\chapter[chap:normalization]{Normalization}
% A helper to print a single example in the half the page width. The example
% text should be in a buffer whose name is given in an argument.
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\define[1]\example{
\framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
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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 (higher-order expressions, limited polymorphism using type
classes, etc.) and because Core can describe expressions that do not
have a direct hardware interpretation.
\section{Normal form}
The transformations described here have a well-defined goal: to bring the
program in a well-defined form that is directly translatable to
\VHDL, while fully preserving the semantics of the program. We refer
to this form as the \emph{normal form} of the program. The formal
definition of this normal form is quite simple:
\placedefinition[force]{}{\startboxed A program is in \emph{normal form} if none of the
transformations from this chapter apply.\stopboxed}
Of course, this is an \quote{easy} definition of the normal form, since our
program will end up in normal form automatically. The more interesting part is
to see if this normal form actually has the properties we would like it to
have.
But, before getting into more definitions and details about this normal
form, let us try to get a feeling for it first. The easiest way to do this
is by describing the things that are unwanted in the intended normal form.
\startitemize
\item Any \emph{polymorphism} must be removed. When laying down hardware, we
cannot generate any signals that can have multiple types. All types must be
completely known to generate hardware.
\item All \emph{higher-order} constructions must be removed. We cannot
generate a hardware signal that contains a function, so all values,
arguments and return values used must be first order.
\item All complex \emph{nested scopes} must be removed. In the \small{VHDL}
description, every signal is in a single scope. Also, full expressions are
not supported everywhere (in particular port maps can only map signal
names and constants, not complete expressions). To make the \small{VHDL}
generation easy, a separate binder must be bound to ever application or
other expression.
\stopitemize
\startbuffer[MulSum]
alu :: Bit -> Word -> Word -> Word
alu = λa.λb.λc.
let
mul = (*) a b
sum = (+) mul c
in
sum
\stopbuffer
\startuseMPgraphic{MulSum}
save a, b, c, mul, add, sum;
% I/O ports
newCircle.a(btex $a$ etex) "framed(false)";
newCircle.b(btex $b$ etex) "framed(false)";
newCircle.c(btex $c$ etex) "framed(false)";
newCircle.sum(btex $sum$ etex) "framed(false)";
% Components
newCircle.mul(btex * etex);
newCircle.add(btex + etex);
a.c - b.c = (0cm, 2cm);
b.c - c.c = (0cm, 2cm);
add.c = c.c + (2cm, 0cm);
mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
sum.c = add.c + (2cm, 0cm);
c.c = origin;
% Draw objects and lines
drawObj(a, b, c, mul, add, sum);
ncarc(a)(mul) "arcangle(15)";
ncarc(b)(mul) "arcangle(-15)";
ncline(c)(add);
ncline(mul)(add);
ncline(add)(sum);
\stopuseMPgraphic
\placeexample[][ex:MulSum]{Simple architecture consisting of a
multiplier and a subtractor.}
\startcombination[2*1]
{\typebufferlam{MulSum}}{Core description in normal form.}
{\boxedgraphic{MulSum}}{The architecture described by the normal form.}
\stopcombination
\todo{Intermezzo: functions vs plain values}
A very simple example of a program in normal form is given in
\in{example}[ex:MulSum]. As you can see, all arguments to the function (which
will become input ports in the generated \VHDL) are at the outer level.
This means that the body of the inner lambda abstraction is never a
function, but always a plain value.
As the body of the inner lambda abstraction, we see a single (recursive)
let expression, that binds two variables (\lam{mul} and \lam{sum}). These
variables will be signals in the generated \VHDL, bound to the output port
of the \lam{*} and \lam{+} components.
The final line (the \quote{return value} of the function) selects the
\lam{sum} signal to be the output port of the function. This \quote{return
value} can always only be a variable reference, never a more complex
expression.
\todo{Add generated VHDL}
\in{Example}[ex:MulSum] showed a function that just applied two
other functions (multiplication and addition), resulting in a simple
architecture with two components and some connections. There is of
course also some mechanism for choice in the normal form. In a
normal Core program, the \emph{case} expression can be used in a few
different ways to describe choice. In normal form, this is limited
to a very specific form.
\in{Example}[ex:AddSubAlu] shows an example describing a
simple \small{ALU}, which chooses between two operations based on an opcode
bit. The main structure is similar to \in{example}[ex:MulSum], but this
time the \lam{res} variable is bound to a case expression. This case
expression scrutinizes the variable \lam{opcode} (and scrutinizing more
complex expressions is not supported). The case expression can select a
different variable based on the constructor of \lam{opcode}.
\refdef{case expression}
\startbuffer[AddSubAlu]
alu :: Bit -> Word -> Word -> Word
alu = λopcode.λa.λb.
let
res1 = (+) a b
res2 = (-) a b
res = case opcode of
Low -> res1
High -> res2
in
res
\stopbuffer
\startuseMPgraphic{AddSubAlu}
save opcode, a, b, add, sub, mux, res;
% I/O ports
newCircle.opcode(btex $opcode$ etex) "framed(false)";
newCircle.a(btex $a$ etex) "framed(false)";
newCircle.b(btex $b$ etex) "framed(false)";
newCircle.res(btex $res$ etex) "framed(false)";
% Components
newCircle.add(btex + etex);
newCircle.sub(btex - etex);
newMux.mux;
opcode.c - a.c = (0cm, 2cm);
add.c - a.c = (4cm, 0cm);
sub.c - b.c = (4cm, 0cm);
a.c - b.c = (0cm, 3cm);
mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
res.c - mux.c = (1.5cm, 0cm);
b.c = origin;
% Draw objects and lines
drawObj(opcode, a, b, res, add, sub, mux);
ncline(a)(add) "posA(e)";
ncline(b)(sub) "posA(e)";
nccurve(a)(sub) "posA(e)", "angleA(0)";
nccurve(b)(add) "posA(e)", "angleA(0)";
nccurve(add)(mux) "posB(inpa)", "angleB(0)";
nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
ncline(mux)(res) "posA(out)";
\stopuseMPgraphic
\placeexample[][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
\startcombination[2*1]
{\typebufferlam{AddSubAlu}}{Core description in normal form.}
{\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
\stopcombination
As a more complete example, consider
\in{example}[ex:NormalComplete]. This example shows everything that
is allowed in normal form, except for built-in higher-order functions
(like \lam{map}). The graphical version of the architecture contains
a slightly simplified version, since the state tuple packing and
unpacking have been left out. Instead, two separate registers are
drawn. Most synthesis tools will further optimize this architecture by
removing the multiplexers at the register input and instead use the write
enable port of the register (when it is available), but we want to show
the architecture as close to the description as possible.
As you can see from the previous examples, the generation of the final
architecture from the normal form is straightforward. In each of the
examples, there is a direct match between the normal form structure,
the generated VHDL and the architecture shown in the images.
\startbuffer[NormalComplete]
regbank :: Bit
-> Word
-> State (Word, Word)
-> (State (Word, Word), Word)
-- All arguments are an initial lambda
-- (address, data, packed state)
regbank = λa.λd.λsp.
-- There are nested let expressions at top level
let
-- Unpack the state by coercion (\eg, cast from
-- State (Word, Word) to (Word, Word))
s = sp ▶ (Word, Word)
-- Extract both registers from the state
r1 = case s of (a, b) -> a
r2 = case s of (a, b) -> b
-- Calling some other user-defined function.
d' = foo d
-- Conditional connections
out = case a of
High -> r1
Low -> r2
r1' = case a of
High -> d'
Low -> r1
r2' = case a of
High -> r2
Low -> d'
-- Packing a tuple
s' = (,) r1' r2'
-- pack the state by coercion (\eg, cast from
-- (Word, Word) to State (Word, Word))
sp' = s' ▶ State (Word, Word)
-- Pack our return value
res = (,) sp' out
in
-- The actual result
res
\stopbuffer
\startuseMPgraphic{NormalComplete}
save a, d, r, foo, muxr, muxout, out;
% I/O ports
newCircle.a(btex \lam{a} etex) "framed(false)";
newCircle.d(btex \lam{d} etex) "framed(false)";
newCircle.out(btex \lam{out} etex) "framed(false)";
% Components
%newCircle.add(btex + etex);
newBox.foo(btex \lam{foo} etex);
newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
newMux.muxr1;
% Reflect over the vertical axis
reflectObj(muxr1)((0,0), (0,1));
newMux.muxr2;
newMux.muxout;
rotateObj(muxout)(-90);
d.c = foo.c + (0cm, 1.5cm);
a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
muxr1.c = r1.c + (0cm, 2cm);
muxr2.c = r2.c + (0cm, 2cm);
r2.c = r1.c + (4cm, 0cm);
r1.c = origin;
muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
out.c = muxout.c - (0cm, 1.5cm);
% % Draw objects and lines
drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
ncline(d)(foo);
nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
% Connect port a
nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
ncline(muxout)(out) "posA(out)";
\stopuseMPgraphic
\todo{Don't split registers in this image?}
\placeexample[][ex:NormalComplete]{Simple architecture consisting of an adder and a
subtractor.}
\startcombination[2*1]
{\typebufferlam{NormalComplete}}{Core description in normal form.}
{\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
\stopcombination
\subsection[sec:normalization:intendednormalform]{Intended normal form definition}
Now we have some intuition for the normal form, we can describe how we want
the normal form to look like in a slightly more formal manner. The
EBNF-like description in \in{definition}[def:IntendedNormal] captures
most of the intended structure (and generates a subset of \GHC's Core
format).
There are two things missing from this definition: cast expressions are
sometimes allowed by the prototype, but not specified here and the below
definition allows uses of state that cannot be translated to \VHDL\
properly. These two problems are discussed in
\in{section}[sec:normalization:castproblems] and
\in{section}[sec:normalization:stateproblems] respectively.
Some clauses have an expression listed behind them in parentheses.
These are conditions that need to apply to the clause. The
predicates used there (\lam{lvar()}, \lam{representable()},
\lam{gvar()}) will be defined in
\in{section}[sec:normalization:predicates].
An expression is in normal form if it matches the first
definition, \emph{normal}.
\todo{Fix indentation}
\startbuffer[IntendedNormal]
\italic{normal} := \italic{lambda}
\italic{lambda} := λvar.\italic{lambda} (representable(var))
| \italic{toplet}
\italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
\italic{binding} := var = \italic{rhs} (representable(rhs))
-- State packing and unpacking by coercion
| var0 = var1 ▶ State ty (lvar(var1))
| var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
\italic{rhs} := \italic{userapp}
| \italic{builtinapp}
-- Extractor case
| case var of C a0 ... an -> ai (lvar(var))
-- Selector case
| case var of (lvar(var))
[ DEFAULT -> var ] (lvar(var))
C0 w0,0 ... w0,n -> var0
\vdots
Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
\italic{userapp} := \italic{userfunc}
| \italic{userapp} {userarg}
\italic{userfunc} := var (gvar(var))
\italic{userarg} := var (lvar(var))
\italic{builtinapp} := \italic{builtinfunc}
| \italic{builtinapp} \italic{builtinarg}
\italic{built-infunc} := var (bvar(var))
\italic{built-inarg} := var (representable(var) ∧ lvar(var))
| \italic{partapp} (partapp :: a -> b)
| \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
\italic{partapp} := \italic{userapp}
| \italic{builtinapp}
\stopbuffer
\placedefinition[][def:IntendedNormal]{Definition of the intended normal form using an \small{EBNF}-like syntax.}
{\defref{intended normal form definition}
\typebufferlam{IntendedNormal}}
When looking at such a program from a hardware perspective, the top
level lambda abstractions (\italic{lambda}) define the input ports.
Lambda abstractions cannot appear anywhere else. The variable reference
in the body of the recursive let expression (\italic{toplet}) is the
output port. Most binders bound by the let expression define a
component instantiation (\italic{userapp}), where the input and output
ports are mapped to local signals (\italic{userarg}). Some of the others
use a built-in construction (\eg\ the \lam{case} expression) or call a
built-in function (\italic{builtinapp}) such as \lam{+} or \lam{map}.
For these, a hard-coded \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 describes a transformation that applies to \lam{} and transforms it into \lam{}, assuming
that all conditions are satisfied. In this format, there are a number of placeholders
in pointy brackets, most of which should be rather obvious in their meaning.
Nevertheless, we will more precisely specify their meaning below:
\startdesc{} The expression pattern that will be matched
against (sub-expressions 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} satisfied, 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 by 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 be added to the context.
This is a way to let a transformation create new top level
functions.
Each addition has the form \lam{binder = template}. As above, any
placeholder in the addition is replaced with the value bound to it, and any
binder placeholder that has no value bound to it yet will be bound to (and
replaced with) a fresh binder.
\stopdesc
To understand this notation better, the step by step application of
the η-expansion transformation to a simple \small{ALU} will be
shown. Consider η-expansion, which is a common transformation from
lambda calculus, described using above notation as follows:
\starttrans
E \lam{E :: a -> b}
-------------- \lam{E} does not occur on a function position in an application
λx.E x \lam{E} is not a lambda abstraction.
\stoptrans
η-expansion is a well known transformation from lambda calculus. What
this transformation does, is take any expression that has a function type
and turn it into a lambda expression (giving an explicit name to the
argument). There are some extra conditions that ensure that this
transformation does not apply infinitely (which are not necessarily part
of the conventional definition of η-expansion).
Consider the following function, in Core notation, which is a fairly obvious way to specify a
simple \small{ALU} (Note that it is not yet in normal form, but
\in{example}[ex:AddSubAlu] shows the normal form of this function).
The parentheses around the \lam{+} and \lam{-} operators are
commonly used in Haskell to show that the operators are used as
normal functions, instead of \emph{infix} operators (\eg, the
operators appear before their arguments, instead of in between).
\startlambda
alu :: Bit -> Word -> Word -> Word
alu = λopcode. case opcode of
Low -> (+)
High -> (-)
\stoplambda
There are a few sub-expressions in this function to which we could possibly
apply the transformation. Since the pattern of the transformation is only
the placeholder \lam{E}, any expression will match that. Whether the
transformation applies to an expression is thus solely decided by the
conditions to the right of the transformation.
We will look at each expression in the function in a top down manner. The
first expression is the entire expression the function is bound to.
\startlambda
λopcode. case opcode of
Low -> (+)
High -> (-)
\stoplambda
As said, the expression pattern matches this. The type of this expression is
\lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
Since this expression is at top level, it does not occur at a function
position of an application. However, The expression is a lambda abstraction,
so this transformation does not apply.
The next expression we could apply this transformation to, is the body of
the lambda abstraction:
\startlambda
case opcode of
Low -> (+)
High -> (-)
\stoplambda
The type of this expression is \lam{Word -> Word -> Word}, which again
matches \lam{a -> b}. The expression is the body of a lambda expression, so
it does not occur at a function position of an application. Finally, the
expression is not a lambda abstraction but a case expression, so all the
conditions match. There are no context conditions to match, so the
transformation applies.
By now, the placeholder \lam{E} is bound to the entire expression. The
placeholder \lam{x}, which occurs in the replacement template, is not bound
yet, so we need to generate a fresh binder for that. Let us use the binder
\lam{a}. This results in the following replacement expression:
\startlambda
λa.(case opcode of
Low -> (+)
High -> (-)) a
\stoplambda
Continuing with this expression, we see that the transformation does not
apply again (it is a lambda expression). Next we look at the body of this
lambda abstraction:
\startlambda
(case opcode of
Low -> (+)
High -> (-)) a
\stoplambda
Here, the transformation does apply, binding \lam{E} to the entire
expression (which has type \lam{Word -> Word}) and binding \lam{x}
to the fresh binder \lam{b}, resulting in the replacement:
\startlambda
λb.(case opcode of
Low -> (+)
High -> (-)) a b
\stoplambda
The transformation does not apply to this lambda abstraction, so we
look at its body. For brevity, we will put the case expression on one line from
now on.
\startlambda
(case opcode of Low -> (+); High -> (-)) a b
\stoplambda
The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
and the transformation does not apply. Next, we have two options for the
next expression to look at: the function position and argument position of
the application. The expression in the argument position is \lam{b}, which
has type \lam{Word}, so the transformation does not apply. The expression in
the function position is:
\startlambda
(case opcode of Low -> (+); High -> (-)) a
\stoplambda
Obviously, the transformation does not apply here, since it occurs in
function position (which makes the second condition false). In the same
way the transformation does not apply to both components of this
expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
we will skip to the components of the case expression: the scrutinee and
both alternatives. Since the opcode is not a function, it does not apply
here.
The first alternative is \lam{(+)}. This expression has a function type
(the operator still needs two arguments). It does not occur in function
position of an application and it is not a lambda expression, so the
transformation applies.
We look at the \lam{} pattern, which is \lam{E}.
This means we bind \lam{E} to \lam{(+)}. We then replace the expression
with the \lam{}, replacing all occurrences 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-expansion until it does no longer apply is:
\startlambda
alu :: Bit -> Word -> Word -> Word
alu = λopcode.λa.b. (case opcode of
Low -> λa1.λb1 (+) a1 b1
High -> λa2.λb2 (-) a2 b2) a b
\stoplambda
\subsection{Transformation application}
In this chapter we define a number of transformations, but how will we apply
these? As stated before, our normal form is reached as soon as no
transformation applies anymore. This means our application strategy is to
simply apply any transformation that applies, and continuing to do that with
the result of each transformation.
In particular, we define no particular order of transformations. Since
transformation order should not influence the resulting normal form,
this leaves the implementation free to choose any application order that
results in an efficient implementation. Unfortunately this is not
entirely true for the current set of transformations. See
\in{section}[sec:normalization:non-determinism] for a discussion of this
problem.
When applying a single transformation, we try to apply it to every (sub)expression
in a function, not just the top level function body. This allows us to
keep the transformation descriptions concise and powerful.
\subsection{Definitions}
A \emph{global variable} is any variable (binder) that is bound at the
top level of a program, or an external module. A \emph{local variable} is any
other variable (\eg, variables local to a function, which can be bound by
lambda abstractions, let expressions and pattern matches of case
alternatives). This is a slightly different notion of global versus
local than what \small{GHC} uses internally, but for our purposes
the distinction \GHC\ makes is not useful.
\defref{global variable} \defref{local variable}
A \emph{hardware representable} (or just \emph{representable}) type or value
is (a value of) a type that we can generate a signal for in hardware. For
example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
not run-time representable notably include (but are not limited to): types,
dictionaries, functions.
\defref{representable}
A \emph{built-in function} is a function supplied by the Cλash
framework, whose implementation is not used to generate \VHDL. This is
either because it is no valid Cλash (like most list functions that need
recursion) or because a Cλash implementation would be unwanted (for the
addition operator, for example, we would rather use the \VHDL addition
operator to let the synthesis tool decide what kind of adder to use
instead of explicitly describing one in Cλash). \defref{built-in
function}
These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
For these functions, Cλash has a \emph{built-in hardware translation},
so calls to these functions can still be translated. Built-in functions
must have a valid Haskell implementation, of course, to allow
simulation.
A \emph{user-defined} function is a function for which no built-in
translation is available and whose definition will thus need to be
translated to Cλash. \defref{user-defined function}
\subsubsection[sec:normalization:predicates]{Predicates}
Here, we define a number of predicates that can be used below to concisely
specify conditions.
\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
global variable. It is false when it references a local variable.
\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
references a local variable, false when it references a global variable.
\emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
\subsection[sec:normalization:uniq]{Binder uniqueness}
A common problem in transformation systems, is binder uniqueness. When not
considering this problem, it is easy to create transformations that mix up
bindings and cause name collisions. Take for example, the following Core
expression:
\startlambda
(λa.λb.λc. a * b * c) x c
\stoplambda
By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
we can simplify this expression to:
\startlambda
(λb.λc. x * b * c) c
\stoplambda
Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
binder. No harm done here. But note that we see multiple occurrences of the
\lam{c} binder. The first is a binding occurrence, to which the second refers.
The last, however refers to \emph{another} instance of \lam{c}, which is
bound somewhere outside of this expression. Now, if we would apply beta
reduction without taking heed of binder uniqueness, we would get:
\startlambda
λc. x * c * c
\stoplambda
This is obviously not what was supposed to happen! The root of this problem is
the reuse of binders: identical binders can be bound in different,
but overlapping scopes. Any variable reference in those
overlapping scopes then refers to the variable bound in the inner
(smallest) scope. There is not way to refer to the variable in the
outer scope. This effect is usually referred to as
\emph{shadowing}: when a binder is bound in a scope where the
binder already had a value, the inner binding is said to
\emph{shadow} the outer binding. In the example above, the \lam{c}
binder was bound outside of the expression and in the inner lambda
expression. Inside that lambda expression, only the inner \lam{c}
can be accessed.
There are a number of ways to solve this. \small{GHC} has isolated this
problem to their binder substitution code, which performs \emph{de-shadowing}
during its expression traversal. This means that any binding that shadows
another binding on a higher level is replaced by a new binder that does not
shadow any other binding. This non-shadowing invariant is enough to prevent
binder uniqueness problems in \small{GHC}.
In our transformation system, maintaining this non-shadowing invariant is
a bit harder to do (mostly due to implementation issues, the prototype
does not use \small{GHC}'s substitution code). Also, the following points
can be observed.
\startitemize
\item De-shadowing does not guarantee overall uniqueness. For example, the
following (slightly contrived) expression shows the identifier \lam{x} bound in
two separate 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 sub-expression will never cause any binder conflicts. If we have
some way to generate fresh binders, introducing new sub-expressions will not
cause any problems either. The only way to cause conflicts is thus to
duplicate an existing sub-expression.
\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.
\item Whenever a new binder must be generated, we generate a fresh binder that
is guaranteed to be different from \emph{all binders generated so far}. This
can thus never introduce duplication and will maintain the invariant.
\item Whenever (a part of) an expression is duplicated (for example when
inlining), all binders in the expression are replaced with fresh binders
(using the same method as at the start of normalization). These fresh binders
can never introduce duplication, so this will maintain the invariant.
\item Whenever we move part of an expression around within the function, there
is no need to do anything special. There is obviously no way to introduce
duplication by moving expressions around. Since we know that each of the
binders is already unique, there is no way to introduce (incorrect) shadowing
either.
\stopitemize
\section{Transform passes}
In this section we describe the actual transforms.
Each transformation will be described informally first, explaining
the need for and goal of the transformation. Then, we will formally define
the transformation using the syntax introduced in
\in{section}[sec:normalization:transformation].
\subsection{General cleanup}
\placeintermezzo{}{
\defref{substitution notation}
\startframedtext[width=8cm,background=box,frame=no]
\startalignment[center]
{\tfa Substitution notation}
\stopalignment
\blank[medium]
In some of the transformations in this chapter, we need to perform
substitution on an expression. Substitution means replacing every
occurrence of some expression (usually a variable reference) with
another expression.
There have been a lot of different notations used in literature for
specifying substitution. The notation that will be used in this report
is the following:
\startlambda
E[A=>B]
\stoplambda
This means expression \lam{E} with all occurrences of \lam{A} replaced
with \lam{B}.
\stopframedtext
}
These transformations are general cleanup transformations, that aim to
make expressions simpler. These transformations usually clean up the
mess left behind by other transformations or clean up expressions to
expose new transformation opportunities for other transformations.
Most of these transformations are standard optimizations in other
compilers as well. However, in our compiler, most of these are not just
optimizations, but they are required to get our program into intended
normal form.
\subsubsection[sec:normalization:beta]{β-reduction}
β-reduction is a well known transformation from lambda calculus, where it is
the main reduction step. It reduces applications of lambda abstractions,
removing both the lambda abstraction and the application.
In our transformation system, this step helps to remove unwanted lambda
abstractions (basically all but the ones at the top level). Other
transformations (application propagation, non-representable inlining) make
sure that most lambda abstractions will eventually be reducible by
β-reduction.
Note that β-reduction also works on type lambda abstractions and type
applications as well. This means the substitution below also works on
type variables, in the case that the binder is a type variable and the
expression applied to is a type.
\starttrans
(λx.E) M
-----------------
E[x=>M]
\stoptrans
% And an example
\startbuffer[from]
(λa. 2 * a) (2 * b)
\stopbuffer
\startbuffer[to]
2 * (2 * b)
\stopbuffer
\transexample{beta}{β-reduction}{from}{to}
\startbuffer[from]
(λt.λa::t. a) @Int
\stopbuffer
\startbuffer[to]
(λa::Int. a)
\stopbuffer
\transexample{beta-type}{β-reduction for type abstractions}{from}{to}
\subsubsection{Unused let binding removal}
This transformation removes let bindings that are never used.
Occasionally, \GHC's desugarer introduces some unused let bindings.
This normalization pass should really be not be necessary to get
into intended normal form (since the intended normal form
definition \refdef{intended normal form definition} does not
require that every binding is used), but in practice the
desugarer or simplifier emits some bindings that cannot be
normalized (e.g., calls to a
\hs{Control.Exception.Base.patError}) but are not used anywhere
either. To prevent the \VHDL\ generation from breaking on these
artifacts, this transformation removes them.
\todo{Do not use old-style numerals in transformations}
\starttrans
letrec
a0 = E0
\vdots
ai = Ei
\vdots
an = En
in
M \lam{ai} does not occur free in \lam{M}
---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej})
letrec
a0 = E0
\vdots
ai-1 = Ei-1
ai+1 = Ei+1
\vdots
an = En
in
M
\stoptrans
% And an example
\startbuffer[from]
let
x = 1
in
2
\stopbuffer
\startbuffer[to]
let
in
2
\stopbuffer
\transexample{unusedlet}{Unused let binding removal}{from}{to}
\subsubsection{Empty let removal}
This transformation is simple: it removes recursive lets that have no bindings
(which usually occurs when unused let binding removal removes the last
binding from it).
Note that there is no need to define this transformation for
non-recursive lets, since they always contain exactly one binding.
\starttrans
letrec in M
--------------
M
\stoptrans
% And an example
\startbuffer[from]
let
in
2
\stopbuffer
\startbuffer[to]
2
\stopbuffer
\transexample{emptylet}{Empty let removal}{from}{to}
\subsubsection[sec:normalization:simplelet]{Simple let binding removal}
This transformation inlines simple let bindings, that bind some
binder to some other binder instead of a more complex expression (\ie\
a = b).
This transformation is not needed to get an expression into intended
normal form (since these bindings are part of the intended normal
form), but makes the resulting \small{VHDL} a lot shorter.
\refdef{substitution notation}
\starttrans
letrec
a0 = E0
\vdots
ai = b
\vdots
an = En
in
M
----------------------------- \lam{b} is a variable reference
letrec \lam{ai} ≠ \lam{b}
a0 = E0 [ai=>b]
\vdots
ai-1 = Ei-1 [ai=>b]
ai+1 = Ei+1 [ai=>b]
\vdots
an = En [ai=>b]
in
M[ai=>b]
\stoptrans
\todo{example}
\subsubsection{Cast propagation / simplification}
This transform pushes casts down into the expression as far as
possible. This transformation has been added to make a few
specific corner cases work, but it is not clear yet if this
transformation handles cast expressions completely or in the
right way. See \in{section}[sec:normalization:castproblems].
\starttrans
(let binds in E) ▶ T
-------------------------
let binds in (E ▶ T)
\stoptrans
\starttrans
(case S of
p0 -> E0
\vdots
pn -> En
) ▶ T
-------------------------
case S of
p0 -> E0 ▶ T
\vdots
pn -> En ▶ T
\stoptrans
\subsubsection{Top level binding inlining}
\refdef{top level binding}
This transform takes simple top level bindings generated by the
\small{GHC} compiler. \small{GHC} sometimes generates very simple
\quote{wrapper} bindings, which are bound to just a variable
reference, or contain just a (partial) function application with
the type and dictionary arguments filled in (such as the
\lam{(+)} in the example below).
Note that this transformation is completely optional. It is not
required to get any function into intended normal form, but it does help making
the resulting VHDL output easier to read (since it removes components
that do not add any real structure, but do hide away operations and
cause extra clutter).
This transform takes any top level binding generated by \GHC,
whose normalized form contains only a single let binding.
\starttrans
x = λa0 ... λan.let y = E in y
~
x
-------------------------------------- \lam{x} is generated by the compiler
λa0 ... λan.let y = E in y
\stoptrans
\startbuffer[from]
(+) :: Word -> Word -> Word
(+) = GHC.Num.(+) @Word \$dNum
~
(+) a b
\stopbuffer
\startbuffer[to]
GHC.Num.(+) @ Alu.Word \$dNum a b
\stopbuffer
\transexample{toplevelinline}{Top level binding inlining}{from}{to}
\in{Example}[ex:trans:toplevelinline] shows a typical application of
the addition operator generated by \GHC. The type and dictionary
arguments used here are described in
\in{Section}[sec:prototype:coretypes].
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 file names, so it effectively does not
work.}, so the entity would be called \quote{w7aA7f} or
something similarly meaningless and auto-generated).
\subsection{Program structure}
These transformations are aimed at normalizing the overall structure
into the intended form. This means ensuring there is a lambda abstraction
at the top for every argument (input port or current state), putting all
of the other value definitions in let bindings and making the final
return value a simple variable reference.
\subsubsection[sec:normalization:eta]{η-expansion}
This transformation makes sure that all arguments of a function-typed
expression are named, by introducing lambda expressions. When combined with
β-reduction and non-representable binding inlining, all function-typed
expressions should be lambda abstractions or global identifiers.
\starttrans
E \lam{E :: a -> b}
-------------- \lam{E} does not occur on a function position in an application
λx.E x \lam{E} is not a lambda abstraction.
\stoptrans
\startbuffer[from]
foo = λa.case a of
True -> λb.mul b b
False -> id
\stopbuffer
\startbuffer[to]
foo = λa.λx.(case a of
True -> λb.mul b b
False -> λy.id y) x
\stopbuffer
\transexample{eta}{η-expansion}{from}{to}
\subsubsection[sec:normalization:appprop]{Application propagation}
This transformation is meant to propagate application expressions downwards
into expressions as far as possible. This allows partial applications inside
expressions to become fully applied and exposes new transformation
opportunities for other transformations (like β-reduction and
specialization).
Since all binders in our expression are unique (see
\in{section}[sec:normalization:uniq]), there is no risk that we will
introduce unintended shadowing by moving an expression into a lower
scope. Also, since only move expression into smaller scopes (down into
our expression), there is no risk of moving a variable reference out
of the scope in which it is defined.
\starttrans
(letrec binds in E) M
------------------------
letrec binds in E M
\stoptrans
% And an example
\startbuffer[from]
( letrec
val = 1
in
add val
) 3
\stopbuffer
\startbuffer[to]
letrec
val = 1
in
add val 3
\stopbuffer
\transexample{appproplet}{Application propagation for a let expression}{from}{to}
\starttrans
(case x of
p0 -> E0
\vdots
pn -> En) M
-----------------
case x of
p0 -> E0 M
\vdots
pn -> En M
\stoptrans
% And an example
\startbuffer[from]
( case x of
True -> id
False -> neg
) 1
\stopbuffer
\startbuffer[to]
case x of
True -> id 1
False -> neg 1
\stopbuffer
\transexample{apppropcase}{Application propagation for a case expression}{from}{to}
\subsubsection[sec:normalization:letrecurse]{Let recursification}
This transformation makes all non-recursive lets recursive. In the
end, we want a single recursive let in our normalized program, so all
non-recursive lets can be converted. This also makes other
transformations simpler: they only need to be specified for recursive
let expressions (and simply will not apply to non-recursive let
expressions until this transformation has been applied).
\starttrans
let
a = E
in
M
------------------------------------------
letrec
a = E
in
M
\stoptrans
\subsubsection{Let flattening}
This transformation puts nested lets in the same scope, by lifting the
binding(s) of the inner let into the outer let. Eventually, this will
cause all let bindings to appear in the same scope.
This transformation only applies to recursive lets, since all
non-recursive lets will be made recursive (see
\in{section}[sec:normalization:letrecurse]).
Since we are joining two scopes together, there is no risk of moving a
variable reference out of the scope where it is defined.
\starttrans
letrec
a0 = E0
\vdots
ai = (letrec bindings in M)
\vdots
an = En
in
N
------------------------------------------
letrec
a0 = E0
\vdots
ai = M
\vdots
an = En
bindings
in
N
\stoptrans
\startbuffer[from]
letrec
a = 1
b = letrec
x = a
y = c
in
x + y
c = 2
in
b
\stopbuffer
\startbuffer[to]
letrec
a = 1
b = x + y
c = 2
x = a
y = c
in
b
\stopbuffer
\transexample{letflat}{Let flattening}{from}{to}
\subsubsection{Return value simplification}
This transformation ensures that the return value of a function is always a
simple local variable reference.
The basic idea of this transformation is to take the body of a
function and bind it with a let expression (so the body of that let
expression becomes a variable reference that can be used as the output
port). If the body of the function happens to have lambda abstractions
at the top level (which is allowed by the intended normal
form\refdef{intended normal form definition}), we take the body of the
inner lambda instead. If that happens to be a let expression already
(which is allowed by the intended normal form), we take the body of
that let (which is not allowed to be anything but a variable reference
according the the intended normal form).
This transformation uses the context conditions in a special way.
These contexts, like \lam{x = λv1 ... λvn.E}, are above the dotted
line and provide a condition on the environment (\ie\ they require a
certain top level binding to be present). These ensure that
expressions are only transformed when they are in the functions
\quote{return value} directly. This means the context conditions have
to interpreted in the right way: not \quote{if there is any function
\lam{x} that binds \lam{E}, any \lam{E} can be transformed}, but we
mean only the \lam{E} that is bound by \lam{x}).
Be careful when reading the transformations: Not the entire function
from the context is transformed, just a part of it.
Note that the return value is not simplified if it is not representable.
Otherwise, this would cause a loop with the inlining of
unrepresentable bindings in
\in{section}[sec:normalization:nonrepinline]. If the return value is
not representable because it has a function type, η-expansion should
make sure that this transformation will eventually apply. If the
value is not representable for other reasons, the function result
itself is not representable, meaning this function is not translatable
anyway.
\starttrans
x = λv1 ... λvn.E \lam{n} can be zero
~ \lam{E} is representable
E \lam{E} is not a lambda abstraction
--------------------------- \lam{E} is not a let expression
letrec y = E in y \lam{E} is not a local variable reference
\stoptrans
\starttrans
x = λv1 ... λvn.letrec binds in E \lam{n} can be zero
~ \lam{E} is representable
letrec binds in E \lam{E} is not a local variable reference
------------------------------------
letrec binds; y = E in y
\stoptrans
\startbuffer[from]
x = add 1 2
\stopbuffer
\startbuffer[to]
x = letrec y = add 1 2 in y
\stopbuffer
\transexample{retvalsimpl}{Return value simplification}{from}{to}
\startbuffer[from]
x = λa. add 1 a
\stopbuffer
\startbuffer[to]
x = λa. letrec
y = add 1 a
in
y
\stopbuffer
\transexample{retvalsimpllam}{Return value simplification with a lambda abstraction}{from}{to}
\startbuffer[from]
x = letrec
a = add 1 2
in
add a 3
\stopbuffer
\startbuffer[to]
x = letrec
a = add 1 2
y = add a 3
in
y
\stopbuffer
\transexample{retvalsimpllet}{Return value simplification with a let expression}{from}{to}
\subsection[sec:normalization:argsimpl]{Representable arguments simplification}
This section contains just a single transformation that deals with
representable arguments in applications. Non-representable arguments are
handled by the transformations in
\in{section}[sec:normalization:nonrep].
This transformation ensures that all representable arguments will become
references to local variables. This ensures they will become references
to local signals in the resulting \small{VHDL}, which is required due to
limitations in the component instantiation code in \VHDL\ (one can only
assign a signal or constant to an input port). By ensuring that all
arguments are always simple variable references, we always have a signal
available to map to the input ports.
To reduce a complex expression to a simple variable reference, we create
a new let expression around the application, which binds the complex
expression to a new variable. The original function is then applied to
this variable.
\refdef{global variable}
Note that references to \emph{global variables} (like a top level
function without arguments, but also an argumentless data-constructors
like \lam{True}) are also simplified. Only local variables generate
signals in the resulting architecture. Even though argumentless
data-constructors generate constants in generated \VHDL\ code and could be
mapped to an input port directly, they are still simplified to make the
normal form more regular.
\refdef{representable}
\starttrans
M N
-------------------- \lam{N} is representable
letrec x = N in M x \lam{N} is not a local variable reference
\stoptrans
\refdef{local variable}
\startbuffer[from]
add (add a 1) 1
\stopbuffer
\startbuffer[to]
letrec x = add a 1 in add x 1
\stopbuffer
\transexample{argsimpl}{Argument simplification}{from}{to}
\subsection[sec:normalization:built-ins]{Built-in functions}
This section deals with (arguments to) built-in functions. In the
intended normal form definition\refdef{intended normal form definition}
we can see that there are three sorts of arguments a built-in function
can receive.
\startitemize[KR]
\item A representable local variable reference. This is the most
common argument to any function. The argument simplification
transformation described in \in{section}[sec:normalization:argsimpl]
makes sure that \emph{any} representable argument to \emph{any}
function (including built-in functions) is turned into a local variable
reference.
\item (A partial application of) a top level function (either built-in on
user-defined). The function extraction transformation described in
this section takes care of turning every function-typed argument into
(a partial application of) a top level function.
\item Any expression that is not representable and does not have a
function type. Since these can be any expression, there is no
transformation needed. Note that this category is exactly all
expressions that are not transformed by the transformations for the
previous two categories. This means that \emph{any} Core expression
that is used as an argument to a built-in function will be either
transformed into one of the above categories, or end up in this
category. In any case, the result is in normal form.
\stopitemize
As noted, the argument simplification will handle any representable
arguments to a built-in function. The following transformation is needed
to handle non-representable arguments with a function type, all other
non-representable arguments do not need any special handling.
\subsubsection[sec:normalization:funextract]{Function extraction}
This transform deals with function-typed arguments to built-in
functions.
Since built-in functions cannot be specialized (see
\in{section}[sec:normalization:specialize]) to remove the arguments,
these arguments are extracted into a new global function instead. In
other words, we create a new top level function that has exactly the
extracted argument as its body. This greatly simplifies the
translation rules needed for built-in functions, since they only need
to handle (partial applications of) top level functions.
Any free variables occurring in the extracted arguments will become
parameters to the new global function. The original argument is replaced
with a reference to the new function, applied to any free variables from
the original argument.
This transformation is useful when applying higher-order built-in functions
like \hs{map} to a lambda abstraction, for example. In this case, the code
that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
partial applications, not any other expression (such as lambda abstractions or
even more complicated expressions).
\starttrans
M N \lam{M} is (a partial application of) a built-in function.
--------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
M (x f0 ... fn) \lam{N :: a -> b}
~ \lam{N} is not a (partial application of) a top level function
x = λf0 ... λfn.N
\stoptrans
\startbuffer[from]
addList = λb.λxs.map (λa . add a b) xs
\stopbuffer
\startbuffer[to]
addList = λb.λxs.map (f b) xs
~
f = λb.λa.add a b
\stopbuffer
\transexample{funextract}{Function extraction}{from}{to}
Note that the function \lam{f} will still need normalization after
this.
\subsection{Case normalization}
The transformations in this section ensure that case statements end up
in normal form.
\subsubsection{Scrutinee simplification}
This transform ensures that the scrutinee of a case expression is always
a simple variable reference.
\starttrans
case E of
alts
----------------- \lam{E} is not a local variable reference
letrec x = E in
case x of
alts
\stoptrans
\startbuffer[from]
case (foo a) of
True -> a
False -> b
\stopbuffer
\startbuffer[to]
letrec x = foo a in
case x of
True -> a
False -> b
\stopbuffer
\transexample{letflat}{Case normalization}{from}{to}
\placeintermezzo{}{
\defref{wild binders}
\startframedtext[width=7cm,background=box,frame=no]
\startalignment[center]
{\tfa Wild binders}
\stopalignment
\blank[medium]
In a functional expression, a \emph{wild binder} refers to any
binder that is never referenced. This means that even though it
will be bound to a particular value, that value is never used.
The Haskell syntax offers the underscore as a wild binder that
cannot even be referenced (It can be seen as introducing a new,
anonymous, binder every time it is used).
In these transformations, the term wild binder will sometimes be
used to indicate that a binder must not be referenced.
\stopframedtext
}
\subsubsection{Scrutinee binder removal}
This transformation removes (or rather, makes wild) the binder to
which the scrutinee is bound after evaluation. This is done by
replacing the bndr with the scrutinee in all alternatives. To prevent
duplication of work, this transformation is only applied when the
scrutinee is already a simple variable reference (but the previous
transformation ensures this will eventually be the case). The
scrutinee binder itself is replaced by a wild binder (which is no
longer displayed).
Note that one could argue that this transformation can change the
meaning of the Core expression. In the regular Core semantics, a case
expression forces the evaluation of its scrutinee and can be used to
implement strict evaluation. However, in the generated \VHDL,
evaluation is always strict. So the semantics we assign to the Core
expression (which differ only at this particular point), this
transformation is completely valid.
\starttrans
case x of bndr
alts
----------------- \lam{x} is a local variable reference
case x of
alts[bndr=>x]
\stoptrans
\startbuffer[from]
case x of y
True -> y
False -> not y
\stopbuffer
\startbuffer[to]
case x of
True -> x
False -> not x
\stopbuffer
\transexample{scrutbndrremove}{Scrutinee binder removal}{from}{to}
\subsubsection{Case normalization}
This transformation ensures that all case expressions get a form
that is allowed by the intended normal form. This means they
will become one of:
\startitemize
\item An extractor case with a single alternative that picks a field
from a datatype, \eg\ \lam{case x of (a, b) ->
a}.\defref{extractor case}
\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}.\defref{selector case}
\stopitemize
For an arbitrary case, that has \lam{n} alternatives, with
\lam{m} binders in each alternatives, this will result in \lam{m
* n} extractor case expression to get at each variable, \lam{n}
let bindings for each of the alternatives' value and a single
selector case to select the right value out of these.
Technically, the definition of this transformation would require
that the constructor for every alternative has exactly the same
amount (\lam{m}) of arguments, but of course this transformation
also applies when this is not the case.
\starttrans
case E of
C0 v0,0 ... v0,m -> E0
\vdots
Cn vn,0 ... vn,m -> En
--------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
letrec The case expression is not an extractor case
v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
\vdots
v0,m = case E of C0 x0,0 .. x0,m -> x0,m
\vdots
vn,m = case E of Cn xn,0 .. xn,m -> xn,m
y0 = E0
\vdots
yn = En
in
case E of
C0 w0,0 ... w0,m -> y0
\vdots
Cn wn,0 ... wn,m -> yn
\stoptrans
Note that this transformation applies to case expressions with any
scrutinee. If the scrutinee is a complex expression, this might
result in duplication of work (hardware). An extra condition to
only apply this transformation when the scrutinee is already
simple (effectively causing this transformation to be only
applied after the scrutinee simplification transformation) might
be in order.
\startbuffer[from]
case a of
True -> add b 1
False -> add b 2
\stopbuffer
\startbuffer[to]
letrec
x0 = add b 1
x1 = add b 2
in
case a of
True -> x0
False -> x1
\stopbuffer
\transexample{selcasesimpl}{Selector case simplification}{from}{to}
\startbuffer[from]
case a of
(,) b c -> add b c
\stopbuffer
\startbuffer[to]
letrec
b = case a of (,) b c -> b
c = case a of (,) b c -> c
x0 = add b c
in
case a of
(,) w0 w1 -> x0
\stopbuffer
\transexample{excasesimpl}{Extractor case simplification}{from}{to}
\refdef{selector case}
In \in{example}[ex:trans:excasesimpl] the case expression is expanded
into multiple case expressions, including a pretty useless expression
(that is neither a selector or extractor case). This case can be
removed by the Case removal transformation in
\in{section}[sec:transformation:caseremoval].
\subsubsection[sec:transformation:caseremoval]{Case removal}
This transform removes any case expression with a single alternative and
only wild binders.\refdef{wild binders}
These "useless" case expressions are usually leftovers from case simplification
on extractor case (see the previous example).
\starttrans
case x of
C v0 ... vm -> E
---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
E
\stoptrans
\startbuffer[from]
case a of
(,) w0 w1 -> x0
\stopbuffer
\startbuffer[to]
x0
\stopbuffer
\transexample{caserem}{Case removal}{from}{to}
\subsection[sec:normalization:nonrep]{Removing unrepresentable values}
The transformations in this section are aimed at making all the
values used in our expression representable. There are two main
transformations that are applied to \emph{all} unrepresentable let
bindings and function arguments. These are meant to address three
different kinds of unrepresentable values: polymorphic values,
higher-order values and literals. The transformation are described
generically: they apply to all non-representable values. However,
non-representable values that do not fall into one of these three
categories will be moved around by these transformations but are
unlikely to completely disappear. They usually mean the program was not
valid in the first place, because unsupported types were used (for
example, a program using strings).
Each of these three categories will be detailed below, followed by the
actual transformations.
\subsubsection{Removing Polymorphism}
As noted in \in{section}[sec:prototype:coretypes],
polymorphism is made explicit in Core through type and
dictionary arguments. To remove the polymorphism from a
function, we can simply specialize the polymorphic function for
the particular type applied to it. The same goes for dictionary
arguments. To remove polymorphism from let bound values, we
simply inline the let bindings that have a polymorphic type,
which should (eventually) make sure that the polymorphic
expression is applied to a type and/or dictionary, which can
then be removed by β-reduction (\in{section}[sec:normalization:beta]).
Since both type and dictionary arguments are not representable,
\refdef{representable}
the non-representable argument specialization and
non-representable let binding inlining transformations below
take care of exactly this.
There is one case where polymorphism cannot be completely
removed: built-in functions are still allowed to be polymorphic
(Since we have no function body that we could properly
specialize). However, the code that generates \VHDL\ for built-in
functions knows how to handle this, so this is not a problem.
\subsubsection[sec:normalization:defunctionalization]{Defunctionalization}
These transformations remove higher-order expressions from our
program, making all values first-order. The approach used for
defunctionalization uses a combination of specialization, inlining and
some cleanup transformations, was also proposed in parallel research
by Neil Mitchell \cite[mitchell09].
Higher order values are always introduced by lambda abstractions, none
of the other Core expression elements can introduce a function type.
However, other expressions can \emph{have} a function type, when they
have a lambda expression in their body.
For example, the following expression is a higher-order expression
that is not a lambda expression itself:
\refdef{id function}
\startlambda
case x of
High -> id
Low -> λx.x
\stoplambda
The reference to the \lam{id} function shows that we can introduce a
higher-order expression in our program without using a lambda
expression directly. However, inside the definition of the \lam{id}
function, we can be sure that a lambda expression is present.
Looking closely at the definition of our normal form in
\in{section}[sec:normalization:intendednormalform], we can see that
there are three possibilities for higher-order values to appear in our
intended normal form:
\startitemize[KR]
\item[item:toplambda] Lambda abstractions can appear at the highest level of a
top level function. These lambda abstractions introduce the
arguments (input ports / current state) of the function.
\item[item:built-inarg] (Partial applications of) top level functions can appear as an
argument to a built-in function.
\item[item:completeapp] (Partial applications of) top level functions can appear in
function position of an application. Since a partial application
cannot appear anywhere else (except as built-in function arguments),
all partial applications are applied, meaning that all applications
will become complete applications. However, since application of
arguments happens one by one, in the expression:
\startlambda
f 1 2
\stoplambda
the sub-expression \lam{f 1} has a function type. But this is
allowed, since it is inside a complete application.
\stopitemize
We will take a typical function with some higher-order values as an
example. The following function takes two arguments: a \lam{Bit} and a
list of numbers. Depending on the first argument, each number in the
list is doubled, or the list is returned unmodified. For the sake of
the example, no polymorphism is shown. In reality, at least map would
be polymorphic.
\startlambda
λy.let double = λx. x + x in
case y of
Low -> map double
High -> λz. z
\stoplambda
This example shows a number of higher-order values that we cannot
translate to \VHDL\ directly. The \lam{double} binder bound in the let
expression has a function type, as well as both of the alternatives of
the case expression. The first alternative is a partial application of
the \lam{map} built-in function, whereas the second alternative is a
lambda abstraction.
To reduce all higher-order values to one of the above items, a number
of transformations we have already seen are used. The η-expansion
transformation from \in{section}[sec:normalization:eta] ensures all
function arguments are introduced by lambda abstraction on the highest
level of a function. These lambda arguments are allowed because of
\in{item}[item:toplambda] above. After η-expansion, our example
becomes a bit bigger:
\startlambda
λy.λq.(let double = λx. x + x in
case y of
Low -> map double
High -> λz. z
) q
\stoplambda
η-expansion also introduces extra applications (the application of
the let expression to \lam{q} in the above example). These
applications can then propagated down by the application propagation
transformation (\in{section}[sec:normalization:appprop]). In our
example, the \lam{q} and \lam{r} variable will be propagated into the
let expression and then into the case expression:
\startlambda
λy.λq.let double = λx. x + x in
case y of
Low -> map double q
High -> (λz. z) q
\stoplambda
This propagation makes higher-order values become applied (in
particular both of the alternatives of the case now have a
representable type). Completely applied top level functions (like the
first alternative) are now no longer invalid (they fall under
\in{item}[item:completeapp] above). (Completely) applied lambda
abstractions can be removed by β-expansion. For our example,
applying β-expansion results in the following:
\startlambda
λy.λq.let double = λx. x + x in
case y of
Low -> map double q
High -> q
\stoplambda
As you can see in our example, all of this moves applications towards
the higher-order values, but misses higher-order functions bound by
let expressions. The applications cannot be moved towards these values
(since they can be used in multiple places), so the values will have
to be moved towards the applications. This is achieved by inlining all
higher-order values bound by let applications, by the
non-representable binding inlining transformation below. When applying
it to our example, we get the following:
\startlambda
λy.λq.case y of
Low -> map (λx. x + x) q
High -> q
\stoplambda
We have nearly eliminated all unsupported higher-order values from this
expressions. The one that is remaining is the first argument to the
\lam{map} function. Having higher-order arguments to a built-in
function like \lam{map} is allowed in the intended normal form, but
only if the argument is a (partial application) of a top level
function. This is easily done by introducing a new top level function
and put the lambda abstraction inside. This is done by the function
extraction transformation from
\in{section}[sec:normalization:funextract].
\startlambda
λy.λq.case y of
Low -> map func q
High -> q
\stoplambda
This also introduces a new function, that we have called \lam{func}:
\startlambda
func = λx. x + x
\stoplambda
Note that this does not actually remove the lambda, but now it is a
lambda at the highest level of a function, which is allowed in the
intended normal form.
There is one case that has not been discussed yet. What if the
\lam{map} function in the example above was not a built-in function
but a user-defined function? Then extracting the lambda expression
into a new function would not be enough, since user-defined functions
can never have higher-order arguments. For example, the following
expression shows an example:
\startlambda
twice :: (Word -> Word) -> Word -> Word
twice = λf.λa.f (f a)
main = λa.app (λx. x + x) a
\stoplambda
This example shows a function \lam{twice} that takes a function as a
first argument and applies that function twice to the second argument.
Again, we have made the function monomorphic for clarity, even though
this function would be a lot more useful if it was polymorphic. The
function \lam{main} uses \lam{twice} to apply a lambda expression twice.
When faced with a user defined function, a body is available for that
function. This means we could create a specialized version of the
function that only works for this particular higher-order argument
(\ie, we can just remove the argument and call the specialized
function without the argument). This transformation is detailed below.
Applying this transformation to the example gives:
\startlambda
twice' :: Word -> Word
twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
main = λa.app' a
\stoplambda
The \lam{main} function is now in normal form, since the only
higher-order value there is the top level lambda expression. The new
\lam{twice'} function is a bit complex, but the entire original body
of the original \lam{twice} function is wrapped in a lambda
abstraction and applied to the argument we have specialized for
(\lam{λx. x + x}) and the other arguments. This complex expression can
fortunately be effectively reduced by repeatedly applying β-reduction:
\startlambda
twice' :: Word -> Word
twice' = λb.(b + b) + (b + b)
\stoplambda
This example also shows that the resulting normal form might not be as
efficient as we might hope it to be (it is calculating \lam{(b + b)}
twice). This is discussed in more detail in
\in{section}[sec:normalization:duplicatework].
\subsubsection[sec:normalization:literals]{Literals}
There are a limited number of literals available in Haskell and Core.
\refdef{enumerated types} When using (enumerating) algebraic
data-types, a literal is just a reference to the corresponding data
constructor, which has a representable type (the algebraic datatype)
and can be translated directly. This also holds for literals of the
\hs{Bool} Haskell type, which is just an enumerated type.
There is, however, a second type of literal that does not have a
representable type: integer literals. Cλash supports using integer
literals for all three integer types supported (\hs{SizedWord},
\hs{SizedInt} and \hs{RangedWord}). This is implemented using
Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
that converts any \hs{Integer} to the Cλash data-types.
When \GHC\ sees integer literals, it will automatically insert calls to
the \hs{fromInteger} method in the resulting Core expression. For
example, the following expression in Haskell creates a 32 bit unsigned
word with the value 1. The explicit type signature is needed, since
there is no context for \GHC\ to determine the type from otherwise.
\starthaskell
1 :: SizedWord D32
\stophaskell
This Haskell code results in the following Core expression:
\startlambda
fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
\stoplambda
The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
\lam{fromInteger} function will finally convert this into a
\lam{SizedWord D32}.
Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
representable, and cannot be translated directly. Fortunately, there
is no need to translate them, since \lam{fromInteger} is a built-in
function that knows how to handle these values. However, this does
require that the \lam{fromInteger} function is directly applied to
these non-representable literal values, otherwise errors will occur.
For example, the following expression is not in the intended normal
form, since one of the let bindings has an unrepresentable type
(\lam{Integer}):
\startlambda
let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
\stoplambda
By inlining these let-bindings, we can ensure that unrepresentable
literals bound by a let binding end up in an application of the
appropriate built-in function, where they are allowed. Since it is
possible that the application of that function is in a different
function than the definition of the literal value, we will always need
to specialize away any unrepresentable literals that are used as
function arguments. The following two transformations do exactly this.
\subsubsection[sec:normalization:nonrepinline]{Non-representable binding inlining}
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.
As we have seen in the previous sections, inlining these bindings
solves (part of) the polymorphism, higher-order values and
unrepresentable literals in an expression.
\refdef{substitution notation}
\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}{Non-representable binding inlining}{from}{to}
\subsubsection[sec:normalization:specialize]{Function specialization}
This transform removes arguments to user-defined functions that are
not representable at run-time. This is done by creating a
\emph{specialized} version of the function that only works for one
particular value of that argument (in other words, the argument can be
removed).
Specialization means to create a specialized version of the called
function, with one argument already filled in. As a simple example, in
the following program (this is not actual Core, since it directly uses
a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
\startlambda
f = λa.λb.a + b
inc = λa.f a 1
\stoplambda
We could specialize the function \lam{f} against the literal argument
1, with the following result:
\startlambda
f' = λa.a + 1
inc = λa.f' a
\stoplambda
In some way, this transformation is similar to β-reduction, but it
operates across function boundaries. It is also similar to
non-representable let binding inlining above, since it sort of
\quote{inlines} an expression into a called function.
Special care must be taken when the argument 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.
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 run-time
representable cannot be brought into normal form by this transform. We
rely on an inlining or β-reduction transformation to replace such a
variable with an expression we can propagate again.
\starttrans
x = E
~
x Y0 ... Yi ... Yn \lam{Yi} is not representable
--------------------------------------------- \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}
~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
λf0 ... λfm.
λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
E y0 ... yi-1 Yi yi+1 ... yn
\stoptrans
This is a bit of a complex transformation. It transforms an
application of the function \lam{x}, where one of the arguments
(\lam{Y_i}) is not representable. A new
function \lam{x'} is created that wraps the body of the old function.
The body of the new function becomes a number of nested lambda
abstractions, one for each of the original arguments that are left
unchanged.
The ith argument is replaced with the free variables of
\lam{Y_i}. Note that we reuse the same binders as those used in
\lam{Y_i}, since we can then just use \lam{Y_i} inside the new
function body and have all of the variables it uses be in scope.
The argument that we are specializing for, \lam{Y_i}, is put inside
the new function body. The old function body is applied to it. Since
we use this new function only in place of an application with that
particular argument \lam{Y_i}, behavior should not change.
Note that the types of the arguments of our new function are taken
from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
means that any polymorphism in the arguments is removed, even when the
corresponding explicit type lambda is not removed
yet.
\todo{Examples. Perhaps reference the previous sections}
\section{Unsolved problems}
The above system of transformations has been implemented in the prototype
and seems to work well to compile simple and more complex examples of
hardware descriptions. \todo{Ref christiaan?} However, this normalization
system has not seen enough review and work to be complete and work for
every Core expression that is supplied to it. A number of problems
have already been identified and are discussed in this section.
\subsection[sec:normalization:duplicatework]{Work duplication}
A possible problem of β-reduction is that it could duplicate work.
When the expression applied is not a simple variable reference, but
requires calculation and the binder the lambda abstraction binds to
is used more than once, more hardware might be generated than strictly
needed.
As an example, consider the expression:
\startlambda
(λx. x + x) (a * b)
\stoplambda
When applying β-reduction to this expression, we get:
\startlambda
(a * b) + (a * b)
\stoplambda
which of course calculates \lam{(a * b)} twice.
A possible solution to this would be to use the following alternative
transformation, which is of course no longer normal β-reduction. The
following transformation has not been tested in the prototype, but is
given here for future reference:
\starttrans
(λx.E) M
-----------------
letrec x = M in E
\stoptrans
This does not seem like much of an improvement, but it does get rid of
the lambda expression (and the associated higher-order value), while
at the same time introducing a new let binding. Since the result of
every application or case expression must be bound by a let expression
in the intended normal form anyway, this is probably not a problem. If
the argument happens to be a variable reference, then simple let
binding removal (\in{section}[sec:normalization:simplelet]) will
remove it, making the result identical to that of the original
β-reduction transformation.
When also applying argument simplification to the above example, we
get the following expression:
\startlambda
let y = (a * b)
z = (a * b)
in y + z
\stoplambda
Looking at this, we could imagine an alternative approach: create a
transformation that removes let bindings that bind identical values.
In the above expression, the \lam{y} and \lam{z} variables could be
merged together, resulting in the more efficient expression:
\startlambda
let y = (a * b) in y + y
\stoplambda
\subsection[sec:normalization:non-determinism]{Non-determinism}
As an example, again consider the following expression:
\startlambda
(λx. x + x) (a * b)
\stoplambda
We can apply both β-reduction (\in{section}[sec:normalization:beta])
as well as argument simplification
(\in{section}[sec:normalization:argsimpl]) to this expression.
When applying argument simplification first and then β-reduction, we
get the following expression:
\startlambda
let y = (a * b) in y + y
\stoplambda
When applying β-reduction first and then argument simplification, we
get the following expression:
\startlambda
let y = (a * b)
z = (a * b)
in y + z
\stoplambda
As you can see, this is a different expression. This means that the
order of expressions, does in fact change the resulting normal form,
which is something that we would like to avoid. In this particular
case one of the alternatives is even clearly more efficient, so we
would of course like the more efficient form to be the normal form.
For this particular problem, the solutions for duplication of work
seem from the previous section seem to fix the determinism of our
transformation system as well. However, it is likely that there are
other occurrences of this problem.
\subsection[sec:normalization:castproblems]{Casts}
We do not fully understand the use of cast expressions in Core, so
there are probably expressions involving cast expressions that cannot
be brought into intended normal form by this transformation system.
The uses of casts in the Core system should be investigated more and
transformations will probably need updating to handle them in all
cases.
\subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
Currently, the intended normal form definition\refdef{intended
normal form definition} offers enough freedom to describe all
valid stateful descriptions, but is not limiting enough. It is
possible to write descriptions which are in intended normal
form, but cannot be translated into \VHDL\ in a meaningful way
(\eg, a function that swaps two substates in its result, or a
function that changes a sub-state itself instead of passing it to
a sub-function).
It is now up to the programmer to not do anything funny with
these state values, whereas the normalization just tries not to
mess up the flow of state values. In practice, there are
situations where a Core program that \emph{could} be a valid
stateful description is not translatable by the prototype. This
most often happens when statefulness is mixed with pattern
matching, causing a state input to be unpacked multiple times or
be unpacked and repacked only in some of the code paths.
Without going into detail about the exact problems (of which
there are probably more than have shown up so far), it seems
unlikely that these problems can be solved entirely by just
improving the \VHDL\ state generation in the final stage. The
normalization stage seems the best place to apply the rewriting
needed to support more complex stateful descriptions. This does
of course mean that the intended normal form definition must be
extended as well to be more specific about how state handling
should look like in normal form.
\in{Section}[sec:prototype:statelimits] already contains a
tight description of the limitations on the use of state
variables, which could be adapted into the intended normal form.
\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 sub-questions:
\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 (see
\in{section}[sec:normalization:non-determinism] and
\in{section}[sec:normalization:stateproblems]) 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 also 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 will look at
transformation systems from a graph perspective. We will first define
the graph view and then illustrate it using a simple example from lambda
calculus (which is a different system than the Cλash normalization
system). 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 the graph for Cλash 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
this 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?}
In such a graph a node (expression) is in normal form if it has no
outgoing edges (meaning no transformation applies to it). The set of
nodes without outgoing edges is called the \emph{normal set}. Similarly,
the set of nodes containing expressions in intended normal form
\refdef{intended normal form definition} 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 walk (sequence of
edges, or transformations) 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 have limited the possible
expressions. In complete 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.
\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. A definition of the operational semantics of \GHC's Core
language is available \cite[sulzmann07], but this does not seem
sufficient for our goals (but it is a good start).
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
sub-expression as well, there is a constraint on our meaning
definition: the meaning of an expression should depend only on the
meaning of sub-expressions, 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 sub-expression 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}
\subsection{Determinism}
A well-known technique for proving determinism in lambda calculus
and other reduction systems, is using the Church-Rosser property
\cite[church36]. A reduction system has the CR property if and only if:
\placedefinition[here]{Church-Rosser theorem}
{\lam{\forall A, B, C \exists D (A ->> B ∧ A ->> C => B ->> D ∧ C ->> D)}}
Here, \lam{A ->> B} means \lam{A} \emph{reduces to} \lam{B}. In
other words, there is a set of transformations that can transform
\lam{A} to \lam{B}. \lam{=>} is used to mean \emph{implies}.
For a transformation system holding the Church-Rosser property, it
is easy to show that it is in fact deterministic. Showing that this
property actually holds is a harder problem, but has been
done for some reduction systems in the lambda calculus
\cite[klop80]\ \cite[barendregt84]. Doing the same for our
transformation system is probably more complicated, but not
impossible.
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