\chapter[chap:normalization]{Normalization}
% A helper to print a single example in the half the page width. The example
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% 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]
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\define[3]\transexample{
\placeexample[here]{#1}
\startcombination[2*1]
{\example{#2}}{Original program}
{\example{#3}}{Transformed program}
\stopcombination
}
The first step in the core to \small{VHDL} translation process, is normalization. We
aim to bring the core description into a simpler form, which we can
subsequently translate into \small{VHDL} easily. This normal form is needed because
the full core language is more expressive than \small{VHDL} in some areas and because
core can describe expressions that do not have a direct hardware
interpretation.
TODO: Describe core properties not supported in \small{VHDL}, and describe how the
\small{VHDL} we want to generate should look like.
\section{Normal form}
The transformations described here have a well-defined goal: To bring the
program in a well-defined form that is directly translatable to hardware,
while fully preserving the semantics of the program. We refer to this form as
the \emph{normal form} of the program. The formal definition of this normal
form is quite simple:
\placedefinition{}{A program is in \emph{normal form} if none of the
transformations from this chapter apply.}
Of course, this is an \quote{easy} definition of the normal form, since our
program will end up in normal form automatically. The more interesting part is
to see if this normal form actually has the properties we would like it to
have.
But, before getting into more definitions and details about this normal form,
let's try to get a feeling for it first. The easiest way to do this is by
describing the things we want to not have in a normal form.
\startitemize
\item Any \emph{polymorphism} must be removed. When laying down hardware, we
can't generate any signals that can have multiple types. All types must be
completely known to generate hardware.
\item Any \emph{higher order} constructions must be removed. We can't
generate a hardware signal that contains a function, so all values,
arguments and returns values used must be first order.
\item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
description, every signal is in a single scope. Also, full expressions are
not supported everywhere (in particular port maps can only map signal names,
not expressions). To make the \small{VHDL} generation easy, all values must be bound
on the \quote{top level}.
\stopitemize
TODO: Intermezzo: functions vs plain values
A very simple example of a program in normal form is given in
\in{example}[ex:MulSum]. As you can see, all arguments to the function (which
will become input ports in the final hardware) are at the top. This means that
the body of the final lambda abstraction is never a function, but always a
plain value.
After the lambda abstractions, we see a single let expression, that binds two
variables (\lam{mul} and \lam{sum}). These variables will be signals in the
final hardware, bound to the output port of the \lam{*} and \lam{+}
components.
The final line (the \quote{return value} of the function) selects the
\lam{sum} signal to be the output port of the function. This \quote{return
value} can always only be a variable reference, never a more complex
expression.
\startbuffer[MulSum]
alu :: Bit -> Word -> Word -> Word
alu = λa.λb.λc.
let
mul = (*) a b
sum = (+) mul c
in
sum
\stopbuffer
\startuseMPgraphic{MulSum}
save a, b, c, mul, add, sum;
% I/O ports
newCircle.a(btex $a$ etex) "framed(false)";
newCircle.b(btex $b$ etex) "framed(false)";
newCircle.c(btex $c$ etex) "framed(false)";
newCircle.sum(btex $res$ etex) "framed(false)";
% Components
newCircle.mul(btex - etex);
newCircle.add(btex + etex);
a.c - b.c = (0cm, 2cm);
b.c - c.c = (0cm, 2cm);
add.c = c.c + (2cm, 0cm);
mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
sum.c = add.c + (2cm, 0cm);
c.c = origin;
% Draw objects and lines
drawObj(a, b, c, mul, add, sum);
ncarc(a)(mul) "arcangle(15)";
ncarc(b)(mul) "arcangle(-15)";
ncline(c)(add);
ncline(mul)(add);
ncline(add)(sum);
\stopuseMPgraphic
\placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
subtractor.}
\startcombination[2*1]
{\typebufferlam{MulSum}}{Core description in normal form.}
{\boxedgraphic{MulSum}}{The architecture described by the normal form.}
\stopcombination
The previous example described composing an architecture by calling other
functions (operators), resulting in a simple architecture with component and
connection. There is of course also some mechanism for choice in the normal
form. In a normal Core program, the \emph{case} expression can be used in a
few different ways to describe choice. In normal form, this is limited to a
very specific form.
\in{Example}[ex:AddSubAlu] shows an example describing a
simple \small{ALU}, which chooses between two operations based on an opcode
bit. The main structure is the same as in \in{example}[ex:MulSum], but this
time the \lam{res} variable is bound to a case expression. This case
expression scrutinizes the variable \lam{opcode} (and scrutinizing more
complex expressions is not supported). The case expression can select a
different variable based on the constructor of \lam{opcode}.
\startbuffer[AddSubAlu]
alu :: Bit -> Word -> Word -> Word
alu = λopcode.λa.λb.
let
res1 = (+) a b
res2 = (-) a b
res = case opcode of
Low -> res1
High -> res2
in
res
\stopbuffer
\startuseMPgraphic{AddSubAlu}
save opcode, a, b, add, sub, mux, res;
% I/O ports
newCircle.opcode(btex $opcode$ etex) "framed(false)";
newCircle.a(btex $a$ etex) "framed(false)";
newCircle.b(btex $b$ etex) "framed(false)";
newCircle.res(btex $res$ etex) "framed(false)";
% Components
newCircle.add(btex + etex);
newCircle.sub(btex - etex);
newMux.mux;
opcode.c - a.c = (0cm, 2cm);
add.c - a.c = (4cm, 0cm);
sub.c - b.c = (4cm, 0cm);
a.c - b.c = (0cm, 3cm);
mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
res.c - mux.c = (1.5cm, 0cm);
b.c = origin;
% Draw objects and lines
drawObj(opcode, a, b, res, add, sub, mux);
ncline(a)(add) "posA(e)";
ncline(b)(sub) "posA(e)";
nccurve(a)(sub) "posA(e)", "angleA(0)";
nccurve(b)(add) "posA(e)", "angleA(0)";
nccurve(add)(mux) "posB(inpa)", "angleB(0)";
nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
ncline(mux)(res) "posA(out)";
\stopuseMPgraphic
\placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
\startcombination[2*1]
{\typebufferlam{AddSubAlu}}{Core description in normal form.}
{\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
\stopcombination
As a more complete example, consider \in{example}[ex:NormalComplete]. This
example contains everything that is supported in normal form, with the
exception of builtin higher order functions. The graphical version of the
architecture contains a slightly simplified version, since the state tuple
packing and unpacking have been left out. Instead, two seperate registers are
drawn. Also note that most synthesis tools will further optimize this
architecture by removing the multiplexers at the register input and replace
them with some logic in the clock inputs, but we want to show the architecture
as close to the description as possible.
\startbuffer[NormalComplete]
regbank :: Bit
-> Word
-> State (Word, Word)
-> (State (Word, Word), Word)
-- All arguments are an inital lambda
regbank = λa.λd.λsp.
-- There are nested let expressions at top level
let
-- Unpack the state by coercion (\eg, cast from
-- State (Word, Word) to (Word, Word))
s = sp :: (Word, Word)
-- Extract both registers from the state
r1 = case s of (fst, snd) -> fst
r2 = case s of (fst, snd) -> snd
-- Calling some other user-defined function.
d' = foo d
-- Conditional connections
out = case a of
High -> r1
Low -> r2
r1' = case a of
High -> d'
Low -> r1
r2' = case a of
High -> r2
Low -> d'
-- Packing a tuple
s' = (,) r1' r2'
-- pack the state by coercion (\eg, cast from
-- (Word, Word) to State (Word, Word))
sp' = s' :: State (Word, Word)
-- Pack our return value
res = (,) sp' out
in
-- The actual result
res
\stopbuffer
\startuseMPgraphic{NormalComplete}
save a, d, r, foo, muxr, muxout, out;
% I/O ports
newCircle.a(btex \lam{a} etex) "framed(false)";
newCircle.d(btex \lam{d} etex) "framed(false)";
newCircle.out(btex \lam{out} etex) "framed(false)";
% Components
%newCircle.add(btex + etex);
newBox.foo(btex \lam{foo} etex);
newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
newMux.muxr1;
% Reflect over the vertical axis
reflectObj(muxr1)((0,0), (0,1));
newMux.muxr2;
newMux.muxout;
rotateObj(muxout)(-90);
d.c = foo.c + (0cm, 1.5cm);
a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
muxr1.c = r1.c + (0cm, 2cm);
muxr2.c = r2.c + (0cm, 2cm);
r2.c = r1.c + (4cm, 0cm);
r1.c = origin;
muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
out.c = muxout.c - (0cm, 1.5cm);
% % Draw objects and lines
drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
ncline(d)(foo);
nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
% Connect port a
nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
ncline(muxout)(out) "posA(out)";
\stopuseMPgraphic
\placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
subtractor.}
\startcombination[2*1]
{\typebufferlam{NormalComplete}}{Core description in normal form.}
{\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
\stopcombination
\subsection{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?
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
~
--------------------------
~
\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{} 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{}
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{}
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, this
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{original
expression} 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 functiosn.
Each addition has the form \lam{binder = template}. As above, any
placeholder in the addition is replaced with the value bound to it, and any
binder placeholder that has no value bound to it yet will be bound to (and
replaced with) a fresh binder.
\stopdesc
As an example, we'll look at η-abstraction:
\starttrans
E \lam{E :: a -> b}
-------------- \lam{E} does not occur on a function position in an application
λx.E x \lam{E} is not a lambda abstraction.
\stoptrans
Consider the following function, which is a fairly obvious way to specify a
simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
function):
\startlambda
alu :: Bit -> Word -> Word -> Word
alu = λopcode. case opcode of
Low -> (+)
High -> (-)
\stoplambda
There are a few subexpressions in this function to which we could possibly
apply the transformation. Since the pattern of the transformation is only
the placeholder \lam{E}, any expression will match that. Whether the
transformation applies to an expression is thus solely decided by the
conditions to the right of the transformation.
We will look at each expression in the function in a top down manner. The
first expression is the entire expression the function is bound to.
\startlambda
λopcode. case opcode of
Low -> (+)
High -> (-)
\stoplambda
As said, the expression pattern matches this. The type of this expression is
\lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
Since this expression is at top level, it does not occur at a function
position of an application. However, The expression is a lambda abstraction,
so this transformation does not apply.
The next expression we could apply this transformation to, is the body of
the lambda abstraction:
\startlambda
case opcode of
Low -> (+)
High -> (-)
\stoplambda
The type of this expression is \lam{Word -> Word -> Word}, which again
matches \lam{a -> b}. The expression is the body of a lambda expression, so
it does not occur at a function position of an application. Finally, the
expression is not a lambda abstraction but a case expression, so all the
conditions match. There are no context conditions to match, so the
transformation applies.
By now, the placeholder \lam{E} is bound to the entire expression. The
placeholder \lam{x}, which occurs in the replacement template, is not bound
yet, so we need to generate a fresh binder for that. Let's use the binder
\lam{a}. This results in the following replacement expression:
\startlambda
λa.(case opcode of
Low -> (+)
High -> (-)) a
\stoplambda
Continuing with this expression, we see that the transformation does not
apply again (it is a lambda expression). Next we look at the body of this
labmda abstraction:
\startlambda
(case opcode of
Low -> (+)
High -> (-)) a
\stoplambda
Here, the transformation does apply, binding \lam{E} to the entire
expression and \lam{x} to the fresh binder \lam{b}, resulting in the
replacement:
\startlambda
λb.(case opcode of
Low -> (+)
High -> (-)) a b
\stoplambda
Again, the transformation does not apply to this lambda abstraction, so we
look at its body. For brevity, we'll put the case statement on one line from
now on.
\startlambda
(case opcode of Low -> (+); High -> (-)) a b
\stoplambda
The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
and the transformation does not apply. Next, we have two options for the
next expression to look at: The function position and argument position of
the application. The expression in the argument position is \lam{b}, which
has type \lam{Word}, so the transformation does not apply. The expression in
the function position is:
\startlambda
(case opcode of Low -> (+); High -> (-)) a
\stoplambda
Obviously, the transformation does not apply here, since it occurs in
function position. In the same way the transformation does not apply to both
components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
and \lam{a}), so we'll skip to the components of the case expression: The
scrutinee and both alternatives. Since the opcode is not a function, it does
not apply here, and we'll leave both alternatives as an exercise to the
reader. The final function, after all these transformations becomes:
\startlambda
alu :: Bit -> Word -> Word -> Word
alu = λopcode.λa.b. (case opcode of
Low -> λa1.λb1 (+) a1 b1
High -> λa2.λb2 (-) a2 b2) a b
\stoplambda
In this case, the transformation does not apply anymore, though this might
not always be the case (e.g., the application of a transformation on a
subexpression might open up possibilities to apply the transformation
further up in the expression).
\subsection{Transformation application}
In this chapter we define a number of transformations, but how will we apply
these? As stated before, our normal form is reached as soon as no
transformation applies anymore. This means our application strategy is to
simply apply any transformation that applies, and continuing to do that with
the result of each transformation.
In particular, we define no particular order of transformations. Since
transformation order should not influence the resulting normal form (see TODO
ref), this leaves the implementation free to choose any application order that
results in an efficient implementation.
When applying a single transformation, we try to apply it to every (sub)expression
in a function, not just the top level function. This allows us to keep the
transformation descriptions concise and powerful.
\subsection{Definitions}
In the following sections, we will be using a number of functions and
notations, which we will define here.
TODO: Define substitution
\subsubsection{Other concepts}
A \emph{global variable} is any variable that is bound at the
top level of a program, or an external module. A \emph{local variable} is any
other variable (\eg, variables local to a function, which can be bound by
lambda abstractions, let expressions and pattern matches of case
alternatives). Note that this is a slightly different notion of global versus
local than what \small{GHC} uses internally.
\defref{global variable} \defref{local variable}
A \emph{hardware representable} (or just \emph{representable}) type or value
is (a value of) a type that we can generate a signal for in hardware. For
example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
not runtime representable notably include (but are not limited to): Types,
dictionaries, functions.
\defref{representable}
A \emph{builtin function} is a function supplied by the Cλash framework, whose
implementation is not valid Cλash. The implementation is of course valid
Haskell, for simulation, but it is not expressable in Cλash.
\defref{builtin function} \defref{user-defined function}
For these functions, Cλash has a \emph{builtin hardware translation}, so calls
to these functions can still be translated. These are functions like
\lam{map}, \lam{hwor} and \lam{length}.
A \emph{user-defined} function is a function for which we do have a Cλash
implementation available.
\subsubsection{Functions}
Here, we define a number of functions that can be used below to concisely
specify conditions.
\refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
global variable. It is false when it references a local variable.
\refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
references a local variable, false when it references a global variable.
\refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
\emph{expr} or \emph{var} is \emph{representable}.
\subsection{Binder uniqueness}
A common problem in transformation systems, is binder uniqueness. When not
considering this problem, it is easy to create transformations that mix up
bindings and cause name collisions. Take for example, the following core
expression:
\startlambda
(λa.λb.λc. a * b * c) x c
\stoplambda
By applying β-reduction (see below) once, we can simplify this expression to:
\startlambda
(λb.λc. x * b * c) c
\stoplambda
Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
binder. No harm done here. But note that we see multiple occurences of the
\lam{c} binder. The first is a binding occurence, to which the second refers.
The last, however refers to \emph{another} instance of \lam{c}, which is
bound somewhere outside of this expression. Now, if we would apply beta
reduction without taking heed of binder uniqueness, we would get:
\startlambda
λc. x * c * c
\stoplambda
This is obviously not what was supposed to happen! The root of this problem is
the reuse of binders: Identical binders can be bound in different scopes, such
that only the inner one is \quote{visible} in the inner expression. In the example
above, the \lam{c} binder was bound outside of the expression and in the inner
lambda expression. Inside that lambda expression, only the inner \lam{c} is
visible.
There are a number of ways to solve this. \small{GHC} has isolated this
problem to their binder substitution code, which performs \emph{deshadowing}
during its expression traversal. This means that any binding that shadows
another binding on a higher level is replaced by a new binder that does not
shadow any other binding. This non-shadowing invariant is enough to prevent
binder uniqueness problems in \small{GHC}.
In our transformation system, maintaining this non-shadowing invariant is
a bit harder to do (mostly due to implementation issues, the prototype doesn't
use \small{GHC}'s subsitution code). Also, we can observe the following
points.
\startitemize
\item Deshadowing does not guarantee overall uniqueness. For example, the
following (slightly contrived) expression shows the identifier \lam{x} bound in
two seperate places (and to different values), even though no shadowing
occurs.
\startlambda
(let x = 1 in x) + (let x = 2 in x)
\stoplambda
\item In our normal form (and the resulting \small{VHDL}), all binders
(signals) will end up in the same scope. To allow this, all binders within the
same function should be unique.
\item When we know that all binders in an expression are unique, moving around
or removing a subexpression will never cause any binder conflicts. If we have
some way to generate fresh binders, introducing new subexpressions will not
cause any problems either. The only way to cause conflicts is thus to
duplicate an existing subexpression.
\stopitemize
Given the above, our prototype maintains a unique binder invariant. This
meanst that in any given moment during normalization, all binders \emph{within
a single function} must be unique. To achieve this, we apply the following
technique.
TODO: Define fresh binders and unique supplies
\startitemize
\item Before starting normalization, all binders in the function are made
unique. This is done by generating a fresh binder for every binder used. This
also replaces binders that did not pose any conflict, but it does ensure that
all binders within the function are generated by the same unique supply. See
(TODO: ref fresh binder).
\item Whenever a new binder must be generated, we generate a fresh binder that
is guaranteed to be different from \emph{all binders generated so far}. This
can thus never introduce duplication and will maintain the invariant.
\item Whenever (part of) an expression is duplicated (for example when
inlining), all binders in the expression are replaced with fresh binders
(using the same method as at the start of normalization). These fresh binders
can never introduce duplication, so this will maintain the invariant.
\item Whenever we move part of an expression around within the function, there
is no need to do anything special. There is obviously no way to introduce
duplication by moving expressions around. Since we know that each of the
binders is already unique, there is no way to introduce (incorrect) shadowing
either.
\stopitemize
\section{Transform passes}
In this section we describe the actual transforms. Here we're using
the core language in a notation that resembles lambda calculus.
Each of these transforms is meant to be applied to every (sub)expression
in a program, for as long as it applies. Only when none of the
transformations can be applied anymore, the program is in normal form (by
definition). We hope to be able to prove that this form will obey all of the
constraints defined above, but this has yet to happen (though it seems likely
that it will).
Each of the transforms will be described informally first, explaining
the need for and goal of the transform. Then, a formal definition is
given, using a familiar syntax from the world of logic. Each transform
is specified as a number of conditions (above the horizontal line) and a
number of conclusions (below the horizontal line). The details of using
this notation are still a bit fuzzy, so comments are welcom.
\subsection{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
\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}
TODO: Generalize this section into 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
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]
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?