same class contain
        different methods, of course.

        A dictionary is introduced by \small{GHC} whenever it encounters an
        instance declaration. This dictionary, as well as the binder
        introduced by a lambda that introduces a dictionary, have the
        predicate type as their type. These binders are usually named starting
        with a \lam{\$}. Usually the name of the type concerned is not
        reflected in the name of the dictionary, but the name of the type
        class is. The Haskell expression \hs{show True} thus becomes:

        \startlambda
        show @Bool \$dShow True
        \stoplambda
      \stopdesc

      Using this set of types, all types in basic Haskell can be represented.
      \todo{Overview of polymorphism with more examples (or move examples
      here)}
        
  \section[sec:prototype:statetype]{State annotations in Haskell}
    As noted in \in{section}[sec:description:stateann], Cλash needs some
    way to let the programmer explicitly specify which of a function's
    arguments and which part of a function's result represent the
    function's state.

    Using the Haskell type systems, there are a few ways we can tackle this.

    \subsection{Type synonyms}
      Haskell provides type synonyms as a way to declare a new type that is
      equal to an existing type (or rather, a new name for an existing type).
      This allows both the original type and the synonym to be used
      interchangeably in a Haskell program. This means no explicit conversion
      is needed. For example, a simple accumulator would become:

      \starthaskell
      -- This type synonym would become part of Cλash, it is shown here
      -- just for clarity.
      type State s = s

      acc :: Word -> State Word -> (State Word, Word)
      acc i s = let sum = s + i in (sum, sum)
      \stophaskell

      This looks nice in Haskell, but turns out to be hard to implement. There
      is no explicit conversion in Haskell, but not in Core either. This
      means the type of a value might be shown as \hs{State Word} in
      some places, but \hs{Word} in others (and this can even change due
      to transformations). Since every binder has an explicit type
      associated with it, the type of every function type will be
      properly preserved and could be used to track down the
      statefulness of each value by the compiler. However, this would make
      the implementation a lot more complicated than when using type
      renamings as described in the next section.

    % Use \type instead of \hs here, since the latter breaks inside
    % section headings.
    \subsection{Type renaming (\type{newtype})}
      Haskell also supports type renamings as a way to declare a new type that
      has the same (run-time) representation as an existing type (but is in
      fact a different type to the type-checker). With type renaming,
      explicit conversion between values of the two types is needed. The
      accumulator would then become:

      \starthaskell
      -- This type renaming would become part of Cλash, it is shown here
      -- just for clarity.
      newtype State s = State s

      acc :: Word -> State Word -> (State Word, Word)
      acc i (State s) = let sum = s + i in (State sum, sum)
      \stophaskell

      The \hs{newtype} line declares a new type \hs{State} that has one type
      argument, \hs{s}. This type contains one \quote{constructor} \hs{State}
      with a single argument of type \hs{s}. It is customary to name the
      constructor the same as the type, which is allowed (since types can
      never cause name collisions with values). The difference with the type
      synonym example is in the explicit conversion between the \hs{State
      Word} and \hs{Word} types by pattern matching and by using the explicit
      the \hs{State} constructor.

      This explicit conversion makes the \VHDL\ generation easier: whenever we
      remove (unpack) the \hs{State} type, this means we are accessing the
      current state (\ie, accessing the register output). Whenever we are
      adding (packing) the \hs{State} type, we are producing a new value for
      the state (\ie, providing the register input).

      When dealing with nested states (a stateful function that calls stateful
      functions, which might call stateful functions, etc.) the state type
      could quickly grow complex because of all the \hs{State} type constructors
      needed. For example, consider the following state type (this is just the
      state type, not the entire function type):

      \starthaskell
      State (State Bit, State (State Word, Bit), Word)
      \stophaskell

      We cannot leave all these \hs{State} type constructors out, since that
      would change the type (unlike when using type synonyms). However, when
      using type synonyms to hide away sub-states (see
      \in{section}[sec:prototype:sub-statesynonyms] below), this
      disadvantage should be limited.

      \subsubsection{Different input and output types}
        An alternative could be to use different types for input and output
        state (\ie\ current and updated state). The accumulator example would
        then become something like:

        \starthaskell
        -- These type renamings would become part of Cλash, it is shown
        -- here just for clarity.
        newtype StateIn s = StateIn s
        newtype StateOut s = StateOut s

        acc :: Word -> StateIn Word -> (StateIn Word, Word)
        acc i (StateIn s) = let sum = s + i in (StateIn sum, sum)
        \stophaskell
        
        This could make the implementation easier and the hardware
        descriptions less error-prone (you can no longer \quote{forget} to
        unpack and repack a state variable and just return it directly, which
        can be a problem in the current prototype). However, it also means we
        need twice as many type synonyms to hide away sub-states, making this
        approach a bit cumbersome. It also makes it harder to compare input
        and output state types, possible reducing the type-safety of the
        descriptions.

    \subsection[sec:prototype:sub-statesynonyms]{Type synonyms for sub-states}
      As noted above, when using nested (hierarchical) states, the state types
      of the \quote{upper} functions (those that call other functions, which
      call other functions, etc.) quickly become complicated. Also, when the
      state type of one of the \quote{lower} functions changes, the state
      types of all the upper functions changes as well. If the state type for
      each function is explicitly and completely specified, this means that a
      lot of code needs updating whenever a state type changes.

      To prevent this, it is recommended (but not enforced) to use a type
      synonym for the state type of every function. Every function calling
      other functions will then use the state type synonym of the called
      functions in its own type, requiring no code changes when the state type
      of a called function changes. This approach is used in
      \in{example}[ex:AvgState] below. The \hs{AccState} and \hs{AvgState}
      are examples of such state type synonyms.

    \subsection{Chosen approach}
      To keep implementation simple, the current prototype uses the type
      renaming approach, with a single type for both input and output
      states. In the future, it might be worthwhile to revisit this
      approach if more complicated flow analysis is implemented for
      state variables. This analysis is needed to add proper error
      checking anyway and might allow the use of type synonyms without
      losing any expressivity.

      \subsubsection{Example}
        As an example of the used approach, a simple averaging circuit
        is shown in \in{example}[ex:AvgState]. This circuit lets the
        accumulation of the inputs be done by a sub-component, \hs{acc},
        but keeps a count of value accumulated in its own
        state.\footnote{Currently, the prototype is not able to compile
        this example, since there is no built-in function for division.}
        
        \startbuffer[AvgState]
          -- This type renaming would become part of Cλash, it is shown
          -- here just for clarity
          newtype State s = State s

          -- The accumulator state type
          type AccState = State Word
          -- The accumulator
          acc :: Word -> AccState -> (AccState, Word)
          acc i (State s) = let sum = s + i in (State sum, sum)

          -- The averaging circuit state type
          type AvgState = State (AccState, Word)
          -- The averaging circuit
          avg :: Word -> AvgState -> (AvgState, Word)
          avg i (State s) = (State s', o)
            where
              (accs, count) = s
              -- Pass our input through the accumulator, which outputs a sum
              (accs', sum) = acc i accs
              -- Increment the count (which will be our new state)
              count' = count + 1
              -- Compute the average
              o = sum / count'
              s' = (accs', count')
        \stopbuffer

        \placeexample[here][ex:AvgState]{Simple stateful averaging circuit.}
          %\startcombination[2*1]
            {\typebufferhs{AvgState}}%{Haskell description using function applications.}
          %  {\boxedgraphic{AvgState}}{The architecture described by the Haskell description.}
          %\stopcombination
        \todo{Picture}

  \section{\VHDL\ generation for state}  
    Now its clear how to put state annotations in the Haskell source,
    there is the question of how to implement this state translation. As
    we have seen in \in{section}[sec:prototype:design], the translation to
    \VHDL\ happens as a simple, final step in the compilation process.
    This step works on a Core expression in normal form. The specifics
    of normal form will be explained in
    \in{chapter}[chap:normalization], but the examples given should be
    easy to understand using the definition of Core given above. The
    conversion to and from the \hs{State} type is done using the cast
    operator, \lam{▶}.

        \startbuffer[AvgStateNormal]
          acc = λi.λspacked.
            let
              -- Remove the State newtype
              s = spacked ▶ Word
              sum = s + i
              -- Add the State newtype again
              spacked' = sum ▶ State Word
              res = (spacked', sum)
            in
              res

          avg = λi.λspacked.
            let
              s = spacked ▶ (AccState, Word)
              accs = case s of (a, b) -> a
              count = case s of (c, d) -> d
              accres = acc i accs
              accs' = case accres of (e, f) -> e
              sum = case accres of (g, h) -> h
              count' = count + 1
              o = sum / count'
              s' = (accs', count')
              spacked' = s' ▶ State (AccState, Word)
              res = (spacked', o)
            in
              res
        \stopbuffer

        \placeexample[here][ex:AvgStateNormal]{Normalized version of \in{example}[ex:AvgState]}
            {\typebufferlam{AvgStateNormal}}

    \subsection[sec:prototype:statelimits]{State in normal form}
      Before describing how to translate state from normal form to
      \VHDL, we will first see how state handling looks in normal form.
      How must their use be limited to guarantee that proper \VHDL\ can
      be generated?

      We will formulate a number of rules about what operations are
      allowed with state variables. These rules apply to the normalized Core
      representation, but will in practice apply to the original Haskell
      hardware description as well. Ideally, these rules would become part
      of the intended normal form definition \refdef{intended normal form
      definition}, but this is not the case right now. This can cause some
      problems, which are detailed in
      \in{section}[sec:normalization:stateproblems].

      In these rules we use the terms \emph{state variable} to refer to any
      variable that has a \lam{State} type. A \emph{state-containing
      variable} is any variable whose type contains a \lam{State} type,
      but is not one itself (like \lam{(AccState, Word)} in the example,
      which is a tuple type, but contains \lam{AccState}, which is again
      equal to \lam{State Word}).

      We also use a distinction between \emph{input} and \emph{output
      (state) variables} and \emph{sub-state variables}, which will be
      defined in the rules themselves.

      These rules describe everything that can be done with state
      variables and state-containing variables. Everything else is
      invalid. For every rule, the corresponding part of
      \in{example}[ex:AvgStateNormal] is shown.

      \startdesc{State variables can appear as an argument.}
        \startlambda
          avg = λi.λspacked. ...
        \stoplambda

        Any lambda that binds a variable with a state type, creates a new
        input state variable.
      \stopdesc

      \startdesc{Input state variables can be unpacked.}
        \startlambda
          s = spacked ▶ (AccState, Word)
        \stoplambda

        An input state variable may be unpacked using a cast operation. This
        removes the \lam{State} type renaming and the result has no longer a
        \lam{State} type.

        If the result of this unpacking does not have a state type and does
        not contain state variables, there are no limitations on its
        use (this is the function's own state).  Otherwise if it does
        not have a state type but does contain sub-states, we refer to it
        as a \emph{state-containing input variable} and the limitations
        below apply. If it has a state type itself, we refer to it as an
        \emph{input sub-state variable} and the below limitations apply
        as well.

        It may seem strange to consider a variable that still has a state
        type directly after unpacking, but consider the case where a
        function does not have any state of its own, but does call a single
        stateful function. This means it must have a state argument that
        contains just a sub-state. The function signature of such a function
        could look like:

        \starthaskell
          type FooState = State AccState
        \stophaskell

        Which is of course equivalent to \lam{State (State Word)}.
      \stopdesc

      \startdesc{Variables can be extracted from state-containing input variables.}
        \startlambda
          accs = case s of (a, b) -> a
        \stoplambda

        A state-containing input variable is typically a tuple containing
        multiple elements (like the current function's state, sub-states or
        more tuples containing sub-states). All of these can be extracted
        from an input variable using an extractor case (or possibly
        multiple, when the input variable is nested).

        If the result has no state type and does not contain any state
        variables either, there are no further limitations on its use
        (this is the function's own state). If the result has no state
        type but does contain state variables we refer to it as a
        \emph{state-containing input variable} and this limitation keeps
        applying. If the variable has a state type itself, we refer to
        it as an \emph{input sub-state variable} and below limitations
        apply.

      \startdesc{Input sub-state variables can be passed to functions.} 
        \startlambda
          accres = acc i accs
          accs' = case accres of (e, f) -> e
        \stoplambda
        
        An input sub-state variable can (only) be passed to a function.
        Additionally, every input sub-state variable must be used in exactly
        \emph{one} application, no more and no less.

        The function result should contain exactly one state variable, which
        can be extracted using (multiple) case expressions. The extracted
        state variable is referred to the \emph{output sub-state}

        The type of this output sub-state must be identical to the type of
        the input sub-state passed to the function.
      \stopdesc

      \startdesc{Variables can be inserted into a state-containing output variable.}
        \startlambda
          s' = (accs', count')
        \stoplambda
        
        A function's output state is usually a tuple containing its own
        updated state variables and all output sub-states. This result is
        built up using any single-constructor algebraic datatype
        (possibly nested).

        The result of these expressions is referred to as a
        \emph{state-containing output variable}, which are subject to these
        limitations.
      \stopdesc

      \startdesc{State containing output variables can be packed.}
        \startlambda
          spacked' = s' ▶ State (AccState, Word)
        \stoplambda

        As soon as all a functions own update state and output sub-state
        variables have been joined together, the resulting
        state-containing output variable can be packed into an output
        state variable. Packing is done by casting to a state type.
      \stopdesc

      \startdesc{Output state variables can appear as (part of) a function result.}
        \startlambda
          avg = λi.λspacked.
            let
            \vdots
            res = (spacked', o)
          in
            res
        \stoplambda
        When the output state is packed, it can be returned as a part
        of the function result. Nothing else can be done with this
        value (or any value that contains it).
      \stopdesc

      There is one final limitation that is hard to express in the above
      itemization. Whenever sub-states are extracted from the input state
      to be passed to functions, the corresponding output sub-states
      should be inserted into the output state in the same way. In other
      words, each pair of corresponding sub-states in the input and
      output states should be passed to / returned from the same called
      function.

      The prototype currently does not check much of the above
      conditions. This means that if the conditions are violated,
      sometimes a compile error is generated, but in other cases output
      can be generated that is not valid \VHDL\ or at the very least does
      not correspond to the input.

    \subsection{Translating to \VHDL}
      As noted above, the basic approach when generating \VHDL\ for stateful
      functions is to generate a single register for every stateful function.
      We look around the normal form to find the let binding that removes the
      \lam{State} type renaming (using a cast). We also find the let binding that
      adds a \lam{State} type. These are connected to the output and the input
      of the generated let binding respectively. This means that there can
      only be one let binding that adds and one that removes the \lam{State}
      type. It is easy to violate this constraint. This problem is detailed in
      \in{section}[sec:normalization:stateproblems].

      This approach seems simple enough, but will this also work for more
      complex stateful functions involving sub-states?  Observe that any
      component of a function's state that is a sub-state, \ie\ passed on as
      the state of another function, should have no influence on the
      hardware generated for the calling function. Any state-specific
      \small{VHDL} for this component can be generated entirely within the
      called function.  So, we can completely ignore sub-states when
      generating \VHDL\ for a function.
      
      From this observation it might seem logical to remove the
      sub-states from a function's states altogether and leave only the
      state components which are actual states of the current function.
      While doing this would not remove any information needed to
      generate \small{VHDL} from the function, it would cause the
      function definition to become invalid (since we will not have any
      sub-state to pass to the functions anymore).  We could solve the
      syntactic problems by passing \type{undefined} for state
      variables, but that would still break the code on the semantic
      level (\ie, the function would no longer be semantically
      equivalent to the original input).

      To keep the function definition correct until the very end of the
      process, we will not deal with (sub)states until we get to the
      \small{VHDL} generation.  Then, we are translating from Core to
      \small{VHDL}, and we can simply generate no \VHDL for sub-states,
      effectively removing them altogether.

      But, how will we know what exactly is a sub-state? Since any state
      argument or return value that represents state must be of the
      \type{State} type, we can look at the type of a value. However, we
      must be careful to ignore only \emph{sub-states}, and not a
      function's own state.

      For \in{example}[ex:AvgStateNormal] above, we should generate a register
      with its output connected to \lam{s} and its input connected
      to \lam{s'}. However, \lam{s'} is build up from both \lam{accs'} and
      \lam{count'}, while only \lam{count'} should end up in the register.
      \lam{accs'} is a sub-state for the \lam{acc} function, for which a
      register will be created when generating \VHDL\ for the \lam{acc}
      function.

      Fortunately, the \lam{accs'} variable (and any other sub-state) has a
      property that we can easily check: it has a \lam{State} type. This
      means that whenever \VHDL\ is generated for a tuple (or other
      algebraic type), we can simply leave out all elements that have a
      \lam{State} type. This will leave just the parts of the state that
      do not have a \lam{State} type themselves, like \lam{count'},
      which is exactly a function's own state. This approach also means
      that the state part of the result (\eg\ \lam{s'} in \lam{res}) is
      automatically excluded when generating the output port, which is
      also required.

      We can formalize this translation a bit, using the following
      rules.

      \startitemize
        \item A state unpack operation should not generate any \small{VHDL}.
        The binder to which the unpacked state is bound should still be
        declared, this signal will become the register and will hold the
        current state.
        \item A state pack operation should not generate any \small{VHDL}.
        The binder to which the packed state is bound should not be
        declared. The binder that is packed is the signal that will hold the
        new state.
        \item Any values of a State type should not be translated to
        \small{VHDL}. In particular, State elements should be removed from
        tuples (and other data-types) and arguments with a state type should
        not generate ports.
        \item To make the state actually work, a simple \small{VHDL}
        (sequential) process should be generated. This process updates
        the state at every clock cycle, by assigning the new state to the
        current state. This will be recognized by synthesis tools as a
        register specification.
      \stopitemize

      When applying these rules to the function \lam{avg} from
      \in{example}[ex:AvgStateNormal], we be left with the description
      in \in{example}[ex:AvgStateRemoved]. All the parts that do not
      generate any \VHDL\ directly are crossed out, leaving just the
      actual flow of values in the final hardware. To illustrate the
      change of the types of \lam{s} and \lam{s'}, their types are also
      shown.
      
      \startbuffer[AvgStateRemoved]
        avg = iλ.λ--spacked.--
          let 
            s :: (--AccState,-- Word)
            s = --spacked ▶ (AccState, Word)--
            --accs = case s of (a, b) -> a--
            count = case s of (--c,-- d) -> d
            accres = acc i --accs--
            --accs' = case accres of (e, f) -> e--
            sum = case accres of (--g,-- h) -> h
            count' = count + 1
            o = sum / count'
            s' :: (--AccState,-- Word)
            s' = (--accs',-- count')
            --spacked' = s' ▶ State (AccState, Word)--
            res = (--spacked',-- o)
          in
            res
      \stopbuffer
      \placeexample[here][ex:AvgStateRemoved]{Normalized version of \in{example}[ex:AvgState] with ignored parts crossed out}
          {\typebufferlam{AvgStateRemoved}}
              
      When we actually leave out the crossed out parts, we get a slightly
      weird program: there is a variable \lam{s} which has no value, and there
      is a variable \lam{s'} that is never used. But together, these two will form
      the state process of the function. \lam{s} contains the "current" state,
      \lam{s'} is assigned the "next" state. So, at the end of each clock
      cycle, \lam{s'} should be assigned to \lam{s}.

      As an illustration of the result of this function,
      \in{example}[ex:AccStateVHDL] and \in{example}[ex:AvgStateVHDL] show the the \VHDL\ that is
      generated by Cλash from the examples is this section.

      \startbuffer[AvgStateVHDL]
        entity avgComponent_0 is
             port (\izAlE2\ : in \unsigned_31\;
                   \foozAo1zAo12\ : out \(,)unsigned_31\;
                   clock : in std_logic;
                   resetn : in std_logic);
        end entity avgComponent_0;


        architecture structural of avgComponent_0 is
             signal \szAlG2\ : \(,)unsigned_31\;
             signal \countzAlW2\ : \unsigned_31\;
             signal \dszAm62\ : \(,)unsigned_31\;
             signal \sumzAmk3\ : \unsigned_31\;
             signal \reszAnCzAnM2\ : \unsigned_31\;
             signal \foozAnZzAnZ2\ : \unsigned_31\;
             signal \reszAnfzAnj3\ : \unsigned_31\;
             signal \s'zAmC2\ : \(,)unsigned_31\;
        begin
             \countzAlW2\ <= \szAlG2\.A;

             \comp_ins_dszAm62\ : entity accComponent_1
                                       port map (\izAob3\ => \izAlE2\,
                                                 \foozAoBzAoB2\ => \dszAm62\,
                                                 clock => clock,
                                                 resetn => resetn);

             \sumzAmk3\ <= \dszAm62\.A;

             \reszAnCzAnM2\ <= to_unsigned(1, 32);

             \foozAnZzAnZ2\ <= \countzAlW2\ + \reszAnCzAnM2\;

             \reszAnfzAnj3\ <= \sumzAmk3\ * \foozAnZzAnZ2\;

             \s'zAmC2\.A <= \foozAnZzAnZ2\;

             \foozAo1zAo12\.A <= \reszAnfzAnj3\;

             state : process (clock, resetn)
             begin
                  if resetn = '0' then
                  elseif rising_edge(clock) then
                       \szAlG2\ <= \s'zAmC2\;
                  end if;
             end process state;
        end architecture structural;
      \stopbuffer

      \startbuffer[AvgStateTypes]
        package types is
             subtype \unsigned_31\ is unsigned (0 to 31);
                  
             type \(,)unsigned_31\ is record 
                   A : \unsigned_31\;
             end record;
        end package types;
      \stopbuffer

      \startbuffer[AccStateVHDL]
        entity accComponent_1 is
             port (\izAob3\ : in \unsigned_31\;
                   \foozAoBzAoB2\ : out \(,)unsigned_31\;
                   clock : in std_logic;
                   resetn : in std_logic);
        end entity accComponent_1;

        architecture structural of accComponent_1 is
             signal \szAod3\ : \unsigned_31\;
             signal \reszAonzAor3\ : \unsigned_31\;
        begin
             \reszAonzAor3\ <= \szAod3\ + \izAob3\;
             
             \foozAoBzAoB2\.A <= \reszAonzAor3\;
             
             state : process (clock, resetn)
             begin
                  if resetn = '0' then
                  elseif rising_edge(clock) then
                       \szAod3\ <= \reszAonzAor3\;
                  end if;
             end process state;
        end architecture structural;
      \stopbuffer 
    
      \placeexample[][ex:AvgStateTypes]{\VHDL\ types generated for \hs{acc} and \hs{avg} from \in{example}[ex:AvgState]}
          {\typebuffervhdl{AvgStateTypes}}
      \placeexample[][ex:AccStateVHDL]{\VHDL\ generated for \hs{acc} from \in{example}[ex:AvgState]}
          {\typebuffervhdl{AccStateVHDL}}
      \placeexample[][ex:AvgStateVHDL]{\VHDL\ generated for \hs{avg} from \in{example}[ex:AvgState]}
          {\typebuffervhdl{AvgStateVHDL}}
  \section{Prototype implementation}
    The prototype has been implemented using Haskell as its
    implementation language, just like \GHC. This allows the prototype
    do directly use parts of \GHC\ through the \small{API} it exposes
    (which essentially taps directly into the internals of \GHC, making
    this \small{API} not really a stable interface).

    Cλash can be run from a separate library, but has also been
    integrated into \type{ghci} \cite[baaij09]. The latter does requires
    a custom \GHC\ build, however.

    The latest version and all history of the Cλash code can be browsed
    on-line or retrieved using the \type{git} program.

    http://git.stderr.nl/gitweb?p=matthijs/projects/cλash.git

%    \subsection{Initial state}
%      How to specify the initial state? Cannot be done inside a hardware
%      function, since the initial state is its own state argument for the first
%      call (unless you add an explicit, synchronous reset port).
%
%      External init state is natural for simulation.
%
%      External init state works for hardware generation as well.
%
%      Implementation issues: state splitting, linking input to output state,
%      checking usage constraints on state variables.
%
%
% vim: set sw=2 sts=2 expandtab: