\chapter[chap:normalization]{Normalization}
+ % A helper to print a single example in the half the page width. The example
+ % text should be in a buffer whose name is given in an argument.
+ %
+ % The align=right option really does left-alignment, but without the program
+ % will end up on a single line. The strut=no option prevents a bunch of empty
+ % space at the start of the frame.
+ \define[1]\example{
+ \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
+ \setuptyping[option=LAM,style=sans,before=,after=,strip=auto]
+ \typebuffer[#1]
+ \setuptyping[option=none,style=\tttf,strip=auto]
+ }
+ }
-% A helper to print a single example in the half the page width. The example
-% text should be in a buffer whose name is given in an argument.
-%
-% The align=right option really does left-alignment, but without the program
-% will end up on a single line. The strut=no option prevents a bunch of empty
-% space at the start of the frame.
-\define[1]\example{
- \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
- \setuptyping[option=LAM,style=sans,before=,after=]
- \typebuffer[#1]
- \setuptyping[option=none,style=\tttf]
+ \define[3]\transexample{
+ \placeexample[here]{#1}
+ \startcombination[2*1]
+ {\example{#2}}{Original program}
+ {\example{#3}}{Transformed program}
+ \stopcombination
}
-}
-
-
-% 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.
-
-\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.
-
-\subsection{Definitions}
-In the following sections, we will be using a number of functions and
-notations, which we will define here.
-
-\subsubsection{Transformations}
-The most important notation is the one for transformation, which looks like
-the following:
-
-\starttrans
-context conditions
-~
-from
------------------------- expression conditions
-to
-~
-context additions
-\stoptrans
-
-Here, we describe a transformation. The most import parts are \lam{from} and
-\lam{to}, which describe the Core expresssion that should be matched and the
-expression that it should be replaced with. This matching can occur anywhere
-in function that is being normalized, so it applies to any subexpression as
-well.
-
-The \lam{expression conditions} list a number of conditions on the \lam{from}
-expression that must hold for the transformation to apply.
-
-Furthermore, there is some way to look into the environment (\eg, other top
-level bindings). The \lam{context conditions} part specifies any number of
-top level bindings that must be present for the transformation to apply.
-Usually, this lists a top level binding that binds an identfier that is also
-used in the \lam{from} expression, allowing us to "access" the value of a top
-level binding in the \lam{to} expression (\eg, for inlining).
-
-Finally, there is a way to influence the environment. The \lam{context
-additions} part lists any number of new top level bindings that should be
-added.
-
-If there are no \lam{context conditions} or \lam{context additions}, they can
-be left out alltogether, along with the separator \lam{~}.
-
-TODO: Example
-
-\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 local variable is any other
-variable (\eg, variables local to a function, which can be bound by lambda
-abstractions, let expressions and case expressions).
-
-A \emph{hardware representable} type is 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.
-
-A \emph{builtin function} is a function for which a builtin
-hardware translation is available, because its actual definition is not
-translatable. A user-defined function is any other function.
-
-\subsubsection{Functions}
-Here, we define a number of functions 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 inverse of \emph{gvar}; it is true when \emph{expr}
-references a local variable, false when it references a global variable.
-
-\emph{representable(expr)} or \emph{representable(var)} is true when
-\emph{expr} or \emph{var} has a type that is representable at runtime.
-
-\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.
-
-TODO: Formally describe the "apply to every (sub)expression" in terms of
-rules with full transformations in the conditions.
-
-\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 :: * -> *}
--------------- \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{Extended β-reduction}
-This transformation is meant to propagate application expressions downwards
-into expressions as far as possible. In lambda calculus, this reduction
-is known as β-reduction, but it is of course only defined for
-applications of lambda abstractions. We extend this reduction to also
-work for the rest of core (case and let expressions).
-
-For let expressions:
-\starttrans
-let binds in E) M
------------------
-let binds in E M
-\stoptrans
-
-For case statements:
-\starttrans
-(case x of
- p1 -> E1
- \vdots
- pn -> En) M
------------------
-case x of
- p1 -> E1 M
- \vdots
- pn -> En M
-\stoptrans
-
-For lambda expressions:
-\starttrans
-(λx.E) M
------------------
-E[M/x]
-\stoptrans
-
-% And an example
-\startbuffer[from]
-( let a = (case x of
+
+ 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 \VHDL, and describe how the
+ \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{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 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} = letrec [\italic{binding}...] in 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? This is
+ no longer true, btw}
+
+ 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,
+ \todo{This is not really true, but would like it to be...} 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.
+
+ \todo{Define substitution (notation)}
+
+ \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
+ \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 (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{General cleanup}
+ 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 normal
+ form.
+
+ \subsubsection{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.
+
+ \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}
+
+ \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).
+
+ \starttrans
+ letrec in M
+ --------------
+ M
+ \stoptrans
+
+ \todo{Example}
+
+ \subsubsection{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
+ letrec
+ a0 = E0
+ \vdots
+ ai = b
+ \vdots
+ an = En
+ in
+ M
+ ----------------------------- \lam{b} is a variable reference
+ letrec
+ a0 = E0 [b/ai]
+ \vdots
+ ai-1 = Ei-1 [b/ai]
+ ai+1 = Ei+1 [b/ai]
+ \vdots
+ an = En [b/ai]
+ in
+ M[b/ai]
+ \stoptrans
+
+ \todo{example}
+
+ \subsubsection{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 unused 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
+ letrec
+ a0 = E0
+ \vdots
+ ai = Ei
+ \vdots
+ an = En
+ in
+ M \lam{a} does not occur free in \lam{M}
+ ---------------------------- \forall j, 0 <= j <= n, j ≠ i (\lam{a} 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
+
+ \todo{Example}
+
+ \subsubsection{Cast propagation / simplification}
+ This transform pushes casts down into the expression as far as possible.
+ Since its exact role and need is not clear yet, this transformation is
+ not yet specified.
+
+ \todo{Cast propagation}
+
+ \subsubsection{Top level binding inlining}
+ 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 a partial application to constants or other variable
+ references.
+
+ Note that this transformation is completely optional. It is not
+ required to get any function into normal form, but it does help making
+ the resulting VHDL output easier to read (since it removes a bunch of
+ components that are really boring).
+
+ This transform takes any top level binding generated by the compiler,
+ 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{Top level binding inlining}{from}{to}
+
+ Without this transformation, the (+) function would generate an
+ architecture 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 + is 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 doesn't work}, so the entity would be called
+ \quote{w7aA7f} or something similarly unreadable 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), putting all of the other
+ value definitions in let bindings and making the final return value a
+ simple variable reference.
+
+ \subsubsection{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 non-representable binding inlining, 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}
+
+ \subsubsection{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).
+
+ \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{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
- b = (let y = 3 in add y) 2
- in
- (λz.add 1 z)
-) 3
-\stopbuffer
-
-\startbuffer[to]
-let a = case x of
- True -> id 1
- False -> neg 1
- b = let y = 3 in add y 2
-in
- add 1 3
-\stopbuffer
-
-\transexample{Extended β-reduction}{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.
-
- These arguments cannot be preserved in the program, since we
- cannot represent them as input or output ports in the resulting
- \small{VHDL}. To remove them, we create a specialized version of the
- called function with these arguments filled in. This is done by
- the argument propagation transform.
-
- Typically, these arguments are type and dictionary arguments that are
- used to make functions polymorphic. By propagating these arguments, we
- are essentially doing the same which GHC does when it specializes
- functions: Creating multiple variants of the same function, one for
- each type for which it is used. Other common non-representable
- arguments are functions, e.g. when calling a higher order function
- with another function or a lambda abstraction as an argument.
-
- The reason for doing this is similar to the reasoning provided for
- the inlining of non-representable let bindings above. In fact, this
- argument propagation could be viewed as a form of cross-function
- inlining.
-\stopitemize
-
-TODO: Check the following itemization.
-
-When looking at the arguments of a builtin function, we can divide them
-into categories:
-
-\startitemize
- \item Arguments of a runtime representable type.
-
- As we have seen with user-defined functions, these arguments can
- always be reduced to a simple variable reference, by the
- argument extraction transform. Performing this transform for
- builtin functions as well, means that the translation of builtin
- functions can be limited to signal references, instead of
- needing to support all possible expressions.
-
- \item Arguments of a function type.
-
- These arguments are functions passed to higher order builtins,
- like \lam{map} and \lam{foldl}. Since implementing these
- functions for arbitrary function-typed expressions (\eg, lambda
- expressions) is rather comlex, we reduce these arguments to
- (partial applications of) global functions.
-
- We can still support arbitrary expressions from the user code,
- by creating a new global function containing that expression.
- This way, we can simply replace the argument with a reference to
- that new function. However, since the expression can contain any
- number of free variables we also have to include partial
- applications in our normal form.
-
- This category of arguments is handled by the function extraction
- transform.
- \item Other unrepresentable arguments.
-
- These arguments can take a few different forms:
- \startdesc{Type arguments}
- In the core language, type arguments can only take a single
- form: A type wrapped in the Type constructor. Also, there is
- nothing that can be done with type expressions, except for
- applying functions to them, so we can simply leave type
- arguments as they are.
- \stopdesc
- \startdesc{Dictionary arguments}
- In the core language, dictionary arguments are used to find
- operations operating on one of the type arguments (mostly for
- finding class methods). Since we will not actually evaluatie
- the function body for builtin functions and can generate
- code for builtin functions by just looking at the type
- arguments, these arguments can be ignored and left as they
- are.
- \stopdesc
- \startdesc{Type level arguments}
- Sometimes, we want to pass a value to a builtin function, but
- we need to know the value at compile time. Additionally, the
- value has an impact on the type of the function. This is
- encoded using type-level values, where the actual value of the
- argument is not important, but the type encodes some integer,
- for example. Since the value is not important, the actual form
- of the expression does not matter either and we can leave
- these arguments as they are.
- \stopdesc
- \startdesc{Other arguments}
- Technically, there is still a wide array of arguments that can
- be passed, but does not fall into any of the above categories.
- However, none of the supported builtin functions requires such
- an argument. This leaves use with passing unsupported types to
- a function, such as calling \lam{head} on a list of functions.
-
- In these cases, it would be impossible to generate hardware
- for such a function call anyway, so we can ignore these
- arguments.
-
- The only way to generate hardware for builtin functions with
- arguments like these, is to expand the function call into an
- equivalent core expression (\eg, expand map into a series of
- function applications). But for now, we choose to simply not
- support expressions like these.
- \stopdesc
-
- From the above, we can conclude that we can simply ignore these
- other unrepresentable arguments and focus on the first two
- categories instead.
-\stopitemize
-
-\subsubsection{Argument simplification}
-This transform deals with arguments to functions that
-are of a runtime representable type. It ensures that they will all become
-references to global variables, or local signals in the resulting \small{VHDL}.
-
-TODO: It seems we can map an expression to a port, not only a signal.
-Perhaps this makes this transformation not needed?
-TODO: Say something about dataconstructors (without arguments, like True
-or False), which are variable references of a runtime representable
-type, but do not result in a signal.
-
-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.
-
-\starttrans
-M N
--------------------- \lam{N} is of a representable type
-let x = N in M x \lam{N} is not a local variable reference
-\stoptrans
-
-\startbuffer[from]
-add (add a 1) 1
-\stopbuffer
-
-\startbuffer[to]
-let x = add a 1 in add x 1
-\stopbuffer
-
-\transexample{Argument extraction}{from}{to}
-
-\subsubsection{Function extraction}
-This transform deals with function-typed arguments to builtin functions.
-Since these arguments cannot be propagated, we choose to extract them
-into a new global function instead.
-
-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 builtin 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) builtin function.
---------------------- \lam{f0 ... fn} = 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]
-map (λa . add a b) xs
-
-map (add b) ys
-\stopbuffer
-
-\startbuffer[to]
-x0 = λb.λa.add a b
-~
-map x0 xs
-
-x1 = λb.add b
-map x1 ys
-\stopbuffer
-
-\transexample{Function extraction}{from}{to}
-
-\subsubsection{Argument propagation}
-This transform deals with arguments to user-defined functions that are
-not representable at runtime. This means these arguments cannot be
-preserved in the final form and most be {\em propagated}.
-
-Propagation means to create a specialized version of the called
-function, with the propagated argument already filled in. As a simple
-example, in the following program:
-
-\startlambda
-f = λa.λb.a + b
-inc = λa.f a 1
-\stoplambda
-
-we could {\em propagate} the constant argument 1, with the following
-result:
-
-\startlambda
-f' = λa.a + 1
-inc = λa.f' a
-\stoplambda
-
-Special care must be taken when the to-be-propagated expression has any
-free variables. If this is the case, the original argument should not be
-removed alltogether, but replaced by all the free variables of the
-expression. In this way, the original expression can still be evaluated
-inside the new function. Also, this brings us closer to our goal: All
-these free variables will be simple variable references.
-
-To prevent us from propagating the same argument over and over, a simple
-local variable reference is not propagated (since is has exactly one
-free variable, itself, we would only replace that argument with itself).
-
-This shows that any free local variables that are not runtime representable
-cannot be brought into normal form by this transform. We rely on an
-inlining transformation to replace such a variable with an expression we
-can propagate again.
-
-\starttrans
-x = E
-~
-x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
---------------------------------------------- \lam{Yi} is not a local variable reference
-x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
-~
-x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
- E y0 ... yi-1 Yi yi+1 ... yn
-
-\stoptrans
-
-TODO: Example
-
-\subsection{Cast propagation / simplification}
-This transform pushes casts down into the expression as far as possible. Since
-its exact role and need is not clear yet, this transformation is not yet
-specified.
-
-\subsection{Return value simplification}
-This transformation ensures that the return value of a function is always a
-simple local variable reference.
-
-Currently implemented using lambda simplification, let simplification, and
-top simplification. Should change into something like the following, which
-works only on the result of a function instead of any subexpression. This is
-achieved by the contexts, like \lam{x = E}, though this is strictly not
-correct (you could read this as "if there is any function \lam{x} that binds
-\lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
-is bound by \lam{x}. This might need some extra notes or something).
-
-\starttrans
-x = E \lam{E} is representable
-~ \lam{E} is not a lambda abstraction
-E \lam{E} is not a let expression
---------------------------- \lam{E} is not a local variable reference
-let x = E in x
-\stoptrans
-
-\starttrans
-x = λv0 ... λvn.E
-~ \lam{E} is representable
-E \lam{E} is not a let expression
---------------------------- \lam{E} is not a local variable reference
-let x = E in x
-\stoptrans
-
-\starttrans
-x = λv0 ... λvn.let ... in E
-~ \lam{E} is representable
-E \lam{E} is not a local variable reference
----------------------------
-let x = E in x
-\stoptrans
-
-\startbuffer[from]
-x = add 1 2
-\stopbuffer
-
-\startbuffer[to]
-x = let x = add 1 2 in x
-\stopbuffer
-
-\transexample{Return value simplification}{from}{to}
+ ) 1
+ \stopbuffer
+
+ \startbuffer[to]
+ case x of
+ True -> id 1
+ False -> neg 1
+ \stopbuffer
+
+ \transexample{Application propagation for a case expression}{from}{to}
+
+ \subsubsection{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 can simply assume all lets are
+ recursive.
+
+ \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 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).
+
+ \starttrans
+ letrec
+ \vdots
+ x = (letrec bindings in M)
+ \vdots
+ in
+ N
+ ------------------------------------------
+ letrec
+ \vdots
+ bindings
+ x = M
+ \vdots
+ in
+ N
+ \stoptrans
+
+ \startbuffer[from]
+ letrec
+ a = letrec
+ x = 1
+ y = 2
+ in
+ x + y
+ in
+ a
+ \stopbuffer
+ \startbuffer[to]
+ letrec
+ x = 1
+ y = 2
+ a = x + y
+ in
+ a
+ \stopbuffer
+
+ \transexample{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.
+
+ Currently implemented using lambda simplification, let simplification, and
+ top simplification. Should change into something like the following, which
+ works only on the result of a function instead of any subexpression. This is
+ achieved by the contexts, like \lam{x = E}, though this is strictly not
+ correct (you could read this as "if there is any function \lam{x} that binds
+ \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
+ is bound by \lam{x}. This might need some extra notes or something).
+
+ Note that the return value is not simplified if its not representable.
+ Otherwise, this would cause a direct loop with the inlining of
+ unrepresentable bindings, of course. If the return value is not
+ representable because it has a function type, η-abstraction 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 representable anyway!
+
+ \starttrans
+ x = E \lam{E} is representable
+ ~ \lam{E} is not a lambda abstraction
+ E \lam{E} is not a let expression
+ --------------------------- \lam{E} is not a local variable reference
+ letrec x = E in x
+ \stoptrans
+
+ \starttrans
+ x = λv0 ... λvn.E
+ ~ \lam{E} is representable
+ E \lam{E} is not a let expression
+ --------------------------- \lam{E} is not a local variable reference
+ letrec x = E in x
+ \stoptrans
+
+ \starttrans
+ x = λv0 ... λvn.let ... in E
+ ~ \lam{E} is representable
+ E \lam{E} is not a local variable reference
+ ---------------------------
+ letrec x = E in x
+ \stoptrans
+
+ \startbuffer[from]
+ x = add 1 2
+ \stopbuffer
+
+ \startbuffer[to]
+ x = letrec x = add 1 2 in x
+ \stopbuffer
+
+ \transexample{Return value simplification}{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.
+
+ These arguments cannot be preserved in the program, since we
+ cannot represent them as input or output ports in the resulting
+ \small{VHDL}. To remove them, we create a specialized version of the
+ called function with these arguments filled in. This is done by
+ the argument propagation transform.
+
+ Typically, these arguments are type and dictionary arguments that are
+ used to make functions polymorphic. By propagating these arguments, we
+ are essentially doing the same which GHC does when it specializes
+ functions: Creating multiple variants of the same function, one for
+ each type for which it is used. Other common non-representable
+ arguments are functions, e.g. when calling a higher order function
+ with another function or a lambda abstraction as an argument.
+
+ The reason for doing this is similar to the reasoning provided for
+ the inlining of non-representable let bindings above. In fact, this
+ argument propagation could be viewed as a form of cross-function
+ inlining.
+ \stopitemize
+
+ \todo{Check the following itemization.}
+
+ When looking at the arguments of a builtin function, we can divide them
+ into categories:
+
+ \startitemize
+ \item Arguments of a runtime representable type.
+
+ As we have seen with user-defined functions, these arguments can
+ always be reduced to a simple variable reference, by the
+ argument extraction transform. Performing this transform for
+ builtin functions as well, means that the translation of builtin
+ functions can be limited to signal references, instead of
+ needing to support all possible expressions.
+
+ \item Arguments of a function type.
+
+ These arguments are functions passed to higher order builtins,
+ like \lam{map} and \lam{foldl}. Since implementing these
+ functions for arbitrary function-typed expressions (\eg, lambda
+ expressions) is rather comlex, we reduce these arguments to
+ (partial applications of) global functions.
+
+ We can still support arbitrary expressions from the user code,
+ by creating a new global function containing that expression.
+ This way, we can simply replace the argument with a reference to
+ that new function. However, since the expression can contain any
+ number of free variables we also have to include partial
+ applications in our normal form.
+
+ This category of arguments is handled by the function extraction
+ transform.
+ \item Other unrepresentable arguments.
+
+ These arguments can take a few different forms:
+ \startdesc{Type arguments}
+ In the core language, type arguments can only take a single
+ form: A type wrapped in the Type constructor. Also, there is
+ nothing that can be done with type expressions, except for
+ applying functions to them, so we can simply leave type
+ arguments as they are.
+ \stopdesc
+ \startdesc{Dictionary arguments}
+ In the core language, dictionary arguments are used to find
+ operations operating on one of the type arguments (mostly for
+ finding class methods). Since we will not actually evaluatie
+ the function body for builtin functions and can generate
+ code for builtin functions by just looking at the type
+ arguments, these arguments can be ignored and left as they
+ are.
+ \stopdesc
+ \startdesc{Type level arguments}
+ Sometimes, we want to pass a value to a builtin function, but
+ we need to know the value at compile time. Additionally, the
+ value has an impact on the type of the function. This is
+ encoded using type-level values, where the actual value of the
+ argument is not important, but the type encodes some integer,
+ for example. Since the value is not important, the actual form
+ of the expression does not matter either and we can leave
+ these arguments as they are.
+ \stopdesc
+ \startdesc{Other arguments}
+ Technically, there is still a wide array of arguments that can
+ be passed, but does not fall into any of the above categories.
+ However, none of the supported builtin functions requires such
+ an argument. This leaves use with passing unsupported types to
+ a function, such as calling \lam{head} on a list of functions.
+
+ In these cases, it would be impossible to generate hardware
+ for such a function call anyway, so we can ignore these
+ arguments.
+
+ The only way to generate hardware for builtin functions with
+ arguments like these, is to expand the function call into an
+ equivalent core expression (\eg, expand map into a series of
+ function applications). But for now, we choose to simply not
+ support expressions like these.
+ \stopdesc
+
+ From the above, we can conclude that we can simply ignore these
+ other unrepresentable arguments and focus on the first two
+ categories instead.
+ \stopitemize
+
+ \subsubsection{Argument simplification}
+ This transform deals with arguments to functions that
+ are of a runtime representable type. It ensures that they will all become
+ references to global variables, or local signals in the resulting \small{VHDL}.
+
+ \todo{It seems we can map an expression to a port, not only a signal.}
+ Perhaps this makes this transformation not needed?
+ \todo{Say something about dataconstructors (without arguments, like True
+ or False), which are variable references of a runtime representable
+ type, but do not result in a signal.}
+
+ 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.
+
+ \starttrans
+ M N
+ -------------------- \lam{N} is of a representable type
+ letrec x = N in M x \lam{N} is not a local variable reference
+ \stoptrans
+
+ \startbuffer[from]
+ add (add a 1) 1
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec x = add a 1 in add x 1
+ \stopbuffer
+
+ \transexample{Argument extraction}{from}{to}
+
+ \subsubsection{Function extraction}
+ This transform deals with function-typed arguments to builtin functions.
+ Since these arguments cannot be propagated, we choose to extract them
+ into a new global function instead.
+
+ 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 builtin 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) builtin function.
+ --------------------- \lam{f0 ... fn} = 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]
+ map (λa . add a b) xs
+
+ map (add b) ys
+ \stopbuffer
+
+ \startbuffer[to]
+ map (x0 b) xs
+
+ map x1 ys
+ ~
+ x0 = λb.λa.add a b
+ x1 = λb.add b
+ \stopbuffer
+
+ \transexample{Function extraction}{from}{to}
+
+ Note that \lam{x0} and {x1} will still need normalization after this.
+
+ \subsubsection{Argument propagation}
+ \fxnote{This section should be generalized and describe
+ specialization, so other transformations can refer to this (since
+ specialization is really used in multiple categories).}
+
+ This transform deals with arguments to user-defined functions that are
+ not representable at runtime. This means these arguments cannot be
+ preserved in the final form and most be {\em propagated}.
+
+ Propagation means to create a specialized version of the called
+ function, with the propagated argument already filled in. As a simple
+ example, in the following program:
+
+ \startlambda
+ f = λa.λb.a + b
+ inc = λa.f a 1
+ \stoplambda
+
+ We could {\em propagate} the constant argument 1, with the following
+ result:
+
+ \startlambda
+ f' = λa.a + 1
+ inc = λa.f' a
+ \stoplambda
+
+ Special care must be taken when the to-be-propagated expression has any
+ free variables. If this is the case, the original argument should not be
+ removed alltogether, but replaced by all the free variables of the
+ expression. In this way, the original expression can still be evaluated
+ inside the new function. Also, this brings us closer to our goal: All
+ these free variables will be simple variable references.
+
+ To prevent us from propagating the same argument over and over, a simple
+ local variable reference is not propagated (since is has exactly one
+ free variable, itself, we would only replace that argument with itself).
+
+ This shows that any free local variables that are not runtime representable
+ cannot be brought into normal form by this transform. We rely on an
+ inlining transformation to replace such a variable with an expression we
+ can propagate again.
+
+ \starttrans
+ x = E
+ ~
+ x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
+ --------------------------------------------- \lam{Yi} is not a local variable reference
+ x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
+ ~
+ x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
+ E y0 ... yi-1 Yi yi+1 ... yn
+
+ \stoptrans
+
+ \todo{Example}
+
+ \subsection{Case simplification}
+ \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 E 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{Let flattening}{from}{to}
+
+
+ \subsubsection{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)
+ letrec
+ 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
+
+ \fxnote{This transformation specified like this is complicated and misses
+ conditions to prevent looping with itself. Perhaps it should be split 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]
+ 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{Extractor case simplification}{from}{to}
+
+ \subsubsection{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{Removing polymorphism}
+ Reference type-specialization (== argument propagation)
+
+ Reference polymporphic binding inlining (== non-representable binding
+ inlining).
+
+ \subsection{Defunctionalization}
+ These transformations remove most higher order expressions from our
+ program, making it completely first-order (the only exception here is for
+ arguments to builtin functions, since we can't specialize builtin
+ function. \todo{Talk more about this somewhere}
+
+ Reference higher-order-specialization (== argument propagation)
+
+ \subsubsection{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
+ letrec
+ a0 = E0
+ \vdots
+ ai = Ei
+ \vdots
+ an = En
+ in
+ M
+ -------------------------- \lam{Ei} has a non-representable type.
+ letrec
+ a0 = E0 [Ei/ai]
+ \vdots
+ ai-1 = Ei-1 [Ei/ai]
+ ai+1 = Ei+1 [Ei/ai]
+ \vdots
+ an = En [Ei/ai]
+ in
+ M[Ei/ai]
+ \stoptrans
+
+ \startbuffer[from]
+ letrec
+ a = smallInteger 10
+ inc = λb -> add b 1
+ inc' = add 1
+ x = fromInteger a
+ in
+ inc (inc' x)
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec
+ x = fromInteger (smallInteger 10)
+ in
+ (λb -> add b 1) (add 1 x)
+ \stopbuffer
+
+ \transexample{None representable binding inlining}{from}{to}
+
+
+ \section{Provable properties}
+ When looking at the system of transformations outlined above, there are a
+ number of questions that we can ask ourselves. The main question is of course:
+ \quote{Does our system work as intended?}. We can split this question into a
+ number of subquestions:
+
+ \startitemize[KR]
+ \item[q:termination] Does our system \emph{terminate}? Since our system will
+ keep running as long as transformations apply, there is an obvious risk that
+ it will keep running indefinitely. One transformation produces a result that
+ is transformed back to the original by another transformation, for example.
+ \item[q:soundness] Is our system \emph{sound}? Since our transformations
+ continuously modify the expression, there is an obvious risk that the final
+ normal form will not be equivalent to the original program: Its meaning could
+ have changed.
+ \item[q:completeness] Is our system \emph{complete}? Since we have a complex
+ system of transformations, there is an obvious risk that some expressions will
+ not end up in our intended normal form, because we forgot some transformation.
+ In other words: Does our transformation system result in our intended normal
+ form for all possible inputs?
+ \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
+ no particular order in which the transformation should be applied, there is an
+ obvious risk that different transformation orderings will result in
+ \emph{different} normal forms. They might still both be intended normal forms
+ (if our system is \emph{complete}) and describe correct hardware (if our
+ system is \emph{sound}), so this property is less important than the previous
+ three: The translator would still function properly without it.
+ \stopitemize
+
+ \subsection{Graph representation}
+ Before looking into how to prove these properties, we'll look at our
+ transformation system from a graph perspective. The nodes of the graph are
+ all possible Core expressions. The (directed) edges of the graph are
+ transformations. When a transformation α applies to an expression \lam{A} to
+ produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
+ node for \lam{B}, labeled α.
+
+ \startuseMPgraphic{TransformGraph}
+ save a, b, c, d;
+
+ % Nodes
+ newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
+ newCircle.b(btex \lam{λy. (+) 1 y} etex);
+ newCircle.c(btex \lam{(λx.(+) x) 1} etex);
+ newCircle.d(btex \lam{(+) 1} etex);
+
+ b.c = origin;
+ c.c = b.c + (4cm, 0cm);
+ a.c = midpoint(b.c, c.c) + (0cm, 4cm);
+ d.c = midpoint(b.c, c.c) - (0cm, 3cm);
+
+ % β-conversion between a and b
+ ncarc.a(a)(b) "name(bred)";
+ ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
+ ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
+ ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
+
+ % η-conversion between a and c
+ ncarc.a(a)(c) "name(ered)";
+ ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
+ ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
+ ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
+
+ % η-conversion between b and d
+ ncarc.b(b)(d) "name(ered)";
+ ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
+ ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
+ ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
+
+ % β-conversion between c and d
+ ncarc.c(c)(d) "name(bred)";
+ ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
+ ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
+ ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
+
+ % Draw objects and lines
+ drawObj(a, b, c, d);
+ \stopuseMPgraphic
+
+ \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus
+ system with β and η reduction (solid lines) and expansion (dotted lines).}
+ \boxedgraphic{TransformGraph}
+
+ Of course our graph is unbounded, since we can construct an infinite amount of
+ Core expressions. Also, there might potentially be multiple edges between two
+ given nodes (with different labels), though seems unlikely to actually happen
+ in our system.
+
+ See \in{example}[ex:TransformGraph] for the graph representation of a very
+ simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
+ y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
+ transformation system consists of β-reduction and η-reduction (solid edges) or
+ β-reduction and η-reduction (dotted edges).
+
+ \todo{Define β-reduction and η-reduction?}
+
+ Note that the normal form of such a system consists of the set of nodes
+ (expressions) without outgoing edges, since those are the expression to which
+ no transformation applies anymore. We call this set of nodes the \emph{normal
+ set}.
+
+ From such a graph, we can derive some properties easily:
+ \startitemize[KR]
+ \item A system will \emph{terminate} if there is no path of infinite length
+ in the graph (this includes cycles).
+ \item Soundness is not easily represented in the graph.
+ \item A system is \emph{complete} if all of the nodes in the normal set have
+ the intended normal form. The inverse (that all of the nodes outside of
+ the normal set are \emph{not} in the intended normal form) is not
+ strictly required.
+ \item A system is deterministic if all paths from a node, which end in a node
+ in the normal set, end at the same node.
+ \stopitemize
+
+ When looking at the \in{example}[ex:TransformGraph], we see that the system
+ terminates for both the reduction and expansion systems (but note that, for
+ expansion, this is only true because we've limited the possible expressions!
+ In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y)
+ 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1}
+ etc.)
+
+ If we would consider the system with both expansion and reduction, there would
+ no longer be termination, since there would be cycles all over the place.
+
+ The reduction and expansion systems have a normal set of containing just
+ \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
+ either system end up in these normal forms, both systems are \emph{complete}.
+ Also, since there is only one normal form, it must obviously be
+ \emph{deterministic} as well.
+
+ \subsection{Termination}
+ Approach: Counting.
+
+ Church-Rosser?
+
+ \subsection{Soundness}
+ Needs formal definition of semantics.
+ Prove for each transformation seperately, implies soundness of the system.
+
+ \subsection{Completeness}
+ Show that any transformation applies to every Core expression that is not
+ in normal form. To prove: no transformation applies => in intended form.
+ Show the reverse: Not in intended form => transformation applies.
+
+ \subsection{Determinism}
+ How to prove this?
projects / matthijs / master-project / report.git / blobdiff