-}
-
-
-% A transformation example
-\definefloat[example][examples]
-\setupcaption[example][location=top] % Put captions on top
-
-\define[3]\transexample{
- \placeexample[here]{#1}
- \startcombination[2*1]
- {\example{#2}}{Original program}
- {\example{#3}}{Transformed program}
- \stopcombination
-}
-%
-%\define[3]\transexampleh{
-%% \placeexample[here]{#1}
-%% \startcombination[1*2]
-%% {\example{#2}}{Original program}
-%% {\example{#3}}{Transformed program}
-%% \stopcombination
-%}
-
-The first step in the core to \small{VHDL} translation process, is normalization. We
-aim to bring the core description into a simpler form, which we can
-subsequently translate into \small{VHDL} easily. This normal form is needed because
-the full core language is more expressive than \small{VHDL} in some areas and because
-core can describe expressions that do not have a direct hardware
-interpretation.
-
-TODO: Describe core properties not supported in \small{VHDL}, and describe how the
-\small{VHDL} we want to generate should look like.
-
-\section{Normal form}
-The transformations described here have a well-defined goal: To bring the
-program in a well-defined form that is directly translatable to hardware,
-while fully preserving the semantics of the program. We refer to this form as
-the \emph{normal form} of the program. The formal definition of this normal
-form is quite simple:
-
-\placedefinition{}{A program is in \emph{normal form} if none of the
-transformations from this chapter apply.}
-
-Of course, this is an \quote{easy} definition of the normal form, since our
-program will end up in normal form automatically. The more interesting part is
-to see if this normal form actually has the properties we would like it to
-have.
-
-But, before getting into more definitions and details about this normal form,
-let's try to get a feeling for it first. The easiest way to do this is by
-describing the things we want to not have in a normal form.
-
-\startitemize
- \item Any \emph{polymorphism} must be removed. When laying down hardware, we
- can't generate any signals that can have multiple types. All types must be
- completely known to generate hardware.
-
- \item Any \emph{higher order} constructions must be removed. We can't
- generate a hardware signal that contains a function, so all values,
- arguments and returns values used must be first order.
-
- \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
- description, every signal is in a single scope. Also, full expressions are
- not supported everywhere (in particular port maps can only map signal names,
- not expressions). To make the \small{VHDL} generation easy, all values must be bound
- on the \quote{top level}.
-\stopitemize
-
-TODO: Intermezzo: functions vs plain values
-
-A very simple example of a program in normal form is given in
-\in{example}[ex:MulSum]. As you can see, all arguments to the function (which
-will become input ports in the final hardware) are at the top. This means that
-the body of the final lambda abstraction is never a function, but always a
-plain value.
-
-After the lambda abstractions, we see a single let expression, that binds two
-variables (\lam{mul} and \lam{sum}). These variables will be signals in the
-final hardware, bound to the output port of the \lam{*} and \lam{+}
-components.
-
-The final line (the \quote{return value} of the function) selects the
-\lam{sum} signal to be the output port of the function. This \quote{return
-value} can always only be a variable reference, never a more complex
-expression.
-
-\startbuffer[MulSum]
-alu :: Bit -> Word -> Word -> Word
-alu = λa.λb.λc.
- let
- mul = (*) a b
- sum = (+) mul c
- in
- sum
-\stopbuffer
-
-\startuseMPgraphic{MulSum}
- save a, b, c, mul, add, sum;
-
- % I/O ports
- newCircle.a(btex $a$ etex) "framed(false)";
- newCircle.b(btex $b$ etex) "framed(false)";
- newCircle.c(btex $c$ etex) "framed(false)";
- newCircle.sum(btex $res$ etex) "framed(false)";
-
- % Components
- newCircle.mul(btex - etex);
- newCircle.add(btex + etex);
-
- a.c - b.c = (0cm, 2cm);
- b.c - c.c = (0cm, 2cm);
- add.c = c.c + (2cm, 0cm);
- mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
- sum.c = add.c + (2cm, 0cm);
- c.c = origin;
-
- % Draw objects and lines
- drawObj(a, b, c, mul, add, sum);
-
- ncarc(a)(mul) "arcangle(15)";
- ncarc(b)(mul) "arcangle(-15)";
- ncline(c)(add);
- ncline(mul)(add);
- ncline(add)(sum);
-\stopuseMPgraphic
-
-\placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
-subtractor.}
- \startcombination[2*1]
- {\typebufferlam{MulSum}}{Core description in normal form.}
- {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
- \stopcombination
-
-The previous example described composing an architecture by calling other
-functions (operators), resulting in a simple architecture with component and
-connection. There is of course also some mechanism for choice in the normal
-form. In a normal Core program, the \emph{case} expression can be used in a
-few different ways to describe choice. In normal form, this is limited to a
-very specific form.
-
-\in{Example}[ex:AddSubAlu] shows an example describing a
-simple \small{ALU}, which chooses between two operations based on an opcode
-bit. The main structure is the same as in \in{example}[ex:MulSum], but this
-time the \lam{res} variable is bound to a case expression. This case
-expression scrutinizes the variable \lam{opcode} (and scrutinizing more
-complex expressions is not supported). The case expression can select a
-different variable based on the constructor of \lam{opcode}.
-
-\startbuffer[AddSubAlu]
-alu :: Bit -> Word -> Word -> Word
-alu = λopcode.λa.λb.
- let
- res1 = (+) a b
- res2 = (-) a b
- res = case opcode of
- Low -> res1
- High -> res2
- in
- res
-\stopbuffer
-
-\startuseMPgraphic{AddSubAlu}
- save opcode, a, b, add, sub, mux, res;
-
- % I/O ports
- newCircle.opcode(btex $opcode$ etex) "framed(false)";
- newCircle.a(btex $a$ etex) "framed(false)";
- newCircle.b(btex $b$ etex) "framed(false)";
- newCircle.res(btex $res$ etex) "framed(false)";
- % Components
- newCircle.add(btex + etex);
- newCircle.sub(btex - etex);
- newMux.mux;
-
- opcode.c - a.c = (0cm, 2cm);
- add.c - a.c = (4cm, 0cm);
- sub.c - b.c = (4cm, 0cm);
- a.c - b.c = (0cm, 3cm);
- mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
- res.c - mux.c = (1.5cm, 0cm);
- b.c = origin;
-
- % Draw objects and lines
- drawObj(opcode, a, b, res, add, sub, mux);
-
- ncline(a)(add) "posA(e)";
- ncline(b)(sub) "posA(e)";
- nccurve(a)(sub) "posA(e)", "angleA(0)";
- nccurve(b)(add) "posA(e)", "angleA(0)";
- nccurve(add)(mux) "posB(inpa)", "angleB(0)";
- nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
- nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
- ncline(mux)(res) "posA(out)";
-\stopuseMPgraphic
-
-\placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
- \startcombination[2*1]
- {\typebufferlam{AddSubAlu}}{Core description in normal form.}
- {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
- \stopcombination
-
-As a more complete example, consider \in{example}[ex:NormalComplete]. This
-example contains everything that is supported in normal form, with the
-exception of builtin higher order functions. The graphical version of the
-architecture contains a slightly simplified version, since the state tuple
-packing and unpacking have been left out. Instead, two seperate registers are
-drawn. Also note that most synthesis tools will further optimize this
-architecture by removing the multiplexers at the register input and replace
-them with some logic in the clock inputs, but we want to show the architecture
-as close to the description as possible.
-
-\startbuffer[NormalComplete]
- regbank :: Bit
- -> Word
- -> State (Word, Word)
- -> (State (Word, Word), Word)
-
- -- All arguments are an inital lambda
- regbank = λa.λd.λsp.
- -- There are nested let expressions at top level
- let
- -- Unpack the state by coercion (\eg, cast from
- -- State (Word, Word) to (Word, Word))
- s = sp :: (Word, Word)
- -- Extract both registers from the state
- r1 = case s of (fst, snd) -> fst
- r2 = case s of (fst, snd) -> snd
- -- Calling some other user-defined function.
- d' = foo d
- -- Conditional connections
- out = case a of
- High -> r1
- Low -> r2
- r1' = case a of
- High -> d'
- Low -> r1
- r2' = case a of
- High -> r2
- Low -> d'
- -- Packing a tuple
- s' = (,) r1' r2'
- -- pack the state by coercion (\eg, cast from
- -- (Word, Word) to State (Word, Word))
- sp' = s' :: State (Word, Word)
- -- Pack our return value
- res = (,) sp' out
- in
- -- The actual result
- res
-\stopbuffer
-
-\startuseMPgraphic{NormalComplete}
- save a, d, r, foo, muxr, muxout, out;
-
- % I/O ports
- newCircle.a(btex \lam{a} etex) "framed(false)";
- newCircle.d(btex \lam{d} etex) "framed(false)";
- newCircle.out(btex \lam{out} etex) "framed(false)";
- % Components
- %newCircle.add(btex + etex);
- newBox.foo(btex \lam{foo} etex);
- newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
- newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
- newMux.muxr1;
- % Reflect over the vertical axis
- reflectObj(muxr1)((0,0), (0,1));
- newMux.muxr2;
- newMux.muxout;
- rotateObj(muxout)(-90);
-
- d.c = foo.c + (0cm, 1.5cm);
- a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
- foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
- muxr1.c = r1.c + (0cm, 2cm);
- muxr2.c = r2.c + (0cm, 2cm);
- r2.c = r1.c + (4cm, 0cm);
- r1.c = origin;
- muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
- out.c = muxout.c - (0cm, 1.5cm);
-
-% % Draw objects and lines
- drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
-
- ncline(d)(foo);
- nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
- nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
- nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
- nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
- nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
- nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
- nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
- nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
- % Connect port a
- nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
- nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
- nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
- ncline(muxout)(out) "posA(out)";
-\stopuseMPgraphic
-
-\placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
-subtractor.}
- \startcombination[2*1]
- {\typebufferlam{NormalComplete}}{Core description in normal form.}
- {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
- \stopcombination
-
-\subsection{Normal form definition}
-Now we have some intuition for the normal form, we can describe how we want
-the normal form to look like in a slightly more formal manner. The following
-EBNF-like description completely captures the intended structure (and
-generates a subset of GHC's core format).
-
-Some clauses have an expression listed in parentheses. These are conditions
-that need to apply to the clause.
-
-\startlambda
-\italic{normal} = \italic{lambda}
-\italic{lambda} = λvar.\italic{lambda} (representable(var))
- | \italic{toplet}
-\italic{toplet} = let \italic{binding} in \italic{toplet}
- | letrec [\italic{binding}] in \italic{toplet}
- | var (representable(varvar))
-\italic{binding} = var = \italic{rhs} (representable(rhs))
- -- State packing and unpacking by coercion
- | var0 = var1 :: State ty (lvar(var1))
- | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
-\italic{rhs} = userapp
- | builtinapp
- -- Extractor case
- | case var of C a0 ... an -> ai (lvar(var))
- -- Selector case
- | case var of (lvar(var))
- DEFAULT -> var0 (lvar(var0))
- C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
-\italic{userapp} = \italic{userfunc}
- | \italic{userapp} {userarg}
-\italic{userfunc} = var (gvar(var))
-\italic{userarg} = var (lvar(var))
-\italic{builtinapp} = \italic{builtinfunc}
- | \italic{builtinapp} \italic{builtinarg}
-\italic{builtinfunc} = var (bvar(var))
-\italic{builtinarg} = \italic{coreexpr}
-\stoplambda
-
--- TODO: Limit builtinarg further
-
--- TODO: There can still be other casts around (which the code can handle,
-e.g., ignore), which still need to be documented here.
-
--- TODO: Note about the selector case. It just supports Bit and Bool
-currently, perhaps it should be generalized in the normal form?
-
-When looking at such a program from a hardware perspective, the top level
-lambda's define the input ports. The value produced by the let expression is
-the output port. Most function applications bound by the let expression
-define a component instantiation, where the input and output ports are mapped
-to local signals or arguments. Some of the others use a builtin
-construction (\eg the \lam{case} statement) or call a builtin function
-(\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
-available.
-
-\section{Transformation notation}
-To be able to concisely present transformations, we use a specific format to
-them. It is a simple format, similar to one used in logic reasoning.
-
-Such a transformation description looks like the following.
-
-\starttrans
-<context conditions>
-~
-<original expression>
--------------------------- <expression conditions>
-<transformed expresssion>
-~
-<context additions>
-\stoptrans
-
-This format desribes a transformation that applies to \lam{original
-expresssion} and transforms it into \lam{transformed expression}, assuming
-that all conditions apply. In this format, there are a number of placeholders
-in pointy brackets, most of which should be rather obvious in their meaning.
-Nevertheless, we will more precisely specify their meaning below:
-
- \startdesc{<original expression>} The expression pattern that will be matched
- against (subexpressions of) the expression to be transformed. We call this a
- pattern, because it can contain \emph{placeholders} (variables), which match
- any expression or binder. Any such placeholder is said to be \emph{bound} to
- the expression it matches. It is convention to use an uppercase latter (\eg
- \lam{M} or \lam{E} to refer to any expression (including a simple variable
- reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
- (references to) binders.
-
- For example, the pattern \lam{a + B} will match the expression
- \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
- \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
- \stopdesc
-
- \startdesc{<expression conditions>}
- These are extra conditions on the expression that is matched. These
- conditions can be used to further limit the cases in which the
- transformation applies, in particular to prevent a transformation from
- causing a loop with itself or another transformation.
-
- Only if these if these conditions are \emph{all} true, this transformation
- applies.
- \stopdesc
-
- \startdesc{<context conditions>}
- These are a number of extra conditions on the context of the function. In
- particular, these conditions can require some other top level function to be
- present, whose value matches the pattern given here. The format of each of
- these conditions is: \lam{binder = <pattern>}.
-
- Typically, the binder is some placeholder bound in the \lam{<original
- expression>}, while the pattern contains some placeholders that are used in
- the \lam{transformed expression}.
-
- Only if a top level binder exists that matches each binder and pattern, this
- transformation applies.
- \stopdesc
-
- \startdesc{<transformed expression>}
- This is the expression template that is the result of the transformation. If, looking
- at the above three items, the transformation applies, the \lam{original
- expression} is completely replaced with the \lam{<transformed expression>}.
- We call this a template, because it can contain placeholders, referring to
- any placeholder bound by the \lam{<original expression>} or the
- \lam{<context conditions>}. The resulting expression will have those
- placeholders replaced by the values bound to them.
-
- Any binder (lowercase) placeholder that has no value bound to it yet will be
- bound to (and replaced with) a fresh binder.
- \stopdesc
-
- \startdesc{<context additions>}
- These are templates for new functions to add to the context. This is a way
- to have a transformation create new top level functiosn.
-
- Each addition has the form \lam{binder = template}. As above, any
- placeholder in the addition is replaced with the value bound to it, and any
- binder placeholder that has no value bound to it yet will be bound to (and
- replaced with) a fresh binder.
- \stopdesc
-
- As an example, we'll look at η-abstraction:
-
-\starttrans
-E \lam{E :: a -> b}
--------------- \lam{E} does not occur on a function position in an application
-λx.E x \lam{E} is not a lambda abstraction.
-\stoptrans
-
- Consider the following function, which is a fairly obvious way to specify a
- simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
- function):
-
-\startlambda
-alu :: Bit -> Word -> Word -> Word
-alu = λopcode. case opcode of
- Low -> (+)
- High -> (-)
-\stoplambda
-
- There are a few subexpressions in this function to which we could possibly
- apply the transformation. Since the pattern of the transformation is only
- the placeholder \lam{E}, any expression will match that. Whether the
- transformation applies to an expression is thus solely decided by the
- conditions to the right of the transformation.
-
- We will look at each expression in the function in a top down manner. The
- first expression is the entire expression the function is bound to.
-
-\startlambda
-λopcode. case opcode of
- Low -> (+)
- High -> (-)
-\stoplambda
-
- As said, the expression pattern matches this. The type of this expression is
- \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
- this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
-
- Since this expression is at top level, it does not occur at a function
- position of an application. However, The expression is a lambda abstraction,
- so this transformation does not apply.
-
- The next expression we could apply this transformation to, is the body of
- the lambda abstraction:
-
-\startlambda
-case opcode of
- Low -> (+)
- High -> (-)
-\stoplambda
-
- The type of this expression is \lam{Word -> Word -> Word}, which again
- matches \lam{a -> b}. The expression is the body of a lambda expression, so
- it does not occur at a function position of an application. Finally, the
- expression is not a lambda abstraction but a case expression, so all the
- conditions match. There are no context conditions to match, so the
- transformation applies.
-
- By now, the placeholder \lam{E} is bound to the entire expression. The
- placeholder \lam{x}, which occurs in the replacement template, is not bound
- yet, so we need to generate a fresh binder for that. Let's use the binder
- \lam{a}. This results in the following replacement expression:
-
-\startlambda
-λa.(case opcode of
- Low -> (+)
- High -> (-)) a
-\stoplambda
-
- Continuing with this expression, we see that the transformation does not
- apply again (it is a lambda expression). Next we look at the body of this
- labmda abstraction:
-
-\startlambda
-(case opcode of
- Low -> (+)
- High -> (-)) a
-\stoplambda
-
- Here, the transformation does apply, binding \lam{E} to the entire
- expression and \lam{x} to the fresh binder \lam{b}, resulting in the
- replacement:
-
-\startlambda
-λb.(case opcode of
- Low -> (+)
- High -> (-)) a b
-\stoplambda
-
- Again, the transformation does not apply to this lambda abstraction, so we
- look at its body. For brevity, we'll put the case statement on one line from
- now on.
-
-\startlambda
-(case opcode of Low -> (+); High -> (-)) a b
-\stoplambda
-
- The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
- and the transformation does not apply. Next, we have two options for the
- next expression to look at: The function position and argument position of
- the application. The expression in the argument position is \lam{b}, which
- has type \lam{Word}, so the transformation does not apply. The expression in
- the function position is:
-
-\startlambda
-(case opcode of Low -> (+); High -> (-)) a
-\stoplambda
-
- Obviously, the transformation does not apply here, since it occurs in
- function position. In the same way the transformation does not apply to both
- components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
- and \lam{a}), so we'll skip to the components of the case expression: The
- scrutinee and both alternatives. Since the opcode is not a function, it does
- not apply here, and we'll leave both alternatives as an exercise to the
- reader. The final function, after all these transformations becomes:
-
-\startlambda
-alu :: Bit -> Word -> Word -> Word
-alu = λopcode.λa.b. (case opcode of
- Low -> λa1.λb1 (+) a1 b1
- High -> λa2.λb2 (-) a2 b2) a b
-\stoplambda
-
- In this case, the transformation does not apply anymore, though this might
- not always be the case (e.g., the application of a transformation on a
- subexpression might open up possibilities to apply the transformation
- further up in the expression).
-
-\subsection{Transformation application}
-In this chapter we define a number of transformations, but how will we apply
-these? As stated before, our normal form is reached as soon as no
-transformation applies anymore. This means our application strategy is to
-simply apply any transformation that applies, and continuing to do that with
-the result of each transformation.
-
-In particular, we define no particular order of transformations. Since
-transformation order should not influence the resulting normal form (see TODO
-ref), this leaves the implementation free to choose any application order that
-results in an efficient implementation.
-
-When applying a single transformation, we try to apply it to every (sub)expression
-in a function, not just the top level function. This allows us to keep the
-transformation descriptions concise and powerful.
-
-\subsection{Definitions}
-In the following sections, we will be using a number of functions and
-notations, which we will define here.
-
-\subsubsection{Other concepts}
-A \emph{global variable} is any variable that is bound at the
-top level of a program, or an external module. A \emph{local variable} is any
-other variable (\eg, variables local to a function, which can be bound by
-lambda abstractions, let expressions and pattern matches of case
-alternatives). Note that this is a slightly different notion of global versus
-local than what \small{GHC} uses internally.
-\defref{global variable} \defref{local variable}
-
-A \emph{hardware representable} (or just \emph{representable}) type or value
-is (a value of) a type that we can generate a signal for in hardware. For
-example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
-not runtime representable notably include (but are not limited to): Types,
-dictionaries, functions.
-\defref{representable}
-
-A \emph{builtin function} is a function supplied by the Cλash framework, whose
-implementation is not valid Cλash. The implementation is of course valid
-Haskell, for simulation, but it is not expressable in Cλash.
-\defref{builtin function} \defref{user-defined function}
-
-For these functions, Cλash has a \emph{builtin hardware translation}, so calls
-to these functions can still be translated. These are functions like
-\lam{map}, \lam{hwor} and \lam{length}.
-
-A \emph{user-defined} function is a function for which we do have a Cλash
-implementation available.
-
-\subsubsection{Functions}
-Here, we define a number of functions that can be used below to concisely
-specify conditions.
-
-\refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
-global variable. It is false when it references a local variable.
-
-\refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
-references a local variable, false when it references a global variable.
-
-\refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
-\emph{expr} or \emph{var} is \emph{representable}.
-
-\subsection{Binder uniqueness}
-A common problem in transformation systems, is binder uniqueness. When not
-considering this problem, it is easy to create transformations that mix up
-bindings and cause name collisions. Take for example, the following core
-expression:
-
-\startlambda
-(λa.λb.λc. a * b * c) x c
-\stoplambda
-
-By applying β-reduction (see below) once, we can simplify this expression to:
-
-\startlambda
-(λb.λc. x * b * c) c
-\stoplambda
-
-Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
-binder. No harm done here. But note that we see multiple occurences of the
-\lam{c} binder. The first is a binding occurence, to which the second refers.
-The last, however refers to \emph{another} instance of \lam{c}, which is
-bound somewhere outside of this expression. Now, if we would apply beta
-reduction without taking heed of binder uniqueness, we would get:
-
-\startlambda
-λc. x * c * c
-\stoplambda
-
-This is obviously not what was supposed to happen! The root of this problem is
-the reuse of binders: Identical binders can be bound in different scopes, such
-that only the inner one is \quote{visible} in the inner expression. In the example
-above, the \lam{c} binder was bound outside of the expression and in the inner
-lambda expression. Inside that lambda expression, only the inner \lam{c} is
-visible.
-
-There are a number of ways to solve this. \small{GHC} has isolated this
-problem to their binder substitution code, which performs \emph{deshadowing}
-during its expression traversal. This means that any binding that shadows
-another binding on a higher level is replaced by a new binder that does not
-shadow any other binding. This non-shadowing invariant is enough to prevent
-binder uniqueness problems in \small{GHC}.
-
-In our transformation system, maintaining this non-shadowing invariant is
-a bit harder to do (mostly due to implementation issues, the prototype doesn't
-use \small{GHC}'s subsitution code). Also, we can observe the following
-points.
-
-\startitemize
-\item Deshadowing does not guarantee overall uniqueness. For example, the
-following (slightly contrived) expression shows the identifier \lam{x} bound in
-two seperate places (and to different values), even though no shadowing
-occurs.
-
-\startlambda
-(let x = 1 in x) + (let x = 2 in x)
-\stoplambda
-
-\item In our normal form (and the resulting \small{VHDL}), all binders
-(signals) will end up in the same scope. To allow this, all binders within the
-same function should be unique.
-
-\item When we know that all binders in an expression are unique, moving around
-or removing a subexpression will never cause any binder conflicts. If we have
-some way to generate fresh binders, introducing new subexpressions will not
-cause any problems either. The only way to cause conflicts is thus to
-duplicate an existing subexpression.
-\stopitemize
-
-Given the above, our prototype maintains a unique binder invariant. This
-meanst that in any given moment during normalization, all binders \emph{within
-a single function} must be unique. To achieve this, we apply the following
-technique.
-
-TODO: Define fresh binders and unique supplies
-
-\startitemize
-\item Before starting normalization, all binders in the function are made
-unique. This is done by generating a fresh binder for every binder used. This
-also replaces binders that did not pose any conflict, but it does ensure that
-all binders within the function are generated by the same unique supply. See
-(TODO: ref fresh binder).
-\item Whenever a new binder must be generated, we generate a fresh binder that
-is guaranteed to be different from \emph{all binders generated so far}. This
-can thus never introduce duplication and will maintain the invariant.
-\item Whenever (part of) an expression is duplicated (for example when
-inlining), all binders in the expression are replaced with fresh binders
-(using the same method as at the start of normalization). These fresh binders
-can never introduce duplication, so this will maintain the invariant.
-\item Whenever we move part of an expression around within the function, there
-is no need to do anything special. There is obviously no way to introduce
-duplication by moving expressions around. Since we know that each of the
-binders is already unique, there is no way to introduce (incorrect) shadowing
-either.
-\stopitemize
-
-\section{Transform passes}
-In this section we describe the actual transforms. Here we're using
-the core language in a notation that resembles lambda calculus.
-
-Each of these transforms is meant to be applied to every (sub)expression
-in a program, for as long as it applies. Only when none of the
-transformations can be applied anymore, the program is in normal form (by
-definition). We hope to be able to prove that this form will obey all of the
-constraints defined above, but this has yet to happen (though it seems likely
-that it will).
-
-Each of the transforms will be described informally first, explaining
-the need for and goal of the transform. Then, a formal definition is
-given, using a familiar syntax from the world of logic. Each transform
-is specified as a number of conditions (above the horizontal line) and a
-number of conclusions (below the horizontal line). The details of using
-this notation are still a bit fuzzy, so comments are welcom.
-
-\subsection{η-abstraction}
-This transformation makes sure that all arguments of a function-typed
-expression are named, by introducing lambda expressions. When combined with
-β-reduction and function inlining below, all function-typed expressions should
-be lambda abstractions or global identifiers.
-
-\starttrans
-E \lam{E :: a -> b}
--------------- \lam{E} is not the first argument of an application.
-λx.E x \lam{E} is not a lambda abstraction.
- \lam{x} is a variable that does not occur free in \lam{E}.
-\stoptrans
-
-\startbuffer[from]
-foo = λa.case a of
- True -> λb.mul b b
- False -> id
-\stopbuffer
-
-\startbuffer[to]
-foo = λa.λx.(case a of
- True -> λb.mul b b
- False -> λy.id y) x
-\stopbuffer
-
-\transexample{η-abstraction}{from}{to}
-
-\subsection{β-reduction}
-β-reduction is a well known transformation from lambda calculus, where it is
-the main reduction step. It reduces applications of labmda abstractions,
-removing both the lambda abstraction and the application.
-
-In our transformation system, this step helps to remove unwanted lambda
-abstractions (basically all but the ones at the top level). Other
-transformations (application propagation, non-representable inlining) make
-sure that most lambda abstractions will eventually be reducable by
-β-reduction.
-
-TODO: Define substitution syntax
-
-\starttrans
-(λx.E) M
------------------
-E[M/x]
-\stoptrans
-
-% And an example
-\startbuffer[from]
-(λa. 2 * a) (2 * b)
-\stopbuffer
-
-\startbuffer[to]
-2 * (2 * b)
-\stopbuffer
-
-\transexample{β-reduction}{from}{to}
-
-\subsection{Application propagation}
-This transformation is meant to propagate application expressions downwards
-into expressions as far as possible. This allows partial applications inside
-expressions to become fully applied and exposes new transformation
-possibilities for other transformations (like β-reduction).
-
-\starttrans
-let binds in E) M
------------------
-let binds in E M
-\stoptrans
-
-% And an example
-\startbuffer[from]
-( let
- val = 1
- in
- add val
-) 3
-\stopbuffer
-
-\startbuffer[to]
-let
- val = 1
-in
- add val 3
-\stopbuffer
-
-\transexample{Application propagation for a let expression}{from}{to}
-
-\starttrans
-(case x of
- p1 -> E1
- \vdots
- pn -> En) M
------------------
-case x of
- p1 -> E1 M
- \vdots
- pn -> En M
-\stoptrans
-
-% And an example
-\startbuffer[from]
-( case x of
- True -> id
- False -> neg
-) 1
-\stopbuffer
-
-\startbuffer[to]
-case x of
- True -> id 1
- False -> neg 1
-\stopbuffer
-
-\transexample{Application propagation for a case expression}{from}{to}
-
-\subsection{Let derecursification}
-This transformation is meant to make lets non-recursive whenever possible.
-This might allow other optimizations to do their work better. TODO: Why is
-this needed exactly?
-
-\subsection{Let flattening}
-This transformation puts nested lets in the same scope, by lifting the
-binding(s) of the inner let into a new let around the outer let. Eventually,
-this will cause all let bindings to appear in the same scope (they will all be
-in scope for the function return value).
-
-Note that this transformation does not try to be smart when faced with
-recursive lets, it will just leave the lets recursive (possibly joining a
-recursive and non-recursive let into a single recursive let). The let
-rederursification transformation will do this instead.
-
-\starttrans
-letnonrec x = (let bindings in M) in N
-------------------------------------------
-let bindings in (letnonrec x = M) in N
-\stoptrans
-
-\starttrans
-letrec
- \vdots
- x = (let bindings in M)
- \vdots
-in
- N
-------------------------------------------
-letrec
- \vdots
- bindings
- x = M
- \vdots
-in
- N
-\stoptrans
-
-\startbuffer[from]
-let
- a = letrec
- x = 1
- y = 2
- in
- x + y
-in
- letrec
- b = let c = 3 in a + c
- d = 4
- in
- d + b
-\stopbuffer
-\startbuffer[to]
-letrec
- x = 1
- y = 2
-in
- let
- a = x + y
- in
- letrec
- c = 3
- b = a + c
- d = 4
- in
- d + b
-\stopbuffer
-
-\transexample{Let flattening}{from}{to}
-
-\subsection{Empty let removal}
-This transformation is simple: It removes recursive lets that have no bindings
-(which usually occurs when let derecursification removes the last binding from
-it).
-
-\starttrans
-letrec in M
---------------
-M
-\stoptrans
-
-\subsection{Simple let binding removal}
-This transformation inlines simple let bindings (\eg a = b).
-
-This transformation is not needed to get into normal form, but makes the
-resulting \small{VHDL} a lot shorter.
-
-\starttrans
-letnonrec
- a = b
-in
- M
------------------
-M[b/a]
-\stoptrans
-
-\starttrans
-letrec
- \vdots
- a = b
- \vdots
-in
- M
------------------
-let
- \vdots [b/a]
- \vdots [b/a]
-in
- M[b/a]
-\stoptrans
-
-\subsection{Unused let binding removal}
-This transformation removes let bindings that are never used. Usually,
-the desugarer introduces some unused let bindings.
-
-This normalization pass should really be unneeded to get into normal form
-(since ununsed bindings are not forbidden by the normal form), but in practice
-the desugarer or simplifier emits some unused bindings that cannot be
-normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
-this transformation makes the resulting \small{VHDL} a lot shorter.
-
-\starttrans
-let a = E in M
----------------------------- \lam{a} does not occur free in \lam{M}
-M
-\stoptrans
-
-\starttrans
-letrec
- \vdots
- a = E
- \vdots
-in
- M
----------------------------- \lam{a} does not occur free in \lam{M}
-letrec
- \vdots
- \vdots
-in
- M
-\stoptrans
-
-\subsection{Non-representable binding inlining}
-This transform inlines let bindings that have a non-representable type. Since
-we can never generate a signal assignment for these bindings (we cannot
-declare a signal assignment with a non-representable type, for obvious
-reasons), we have no choice but to inline the binding to remove it.
-
-If the binding is non-representable because it is a lambda abstraction, it is
-likely that it will inlined into an application and β-reduction will remove
-the lambda abstraction and turn it into a representable expression at the
-inline site. The same holds for partial applications, which can be turned into
-full applications by inlining.
-
-Other cases of non-representable bindings we see in practice are primitive
-Haskell types. In most cases, these will not result in a valid normalized
-output, but then the input would have been invalid to start with. There is one
-exception to this: When a builtin function is applied to a non-representable
-expression, things might work out in some cases. For example, when you write a
-literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
-the following core: \lam{fromInteger (smallInteger 10)}, where for example
-\lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
-non-representable types. TODO: This/these paragraph(s) should probably become a
-separate discussion somewhere else.
-
-\starttrans
-letnonrec a = E in M
--------------------------- \lam{E} has a non-representable type.
-M[E/a]
-\stoptrans
-
-\starttrans
-letrec
- \vdots
- a = E
- \vdots
-in
- M
--------------------------- \lam{E} has a non-representable type.
-letrec
- \vdots [E/a]
- \vdots [E/a]
-in
- M[E/a]
-\stoptrans
-
-\startbuffer[from]
-letrec
- a = smallInteger 10
- inc = λa -> add a 1
- inc' = add 1
- x = fromInteger a
-in
- inc (inc' x)
-\stopbuffer
-
-\startbuffer[to]
-letrec
- x = fromInteger (smallInteger 10)
-in
- (λa -> add a 1) (add 1 x)
-\stopbuffer
-
-\transexample{Let flattening}{from}{to}
-
-\subsection{Compiler generated top level binding inlining}
-TODO
-
-\subsection{Scrutinee simplification}
-This transform ensures that the scrutinee of a case expression is always
-a simple variable reference.
-
-\starttrans
-case E of
- alts
------------------ \lam{E} is not a local variable reference
-let x = E in
- case E of
- alts
-\stoptrans
-
-\startbuffer[from]
-case (foo a) of
- True -> a
- False -> b
-\stopbuffer
-
-\startbuffer[to]
-let x = foo a in
- case x of
- True -> a
- False -> b
-\stopbuffer
-
-\transexample{Let flattening}{from}{to}
-
-
-\subsection{Case simplification}
-This transformation ensures that all case expressions become normal form. This
-means they will become one of:
-\startitemize
-\item An extractor case with a single alternative that picks a single field
-from a datatype, \eg \lam{case x of (a, b) -> a}.
-\item A selector case with multiple alternatives and only wild binders, that
-makes a choice between expressions based on the constructor of another
-expression, \eg \lam{case x of Low -> a; High -> b}.
-\stopitemize
-
-\starttrans
-case E of
- C0 v0,0 ... v0,m -> E0
- \vdots
- Cn vn,0 ... vn,m -> En
---------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
-letnonrec
- v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
- \vdots
- v0,m = case x of C0 v0,0 .. v0,m -> v0,m
- x0 = E0
- \dots
- vn,m = case x of Cn vn,0 .. vn,m -> vn,m
- xn = En
-in
- case E of
- C0 w0,0 ... w0,m -> x0
- \vdots
- Cn wn,0 ... wn,m -> xn
-\stoptrans
-
-TODO: This transformation specified like this is complicated and misses
-conditions to prevent looping with itself. Perhaps we should split it here for
-discussion?
-
-\startbuffer[from]
-case a of
- True -> add b 1
- False -> add b 2
-\stopbuffer
-
-\startbuffer[to]
-letnonrec
- x0 = add b 1
- x1 = add b 2
-in
- case a of
- True -> x0
- False -> x1
-\stopbuffer
-
-\transexample{Selector case simplification}{from}{to}
-
-\startbuffer[from]
-case a of
- (,) b c -> add b c
-\stopbuffer
-\startbuffer[to]
-letnonrec
- b = case a of (,) b c -> b
- c = case a of (,) b c -> c
- x0 = add b c
-in
- case a of
- (,) w0 w1 -> x0
-\stopbuffer
-
-\transexample{Extractor case simplification}{from}{to}
-
-\subsection{Case removal}
-This transform removes any case statements with a single alternative and
-only wild binders.
-
-These "useless" case statements are usually leftovers from case simplification
-on extractor case (see the previous example).
-
-\starttrans
-case x of
- C v0 ... vm -> E
----------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
-E
-\stoptrans
-
-\startbuffer[from]
-case a of
- (,) w0 w1 -> x0
-\stopbuffer
-
-\startbuffer[to]
-x0
-\stopbuffer
-
-\transexample{Case removal}{from}{to}
-
-\subsection{Argument simplification}
-The transforms in this section deal with simplifying application
-arguments into normal form. The goal here is to:
-
-\startitemize
- \item Make all arguments of user-defined functions (\eg, of which
- we have a function body) simple variable references of a runtime
- representable type. This is needed, since these applications will be turned
- into component instantiations.
- \item Make all arguments of builtin functions one of:
- \startitemize
- \item A type argument.
- \item A dictionary argument.
- \item A type level expression.
- \item A variable reference of a runtime representable type.
- \item A variable reference or partial application of a function type.
- \stopitemize
-\stopitemize
-
-When looking at the arguments of a user-defined function, we can
-divide them into two categories:
-\startitemize
- \item Arguments of a runtime representable type (\eg bits or vectors).
-
- These arguments can be preserved in the program, since they can
- be translated to input ports later on. However, since we can
- only connect signals to input ports, these arguments must be
- reduced to simple variables (for which signals will be
- produced). This is taken care of by the argument extraction
- transform.
- \item Non-runtime representable typed arguments.
+
+ The first step in the core to \small{VHDL} translation process, is normalization. We
+ aim to bring the core description into a simpler form, which we can
+ subsequently translate into \small{VHDL} easily. This normal form is needed because
+ the full core language is more expressive than \small{VHDL} in some
+ areas (higher-order expressions, limited polymorphism using type
+ classes, etc.) and because core can describe expressions that do not
+ have a direct hardware interpretation.
+
+ \section{Normal form}
+ The transformations described here have a well-defined goal: to bring the
+ program in a well-defined form that is directly translatable to
+ \VHDL, while fully preserving the semantics of the program. We refer
+ to this form as the \emph{normal form} of the program. The formal
+ definition of this normal form is quite simple:
+
+ \placedefinition[force]{}{\startboxed A program is in \emph{normal form} if none of the
+ transformations from this chapter apply.\stopboxed}
+
+ Of course, this is an \quote{easy} definition of the normal form, since our
+ program will end up in normal form automatically. The more interesting part is
+ to see if this normal form actually has the properties we would like it to
+ have.
+
+ But, before getting into more definitions and details about this normal
+ form, let us try to get a feeling for it first. The easiest way to do this
+ is by describing the things that are unwanted in the intended normal form.
+
+ \startitemize
+ \item Any \emph{polymorphism} must be removed. When laying down hardware, we
+ cannot generate any signals that can have multiple types. All types must be
+ completely known to generate hardware.
+
+ \item All \emph{higher-order} constructions must be removed. We cannot
+ generate a hardware signal that contains a function, so all values,
+ arguments and return values used must be first order.
+
+ \item All complex \emph{nested scopes} must be removed. In the \small{VHDL}
+ description, every signal is in a single scope. Also, full expressions are
+ not supported everywhere (in particular port maps can only map signal
+ names and constants, not complete expressions). To make the \small{VHDL}
+ generation easy, a separate binder must be bound to ever application or
+ other expression.
+ \stopitemize
+
+ \startbuffer[MulSum]
+ alu :: Bit -> Word -> Word -> Word
+ alu = λa.λb.λc.
+ let
+ mul = (*) a b
+ sum = (+) mul c
+ in
+ sum
+ \stopbuffer
+
+ \startuseMPgraphic{MulSum}
+ save a, b, c, mul, add, sum;
+
+ % I/O ports
+ newCircle.a(btex $a$ etex) "framed(false)";
+ newCircle.b(btex $b$ etex) "framed(false)";
+ newCircle.c(btex $c$ etex) "framed(false)";
+ newCircle.sum(btex $sum$ etex) "framed(false)";
+
+ % Components
+ newCircle.mul(btex * etex);
+ newCircle.add(btex + etex);
+
+ a.c - b.c = (0cm, 2cm);
+ b.c - c.c = (0cm, 2cm);
+ add.c = c.c + (2cm, 0cm);
+ mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
+ sum.c = add.c + (2cm, 0cm);
+ c.c = origin;
+
+ % Draw objects and lines
+ drawObj(a, b, c, mul, add, sum);
+
+ ncarc(a)(mul) "arcangle(15)";
+ ncarc(b)(mul) "arcangle(-15)";
+ ncline(c)(add);
+ ncline(mul)(add);
+ ncline(add)(sum);
+ \stopuseMPgraphic
+
+ \placeexample[][ex:MulSum]{Simple architecture consisting of a
+ multiplier and a subtractor.}
+ \startcombination[2*1]
+ {\typebufferlam{MulSum}}{Core description in normal form.}
+ {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
+ \stopcombination
+
+ \todo{Intermezzo: functions vs plain values}
+
+ A very simple example of a program in normal form is given in
+ \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
+ will become input ports in the generated \VHDL) are at the outer level.
+ This means that the body of the inner lambda abstraction is never a
+ function, but always a plain value.
+
+ As the body of the inner lambda abstraction, we see a single (recursive)
+ let expression, that binds two variables (\lam{mul} and \lam{sum}). These
+ variables will be signals in the generated \VHDL, bound to the output port
+ of the \lam{*} and \lam{+} components.
+
+ The final line (the \quote{return value} of the function) selects the
+ \lam{sum} signal to be the output port of the function. This \quote{return
+ value} can always only be a variable reference, never a more complex
+ expression.
+
+ \todo{Add generated VHDL}
+
+ \in{Example}[ex:MulSum] showed a function that just applied two
+ other functions (multiplication and addition), resulting in a simple
+ architecture with two components and some connections. There is of
+ course also some mechanism for choice in the normal form. In a
+ normal Core program, the \emph{case} expression can be used in a few
+ different ways to describe choice. In normal form, this is limited
+ to a very specific form.
+
+ \in{Example}[ex:AddSubAlu] shows an example describing a
+ simple \small{ALU}, which chooses between two operations based on an opcode
+ bit. The main structure is similar to \in{example}[ex:MulSum], but this
+ time the \lam{res} variable is bound to a case expression. This case
+ expression scrutinizes the variable \lam{opcode} (and scrutinizing more
+ complex expressions is not supported). The case expression can select a
+ different variable based on the constructor of \lam{opcode}.
+ \refdef{case expression}
+
+ \startbuffer[AddSubAlu]
+ alu :: Bit -> Word -> Word -> Word
+ alu = λopcode.λa.λb.
+ let
+ res1 = (+) a b
+ res2 = (-) a b
+ res = case opcode of
+ Low -> res1
+ High -> res2
+ in
+ res
+ \stopbuffer
+
+ \startuseMPgraphic{AddSubAlu}
+ save opcode, a, b, add, sub, mux, res;
+
+ % I/O ports
+ newCircle.opcode(btex $opcode$ etex) "framed(false)";
+ newCircle.a(btex $a$ etex) "framed(false)";
+ newCircle.b(btex $b$ etex) "framed(false)";
+ newCircle.res(btex $res$ etex) "framed(false)";
+ % Components
+ newCircle.add(btex + etex);
+ newCircle.sub(btex - etex);
+ newMux.mux;
+
+ opcode.c - a.c = (0cm, 2cm);
+ add.c - a.c = (4cm, 0cm);
+ sub.c - b.c = (4cm, 0cm);
+ a.c - b.c = (0cm, 3cm);
+ mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
+ res.c - mux.c = (1.5cm, 0cm);
+ b.c = origin;
+
+ % Draw objects and lines
+ drawObj(opcode, a, b, res, add, sub, mux);
+
+ ncline(a)(add) "posA(e)";
+ ncline(b)(sub) "posA(e)";
+ nccurve(a)(sub) "posA(e)", "angleA(0)";
+ nccurve(b)(add) "posA(e)", "angleA(0)";
+ nccurve(add)(mux) "posB(inpa)", "angleB(0)";
+ nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
+ nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
+ ncline(mux)(res) "posA(out)";
+ \stopuseMPgraphic
+
+ \placeexample[][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
+ \startcombination[2*1]
+ {\typebufferlam{AddSubAlu}}{Core description in normal form.}
+ {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
+ \stopcombination
+
+ As a more complete example, consider
+ \in{example}[ex:NormalComplete]. This example shows everything that
+ is allowed in normal form, except for built-in higher-order functions
+ (like \lam{map}). The graphical version of the architecture contains
+ a slightly simplified version, since the state tuple packing and
+ unpacking have been left out. Instead, two separate registers are
+ drawn. Most synthesis tools will further optimize this architecture by
+ removing the multiplexers at the register input and instead use the write
+ enable port of the register (when it is available), but we want to show
+ the architecture as close to the description as possible.
+
+ As you can see from the previous examples, the generation of the final
+ architecture from the normal form is straightforward. In each of the
+ examples, there is a direct match between the normal form structure,
+ the generated VHDL and the architecture shown in the images.
+
+ \startbuffer[NormalComplete]
+ regbank :: Bit
+ -> Word
+ -> State (Word, Word)
+ -> (State (Word, Word), Word)
+
+ -- All arguments are an inital lambda (address, data, packed state)
+ regbank = λa.λd.λsp.
+ -- There are nested let expressions at top level
+ let
+ -- Unpack the state by coercion (\eg, cast from
+ -- State (Word, Word) to (Word, Word))
+ s = sp ▶ (Word, Word)
+ -- Extract both registers from the state
+ r1 = case s of (a, b) -> a
+ r2 = case s of (a, b) -> b
+ -- Calling some other user-defined function.
+ d' = foo d
+ -- Conditional connections
+ out = case a of
+ High -> r1
+ Low -> r2
+ r1' = case a of
+ High -> d'
+ Low -> r1
+ r2' = case a of
+ High -> r2
+ Low -> d'
+ -- Packing a tuple
+ s' = (,) r1' r2'
+ -- pack the state by coercion (\eg, cast from
+ -- (Word, Word) to State (Word, Word))
+ sp' = s' ▶ State (Word, Word)
+ -- Pack our return value
+ res = (,) sp' out
+ in
+ -- The actual result
+ res
+ \stopbuffer
+
+ \startuseMPgraphic{NormalComplete}
+ save a, d, r, foo, muxr, muxout, out;
+
+ % I/O ports
+ newCircle.a(btex \lam{a} etex) "framed(false)";
+ newCircle.d(btex \lam{d} etex) "framed(false)";
+ newCircle.out(btex \lam{out} etex) "framed(false)";
+ % Components
+ %newCircle.add(btex + etex);
+ newBox.foo(btex \lam{foo} etex);
+ newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
+ newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
+ newMux.muxr1;
+ % Reflect over the vertical axis
+ reflectObj(muxr1)((0,0), (0,1));
+ newMux.muxr2;
+ newMux.muxout;
+ rotateObj(muxout)(-90);
+
+ d.c = foo.c + (0cm, 1.5cm);
+ a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
+ foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
+ muxr1.c = r1.c + (0cm, 2cm);
+ muxr2.c = r2.c + (0cm, 2cm);
+ r2.c = r1.c + (4cm, 0cm);
+ r1.c = origin;
+ muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
+ out.c = muxout.c - (0cm, 1.5cm);
+
+ % % Draw objects and lines
+ drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
+
+ ncline(d)(foo);
+ nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
+ nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
+ nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
+ nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
+ nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
+ nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
+ nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
+ nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
+ % Connect port a
+ nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
+ nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
+ nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
+ ncline(muxout)(out) "posA(out)";
+ \stopuseMPgraphic
+
+ \todo{Don't split registers in this image?}
+ \placeexample[][ex:NormalComplete]{Simple architecture consisting of an adder and a
+ subtractor.}
+ \startcombination[2*1]
+ {\typebufferlam{NormalComplete}}{Core description in normal form.}
+ {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
+ \stopcombination
+
+
+
+ \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
+ Now we have some intuition for the normal form, we can describe how we want
+ the normal form to look like in a slightly more formal manner. The
+ EBNF-like description in \in{definition}[def:IntendedNormal] captures
+ most of the intended structure (and generates a subset of \GHC's core
+ format).
+
+ There are two things missing from this definition: cast expressions are
+ sometimes allowed by the prototype, but not specified here and the below
+ definition allows uses of state that cannot be translated to \VHDL\
+ properly. These two problems are discussed in
+ \in{section}[sec:normalization:castproblems] and
+ \in{section}[sec:normalization:stateproblems] respectively.
+
+ Some clauses have an expression listed behind them in parentheses.
+ These are conditions that need to apply to the clause. The
+ predicates used there (\lam{lvar()}, \lam{representable()},
+ \lam{gvar()}) will be defined in
+ \in{section}[sec:normalization:predicates].
+
+ An expression is in normal form if it matches the first
+ definition, \emph{normal}.
+
+ \todo{Fix indentation}
+ \startbuffer[IntendedNormal]
+ \italic{normal} := \italic{lambda}
+ \italic{lambda} := λvar.\italic{lambda} (representable(var))
+ | \italic{toplet}
+ \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
+ \italic{binding} := var = \italic{rhs} (representable(rhs))
+ -- State packing and unpacking by coercion
+ | var0 = var1 ▶ State ty (lvar(var1))
+ | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
+ \italic{rhs} := \italic{userapp}
+ | \italic{builtinapp}
+ -- Extractor case
+ | case var of C a0 ... an -> ai (lvar(var))
+ -- Selector case
+ | case var of (lvar(var))
+ [ DEFAULT -> var ] (lvar(var))
+ C0 w0,0 ... w0,n -> var0
+ \vdots
+ Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
+ \italic{userapp} := \italic{userfunc}
+ | \italic{userapp} {userarg}
+ \italic{userfunc} := var (gvar(var))
+ \italic{userarg} := var (lvar(var))
+ \italic{builtinapp} := \italic{builtinfunc}
+ | \italic{builtinapp} \italic{builtinarg}
+ \italic{built-infunc} := var (bvar(var))
+ \italic{built-inarg} := var (representable(var) ∧ lvar(var))
+ | \italic{partapp} (partapp :: a -> b)
+ | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
+ \italic{partapp} := \italic{userapp}
+ | \italic{builtinapp}
+ \stopbuffer
+
+ \placedefinition[][def:IntendedNormal]{Definition of the intended nnormal form using an \small{EBNF}-like syntax.}
+ {\defref{intended normal form definition}
+ \typebufferlam{IntendedNormal}}
+
+ When looking at such a program from a hardware perspective, the top
+ level lambda abstractions (\italic{lambda}) define the input ports.
+ Lambda abstractions cannot appear anywhere else. The variable reference
+ in the body of the recursive let expression (\italic{toplet}) is the
+ output port. Most binders bound by the let expression define a
+ component instantiation (\italic{userapp}), where the input and output
+ ports are mapped to local signals (\italic{userarg}). Some of the others
+ use a built-in construction (\eg\ the \lam{case} expression) or call a
+ built-in function (\italic{builtinapp}) such as \lam{+} or \lam{map}.
+ For these, a hardcoded \small{VHDL} translation is available.
+
+ \section[sec:normalization:transformation]{Transformation notation}
+ To be able to concisely present transformations, we use a specific format
+ for them. It is a simple format, similar to one used in logic reasoning.
+
+ Such a transformation description looks like the following.
+
+ \starttrans
+ <context conditions>
+ ~
+ <original expression>
+ -------------------------- <expression conditions>
+ <transformed expression>
+ ~
+ <context additions>
+ \stoptrans
+
+ This format describes a transformation that applies to \lam{<original
+ expression>} and transforms it into \lam{<transformed expression>}, assuming
+ that all conditions are satisfied. In this format, there are a number of placeholders
+ in pointy brackets, most of which should be rather obvious in their meaning.
+ Nevertheless, we will more precisely specify their meaning below:
+
+ \startdesc{<original expression>} The expression pattern that will be matched
+ against (subexpressions of) the expression to be transformed. We call this a
+ pattern, because it can contain \emph{placeholders} (variables), which match
+ any expression or binder. Any such placeholder is said to be \emph{bound} to
+ the expression it matches. It is convention to use an uppercase letter (\eg\
+ \lam{M} or \lam{E}) to refer to any expression (including a simple variable
+ reference) and lowercase letters (\eg\ \lam{v} or \lam{b}) to refer to
+ (references to) binders.
+
+ For example, the pattern \lam{a + B} will match the expression
+ \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
+ \lam{(2 * w)}), but not \lam{(2 * w) + v}.
+ \stopdesc
+
+ \startdesc{<expression conditions>}
+ These are extra conditions on the expression that is matched. These
+ conditions can be used to further limit the cases in which the
+ transformation applies, commonly to prevent a transformation from
+ causing a loop with itself or another transformation.
+
+ Only if these conditions are \emph{all} satisfied, the transformation
+ applies.
+ \stopdesc
+
+ \startdesc{<context conditions>}
+ These are a number of extra conditions on the context of the function. In
+ particular, these conditions can require some (other) top level function to be
+ present, whose value matches the pattern given here. The format of each of
+ these conditions is: \lam{binder = <pattern>}.
+
+ Typically, the binder is some placeholder bound in the \lam{<original
+ expression>}, while the pattern contains some placeholders that are used in
+ the \lam{transformed expression}.
+
+ Only if a top level binder exists that matches each binder and pattern,
+ the transformation applies.
+ \stopdesc
+
+ \startdesc{<transformed expression>}
+ This is the expression template that is the result of the transformation. If, looking
+ at the above three items, the transformation applies, the \lam{<original
+ expression>} is completely replaced by the \lam{<transformed expression>}.
+ We call this a template, because it can contain placeholders, referring to
+ any placeholder bound by the \lam{<original expression>} or the
+ \lam{<context conditions>}. The resulting expression will have those
+ placeholders replaced by the values bound to them.
+
+ Any binder (lowercase) placeholder that has no value bound to it yet will be
+ bound to (and replaced with) a fresh binder.
+ \stopdesc
+
+ \startdesc{<context additions>}
+ These are templates for new functions to be added to the context.
+ This is a way to let a transformation create new top level
+ functions.
+
+ Each addition has the form \lam{binder = template}. As above, any
+ placeholder in the addition is replaced with the value bound to it, and any
+ binder placeholder that has no value bound to it yet will be bound to (and
+ replaced with) a fresh binder.
+ \stopdesc
+
+ To understand this notation better, the step by step application of
+ the η-expansion transformation to a simple \small{ALU} will be
+ shown. Consider η-expansion, which is a common transformation from
+ labmda calculus, described using above notation as follows:
+
+ \starttrans
+ E \lam{E :: a -> b}
+ -------------- \lam{E} does not occur on a function position in an application
+ λx.E x \lam{E} is not a lambda abstraction.
+ \stoptrans
+
+ η-expansion is a well known transformation from lambda calculus. What
+ this transformation does, is take any expression that has a function type
+ and turn it into a lambda expression (giving an explicit name to the
+ argument). There are some extra conditions that ensure that this
+ transformation does not apply infinitely (which are not necessarily part
+ of the conventional definition of η-expansion).
+
+ Consider the following function, in Core notation, which is a fairly obvious way to specify a
+ simple \small{ALU} (Note that it is not yet in normal form, but
+ \in{example}[ex:AddSubAlu] shows the normal form of this function).
+ The parentheses around the \lam{+} and \lam{-} operators are
+ commonly used in Haskell to show that the operators are used as
+ normal functions, instead of \emph{infix} operators (\eg, the
+ operators appear before their arguments, instead of in between).
+
+ \startlambda
+ alu :: Bit -> Word -> Word -> Word
+ alu = λopcode. case opcode of
+ Low -> (+)
+ High -> (-)
+ \stoplambda
+
+ There are a few subexpressions in this function to which we could possibly
+ apply the transformation. Since the pattern of the transformation is only
+ the placeholder \lam{E}, any expression will match that. Whether the
+ transformation applies to an expression is thus solely decided by the
+ conditions to the right of the transformation.
+
+ We will look at each expression in the function in a top down manner. The
+ first expression is the entire expression the function is bound to.
+
+ \startlambda
+ λopcode. case opcode of
+ Low -> (+)
+ High -> (-)
+ \stoplambda
+
+ As said, the expression pattern matches this. The type of this expression is
+ \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
+ this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
+
+ Since this expression is at top level, it does not occur at a function
+ position of an application. However, The expression is a lambda abstraction,
+ so this transformation does not apply.
+
+ The next expression we could apply this transformation to, is the body of
+ the lambda abstraction:
+
+ \startlambda
+ case opcode of
+ Low -> (+)
+ High -> (-)
+ \stoplambda
+
+ The type of this expression is \lam{Word -> Word -> Word}, which again
+ matches \lam{a -> b}. The expression is the body of a lambda expression, so
+ it does not occur at a function position of an application. Finally, the
+ expression is not a lambda abstraction but a case expression, so all the
+ conditions match. There are no context conditions to match, so the
+ transformation applies.
+
+ By now, the placeholder \lam{E} is bound to the entire expression. The
+ placeholder \lam{x}, which occurs in the replacement template, is not bound
+ yet, so we need to generate a fresh binder for that. Let us use the binder
+ \lam{a}. This results in the following replacement expression:
+
+ \startlambda
+ λa.(case opcode of
+ Low -> (+)
+ High -> (-)) a
+ \stoplambda
+
+ Continuing with this expression, we see that the transformation does not
+ apply again (it is a lambda expression). Next we look at the body of this
+ lambda abstraction:
+
+ \startlambda
+ (case opcode of
+ Low -> (+)
+ High -> (-)) a
+ \stoplambda
+
+ Here, the transformation does apply, binding \lam{E} to the entire
+ expression (which has type \lam{Word -> Word}) and binding \lam{x}
+ to the fresh binder \lam{b}, resulting in the replacement:
+
+ \startlambda
+ λb.(case opcode of
+ Low -> (+)
+ High -> (-)) a b
+ \stoplambda
+
+ The transformation does not apply to this lambda abstraction, so we
+ look at its body. For brevity, we will put the case expression on one line from
+ now on.
+
+ \startlambda
+ (case opcode of Low -> (+); High -> (-)) a b
+ \stoplambda
+
+ The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
+ and the transformation does not apply. Next, we have two options for the
+ next expression to look at: the function position and argument position of
+ the application. The expression in the argument position is \lam{b}, which
+ has type \lam{Word}, so the transformation does not apply. The expression in
+ the function position is:
+
+ \startlambda
+ (case opcode of Low -> (+); High -> (-)) a
+ \stoplambda
+
+ Obviously, the transformation does not apply here, since it occurs in
+ function position (which makes the second condition false). In the same
+ way the transformation does not apply to both components of this
+ expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
+ we will skip to the components of the case expression: the scrutinee and
+ both alternatives. Since the opcode is not a function, it does not apply
+ here.
+
+ The first alternative is \lam{(+)}. This expression has a function type
+ (the operator still needs two arguments). It does not occur in function
+ position of an application and it is not a lambda expression, so the
+ transformation applies.
+
+ We look at the \lam{<original expression>} pattern, which is \lam{E}.
+ This means we bind \lam{E} to \lam{(+)}. We then replace the expression
+ with the \lam{<transformed expression>}, replacing all occurences of
+ \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
+ \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
+ applies the addition operator to \lam{x}).
+
+ The complete function then becomes:
+ \startlambda
+ (case opcode of Low -> λa1.(+) a1; High -> (-)) a
+ \stoplambda
+
+ Now the transformation no longer applies to the complete first alternative
+ (since it is a lambda expression). It does not apply to the addition
+ operator again, since it is now in function position in an application. It
+ does, however, apply to the application of the addition operator, since
+ that is neither a lambda expression nor does it occur in function
+ position. This means after one more application of the transformation, the
+ function becomes:
+
+ \startlambda
+ (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
+ \stoplambda
+
+ The other alternative is left as an exercise to the reader. The final
+ function, after applying η-expansion until it does no longer apply is:
+
+ \startlambda
+ alu :: Bit -> Word -> Word -> Word
+ alu = λopcode.λa.b. (case opcode of
+ Low -> λa1.λb1 (+) a1 b1
+ High -> λa2.λb2 (-) a2 b2) a b
+ \stoplambda
+
+ \subsection{Transformation application}
+ In this chapter we define a number of transformations, but how will we apply
+ these? As stated before, our normal form is reached as soon as no
+ transformation applies anymore. This means our application strategy is to
+ simply apply any transformation that applies, and continuing to do that with
+ the result of each transformation.
+
+ In particular, we define no particular order of transformations. Since
+ transformation order should not influence the resulting normal form,
+ this leaves the implementation free to choose any application order that
+ results in an efficient implementation. Unfortunately this is not
+ entirely true for the current set of transformations. See
+ \in{section}[sec:normalization:non-determinism] for a discussion of this
+ problem.
+
+ When applying a single transformation, we try to apply it to every (sub)expression
+ in a function, not just the top level function body. This allows us to
+ keep the transformation descriptions concise and powerful.
+
+ \subsection{Definitions}
+ A \emph{global variable} is any variable (binder) that is bound at the
+ top level of a program, or an external module. A \emph{local variable} is any
+ other variable (\eg, variables local to a function, which can be bound by
+ lambda abstractions, let expressions and pattern matches of case
+ alternatives). This is a slightly different notion of global versus
+ local than what \small{GHC} uses internally, but for our purposes
+ the distinction \GHC\ makes is not useful.
+ \defref{global variable} \defref{local variable}
+
+ A \emph{hardware representable} (or just \emph{representable}) type or value
+ is (a value of) a type that we can generate a signal for in hardware. For
+ example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
+ not runtime representable notably include (but are not limited to): types,
+ dictionaries, functions.
+ \defref{representable}
+
+ A \emph{built-in function} is a function supplied by the Cλash
+ framework, whose implementation is not used to generate \VHDL. This is
+ either because it is no valid Cλash (like most list functions that need
+ recursion) or because a Cλash implementation would be unwanted (for the
+ addition operator, for example, we would rather use the \VHDL addition
+ operator to let the synthesis tool decide what kind of adder to use
+ instead of explicitly describing one in Cλash). \defref{built-in
+ function}
+
+ These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
+
+ For these functions, Cλash has a \emph{built-in hardware translation},
+ so calls to these functions can still be translated. Built-in functions
+ must have a valid Haskell implementation, of course, to allow
+ simulation.
+
+ A \emph{user-defined} function is a function for which no built-in
+ translation is available and whose definition will thus need to be
+ translated to Cλash. \defref{user-defined function}
+
+ \subsubsection[sec:normalization:predicates]{Predicates}
+ Here, we define a number of predicates that can be used below to concisely
+ specify conditions.
+
+ \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
+ global variable. It is false when it references a local variable.
+
+ \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
+ references a local variable, false when it references a global variable.
+
+ \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
+
+ \subsection[sec:normalization:uniq]{Binder uniqueness}
+ A common problem in transformation systems, is binder uniqueness. When not
+ considering this problem, it is easy to create transformations that mix up
+ bindings and cause name collisions. Take for example, the following core
+ expression:
+
+ \startlambda
+ (λa.λb.λc. a * b * c) x c
+ \stoplambda
+
+ By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
+ we can simplify this expression to:
+
+ \startlambda
+ (λb.λc. x * b * c) c
+ \stoplambda
+
+ Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
+ binder. No harm done here. But note that we see multiple occurences of the
+ \lam{c} binder. The first is a binding occurence, to which the second refers.
+ The last, however refers to \emph{another} instance of \lam{c}, which is
+ bound somewhere outside of this expression. Now, if we would apply beta
+ reduction without taking heed of binder uniqueness, we would get:
+
+ \startlambda
+ λc. x * c * c
+ \stoplambda
+
+ This is obviously not what was supposed to happen! The root of this problem is
+ the reuse of binders: identical binders can be bound in different,
+ but overlapping scopes. Any variable reference in those
+ overlapping scopes then refers to the variable bound in the inner
+ (smallest) scope. There is not way to refer to the variable in the
+ outer scope. This effect is usually referred to as
+ \emph{shadowing}: when a binder is bound in a scope where the
+ binder already had a value, the inner binding is said to
+ \emph{shadow} the outer binding. In the example above, the \lam{c}
+ binder was bound outside of the expression and in the inner lambda
+ expression. Inside that lambda expression, only the inner \lam{c}
+ can be accessed.
+
+ There are a number of ways to solve this. \small{GHC} has isolated this
+ problem to their binder substitution code, which performs \emph{deshadowing}
+ during its expression traversal. This means that any binding that shadows
+ another binding on a higher level is replaced by a new binder that does not
+ shadow any other binding. This non-shadowing invariant is enough to prevent
+ binder uniqueness problems in \small{GHC}.
+
+ In our transformation system, maintaining this non-shadowing invariant is
+ a bit harder to do (mostly due to implementation issues, the prototype
+ does not use \small{GHC}'s subsitution code). Also, the following points
+ can be observed.
+
+ \startitemize
+ \item Deshadowing does not guarantee overall uniqueness. For example, the
+ following (slightly contrived) expression shows the identifier \lam{x} bound in
+ two seperate places (and to different values), even though no shadowing
+ occurs.
+
+ \startlambda
+ (let x = 1 in x) + (let x = 2 in x)
+ \stoplambda
+
+ \item In our normal form (and the resulting \small{VHDL}), all binders
+ (signals) within the same function (entity) will end up in the same
+ scope. To allow this, all binders within the same function should be
+ unique.
+
+ \item When we know that all binders in an expression are unique, moving around
+ or removing a subexpression will never cause any binder conflicts. If we have
+ some way to generate fresh binders, introducing new subexpressions will not
+ cause any problems either. The only way to cause conflicts is thus to
+ duplicate an existing subexpression.
+ \stopitemize
+
+ Given the above, our prototype maintains a unique binder invariant. This
+ means that in any given moment during normalization, all binders \emph{within
+ a single function} must be unique. To achieve this, we apply the following
+ technique.
+
+ \todo{Define fresh binders and unique supplies}
+
+ \startitemize
+ \item Before starting normalization, all binders in the function are made
+ unique. This is done by generating a fresh binder for every binder used. This
+ also replaces binders that did not cause any conflict, but it does ensure that
+ all binders within the function are generated by the same unique supply.
+ \refdef{fresh binder}
+ \item Whenever a new binder must be generated, we generate a fresh binder that
+ is guaranteed to be different from \emph{all binders generated so far}. This
+ can thus never introduce duplication and will maintain the invariant.
+ \item Whenever (a part of) an expression is duplicated (for example when
+ inlining), all binders in the expression are replaced with fresh binders
+ (using the same method as at the start of normalization). These fresh binders
+ can never introduce duplication, so this will maintain the invariant.
+ \item Whenever we move part of an expression around within the function, there
+ is no need to do anything special. There is obviously no way to introduce
+ duplication by moving expressions around. Since we know that each of the
+ binders is already unique, there is no way to introduce (incorrect) shadowing
+ either.
+ \stopitemize
+
+ \section{Transform passes}
+ In this section we describe the actual transforms.
+
+ Each transformation will be described informally first, explaining
+ the need for and goal of the transformation. Then, we will formally define
+ the transformation using the syntax introduced in
+ \in{section}[sec:normalization:transformation].
+
+ \subsection{General cleanup}
+ \placeintermezzo{}{
+ \defref{substitution notation}
+ \startframedtext[width=8cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Substitution notation}
+ \stopalignment
+ \blank[medium]
+
+ In some of the transformations in this chapter, we need to perform
+ substitution on an expression. Substitution means replacing every
+ occurence of some expression (usually a variable reference) with
+ another expression.
+
+ There have been a lot of different notations used in literature for
+ specifying substitution. The notation that will be used in this report
+ is the following:
+
+ \startlambda
+ E[A=>B]
+ \stoplambda
+
+ This means expression \lam{E} with all occurences of \lam{A} replaced
+ with \lam{B}.
+ \stopframedtext
+ }
+
+ These transformations are general cleanup transformations, that aim to
+ make expressions simpler. These transformations usually clean up the
+ mess left behind by other transformations or clean up expressions to
+ expose new transformation opportunities for other transformations.
+
+ Most of these transformations are standard optimizations in other
+ compilers as well. However, in our compiler, most of these are not just
+ optimizations, but they are required to get our program into intended
+ normal form.
+
+ \subsubsection[sec:normalization:beta]{β-reduction}
+ β-reduction is a well known transformation from lambda calculus, where it is
+ the main reduction step. It reduces applications of lambda abstractions,
+ removing both the lambda abstraction and the application.
+
+ In our transformation system, this step helps to remove unwanted lambda
+ abstractions (basically all but the ones at the top level). Other
+ transformations (application propagation, non-representable inlining) make
+ sure that most lambda abstractions will eventually be reducable by
+ β-reduction.
+
+ Note that β-reduction also works on type lambda abstractions and type
+ applications as well. This means the substitution below also works on
+ type variables, in the case that the binder is a type variable and teh
+ expression applied to is a type.
+
+ \starttrans
+ (λx.E) M
+ -----------------
+ E[x=>M]
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ (λa. 2 * a) (2 * b)
+ \stopbuffer
+
+ \startbuffer[to]
+ 2 * (2 * b)
+ \stopbuffer
+
+ \transexample{beta}{β-reduction}{from}{to}
+
+ \startbuffer[from]
+ (λt.λa::t. a) @Int
+ \stopbuffer
+
+ \startbuffer[to]
+ (λa::Int. a)
+ \stopbuffer
+
+ \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
+
+ \subsubsection{Unused let binding removal}
+ This transformation removes let bindings that are never used.
+ Occasionally, \GHC's desugarer introduces some unused let bindings.
+
+ This normalization pass should really be not be necessary to get
+ into intended normal form (since the intended normal form
+ definition \refdef{intended normal form definition} does not
+ require that every binding is used), but in practice the
+ desugarer or simplifier emits some bindings that cannot be
+ normalized (e.g., calls to a
+ \hs{Control.Exception.Base.patError}) but are not used anywhere
+ either. To prevent the \VHDL\ generation from breaking on these
+ artifacts, this transformation removes them.
+
+ \todo{Do not use old-style numerals in transformations}
+ \starttrans
+ letrec
+ a0 = E0
+ \vdots
+ ai = Ei
+ \vdots
+ an = En
+ in
+ M \lam{ai} does not occur free in \lam{M}
+ ---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej})
+ letrec
+ a0 = E0
+ \vdots
+ ai-1 = Ei-1
+ ai+1 = Ei+1
+ \vdots
+ an = En
+ in
+ M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ let
+ x = 1
+ in
+ 2
+ \stopbuffer
+
+ \startbuffer[to]
+ let
+ in
+ 2
+ \stopbuffer
+
+ \transexample{unusedlet}{Unused let binding removal}{from}{to}
+
+ \subsubsection{Empty let removal}
+ This transformation is simple: it removes recursive lets that have no bindings
+ (which usually occurs when unused let binding removal removes the last
+ binding from it).
+
+ Note that there is no need to define this transformation for
+ non-recursive lets, since they always contain exactly one binding.
+
+ \starttrans
+ letrec in M
+ --------------
+ M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ let
+ in
+ 2
+ \stopbuffer
+
+ \startbuffer[to]
+ 2
+ \stopbuffer
+
+ \transexample{emptylet}{Empty let removal}{from}{to}
+
+ \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
+ This transformation inlines simple let bindings, that bind some
+ binder to some other binder instead of a more complex expression (\ie\
+ a = b).
+
+ This transformation is not needed to get an expression into intended
+ normal form (since these bindings are part of the intended normal
+ form), but makes the resulting \small{VHDL} a lot shorter.
+
+ \refdef{substitution notation}
+ \starttrans
+ letrec
+ a0 = E0
+ \vdots
+ ai = b
+ \vdots
+ an = En
+ in
+ M
+ ----------------------------- \lam{b} is a variable reference
+ letrec \lam{ai} ≠ \lam{b}
+ a0 = E0 [ai=>b]
+ \vdots
+ ai-1 = Ei-1 [ai=>b]
+ ai+1 = Ei+1 [ai=>b]
+ \vdots
+ an = En [ai=>b]
+ in
+ M[ai=>b]
+ \stoptrans
+
+ \todo{example}
+
+ \subsubsection{Cast propagation / simplification}
+ This transform pushes casts down into the expression as far as
+ possible. This transformation has been added to make a few
+ specific corner cases work, but it is not clear yet if this
+ transformation handles cast expressions completely or in the
+ right way. See \in{section}[sec:normalization:castproblems].
+
+ \starttrans
+ (let binds in E) ▶ T
+ -------------------------
+ let binds in (E ▶ T)
+ \stoptrans
+
+ \starttrans
+ (case S of
+ p0 -> E0
+ \vdots
+ pn -> En
+ ) ▶ T
+ -------------------------
+ case S of
+ p0 -> E0 ▶ T
+ \vdots
+ pn -> En ▶ T
+ \stoptrans
+
+ \subsubsection{Top level binding inlining}
+ \refdef{top level binding}
+ This transform takes simple top level bindings generated by the
+ \small{GHC} compiler. \small{GHC} sometimes generates very simple
+ \quote{wrapper} bindings, which are bound to just a variable
+ reference, or contain just a (partial) function appliation with
+ the type and dictionary arguments filled in (such as the
+ \lam{(+)} in the example below).
+
+ Note that this transformation is completely optional. It is not
+ required to get any function into intended normal form, but it does help making
+ the resulting VHDL output easier to read (since it removes components
+ that do not add any real structure, but do hide away operations and
+ cause extra clutter).
+
+ This transform takes any top level binding generated by \GHC,
+ whose normalized form contains only a single let binding.
+
+ \starttrans
+ x = λa0 ... λan.let y = E in y
+ ~
+ x
+ -------------------------------------- \lam{x} is generated by the compiler
+ λa0 ... λan.let y = E in y
+ \stoptrans
+
+ \startbuffer[from]
+ (+) :: Word -> Word -> Word
+ (+) = GHC.Num.(+) @Word \$dNum
+ ~
+ (+) a b
+ \stopbuffer
+ \startbuffer[to]
+ GHC.Num.(+) @ Alu.Word \$dNum a b
+ \stopbuffer
+
+ \transexample{toplevelinline}{Top level binding inlining}{from}{to}
+
+ \in{Example}[ex:trans:toplevelinline] shows a typical application of
+ the addition operator generated by \GHC. The type and dictionary
+ arguments used here are described in
+ \in{Section}[section:prototype:polymorphism].
+
+ Without this transformation, there would be a \lam{(+)} entity
+ in the \VHDL\ which would just add its inputs. This generates a
+ lot of overhead in the \VHDL, which is particularly annoying
+ when browsing the generated RTL schematic (especially since most
+ non-alphanumerics, like all characters in \lam{(+)}, are not
+ allowed in \VHDL\ architecture names\footnote{Technically, it is
+ allowed to use non-alphanumerics when using extended
+ identifiers, but it seems that none of the tooling likes
+ extended identifiers in filenames, so it effectively does not
+ work.}, so the entity would be called \quote{w7aA7f} or
+ something similarly meaningless and autogenerated).
+
+ \subsection{Program structure}
+ These transformations are aimed at normalizing the overall structure
+ into the intended form. This means ensuring there is a lambda abstraction
+ at the top for every argument (input port or current state), putting all
+ of the other value definitions in let bindings and making the final
+ return value a simple variable reference.
+
+ \subsubsection[sec:normalization:eta]{η-expansion}
+ This transformation makes sure that all arguments of a function-typed
+ expression are named, by introducing lambda expressions. When combined with
+ β-reduction and non-representable binding inlining, all function-typed
+ expressions should be lambda abstractions or global identifiers.
+
+ \starttrans
+ E \lam{E :: a -> b}
+ -------------- \lam{E} does not occur on a function position in an application
+ λx.E x \lam{E} is not a lambda abstraction.
+ \stoptrans
+
+ \startbuffer[from]
+ foo = λa.case a of
+ True -> λb.mul b b
+ False -> id
+ \stopbuffer
+
+ \startbuffer[to]
+ foo = λa.λx.(case a of
+ True -> λb.mul b b
+ False -> λy.id y) x
+ \stopbuffer
+
+ \transexample{eta}{η-expansion}{from}{to}
+
+ \subsubsection[sec:normalization:appprop]{Application propagation}
+ This transformation is meant to propagate application expressions downwards
+ into expressions as far as possible. This allows partial applications inside
+ expressions to become fully applied and exposes new transformation
+ opportunities for other transformations (like β-reduction and
+ specialization).
+
+ Since all binders in our expression are unique (see
+ \in{section}[sec:normalization:uniq]), there is no risk that we will
+ introduce unintended shadowing by moving an expression into a lower
+ scope. Also, since only move expression into smaller scopes (down into
+ our expression), there is no risk of moving a variable reference out
+ of the scope in which it is defined.
+
+ \starttrans
+ (letrec binds in E) M
+ ------------------------
+ letrec binds in E M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ ( letrec
+ val = 1
+ in
+ add val
+ ) 3
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec
+ val = 1
+ in
+ add val 3
+ \stopbuffer
+
+ \transexample{appproplet}{Application propagation for a let expression}{from}{to}
+
+ \starttrans
+ (case x of
+ p0 -> E0
+ \vdots
+ pn -> En) M
+ -----------------
+ case x of
+ p0 -> E0 M
+ \vdots
+ pn -> En M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ ( case x of
+ True -> id
+ False -> neg
+ ) 1
+ \stopbuffer
+
+ \startbuffer[to]
+ case x of
+ True -> id 1
+ False -> neg 1
+ \stopbuffer
+
+ \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
+
+ \subsubsection[sec:normalization:letrecurse]{Let recursification}
+ This transformation makes all non-recursive lets recursive. In the
+ end, we want a single recursive let in our normalized program, so all
+ non-recursive lets can be converted. This also makes other
+ transformations simpler: they only need to be specified for recursive
+ let expressions (and simply will not apply to non-recursive let
+ expressions until this transformation has been applied).
+
+ \starttrans
+ let
+ a = E
+ in
+ M
+ ------------------------------------------
+ letrec
+ a = E
+ in
+ M
+ \stoptrans
+
+ \subsubsection{Let flattening}
+ This transformation puts nested lets in the same scope, by lifting the
+ binding(s) of the inner let into the outer let. Eventually, this will
+ cause all let bindings to appear in the same scope.
+
+ This transformation only applies to recursive lets, since all
+ non-recursive lets will be made recursive (see
+ \in{section}[sec:normalization:letrecurse]).
+
+ Since we are joining two scopes together, there is no risk of moving a
+ variable reference out of the scope where it is defined.
+
+ \starttrans
+ letrec
+ a0 = E0
+ \vdots
+ ai = (letrec bindings in M)
+ \vdots
+ an = En
+ in
+ N
+ ------------------------------------------
+ letrec
+ a0 = E0
+ \vdots
+ ai = M
+ \vdots
+ an = En
+ bindings
+ in
+ N
+ \stoptrans
+
+ \startbuffer[from]
+ letrec
+ a = 1
+ b = letrec
+ x = a
+ y = c
+ in
+ x + y
+ c = 2
+ in
+ b
+ \stopbuffer
+ \startbuffer[to]
+ letrec
+ a = 1
+ b = x + y
+ c = 2
+ x = a
+ y = c
+ in
+ b
+ \stopbuffer
+
+ \transexample{letflat}{Let flattening}{from}{to}
+
+ \subsubsection{Return value simplification}
+ This transformation ensures that the return value of a function is always a
+ simple local variable reference.
+
+ The basic idea of this transformation is to take the body of a
+ function and bind it with a let expression (so the body of that let
+ expression becomes a variable reference that can be used as the output
+ port). If the body of the function happens to have lambda abstractions
+ at the top level (which is allowed by the intended normal
+ form\refdef{intended normal form definition}), we take the body of the
+ inner lambda instead. If that happens to be a let expression already
+ (which is allowed by the intended normal form), we take the body of
+ that let (which is not allowed to be anything but a variable reference
+ according the the intended normal form).
+
+ This transformation uses the context conditions in a special way.
+ These contexts, like \lam{x = λv1 ... λvn.E}, are above the dotted
+ line and provide a condition on the environment (\ie\ they require a
+ certain top level binding to be present). These ensure that
+ expressions are only transformed when they are in the functions
+ \quote{return value} directly. This means the context conditions have
+ to interpreted in the right way: not \quote{if there is any function
+ \lam{x} that binds \lam{E}, any \lam{E} can be transformed}, but we
+ mean only the \lam{E} that is bound by \lam{x}).
+
+ Be careful when reading the transformations: Not the entire function
+ from the context is transformed, just a part of it.
+
+ Note that the return value is not simplified if it is not representable.
+ Otherwise, this would cause a loop with the inlining of
+ unrepresentable bindings in
+ \in{section}[sec:normalization:nonrepinline]. If the return value is
+ not representable because it has a function type, η-expansion should
+ make sure that this transformation will eventually apply. If the
+ value is not representable for other reasons, the function result
+ itself is not representable, meaning this function is not translatable
+ anyway.
+
+ \starttrans
+ x = λv1 ... λvn.E \lam{n} can be zero
+ ~ \lam{E} is representable
+ E \lam{E} is not a lambda abstraction
+ --------------------------- \lam{E} is not a let expression
+ letrec y = E in y \lam{E} is not a local variable reference
+ \stoptrans
+
+ \starttrans
+ x = λv1 ... λvn.letrec binds in E \lam{n} can be zero
+ ~ \lam{E} is representable
+ letrec binds in E \lam{E} is not a local variable reference
+ ------------------------------------
+ letrec binds; y = E in y
+ \stoptrans
+
+ \startbuffer[from]
+ x = add 1 2
+ \stopbuffer
+
+ \startbuffer[to]
+ x = letrec y = add 1 2 in y
+ \stopbuffer
+
+ \transexample{retvalsimpl}{Return value simplification}{from}{to}
+
+ \startbuffer[from]
+ x = λa. add 1 a
+ \stopbuffer
+
+ \startbuffer[to]
+ x = λa. letrec
+ y = add 1 a
+ in
+ y
+ \stopbuffer
+
+ \transexample{retvalsimpllam}{Return value simplification with a lambda abstraction}{from}{to}
+
+ \startbuffer[from]
+ x = letrec
+ a = add 1 2
+ in
+ add a 3
+ \stopbuffer
+
+ \startbuffer[to]
+ x = letrec
+ a = add 1 2
+ y = add a 3
+ in
+ y
+ \stopbuffer
+
+ \transexample{retvalsimpllet}{Return value simplification with a let expression}{from}{to}
+
+ \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
+ This section contains just a single transformation that deals with
+ representable arguments in applications. Non-representable arguments are
+ handled by the transformations in
+ \in{section}[sec:normalization:nonrep].
+
+ This transformation ensures that all representable arguments will become
+ references to local variables. This ensures they will become references
+ to local signals in the resulting \small{VHDL}, which is required due to
+ limitations in the component instantiation code in \VHDL\ (one can only
+ assign a signal or constant to an input port). By ensuring that all
+ arguments are always simple variable references, we always have a signal
+ available to map to the input ports.
+
+ To reduce a complex expression to a simple variable reference, we create
+ a new let expression around the application, which binds the complex
+ expression to a new variable. The original function is then applied to
+ this variable.
+
+ \refdef{global variable}
+ Note that references to \emph{global variables} (like a top level
+ function without arguments, but also an argumentless dataconstructors
+ like \lam{True}) are also simplified. Only local variables generate
+ signals in the resulting architecture. Even though argumentless
+ dataconstructors generate constants in generated \VHDL\ code and could be
+ mapped to an input port directly, they are still simplified to make the
+ normal form more regular.
+
+ \refdef{representable}
+ \starttrans
+ M N
+ -------------------- \lam{N} is representable
+ letrec x = N in M x \lam{N} is not a local variable reference
+ \stoptrans
+ \refdef{local variable}
+
+ \startbuffer[from]
+ add (add a 1) 1
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec x = add a 1 in add x 1
+ \stopbuffer
+
+ \transexample{argsimpl}{Argument simplification}{from}{to}
+
+ \subsection[sec:normalization:built-ins]{Built-in functions}
+ This section deals with (arguments to) built-in functions. In the
+ intended normal form definition\refdef{intended normal form definition}
+ we can see that there are three sorts of arguments a built-in function
+ can receive.
+
+ \startitemize[KR]
+ \item A representable local variable reference. This is the most
+ common argument to any function. The argument simplification
+ transformation described in \in{section}[sec:normalization:argsimpl]
+ makes sure that \emph{any} representable argument to \emph{any}
+ function (including built-in functions) is turned into a local variable
+ reference.
+ \item (A partial application of) a top level function (either built-in on
+ user-defined). The function extraction transformation described in
+ this section takes care of turning every functiontyped argument into
+ (a partial application of) a top level function.
+ \item Any expression that is not representable and does not have a
+ function type. Since these can be any expression, there is no
+ transformation needed. Note that this category is exactly all
+ expressions that are not transformed by the transformations for the
+ previous two categories. This means that \emph{any} core expression
+ that is used as an argument to a built-in function will be either
+ transformed into one of the above categories, or end up in this
+ categorie. In any case, the result is in normal form.
+ \stopitemize
+
+ As noted, the argument simplification will handle any representable
+ arguments to a built-in function. The following transformation is needed
+ to handle non-representable arguments with a function type, all other
+ non-representable arguments do not need any special handling.
+
+ \subsubsection[sec:normalization:funextract]{Function extraction}
+ This transform deals with function-typed arguments to built-in
+ functions.
+ Since built-in functions cannot be specialized (see
+ \in{section}[sec:normalization:specialize]) to remove the arguments,
+ these arguments are extracted into a new global function instead. In
+ other words, we create a new top level function that has exactly the
+ extracted argument as its body. This greatly simplifies the
+ translation rules needed for built-in functions, since they only need
+ to handle (partial applications of) top level functions.
+
+ Any free variables occuring in the extracted arguments will become
+ parameters to the new global function. The original argument is replaced
+ with a reference to the new function, applied to any free variables from
+ the original argument.
+
+ This transformation is useful when applying higher-order built-in functions
+ like \hs{map} to a lambda abstraction, for example. In this case, the code
+ that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
+ partial applications, not any other expression (such as lambda abstractions or
+ even more complicated expressions).
+
+ \starttrans
+ M N \lam{M} is (a partial aplication of) a built-in function.
+ --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
+ M (x f0 ... fn) \lam{N :: a -> b}
+ ~ \lam{N} is not a (partial application of) a top level function
+ x = λf0 ... λfn.N
+ \stoptrans
+
+ \startbuffer[from]
+ addList = λb.λxs.map (λa . add a b) xs
+ \stopbuffer
+
+ \startbuffer[to]
+ addList = λb.λxs.map (f b) xs
+ ~
+ f = λb.λa.add a b
+ \stopbuffer
+
+ \transexample{funextract}{Function extraction}{from}{to}
+
+ Note that the function \lam{f} will still need normalization after
+ this.
+
+ \subsection{Case normalisation}
+ The transformations in this section ensure that case statements end up
+ in normal form.
+
+ \subsubsection{Scrutinee simplification}
+ This transform ensures that the scrutinee of a case expression is always
+ a simple variable reference.
+
+ \starttrans
+ case E of
+ alts
+ ----------------- \lam{E} is not a local variable reference
+ letrec x = E in
+ case x of
+ alts
+ \stoptrans
+
+ \startbuffer[from]
+ case (foo a) of
+ True -> a
+ False -> b
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec x = foo a in
+ case x of
+ True -> a
+ False -> b
+ \stopbuffer
+
+ \transexample{letflat}{Case normalisation}{from}{to}
+
+
+ \placeintermezzo{}{
+ \defref{wild binders}
+ \startframedtext[width=7cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Wild binders}
+ \stopalignment
+ \blank[medium]
+ In a functional expression, a \emph{wild binder} refers to any
+ binder that is never referenced. This means that even though it
+ will be bound to a particular value, that value is never used.
+
+ The Haskell syntax offers the underscore as a wild binder that
+ cannot even be referenced (It can be seen as introducing a new,
+ anonymous, binder everytime it is used).
+
+ In these transformations, the term wild binder will sometimes be
+ used to indicate that a binder must not be referenced.
+ \stopframedtext
+ }
+
+ \subsubsection{Scrutinee binder removal}
+ This transformation removes (or rather, makes wild) the binder to
+ which the scrutinee is bound after evaluation. This is done by
+ replacing the bndr with the scrutinee in all alternatives. To prevent
+ duplication of work, this transformation is only applied when the
+ scrutinee is already a simple variable reference (but the previous
+ transformation ensures this will eventually be the case). The
+ scrutinee binder itself is replaced by a wild binder (which is no
+ longer displayed).
+
+ Note that one could argue that this transformation can change the
+ meaning of the Core expression. In the regular Core semantics, a case
+ expression forces the evaluation of its scrutinee and can be used to
+ implement strict evaluation. However, in the generated \VHDL,
+ evaluation is always strict. So the semantics we assign to the Core
+ expression (which differ only at this particular point), this
+ transformation is completely valid.
+
+ \starttrans
+ case x of bndr
+ alts
+ ----------------- \lam{x} is a local variable reference
+ case x of
+ alts[bndr=>x]
+ \stoptrans
+
+ \startbuffer[from]
+ case x of y
+ True -> y
+ False -> not y
+ \stopbuffer
+
+ \startbuffer[to]
+ case x of
+ True -> x
+ False -> not x
+ \stopbuffer
+
+ \transexample{scrutbndrremove}{Scrutinee binder removal}{from}{to}
+
+ \subsubsection{Case normalization}
+ This transformation ensures that all case expressions get a form
+ that is allowed by the intended normal form. This means they
+ will become one of:
+
+ \startitemize
+ \item An extractor case with a single alternative that picks a field
+ from a datatype, \eg\ \lam{case x of (a, b) -> a}.
+ \item A selector case with multiple alternatives and only wild binders, that
+ makes a choice between expressions based on the constructor of another
+ expression, \eg\ \lam{case x of Low -> a; High -> b}.
+ \stopitemize
+
+ For an arbitrary case, that has \lam{n} alternatives, with
+ \lam{m} binders in each alternatives, this will result in \lam{m
+ * n} extractor case expression to get at each variable, \lam{n}
+ let bindings for each of the alternatives' value and a single
+ selector case to select the right value out of these.
+
+ Technically, the defintion of this transformation would require
+ that the constructor for every alternative has exactly the same
+ amount (\lam{m}) of arguments, but of course this transformation
+ also applies when this is not the case.
+
+ \starttrans
+ case E of
+ C0 v0,0 ... v0,m -> E0
+ \vdots
+ Cn vn,0 ... vn,m -> En
+ --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
+ letrec The case expression is not an extractor case
+ v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
+ \vdots
+ v0,m = case E of C0 x0,0 .. x0,m -> x0,m
+ \vdots
+ vn,m = case E of Cn xn,0 .. xn,m -> xn,m
+ y0 = E0
+ \vdots
+ yn = En
+ in
+ case E of
+ C0 w0,0 ... w0,m -> y0
+ \vdots
+ Cn wn,0 ... wn,m -> yn
+ \stoptrans
+
+ Note that this transformation applies to case expressions with any
+ scrutinee. If the scrutinee is a complex expression, this might
+ result in duplication of work (hardware). An extra condition to
+ only apply this transformation when the scrutinee is already
+ simple (effectively causing this transformation to be only
+ applied after the scrutinee simplification transformation) might
+ be in order.
+
+ \startbuffer[from]
+ case a of
+ True -> add b 1
+ False -> add b 2
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec
+ x0 = add b 1
+ x1 = add b 2
+ in
+ case a of
+ True -> x0
+ False -> x1
+ \stopbuffer
+
+ \transexample{selcasesimpl}{Selector case simplification}{from}{to}
+
+ \startbuffer[from]
+ case a of
+ (,) b c -> add b c
+ \stopbuffer
+ \startbuffer[to]
+ letrec
+ b = case a of (,) b c -> b
+ c = case a of (,) b c -> c
+ x0 = add b c
+ in
+ case a of
+ (,) w0 w1 -> x0
+ \stopbuffer
+
+ \transexample{excasesimpl}{Extractor case simplification}{from}{to}
+
+ \refdef{selector case}
+ In \in{example}[ex:trans:excasesimpl] the case expression is expanded
+ into multiple case expressions, including a pretty useless expression
+ (that is neither a selector or extractor case). This case can be
+ removed by the Case removal transformation in
+ \in{section}[sec:transformation:caseremoval].
+
+ \subsubsection[sec:transformation:caseremoval]{Case removal}
+ This transform removes any case expression with a single alternative and
+ only wild binders.\refdef{wild binders}
+
+ These "useless" case expressions are usually leftovers from case simplification
+ on extractor case (see the previous example).
+
+ \starttrans
+ case x of
+ C v0 ... vm -> E
+ ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
+ E
+ \stoptrans
+
+ \startbuffer[from]
+ case a of
+ (,) w0 w1 -> x0
+ \stopbuffer
+
+ \startbuffer[to]
+ x0
+ \stopbuffer
+
+ \transexample{caserem}{Case removal}{from}{to}
+
+ \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
+ The transformations in this section are aimed at making all the
+ values used in our expression representable. There are two main
+ transformations that are applied to \emph{all} unrepresentable let
+ bindings and function arguments. These are meant to address three
+ different kinds of unrepresentable values: polymorphic values,
+ higher-order values and literals. The transformation are described
+ generically: they apply to all non-representable values. However,
+ non-representable values that do not fall into one of these three
+ categories will be moved around by these transformations but are
+ unlikely to completely disappear. They usually mean the program was not
+ valid in the first place, because unsupported types were used (for
+ example, a program using strings).
+
+ Each of these three categories will be detailed below, followed by the
+ actual transformations.
+
+ \subsubsection{Removing Polymorphism}
+ As noted in \in{section}[sec:prototype:polymporphism],
+ polymorphism is made explicit in Core through type and
+ dictionary arguments. To remove the polymorphism from a
+ function, we can simply specialize the polymorphic function for
+ the particular type applied to it. The same goes for dictionary
+ arguments. To remove polymorphism from let bound values, we
+ simply inline the let bindings that have a polymorphic type,
+ which should (eventually) make sure that the polymorphic
+ expression is applied to a type and/or dictionary, which can
+ then be removed by β-reduction (\in{section}[sec:normalization:beta]).
+
+ Since both type and dictionary arguments are not representable,
+ \refdef{representable}
+ the non-representable argument specialization and
+ non-representable let binding inlining transformations below
+ take care of exactly this.
+
+ There is one case where polymorphism cannot be completely
+ removed: built-in functions are still allowed to be polymorphic
+ (Since we have no function body that we could properly
+ specialize). However, the code that generates \VHDL\ for built-in
+ functions knows how to handle this, so this is not a problem.
+
+ \subsubsection[sec:normalization:defunctionalization]{Defunctionalization}
+ These transformations remove higher-order expressions from our
+ program, making all values first-order. The approach used for
+ defunctionalization uses a combination of specialization, inlining and
+ some cleanup transformations, was also proposed in parallel research
+ by Neil Mitchell \cite[mitchell09].
+
+ Higher order values are always introduced by lambda abstractions, none
+ of the other Core expression elements can introduce a function type.
+ However, other expressions can \emph{have} a function type, when they
+ have a lambda expression in their body.
+
+ For example, the following expression is a higher-order expression
+ that is not a lambda expression itself:
+
+ \refdef{id function}
+ \startlambda
+ case x of
+ High -> id
+ Low -> λx.x
+ \stoplambda
+
+ The reference to the \lam{id} function shows that we can introduce a
+ higher-order expression in our program without using a lambda
+ expression directly. However, inside the definition of the \lam{id}
+ function, we can be sure that a lambda expression is present.
+
+ Looking closely at the definition of our normal form in
+ \in{section}[sec:normalization:intendednormalform], we can see that
+ there are three possibilities for higher-order values to appear in our
+ intended normal form:
+
+ \startitemize[KR]
+ \item[item:toplambda] Lambda abstractions can appear at the highest level of a
+ top level function. These lambda abstractions introduce the
+ arguments (input ports / current state) of the function.
+ \item[item:built-inarg] (Partial applications of) top level functions can appear as an
+ argument to a built-in function.
+ \item[item:completeapp] (Partial applications of) top level functions can appear in
+ function position of an application. Since a partial application
+ cannot appear anywhere else (except as built-in function arguments),
+ all partial applications are applied, meaning that all applications
+ will become complete applications. However, since application of
+ arguments happens one by one, in the expression:
+ \startlambda
+ f 1 2
+ \stoplambda
+ the subexpression \lam{f 1} has a function type. But this is
+ allowed, since it is inside a complete application.
+ \stopitemize
+
+ We will take a typical function with some higher-order values as an
+ example. The following function takes two arguments: a \lam{Bit} and a
+ list of numbers. Depending on the first argument, each number in the
+ list is doubled, or the list is returned unmodified. For the sake of
+ the example, no polymorphism is shown. In reality, at least map would
+ be polymorphic.
+
+ \startlambda
+ λy.let double = λx. x + x in
+ case y of
+ Low -> map double
+ High -> λz. z
+ \stoplambda
+
+ This example shows a number of higher-order values that we cannot
+ translate to \VHDL\ directly. The \lam{double} binder bound in the let
+ expression has a function type, as well as both of the alternatives of
+ the case expression. The first alternative is a partial application of
+ the \lam{map} built-in function, whereas the second alternative is a
+ lambda abstraction.
+
+ To reduce all higher-order values to one of the above items, a number
+ of transformations we have already seen are used. The η-expansion
+ transformation from \in{section}[sec:normalization:eta] ensures all
+ function arguments are introduced by lambda abstraction on the highest
+ level of a function. These lambda arguments are allowed because of
+ \in{item}[item:toplambda] above. After η-expansion, our example
+ becomes a bit bigger:
+
+ \startlambda
+ λy.λq.(let double = λx. x + x in
+ case y of
+ Low -> map double
+ High -> λz. z
+ ) q
+ \stoplambda
+
+ η-expansion also introduces extra applications (the application of
+ the let expression to \lam{q} in the above example). These
+ applications can then propagated down by the application propagation
+ transformation (\in{section}[sec:normalization:appprop]). In our
+ example, the \lam{q} and \lam{r} variable will be propagated into the
+ let expression and then into the case expression:
+
+ \startlambda
+ λy.λq.let double = λx. x + x in
+ case y of
+ Low -> map double q
+ High -> (λz. z) q
+ \stoplambda
+
+ This propagation makes higher-order values become applied (in
+ particular both of the alternatives of the case now have a
+ representable type). Completely applied top level functions (like the
+ first alternative) are now no longer invalid (they fall under
+ \in{item}[item:completeapp] above). (Completely) applied lambda
+ abstractions can be removed by β-expansion. For our example,
+ applying β-expansion results in the following:
+
+ \startlambda
+ λy.λq.let double = λx. x + x in
+ case y of
+ Low -> map double q
+ High -> q
+ \stoplambda
+
+ As you can see in our example, all of this moves applications towards
+ the higher-order values, but misses higher-order functions bound by
+ let expressions. The applications cannot be moved towards these values
+ (since they can be used in multiple places), so the values will have
+ to be moved towards the applications. This is achieved by inlining all
+ higher-order values bound by let applications, by the
+ non-representable binding inlining transformation below. When applying
+ it to our example, we get the following: