[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: What
    (functional) language will we use 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 backend for it. This has
      the advantage that existing tools can be reused, which will speed up
      development.
    \stopitemize

    \todo{Sidenote: No EDSL}

    Considering that we required a prototype which should be working quickly,
    and that implementing parsers, semantic checkers and especially
    typcheckers 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 don't. 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 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? Since
    our prototype won't be able to program FPGA's directly, we'll have to have
    output our hardware in some format that can be later processed and
    programmed by other tools.

    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. \todo{Is this still true? Reference:
    http://delivery.acm.org/10.1145/80000/74534/p803-li.pdf?key1=74534\&key2=8370537521\&coll=GUIDE\&dl=GUIDE\&CFID=61207158\&CFTOKEN=61908473}
   
    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'd 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 years of experience in this
    field, 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).

    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 behavioural descriptions
    (which might not be supported by all tools). This ensures that any tool
    that works with \VHDL will understand our output (most tools don't 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{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. It's
    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 Fronted 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}{\useMPgraphic{ghc-pipeline}}

    \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
      typechecker.
    \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
      subexpression 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 a number of places where we can start our work.
    Assuming that we don't want to deal with (or modify) parsing, typechecking
    and other frontend business and that native code isn't 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
    descriiption looks like and which syntax constructs were used. However,
    the full \small{AST} is a very complicated datastructure. 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 datastructure
    (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, a lot of things will take a larger
    expression to express.

    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.

    However, we will use the normal core representation, not the simplified
    core. Reasons for this are detailed below. \todo{Ref}
    
    The final prototype roughly consists of three steps:
    
    \startuseMPgraphic{clash-pipeline}
      % Create objects
      save inp, front, norm, vhdl, out;
      newEmptyBox.inp(0,0);
      newBox.front(btex \small{GHC} frontend + desugarer 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}{\useMPgraphic{clash-pipeline}}

    \startdesc{Frontend}
      This is exactly the frontend and desugarer from the \small{GHC}
      pipeline, that translates Haskell sources to a 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 extra 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].
    
  \section{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, 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'll 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
      x :: T
      \stoplambda
      This is a reference to a binder. It's 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 datatypes
      also become variable references.

      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, ints and doubles. The types of these literals are the
      \quote{primitive} versions, like \lam{Char\#} and \lam{Word\#}, not the
      normal Haskell versions (but there are builtin conversion functions).
    \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.
      
      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 labmda 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.
     
      Note that 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 (where that binder is in scope). This
      allows for sharing of subexpressions (you can use a binder twice) and
      explicit \quote{naming} of arbitrary expressions. Note that the binder
      is not in scope in the value bound to it, so it's not possible to make
      recursive definitions with the normal form of the 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 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 should also be possible to express a recursive let using normal
      lambda calculus, if we use the \emph{least fixed-point operator},
      \lam{Y}. This falls beyond the scope of this report, since it is not
      needed for this research.
    \stopdesc

    \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

      \todo{Define WHNF}

      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}. It then chooses a body depending on
      the constructor of its scrutinee. If none of the constructors 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).
      
      Any arguments to the constructor in the scrutinee are bound to each of the
      binders after the constructor and are in scope only in the corresponding
      body.

      To support strictness, the scrutinee is always evaluated into
      \small{WHNF}, even when there is only a \lam{DEFAULT} alternative. This
      allows aplication 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 statements are less powerful than the full Haskell
      case statements. 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 statements are also used for unpacking of algebraic datatypes, 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
      it's 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 typechecker,
      but look different at first glance. To ensure that, once the typechecker
      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.

      \todo{Move and update this paragraph}
      Note that this syntax is also used sometimes to indicate that a particular
      expression has a particular type, even when no cast expression is
      involved. This is then purely informational, since the only elements that
      are explicitely typed in the Core language are the binder references and
      cast expressions, the types of all other elements are determined at
      runtime.
    \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 shouldn't be generated normally, so these are not
      handled in any way in the prototype.
    \stopdesc

    \startdesc{Type}
      \defref{type expression}
      \startlambda
      @type
      \stoplambda
      It is possibly to use a Core type as 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 for all polymorphic functions, for
      example, the \lam{fst} function:

      \startlambda
      fst :: \forall a. \forall b. (a, b) -> a
      fst = λtup.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 substitued for \lam{a} and \lam{b} in the type of \lam{fst}, so
      the type of \lam{fst} actual type of arguments and result 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 don't (plan to)
      support.

      In Core, every expression is typed. The translation to Core happens
      after the typechecker, so types in Core are always correct as well
      (though you could of course construct invalidly typed expressions).

      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 a 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 :: a -> b
          foo' :: -> a b
          bar :: (a, b, c)
          bar' :: (,,) a b c
        \stoplambda
      \stopdesc

      \startdesc{The forall type}
        \startlambda
          id :: \forall a. a -> a
        \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 = λa.λx.x
        \stoplambda

        Here, the type of the binder \lam{x} is \lam{a}, 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} appleid to the dataconstructor \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 a. a -> a} 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 a. Show s ⇒ s → 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{a} can only contain types that are an \emph{instance} of
        the \emph{type class} \lam{Show}. \refdef{type class}

        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
      interchangedly in a Haskell program. This means no explicit conversion
      is needed either. For example, a simple accumulator would become:

      \starthaskell
      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
      are no explicit conversion in Haskell, but not in Core either. This
      means the type of a value might be show as \hs{AccState} 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 makes the implementation a lot more complicated
      than it currently is using \hs{newtypes}.
    \subsection{Type renaming (\hs{newtype})}
      Haskell also supports type renamings as a way to declare a new type that
      has the same (runtime) representation as an existing type (but is in
      fact a different type to the typechecker). With type renaming, an
      explicit conversion between values of the two types is needed. The
      accumulator would then become:

      \starthaskell
      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 (\eg, accessing the register output). Whenever we are a
      adding (packing) the \hs{State} type, we are producing a new value for
      the state (\eg, 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):

      \starttyping
      State (State Bit, State (State Word, Bit), Word)
      \stoptyping

      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 substates, \todo{see 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
        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 errorprone (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 substates, making this
        approach a bit cumbersome. It also makes it harder to copmare input
        and output state types, possible reducing the type safety of the
        descriptions.

    \subsection{Type synonyms for substates}
      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 becomes 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.
      \todo{}

      \subsubsection{Example}
        As an example of the used approach, there is a simple averaging circuit in
        \in{example}[ex:AvgState]. This circuit lets the accumulation of the
        inputs be done by a subcomponent, \hs{acc}, but keeps a count of value
        accumulated in its own state.
        
        
        \startbuffer[AvgState]
          -- The state type annotation
          newtype State s = State s

          -- The accumulator state type
          type AccState = State Bit
          -- 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 = (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}

        And the normalized, core-like versions:

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

          avg = λi.λspacked.
            let
              s = spacked ▶ (AccState, Word)
              accs = case s of (accs, _) -> accs
              count = case s of (_, count) -> count
              accres = acc i accs
              accs' = case accres of (accs', _) -> accs'
              sum = case accres of (_, sum) -> sum
              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{What can you do with state?}
        Now that we've seen how state variables are typically used, the
        question rises of what can be done with these state variables exactly?
        What limitations are there on their use to guarantee that proper \VHDL
        can be generated?

        We will try to 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 state variable to refer to any
        variable that has a \lam{State} type. A 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 input and output (state) variables,
        which will be defined in the rules themselves.

        \startdesc{State variables can appear as an argument.}
          \startlam
            λspacked. ...
          \stoplam

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

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

          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.
          Otherwise if it does not have a state type but does contain
          substates, 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 substate 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 substate. 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.}
          \startlam
            accs = case s of (accs, _) -> accs
          \stoplam

          A state-containing input variable is typically a tuple containing
          multiple elements (like the current function's state, substates or
          more tuples containing substates). 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. 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 substate variable} and below
          limitations apply.

        \startdesc{Input substate variables can be passed to functions.} 
          \startlam
            accres = acc i accs
            accs' = case accres of (accs', _) -> accs'
          \stoplam
          
          An input substate variable can (only) be passed to a function.
          Additionally, every input substate 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 statements. The extracted
          state variable is referred to the \emph{output substate}

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

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

          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}
          As soon as all a functions own update state and output substate
          variables have been joined together, the va...
          todo{}
              spacked' = s' ▶ State (AccState, Word)

        \desc{State variables can appear as (part of) a function result.} When
        generating \small{VHDL}, we can completely ignore any part of a
        function result that has a state type. If the entire result is a state
        type, this will mean the entity will not have an output port.
        Otherwise, the state elements will be removed from the type of the
        output port.

      \subsection{Translating to \VHDL}
        \in{Example}[ex:AvgStateNormal] shows the normalized Core version of the
        same description (note that _ is a wild binder). \refdef{wild binder}
        The normal form is used to translated 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} newtype (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 substates?  Observe that any
        component of a function's state that is a substate, \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 substates when
        generating \VHDL for a function.
        
        From this observation, we might think to remove the substates from a
        function's states alltogether, 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 won't have any substate 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 ignore substates, effectively removing
        the substate components alltogether.

        There are a few important points when ignoring substates.

        First, we have to have some definition of "substate". 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{substates}, and not a function's
        own state.

        In \in{example}[ex:AvgStateNorm] above, we should generate a register
        connected 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 substate 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 substate) has a
        property that we can easily check: It has a \lam{State} type
        annotation. 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 is automatically excluded when generating
        the output port, which is also required.










        Now, we know how to handle each use of a state variable separately. If we
        look at the whole, we can conclude the following:

        \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 datatypes) and arguments with a state type should
          not generate ports.
          \item To make the state actually work, a simple \small{VHDL} proc
          should be generated. This proc updates the state at every
          clockcycle, 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 example program (in normal form), we will
        get the following result. All the parts that don't generate any value are
        crossed out, leaving some very boring assignments here and there.
        
      
        \starthaskell
          avg i --spacked-- = res
            where
              s = --case spacked of (State s) -> s--
              --accums = case s of (accums, \_) -> accums--
              count = case s of (--\_,-- count) -> count
              accumres = accum i --accums--
              --accums' = case accumres of (accums', \_) -> accums'--
              sum = case accumres of (--\_,-- sum) -> sum
              count' = count + 1
              o = sum / count'
              s' = (--accums',-- count')
              --spacked' = State s'--
              res = (--spacked',-- o)
        \stophaskell
                
        When we would really leave out the crossed out parts, we get a slightly
        weird program: There is a variable \type{s} which has no value, and there
        is a variable \type{s'} that is never used. Together, these two will form
        the state proc of the function. \type{s} contains the "current" state,
        \type{s'} is assigned the "next" state. So, at the end of each clock
        cycle, \type{s'} should be assigned to \type{s}.

        Note that the definition of \type{s'} is not removed, even though one
        might think it as having a state type. Since the state type has a single
        argument constructor \type{State}, some type that should be the resulting
        state should always be explicitly packed with the State constructor,
        allowing us to remove the packed version, but still generate \small{VHDL} for the
        unpacked version (of course with any substates removed).
        
        As you can see, the definition of \type{s'} is still present, since it
        does not have a state type (The State constructor. The \type{accums'} substate has been removed,
        leaving us just with the state of \type{avg} itself.
    \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.

      \todo{Implementation issues: Separate compilation, simplified core.}

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