\chapter[chap:normalization]{Normalization}
% A helper to print a single example in the half the page width. The example
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\define[1]\example{
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\define[4]\transexample{
\placeexample[here][ex:trans:#1]{#2}
\startcombination[2*1]
{\example{#3}}{Original program}
{\example{#4}}{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 \VHDL, and describe how the
\VHDL we want to generate should look like.}
\section{Normal form}
\todo{Refresh or refer to distinct hardware per application principle}
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 and constants, not complete expressions). To make the \small{VHDL}
generation easy, a separate binder must be bound to ever application or
other expression.
\stopitemize
\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 outer level.
This means that the body of the inner lambda abstraction is never a
function, but always a plain value.
As the body of the inner lambda abstraction, we see a single (recursive)
let expression, that binds two variables (\lam{mul} and \lam{sum}). These
variables will be signals in the 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.
\todo{Add generated VHDL}
\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 a
multiplier and a subtractor.}
\startcombination[2*1]
{\typebufferlam{MulSum}}{Core description in normal form.}
{\boxedgraphic{MulSum}}{The architecture described by the normal form.}
\stopcombination
The previous example described composing an architecture by calling other
functions (operators), resulting in a simple architecture with components and
connections. There is of course also some mechanism for choice in the normal
form. In a normal Core program, the \emph{case} expression can be used in a
few different ways to describe choice. In normal form, this is limited to a
very specific form.
\in{Example}[ex:AddSubAlu] shows an example describing a
simple \small{ALU}, which chooses between two operations based on an opcode
bit. The main structure is similar to \in{example}[ex:MulSum], but this
time the \lam{res} variable is bound to a case expression. This case
expression scrutinizes the variable \lam{opcode} (and scrutinizing more
complex expressions is not supported). The case expression can select a
different variable based on the constructor of \lam{opcode}.
\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
instead put some gates in front of the register's clock input, but we want
to show the architecture as close to the description as possible.
As you can see from the previous examples, the generation of the final
architecture from the normal form is straightforward. In each of the
examples, there is a direct match between the normal form structure,
the generated VHDL and the architecture shown in the images.
\startbuffer[NormalComplete]
regbank :: Bit
-> Word
-> State (Word, Word)
-> (State (Word, Word), Word)
-- All arguments are an inital lambda (address, data, packed state)
regbank = λa.λd.λsp.
-- There are nested let expressions at top level
let
-- Unpack the state by coercion (\eg, cast from
-- State (Word, Word) to (Word, Word))
s = sp :: (Word, Word)
-- Extract both registers from the state
r1 = case s of (a, b) -> a
r2 = case s of (a, b) -> b
-- Calling some other user-defined function.
d' = foo d
-- Conditional connections
out = case a of
High -> r1
Low -> r2
r1' = case a of
High -> d'
Low -> r1
r2' = case a of
High -> r2
Low -> d'
-- Packing a tuple
s' = (,) r1' r2'
-- pack the state by coercion (\eg, cast from
-- (Word, Word) to State (Word, Word))
sp' = s' :: State (Word, Word)
-- Pack our return value
res = (,) sp' out
in
-- The actual result
res
\stopbuffer
\startuseMPgraphic{NormalComplete}
save a, d, r, foo, muxr, muxout, out;
% I/O ports
newCircle.a(btex \lam{a} etex) "framed(false)";
newCircle.d(btex \lam{d} etex) "framed(false)";
newCircle.out(btex \lam{out} etex) "framed(false)";
% Components
%newCircle.add(btex + etex);
newBox.foo(btex \lam{foo} etex);
newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
newMux.muxr1;
% Reflect over the vertical axis
reflectObj(muxr1)((0,0), (0,1));
newMux.muxr2;
newMux.muxout;
rotateObj(muxout)(-90);
d.c = foo.c + (0cm, 1.5cm);
a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
muxr1.c = r1.c + (0cm, 2cm);
muxr2.c = r2.c + (0cm, 2cm);
r2.c = r1.c + (4cm, 0cm);
r1.c = origin;
muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
out.c = muxout.c - (0cm, 1.5cm);
% % Draw objects and lines
drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
ncline(d)(foo);
nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
% Connect port a
nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
ncline(muxout)(out) "posA(out)";
\stopuseMPgraphic
\todo{Don't split registers in this image?}
\placeexample[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.
\defref{intended normal form definition}
\todo{Fix indentation}
\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 variable referenc in the body of
the recursive 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} expression) or call a builtin function (\eg \lam{+} or
\lam{map}). For these, a hardcoded \small{VHDL} translation is
available.
\section[sec:normalization:transformation]{Transformation notation}
To be able to concisely present transformations, we use a specific format
for them. It is a simple format, similar to one used in logic reasoning.
Such a transformation description looks like the following.
\starttrans
~
--------------------------
~
\stoptrans
This format desribes a transformation that applies to \lam{} and transforms it into \lam{}, 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{} The expression pattern that will be matched
against (subexpressions of) the expression to be transformed. We call this a
pattern, because it can contain \emph{placeholders} (variables), which match
any expression or binder. Any such placeholder is said to be \emph{bound} to
the expression it matches. It is convention to use an uppercase letter (\eg
\lam{M} or \lam{E}) to refer to any expression (including a simple variable
reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
(references to) binders.
For example, the pattern \lam{a + B} will match the expression
\lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
\lam{(2 * w)}), but not \lam{(2 * w) + v}.
\stopdesc
\startdesc{}
These are extra conditions on the expression that is matched. These
conditions can be used to further limit the cases in which the
transformation applies, commonly to prevent a transformation from
causing a loop with itself or another transformation.
Only if these conditions are \emph{all} true, the transformation
applies.
\stopdesc
\startdesc{}
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 = }.
Typically, the binder is some placeholder bound in the \lam{}, while the pattern contains some placeholders that are used in
the \lam{transformed expression}.
Only if a top level binder exists that matches each binder and pattern,
the transformation applies.
\stopdesc
\startdesc{}
This is the expression template that is the result of the transformation. If, looking
at the above three items, the transformation applies, the \lam{} is completely replaced with the \lam{}.
We call this a template, because it can contain placeholders, referring to
any placeholder bound by the \lam{} or the
\lam{}. 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{}
These are templates for new functions to add to the context. This is a way
to have a transformation create new top level functions.
Each addition has the form \lam{binder = template}. As above, any
placeholder in the addition is replaced with the value bound to it, and any
binder placeholder that has no value bound to it yet will be bound to (and
replaced with) a fresh binder.
\stopdesc
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
η-abstraction is a well known transformation from lambda calculus. What
this transformation does, is take any expression that has a function type
and turn it into a lambda expression (giving an explicit name to the
argument). There are some extra conditions that ensure that this
transformation does not apply infinitely (which are not necessarily part
of the conventional definition of η-abstraction).
Consider the following function, which is a fairly obvious way to specify a
simple ALU (Note that \in{example}[ex:AddSubAlu] shows the normal form of this
function). The parentheses around the \lam{+} and \lam{-} operators are
commonly used in Haskell to show that the operators are used as normal
functions, instead of \emph{infix} operators (\eg, the operators appear
before their arguments, instead of in between).
\startlambda
alu :: Bit -> Word -> Word -> Word
alu = λopcode. case opcode of
Low -> (+)
High -> (-)
\stoplambda
There are a few subexpressions in this function to which we could possibly
apply the transformation. Since the pattern of the transformation is only
the placeholder \lam{E}, any expression will match that. Whether the
transformation applies to an expression is thus solely decided by the
conditions to the right of the transformation.
We will look at each expression in the function in a top down manner. The
first expression is the entire expression the function is bound to.
\startlambda
λopcode. case opcode of
Low -> (+)
High -> (-)
\stoplambda
As said, the expression pattern matches this. The type of this expression is
\lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
Since this expression is at top level, it does not occur at a function
position of an application. However, The expression is a lambda abstraction,
so this transformation does not apply.
The next expression we could apply this transformation to, is the body of
the lambda abstraction:
\startlambda
case opcode of
Low -> (+)
High -> (-)
\stoplambda
The type of this expression is \lam{Word -> Word -> Word}, which again
matches \lam{a -> b}. The expression is the body of a lambda expression, so
it does not occur at a function position of an application. Finally, the
expression is not a lambda abstraction but a case expression, so all the
conditions match. There are no context conditions to match, so the
transformation applies.
By now, the placeholder \lam{E} is bound to the entire expression. The
placeholder \lam{x}, which occurs in the replacement template, is not bound
yet, so we need to generate a fresh binder for that. Let'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
lambda 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 (which makes the second condition false). In the same
way the transformation does not apply to both components of this
expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
we'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.
The first alternative is \lam{(+)}. This expression has a function type
(the operator still needs two arguments). It does not occur in function
position of an application and it is not a lambda expression, so the
transformation applies.
We look at the \lam{} pattern, which is \lam{E}.
This means we bind \lam{E} to \lam{(+)}. We then replace the expression
with the \lam{}, replacing all occurences of
\lam{E} with \lam{(+)}. In the \lam{}, the This gives us the replacement expression:
\lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
applies the addition operator to \lam{x}).
The complete function then becomes:
\startlambda
(case opcode of Low -> λa1.(+) a1; High -> (-)) a
\stoplambda
Now the transformation no longer applies to the complete first alternative
(since it is a lambda expression). It does not apply to the addition
operator again, since it is now in function position in an application. It
does, however, apply to the application of the addition operator, since
that is neither a lambda expression nor does it occur in function
position. This means after one more application of the transformation, the
function becomes:
\startlambda
(case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
\stoplambda
The other alternative is left as an exercise to the reader. The final
function, after applying η-abstraction until it does no longer apply is:
\startlambda
alu :: Bit -> Word -> Word -> Word
alu = λopcode.λa.b. (case opcode of
Low -> λa1.λb1 (+) a1 b1
High -> λa2.λb2 (-) a2 b2) a b
\stoplambda
\subsection{Transformation application}
In this chapter we define a number of transformations, but how will we apply
these? As stated before, our normal form is reached as soon as no
transformation applies anymore. This means our application strategy is to
simply apply any transformation that applies, and continuing to do that with
the result of each transformation.
In particular, we define no particular order of transformations. Since
transformation order should not influence the resulting normal form,
\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 body. 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{Concepts}
A \emph{global variable} is any variable (binder) that is bound at the
top level of a program, or an external module. A \emph{local variable} is any
other variable (\eg, variables local to a function, which can be bound by
lambda abstractions, let expressions and pattern matches of case
alternatives). 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. Values 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{Predicates}
Here, we define a number of predicates 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[sec:normalization:uniq]{Binder uniqueness}
A common problem in transformation systems, is binder uniqueness. When not
considering this problem, it is easy to create transformations that mix up
bindings and cause name collisions. Take for example, the following core
expression:
\startlambda
(λa.λb.λc. a * b * c) x c
\stoplambda
By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
we can simplify this expression to:
\startlambda
(λb.λc. x * b * c) c
\stoplambda
Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
binder. No harm done here. But note that we see multiple occurences of the
\lam{c} binder. The first is a binding occurence, to which the second refers.
The last, however refers to \emph{another} instance of \lam{c}, which is
bound somewhere outside of this expression. Now, if we would apply beta
reduction without taking heed of binder uniqueness, we would get:
\startlambda
λc. x * c * c
\stoplambda
This is obviously not what was supposed to happen! The root of this problem is
the reuse of binders: Identical binders can be bound in different 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, the following points can be
observed.
\startitemize
\item Deshadowing does not guarantee overall uniqueness. For example, the
following (slightly contrived) expression shows the identifier \lam{x} bound in
two seperate places (and to different values), even though no shadowing
occurs.
\startlambda
(let x = 1 in x) + (let x = 2 in x)
\stoplambda
\item In our normal form (and the resulting \small{VHDL}), all binders
(signals) within the same function (entity) will end up in the same
scope. To allow this, all binders within the same function should be
unique.
\item When we know that all binders in an expression are unique, moving around
or removing a subexpression will never cause any binder conflicts. If we have
some way to generate fresh binders, introducing new subexpressions will not
cause any problems either. The only way to cause conflicts is thus to
duplicate an existing subexpression.
\stopitemize
Given the above, our prototype maintains a unique binder invariant. This
means that in any given moment during normalization, all binders \emph{within
a single function} must be unique. To achieve this, we apply the following
technique.
\todo{Define fresh binders and unique supplies}
\startitemize
\item Before starting normalization, all binders in the function are made
unique. This is done by generating a fresh binder for every binder used. This
also replaces binders that did not cause any conflict, but it does ensure that
all binders within the function are generated by the same unique supply.
\refdef{fresh binder}
\item Whenever a new binder must be generated, we generate a fresh binder that
is guaranteed to be different from \emph{all binders generated so far}. This
can thus never introduce duplication and will maintain the invariant.
\item Whenever (a part of) an expression is duplicated (for example when
inlining), all binders in the expression are replaced with fresh binders
(using the same method as at the start of normalization). These fresh binders
can never introduce duplication, so this will maintain the invariant.
\item Whenever we move part of an expression around within the function, there
is no need to do anything special. There is obviously no way to introduce
duplication by moving expressions around. Since we know that each of the
binders is already unique, there is no way to introduce (incorrect) shadowing
either.
\stopitemize
\section{Transform passes}
In this section we describe the actual transforms.
Each transformation will be described informally first, explaining
the need for and goal of the transformation. Then, we will formally define
the transformation using the syntax introduced in
\in{section}[sec:normalization:transformation].
\subsection{General cleanup}
These transformations are general cleanup transformations, that aim to
make expressions simpler. These transformations usually clean up the
mess left behind by other transformations or clean up expressions to
expose new transformation opportunities for other transformations.
Most of these transformations are standard optimizations in other
compilers as well. However, in our compiler, most of these are not just
optimizations, but they are required to get our program into intended
normal form.
\subsubsection[sec:normalization:beta]{β-reduction}
β-reduction is a well known transformation from lambda calculus, where it is
the main reduction step. It reduces applications of lambda abstractions,
removing both the lambda abstraction and the application.
In our transformation system, this step helps to remove unwanted lambda
abstractions (basically all but the ones at the top level). Other
transformations (application propagation, non-representable inlining) make
sure that most lambda abstractions will eventually be reducable by
β-reduction.
\starttrans
(λx.E) M
-----------------
E[x=>M]
\stoptrans
% And an example
\startbuffer[from]
(λa. 2 * a) (2 * b)
\stopbuffer
\startbuffer[to]
2 * (2 * b)
\stopbuffer
\transexample{beta}{β-reduction}{from}{to}
\subsubsection{Empty let removal}
This transformation is simple: It removes recursive lets that have no bindings
(which usually occurs when unused let binding removal removes the last
binding from it).
Note that there is no need to define this transformation for
non-recursive lets, since they always contain exactly one binding.
\starttrans
letrec in M
--------------
M
\stoptrans
\todo{Example}
\subsubsection{Simple let binding removal}
This transformation inlines simple let bindings, that bind some
binder to some other binder instead of a more complex expression (\ie
a = b).
This transformation is not needed to get an expression into intended
normal form (since these bindings are part of the intended normal
form), but makes the resulting \small{VHDL} a lot shorter.
\starttrans
letrec
a0 = E0
\vdots
ai = b
\vdots
an = En
in
M
----------------------------- \lam{b} is a variable reference
letrec \lam{ai} ≠ \lam{b}
a0 = E0 [ai=>b]
\vdots
ai-1 = Ei-1 [ai=>b]
ai+1 = Ei+1 [ai=>b]
\vdots
an = En [ai=>b]
in
M[ai=>b]
\stoptrans
\todo{example}
\subsubsection{Unused let binding removal}
This transformation removes let bindings that are never used.
Occasionally, \GHC's desugarer introduces some unused let bindings.
This normalization pass should really be unneeded to get into intended 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.
\todo{Don't use old-style numerals in transformations}
\starttrans
letrec
a0 = E0
\vdots
ai = Ei
\vdots
an = En
in
M \lam{ai} does not occur free in \lam{M}
---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej})
letrec
a0 = E0
\vdots
ai-1 = Ei-1
ai+1 = Ei+1
\vdots
an = En
in
M
\stoptrans
\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 intended 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{toplevelinline}{Top level binding inlining}{from}{to}
\in{Example}[ex:trans:toplevelinline] shows a typical application of
the addition operator generated by \GHC. The type and dictionary
arguments used here are described in
\in{Section}[section:prototype:polymorphism].
Without this transformation, there would be a \lam{(+)} entity
in the \VHDL which would just add its inputs. This generates a
lot of overhead in the \VHDL, which is particularly annoying
when browsing the generated RTL schematic (especially since most
non-alphanumerics, like all characters in \lam{(+)}, are not
allowed in \VHDL architecture names\footnote{Technically, it is
allowed to use non-alphanumerics when using extended
identifiers, but it seems that none of the tooling likes
extended identifiers in filenames, so it effectively 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 or current state), 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{eta}{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).
Since all binders in our expression are unique (see
\in{section}[sec:normalization:uniq]), there is no risk that we will
introduce unintended shadowing by moving an expression into a lower
scope. Also, since only move expression into smaller scopes (down into
our expression), there is no risk of moving a variable reference out
of the scope in which it is defined.
\starttrans
(letrec binds in E) M
------------------------
letrec binds in E M
\stoptrans
% And an example
\startbuffer[from]
( letrec
val = 1
in
add val
) 3
\stopbuffer
\startbuffer[to]
letrec
val = 1
in
add val 3
\stopbuffer
\transexample{appproplet}{Application propagation for a let expression}{from}{to}
\starttrans
(case x of
p1 -> E1
\vdots
pn -> En) M
-----------------
case x of
p1 -> E1 M
\vdots
pn -> En M
\stoptrans
% And an example
\startbuffer[from]
( case x of
True -> id
False -> neg
) 1
\stopbuffer
\startbuffer[to]
case x of
True -> id 1
False -> neg 1
\stopbuffer
\transexample{apppropcase}{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 the outer let. Eventually, this will
cause all let bindings to appear in the same scope.
This transformation only applies to recursive lets, since all
non-recursive lets will be made recursive (see
\in{section}[sec:normalization:letrecurse]).
Since we are joining two scopes together, there is no risk of moving a
variable reference out of the scope where it is defined.
\starttrans
letrec
a0 = E0
\vdots
ai = (letrec bindings in M)
\vdots
an = En
in
N
------------------------------------------
letrec
a0 = E0
\vdots
ai = M
\vdots
an = En
bindings
in
N
\stoptrans
\startbuffer[from]
letrec
a = 1
b = letrec
x = a
y = c
in
x + y
c = 2
in
b
\stopbuffer
\startbuffer[to]
letrec
a = 1
b = x + y
c = 2
x = a
y = c
in
b
\stopbuffer
\transexample{letflat}{Let flattening}{from}{to}
\subsubsection{Return value simplification}
This transformation ensures that the return value of a function is always a
simple local variable reference.
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. 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 translatable 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{retvalsimpl}{Return value simplification}{from}{to}
\todo{More examples}
\subsection{Argument simplification}
The transforms in this section deal with simplifying application
arguments into normal form. The goal here is to:
\todo{This section should only talk about representable arguments. Non
representable arguments are treated by specialization.}
\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. \todo{Move this
bullet to specialization}
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{Move this itemization into a new section about builtin functions}
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}, which is required due to limitations in the component
instantiation code in \VHDL (one can only assign a signal or constant
to an input port). By ensuring that all arguments are always simple
variable references, we always have a signal available to assign to
input ports.
\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.
Note that a reference to a \emph{global variable} (like a top level
function without arguments, but also an argumentless dataconstructors
like \lam{True}) is also simplified. Only local variables generate
signals in the resulting architecture.
\refdef{representable}
\starttrans
M N
-------------------- \lam{N} is representable
letrec x = N in M x \lam{N} is not a local variable reference
\stoptrans
\refdef{local variable}
\startbuffer[from]
add (add a 1) 1
\stopbuffer
\startbuffer[to]
letrec x = add a 1 in add x 1
\stopbuffer
\transexample{argextract}{Argument extraction}{from}{to}
\subsubsection{Function extraction}
\todo{Move to section about builtin functions}
This transform deals with function-typed arguments to builtin
functions. Since builtin functions cannot be specialized to remove
the arguments, we choose to extract these arguments into a new global
function instead. This greatly simplifies the translation rules needed
for builtin functions. \todo{Should we talk about these? Reference
Christiaan?}
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} are all free local variables of \lam{N}
M (x f0 ... fn) \lam{N :: a -> b}
~ \lam{N} is not a (partial application of) a top level function
x = λf0 ... λfn.N
\stoptrans
\todo{Split this example}
\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{funextract}{Function extraction}{from}{to}
Note that \lam{x0} and {x1} will still need normalization after this.
\subsubsection{Argument propagation}
\todo{Rename this section to specialization and move it into a
separate section}
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 completely, 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} are all free local vars of \lam{Yi}
~
x' = λy0 ... λyi-1. λf0 ... λfm. λyi+1 ... λyn .
E y0 ... yi-1 Yi yi+1 ... yn
\stoptrans
\todo{Describe what the formal specification means}
\todo{Note that we don't change the sepcialised function body, only
wrap it}
\todo{Example}
\subsection{Case normalisation}
\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{letflat}{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
\defref{wild binder}
\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 E of C0 v0,0 .. v0,m -> v0,0
\vdots
v0,m = case E of C0 v0,0 .. v0,m -> v0,m
\vdots
vn,m = case E of Cn vn,0 .. vn,m -> vn,m
x0 = E0
\vdots
xn = En
in
case E of
C0 w0,0 ... w0,m -> x0
\vdots
Cn wn,0 ... wn,m -> xn
\stoptrans
\todo{Check the subscripts of this transformation}
Note that this transformation applies to case statements with any
scrutinee. If the scrutinee is a complex expression, this might result
in duplicate hardware. An extra condition to only apply this
transformation when the scrutinee is already simple (effectively
causing this transformation to be only applied after the scrutinee
simplification transformation) might be in order.
\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{selcasesimpl}{Selector case simplification}{from}{to}
\startbuffer[from]
case a of
(,) b c -> add b c
\stopbuffer
\startbuffer[to]
letrec
b = case a of (,) b c -> b
c = case a of (,) b c -> c
x0 = add b c
in
case a of
(,) w0 w1 -> x0
\stopbuffer
\transexample{excasesimpl}{Extractor case simplification}{from}{to}
\refdef{selector case}
In \in{example}[ex:trans:excasesimpl] the case expression is expanded
into multiple case expressions, including a pretty useless expression
(that is neither a selector or extractor case). This case can be
removed by the Case removal transformation in
\in{section}[sec:transformation:caseremoval].
\subsubsection[sec:transformation:caseremoval]{Case removal}
This transform removes any case 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{caserem}{Case removal}{from}{to}
\todo{Move these two sections somewhere? Perhaps not?}
\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}
\todo{Move this section into a new section (together with
specialization?)}
This transform inlines let bindings that are bound to a
non-representable value. 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{Expand on this. This/these paragraph(s)
should probably become a separate discussion somewhere else}
\todo{Can this duplicate work?}
\starttrans
letrec
a0 = E0
\vdots
ai = Ei
\vdots
an = En
in
M
-------------------------- \lam{Ei} has a non-representable type.
letrec
a0 = E0 [ai=>Ei] \vdots
ai-1 = Ei-1 [ai=>Ei]
ai+1 = Ei+1 [ai=>Ei]
\vdots
an = En [ai=>Ei]
in
M[ai=>Ei]
\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{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
\section[sec:normalization:properties]{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. This typically happens when one
transformation produces a result that is transformed back to the original
by another transformation, or when one or more transformations keep
expanding some expression.
\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
Unfortunately, the final transformation system has only been
developed in the final part of the research, leaving no more time
for verifying these properties. In fact, it is likely that the
current transformation system still violates some of these
properties in some cases and should be improved (or extra conditions
on the input hardware descriptions should be formulated).
This is most likely the case with the completeness and determinism
properties, perhaps als the termination property. The soundness
property probably holds, since it is easier to manually verify (each
transformation can be reviewed separately).
Even though no complete proofs have been made, some ideas for
possible proof strategies are shown below.
\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 lambda 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-expansion and η-expansion (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}. The set of nodes containing expressions in intended normal
form \refdef{intended normal form} is called the \emph{intended
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, but can also happen without 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. In other words, our normal set must be a
subset of the intended normal form, but they do not need to be
the same set.
form.
\item A system is deterministic if all paths starting at a particular
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 either, 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 node in the normal set, it must obviously be
\emph{deterministic} as well.
\todo{Add content to these sections}
\subsection{Termination}
In general, proving termination of an arbitrary program is a very
hard problem. \todo{Ref about arbitrary termination} Fortunately,
we only have to prove termination for our specific transformation
system.
A common approach for these kinds of proofs is to associate a
measure with each possible expression in our system. If we can
show that each transformation strictly decreases this measure
(\ie, the expression transformed to has a lower measure than the
expression transformed from). \todo{ref about measure-based
termination proofs / analysis}
A good measure for a system consisting of just β-reduction would
be the number of lambda expressions in the expression. Since every
application of β-reduction removes a lambda abstraction (and there
is always a bounded number of lambda abstractions in every
expression) we can easily see that a transformation system with
just β-reduction will always terminate.
For our complete system, this measure would be fairly complex
(probably the sum of a lot of things). Since the (conditions on)
our transformations are pretty complex, we would need to include
both simple things like the number of let expressions as well as
more complex things like the number of case expressions that are
not yet in normal form.
No real attempt has been made at finding a suitable measure for
our system yet.
\subsection{Soundness}
Soundness is a property that can be proven for each transformation
separately. Since our system only runs separate transformations
sequentially, if each of our transformations leaves the
\emph{meaning} of the expression unchanged, then the entire system
will of course leave the meaning unchanged and is thus
\emph{sound}.
The current prototype has only been verified in an ad-hoc fashion
by inspecting (the code for) each transformation. A more formal
verification would be more appropriate.
To be able to formally show that each transformation properly
preserves the meaning of every expression, we require an exact
definition of the \emph{meaning} of every expression, so we can
compare them. Currently there seems to be no formal definition of
the meaning or semantics of \GHC's core language, only informal
descriptions are available.
It should be possible to have a single formal definition of
meaning for Core for both normal Core compilation by \GHC and for
our compilation to \VHDL. The main difference seems to be that in
hardware every expression is always evaluated, while in software
it is only evaluated if needed, but it should be possible to
assign a meaning to core expressions that assumes neither.
Since each of the transformations can be applied to any
subexpression as well, there is a constraint on our meaning
definition: The meaning of an expression should depend only on the
meaning of subexpressions, not on the expressions themselves. For
example, the meaning of the application in \lam{f (let x = 4 in
x)} should be the same as the meaning of the application in \lam{f
4}, since the argument subexpression has the same meaning (though
the actual expression is different).
\subsection{Completeness}
Proving completeness is probably not hard, but it could be a lot
of work. We have seen above that to prove completeness, we must
show that the normal set of our graph representation is a subset
of the intended normal set.
However, it is hard to systematically generate or reason about the
normal set, since it is defined as any nodes to which no
transformation applies. To determine this set, each transformation
must be considered and when a transformation is added, the entire
set should be re-evaluated. This means it is hard to show that
each node in the normal set is also in the intended normal set.
Reasoning about our intended normal set is easier, since we know
how to generate it from its definition. \refdef{intended normal
form definition}.
Fortunately, we can also prove the complement (which is
equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
\subseteq \overline{A}$): Show that the set of nodes not in
intended normal form is a subset of the set of nodes not in normal
form. In other words, show that for every expression that is not
in intended normal form, that there is at least one transformation
that applies to it (since that means it is not in normal form
either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
\rightarrow x \in C)$).
By systematically reviewing the entire Core language definition
along with the intended normal form definition (both of which have
a similar structure), it should be possible to identify all
possible (sets of) core expressions that are not in intended
normal form and identify a transformation that applies to it.
This approach is especially useful for proving completeness of our
system, since if expressions exist to which none of the
transformations apply (\ie if the system is not yet complete), it
is immediately clear which expressions these are and adding
(or modifying) transformations to fix this should be relatively
easy.
As observed above, applying this approach is a lot of work, since
we need to check every (set of) transformation(s) separately.
\todo{Perhaps do a few steps of the proofs as proof-of-concept}
% vim: set sw=2 sts=2 expandtab: