From: Matthijs Kooijman
Date: Wed, 1 Jul 2009 14:51:35 +0000 (+0200)
Subject: Update and/or remove older text.
XGitTag: finalthesis~316
XGitUrl: https://git.stderr.nl/gitweb?p=matthijs%2Fmasterproject%2Freport.git;a=commitdiff_plain;h=204e3063cbdc2825e3f78ae0261dbf30d4cf38e0
Update and/or remove older text.

diff git a/Core2Core.tex b/Core2Core.tex
index ae9c189..1f258d3 100644
 a/Core2Core.tex
+++ b/Core2Core.tex
@@ 140,42 +140,31 @@ Matthijs Kooijman
\section{Introduction}
As a new approach to translating Core to VHDL, we investigate a number of
transformations on our Core program, which should bring the program into a
welldefined "canonical" form, which is subsequently trivial to translate to
VHDL.

The transformations as presented here are far from complete, but are meant as
an exploration of possible transformations. In the running example below, we
apply each of the transformations exactly once, in sequence. As will be
apparent from the end result, there will be additional transformations needed
to fully reach our goal, and some transformations must be applied more than
once. How exactly to (efficiently) do this, has not been investigated.

Lastly, I hope to be able to state a number of pre and postconditions for
each transformation. If these can be proven for each transformation, and it
can be shown that there exists some ordering of transformations for which the
postcondition implies the canonical form, we can show that the transformations
do indeed transform any program (probably satisfying a number of
preconditions) to the canonical form.
+transforms on our Core program, which should bring the program into a
+welldefined {\em normal} form, which is subsequently trivial to
+translate to VHDL.
+
+The transforms as presented here are far from complete, but are meant as
+an exploration of possible transformations.
\section{Goal}
The transformations described here have a welldefined goal: To bring the
program in a welldefined form that is directly translatable to hardware,
while fully preserving the semantics of the program.
This {\em canonical form} is again a Core program, but with a very specific
structure. A function in canonical form has nested lambda's at the top, which
+This {\em normal form} is again a Core program, but with a very specific
+structure. A function in normal form has nested lambda's at the top, which
produce a let expression. This let expression binds every function application
in the function and produces either a simple identifier or a tuple of
identifiers. Every bound value in the let expression is either a simple
function application or a case expression to extract a single element from a
tuple returned by a function.
+in the function and produces a simple identifier. Every bound value in
+the let expression is either a simple function application or a case
+expression to extract a single element from a tuple returned by a
+function.
An example of a program in canonical form would be:
\starttyping
+\startlambda
 All arguments are an inital lambda
 \x c d >
+ Î»x.Î»c.Î»d.
 There is one let expression at the top level
let
 Calling some other userdefined function.
@@ 196,29 +185,21 @@ An example of a program in canonical form would be:
in
 The actual result
r
\stoptyping

In this and all following programs, the following definitions are assumed to
be available:

\starttyping
data Bit = Low  High

foo :: Int > (Bit, Bit)
add :: Int > Int > Int
sub :: Int > Int > Int
\stoptyping
+\stoplambda
When looking at such a program from a hardware perspective, the top level
lambda's define the input ports. The value produced by the let expression are
the output ports. Each function application bound by the let expression
defines a component instantiation, where the input and output ports are mapped
to local signals or arguments. The tuple extracting case expressions don't map
to

\subsection{Canonical form definition}
Formally, the canonical form is a core program obeying the following
constraints.
+lambda's define the input ports. The value produced by the let expression is
+the output port. Most function applications bound by the let expression
+define a component instantiation, where the input and output ports are mapped
+to local signals or arguments. Some of the others use a builtin
+construction (\eg the \lam{case} statement) or call a builtin function
+(\eg \lam{add} or \lam{sub}). For these, a hardcoded VHDL translation is
+available.
+
+\subsection{Normal definition}
+Formally, the normal form is a core program obeying the following
+constraints. TODO: Update this section, this is probably not completely
+accurate or relevant anymore.
\startitemize[R,inmargin]
\item All top level binds must have the form $\expr{\bind{fun}{lamexpr}}$.
@@ 260,47 +241,24 @@ be any of $x_0$ to $x_n$. A case expression must have a $hardware\;type$.
\item TODO: Many more!
\stopitemize
\section{Transformation passes}
+\section{Transform passes}
In this section we describe the actual transformations. Here we're using
mostly Corelike notation, with a few notable points.

\startitemize
\item A core expression (in contrast with a transformation function, for
example), is enclosed in pipes. For example, $\app{transform}{\expr{\lam{z}{z+1}}}$
is the transform function applied to a lambda core expression.

Note that this notation might not be consistently applied everywhere. In
particular where a noncore function is used inside a core expression, things
get slightly confusing.
\item A bind is written as $\expr{\bind{x}{expr}}$. This means binding the identifier
$x$ to the expression $expr$.
\item In the core examples, the layout rule from Haskell is loosely applied.
It should be evident what was meant from indentation, even though it might nog
be strictly correct.
\stopitemize
+In this section we describe the actual transforms. Here we're using
+the core language in a notation that resembles lambda calculus.
\subsection{Example}
In the descriptions of transformations below, the following (slightly
contrived) example program will be used as the running example. Note that this
the example for the canonical form given above is the same program, but in
canonical form.
+Each of these transforms is meant to be applied to every (sub)expression
+in a program, for as long as it applies. Only when none of the
+expressions can be applied anymore, the program is in normal form. We
+hope to be able to prove that this form will obey all of the constraints
+defined above, but this has yet to happen (though it seems likely that
+it will).
\starttyping
 \x >
 let s = foo x
 in
 case s of
 (a, b) >
 case a of
 High > add
 Low > let
 op' = case b of
 High > sub
 Low > \c d > c
 in
 \c d > op' d c
\stoptyping
+Each of the transforms will be described informally first, explaining
+the need for and goal of the transform. Then, a formal definition is
+given, using a familiar syntax from the world of logic. Each transform
+is specified as a number of conditions (above the horizontal line) and a
+number of conclusions (below the horizontal line). The details of using
+this notation are still a bit fuzzy, so comments are welcom.
\subsection{Î·abstraction}
This transformation makes sure that all arguments of a functiontyped
@@ 642,124 +600,14 @@ translatable. A userdefined function is any other function.
TODO: The above definition looks too complicated... Can we find
something more concise?
\subsection{Introducing main scope}
This transformation is meant to introduce a single let expression that will be
the "main scope". This is the let expression as described under requirement
\ref[letexpr]. This let expression will contain only a single binding, which
binds the original expression to some identifier and then evalutes to just
this identifier (to comply with requirement \in[retexpr]).

Formally, we can describe the transformation as follows.

\transformold{Main scope introduction}
{
\NC \app{transform}{\expr{\bind{f}{expr}}} \NC = \expr{\bind{f}{\app{transform'(expr)}}}\NR
\NR
\NC \app{transform'}{\expr{\lam{v}{expr}}} \NC = \expr{\lam{v}{\app{transform'}{expr}}}\NR
\NC \app{transform'}{\expr{expr}} \NC = \expr{\letexpr{\bind{x}{expr}}{x}} \NR
}

When applying this transformation to our running example, we get the following
program.

\starttyping
 \x c d >
 let r = (let s = foo x
 in
 case s of
 (a, b) >
 case a of
 High > add c d
 Low > let
 op' = case b of
 High > sub
 Low > \c d > c
 in
 op' d c
 )
 in
 r
\stoptyping

\subsection{Scope flattening}
This transformation tries to flatten the topmost let expression in a bind,
{\em i.e.}, bind all identifiers in the same scope, and bind them to simple
expressions only (so simplify deeply nested expressions).

Formally, we can describe the transformation as follows.

\transformold{Main scope introduction} { \NC \app{transform}{\expr{\bind{f}{expr}}} \NC = \expr{\bind{f}{\app{transform'(expr)}}}\NR
\NR
\NC \app{transform'}{\expr{\lam{v}{expr}}} \NC = \expr{\lam{v}{\app{transform'}{expr}}}\NR
\NC \app{transform'}{\expr{\letexpr{binds}{expr}}} \NC = \expr{\letexpr{\app{concat . map . flatten}{binds}}{expr}}\NR
\NR
\NC \app{flatten}{\expr{\bind{x}{\letexpr{binds}{expr}}}} \NC = (\app{concat . map . flatten}{binds}) \cup \{\app{flatten}{\expr{\bind{x}{expr}}}\}\NR
\NC \app{flatten}{\expr{\bind{x}{\case{s}{alts}}}} \NC = \app{concat}{binds'} \cup \{\bind{x}{\case{s}{alts'}}\}\NR
\NC \NC \where{(binds', alts')=\app{unzip.map.(flattenalt s)}{alts}}\NR
\NC \app{\app{flattenalt}{s}}{\expr{\alt{\app{con}{x_0\;..\;x_n}}{expr}}} \NC = (extracts \cup \{\app{flatten}{bind}\}, alt)\NR
\NC \NC \where{extracts =\{\expr{\case{s}{\alt{\app{con}{x_0\;..\;x_n}}{x_0}}},}\NR
\NC \NC \;..\;,\expr{\case{s}{\alt{\app{con}{x_0\;..\;x_n}}{x_n}}}\} \NR
\NC \NC bind = \expr{\bind{y}{expr}}\NR
\NC \NC alt = \expr{\alt{\app{con}{\_\;..\;\_}}{y}}\NR
}

When applying this transformation to our running example, we get the following
program.

\starttyping
 \x c d >
 let s = foo x
 r = case s of
 (_, _) > y
 a = case s of (a, b) > a
 b = case s of (a, b) > b
 y = case a of
 High > g
 Low > h
 g = add c d
 h = op' d c
 op' = case b of
 High > i
 Low > j
 i = sub
 j = \c d > c
 in
 r
\stoptyping
+\subsection{Example sequence}
\subsection{More transformations}
As noted before, the above transformations are not complete. Other needed
transformations include:
\startitemize
\item Inlining of local identifiers with a function type. Since these cannot
be represented in hardware directly, they must be transformed into something
else. Inlining them should always be possible without loss of semantics (TODO:
How true is this?) and can expose new possibilities for other transformations
passes (such as application propagation when inlining {\tt j} above).
\item A variation on inlining local identifiers is the propagation of
function arguments with a function type. This will probably be initiated when
transforming the caller (and not the callee), but it will also deal with
identifiers with a function type that are unrepresentable in hardware.

Special care must be taken here, since the expression that is propagated into
the callee comes from a different scope. The function typed argument must thus
be replaced by any identifiers from the callers scope that the propagated
expression uses.

Note that propagating an argument will change both a function's interface and
implementation. For this to work, a new function should be created instead of
modifying the original function, so any other callers will not be broken.
\item Something similar should happen with return values with function types.
\item Polymorphism must be removed from all userdefined functions. This is
again similar to propagation function typed arguments, since this will also
create duplicates of functions (for a specific type). This is an operation
that is commonly known as "specialization" and already happens in GHC (since
nonpolymorph functions can be a lot faster than generic ones).
\item More builtin expressions should be added and these should be evaluated
by the compiler. For example, map, fold, +.
\stopitemize
+This section lists an example expression, with a sequence of transforms
+applied to it. The exact transforms given here probably don't exactly
+match the transforms given above anymore, but perhaps this can clarify
+the big picture a bit.
Initial example
+TODO: Update or remove this section.
\startlambda
Î»x.