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\fnote{\copyright{} Christian Urban, King's College London, 2019}


% CPS translations
% https://felleisen.org/matthias/4400-s20/lecture17.html

%% pattern matching resources
%% https://www.reddit.com/r/ProgrammingLanguages/comments/g1vno3/beginner_resources_for_compiling_pattern_matching/

\section*{Handout 9 (LLVM, SSA and CPS)}

Reflecting on our two tiny compilers targetting the JVM, the code
generation part was actually not so hard, no? Pretty much just some
post-traversal of the abstract syntax tree, yes? One of the reasons for
this ease is that the JVM is a stack-based virtual machine and it is
therefore not hard to translate deeply-nested arithmetic expressions
into a sequence of instructions manipulating the stack. The problem is
that ``real'' CPUs, although supporting stack operations, are not really
designed to be \emph{stack machines}.  The design of CPUs is more like,
here is a chunk of memory---compiler, or better compiler writers, do
something with it. Consequently, modern compilers need to go the extra
mile in order to generate code that is much easier and faster to process
by CPUs. To make this all tractable for this module, we target the LLVM
Intermediate Language. In this way we can take advantage of the tools
coming with LLVM. For example we do not have to worry about things like
register allocations.\bigskip 

\noindent LLVM\footnote{\url{http://llvm.org}} is a beautiful example
that projects from Academia can make a difference in the World. LLVM
started in 2000 as a project by two researchers at the  University of
Illinois at Urbana-Champaign. At the time the behemoth of compilers was
gcc with its myriad of front-ends for other languages (C++, Fortran,
Ada, Go, Objective-C, Pascal etc). The problem was that gcc morphed over
time into a monolithic gigantic piece of m\ldots ehm software, which you
could not mess about in an afternoon. In contrast, LLVM is designed to
be a modular suite of tools with which you can play around easily and
try out something new. LLVM became a big player once Apple hired one of
the original developers (I cannot remember the reason why Apple did not
want to use gcc, but maybe they were also just disgusted by gcc's big
monolithic codebase). Anyway, LLVM is now the big player and gcc is more
or less legacy. This does not mean that programming languages like C and
C++ are dying out any time soon---they are nicely supported by LLVM.

We will target the LLVM Intermediate Language, or LLVM Intermediate
Representation (short LLVM-IR). The LLVM-IR looks very similar to the
assembly language of Jasmin and Krakatau. It will also allow us to
benefit from the modular structure of the LLVM compiler and let for
example the compiler generate code for different CPUs, like X86 or ARM.
That means we can be agnostic about where our code actually runs. We can
also be ignorant about optimising code and allocating memory
efficiently. 

However, what we have to do for LLVM is to generate code in \emph{Static
Single-Assignment} format (short SSA), because that is what the LLVM-IR
expects from us. A reason why LLVM uses the SSA format, rather than
JVM-like stack instructions, is that stack instructions are difficult to
optimise---you cannot just re-arrange instructions without messing about
with what is calculated on the stack. Also it is hard to find out if all
the calculations on the stack are actually necessary and not by chance
dead code. The JVM has for all these obstacles sophisticated machinery
to make such ``high-level'' code still run fast, but let's say that for
the sake of argument we do not want to rely on it. We want to generate
fast code ourselves. This means we have to work around the intricacies
of what instructions CPUs can actually process fast. This is what the
SSA format is designed for.


The main idea behind the SSA format is to use very simple variable
assignments where every tmp-variable is assigned only once. The assignments
also need to be primitive in the sense that they can be just simple
operations like addition, multiplication, jumps, comparisons and so on.
Say, we have an expression $((1 + a) + (3 + (b * 5)))$, then the
corresponding SSA format is 
 
\begin{lstlisting}[language=LLVMIR,numbers=left]
let tmp0 = add 1 a in   
let tmp1 = mul b 5 in 
let tmp2 = add 3 tmp1 in 
let tmp3 = add tmp0 tmp2 in tmp3 
\end{lstlisting}

\noindent where every variable is used only once (we could not write
\texttt{tmp1 = add 3 tmp1} in Line 3 for example).

There are sophisticated algorithms for imperative languages, like C,
that efficiently transform a high-level program into SSA format. But
we can ignore them here. We want to compile a functional language and
there things get much more interesting than just sophisticated. We
will need to have a look at CPS translations, where the CPS stands for
Continuation-Passing-Style---basically black programming art or
abracadabra programming. So sit tight.

\subsection*{LLVM-IR}

Before we start, let's first have a look at the \emph{LLVM Intermediate
Representation} in more detail. The LLVM-IR is in between the frontends
and backends of the LLVM framework. It allows compilation of multiple
source languages to multiple targets. It is also the place where most of
the target independent optimisations are performed. 

What is good about our toy Fun language is that it basically only
contains expressions (be they arithmetic expressions, boolean
expressions or if-expressions). The exception are function definitions.
Luckily, for them we can use the mechanism of defining functions in the
LLVM-IR (this is similar to using JVM methods for functions in our
earlier compiler). For example the simple Fun program 


\begin{lstlisting}[language=Scala,numbers=none]
def sqr(x) = x * x
\end{lstlisting}

\noindent
can be compiled to the following LLVM-IR function:

\begin{lstlisting}[language=LLVM]
define i32 @sqr(i32 %x) {
   %tmp = mul i32 %x, %x
   ret i32 %tmp
}    
\end{lstlisting}

\noindent First notice that all variable names, in this case \texttt{x}
and \texttt{tmp}, are prefixed with \texttt{\%} in the LLVM-IR.
Temporary variables can be named with an identifier, such as
\texttt{tmp}, or numbers. In contrast, function names, since they are ``global'',
need to be prefixed with an @-symbol. Also, the LLVM-IR is a fully typed
language. The \texttt{i32} type stands for 32-bit integers. There are
also types for 64-bit integers (\texttt{i64}), chars (\texttt{i8}),
floats, arrays and even pointer types. In the code above, \texttt{sqr}
takes an argument of type \texttt{i32} and produces a result of type
\texttt{i32} (the result type is in front of the function name, like in
C). Each arithmetic operation, for example addition and multiplication,
are also prefixed with the type they operate on. Obviously these types
need to match up\ldots{} but since we have in our programs only
integers, for the moment \texttt{i32} everywhere will do. We do not have to generate
any other types, but obviously this is a limitation in our Fun language.
 
There are a few interesting instructions in the LLVM-IR which are quite
different than what we have seen in the JVM. Can you remember the
kerfuffle we had to go through with boolean expressions and negating the
condition? In the LLVM-IR, branching  if-conditions is implemented
differently: there is a separate \texttt{br}-instruction as follows:

\begin{lstlisting}[language=LLVM]
br i1 %var, label %if_br, label %else_br
\end{lstlisting}

\noindent
The type \texttt{i1} stands for booleans. If the variable is true, then
this instruction jumps to the if-branch, which needs an explicit label;
otherwise it jumps to the else-branch, again with its own label. This allows us
to keep the meaning of the boolean expression ``as is'' when compiling if's---thanks god no more negating the boolean.
A value of type boolean is generated in the LLVM-IR by the
\texttt{icmp}-instruction. This instruction is for integers (hence the
\texttt{i}) and takes the comparison operation as argument. For example

\begin{lstlisting}[language=LLVM]
icmp eq  i32 %x, %y     ; for equal
icmp sle i32 %x, %y     ; signed less or equal
icmp slt i32 %x, %y     ; signed less than
icmp ult i32 %x, %y     ; unsigned less than 
\end{lstlisting}

\noindent
Note that in some operations the LLVM-IR distinguishes between signed and 
unsigned representations of integers.

It is also easy to call another function in LLVM-IR: as can be 
seen from Figure~\ref{lli} we can just call a function with the 
instruction \texttt{call} and can also assign the result to 
a variable. The syntax is as follows

\begin{lstlisting}[language=LLVM]
%var = call i32 @foo(...args...)
\end{lstlisting}

\noindent
where the arguments can only be simple variables, not compound
expressions.

Conveniently, you can use the program \texttt{lli}, which comes with
LLVM, to interpret programs written in the LLVM-IR. So you can easily
check whether the code you produced actually works. To get a running
program that does something interesting you need to add some boilerplate
about printing out numbers and a main-function that is the entry point
for the program (see Figure~\ref{lli} for a complete listing). Again
this is very similar to the boilerplate we needed to add in our JVM
compiler. 

You can generate a binary for the program in Figure~\ref{lli} by using
the \texttt{llc}-compiler and then \texttt{gcc}, whereby \texttt{llc} generates
an object file and \texttt{gcc} (that is clang) generates the
executable binary:

\begin{lstlisting}[language=bash,numbers=none]
llc -filetype=obj sqr.ll
gcc sqr.o -o a.out
./a.out
> 25
\end{lstlisting}

\begin{figure}[t]\small 
\lstinputlisting[language=LLVM,numbers=left]{../progs/sqr.ll}
\caption{An LLVM-IR program for calculating the square function. It
calls this function in \texttt{@main} with the argument \texttt{5}. The
code for the \texttt{sqr} function is in Lines 13 -- 16. The main
function calls \texttt{sqr} and then prints out the result. The other
code is boilerplate for printing out integers.\label{lli}}
\end{figure}   


    
\subsection*{Our Own Intermediate Language}

Remember compilers have to solve the problem of bridging the gap between
``high-level'' programs and ``low-level'' hardware. If the gap is too
wide for one step, then a good strategy is to lay a stepping stone
somewhere in between. The LLVM-IR itself is such a stepping stone to
make the task of generating and optimising code easier. Like a real
compiler we will use our own stepping stone which I call the
\emph{K-language}. For what follows recall the various kinds of
expressions in the Fun language. For convenience the Scala code of the
corresponding abstract syntax trees is shown on top of
Figure~\ref{absfun}. Below is the code for the abstract syntax trees in
the K-language. In K, here are two kinds of syntactic entities, namely
\emph{K-values} and \emph{K-expressions}. The central constructor of the
K-language is \texttt{KLet}. For this recall in SSA that arithmetic
expressions such as $((1 + a) + (3 + (b * 5)))$ need to be broken up
into smaller ``atomic'' steps, like so
 
\begin{lstlisting}[language=LLVMIR,numbers=none]
let tmp0 = add 1 a in   
let tmp1 = mul b 5 in 
let tmp2 = add 3 tmp1 in 
let tmp3 = add tmp0 tmp2 in
  tmp3 
\end{lstlisting}

\noindent
Here \texttt{tmp3} will contain the result of what the whole expression
stands for. In each individual step we can only perform an ``atomic''
operation, like addition or multiplication of a number and a variable.
We are not allowed to have for example an if-condition on the right-hand
side of an equals. Such constraints are enforced upon us because of how
the SSA format works in the LLVM-IR. By having in \texttt{KLet} taking
first a string (standing for an intermediate result) and second a value,
we can fulfil this constraint ``by construction''---there is no way we
could write anything else than a value. 

To sum up, K-values are the atomic operations that can be on the
right-hand side of equal-signs. The K-language is restricted such that
it is easy to generate the SSA format for the LLVM-IR. 



\begin{figure}[p]\small
\begin{lstlisting}[language=Scala,numbers=none]
// Fun language (expressions)
abstract class Exp 
abstract class BExp 

case class Call(name: String, args: List[Exp]) extends Exp
case class If(a: BExp, e1: Exp, e2: Exp) extends Exp
case class Write(e: Exp) extends Exp
case class Var(s: String) extends Exp
case class Num(i: Int) extends Exp
case class Aop(o: String, a1: Exp, a2: Exp) extends Exp
case class Sequence(e1: Exp, e2: Exp) extends Exp
case class Bop(o: String, a1: Exp, a2: Exp) extends BExp  



// K-language (K-expressions, K-values)
abstract class KExp
abstract class KVal

case class KVar(s: String) extends KVal
case class KNum(i: Int) extends KVal
case class Kop(o: String, v1: KVal, v2: KVal) extends KVal
case class KCall(o: String, vrs: List[KVal]) extends KVal
case class KWrite(v: KVal) extends KVal

case class KIf(x1: String, e1: KExp, e2: KExp) extends KExp
case class KLet(x: String, v: KVal, e: KExp) extends KExp
case class KReturn(v: KVal) extends KExp
\end{lstlisting}
\caption{Abstract syntax trees for the Fun language.\label{absfun}}
\end{figure}



\subsection*{CPS-Translations}

CPS stands for Continuation-Passing-Style. It is a kind of programming
technique often used in advanced functional programming. Before we delve
into the CPS-translation for our Fun language, let us look at 
CPS-versions of some well-known functions. Consider

\begin{lstlisting}[language=Scala, numbers=none]
def fact(n: Int) : Int = 
  if (n == 0) 1 else n * fact(n - 1) 
\end{lstlisting}

\noindent 
This is clearly the usual factorial function. But now consider the
following version of the factorial function:

\begin{lstlisting}[language=Scala, numbers=none]
def factC(n: Int, ret: Int => Int) : Int = 
  if (n == 0) ret(1) 
  else factC(n - 1, x => ret(n * x)) 

factC(3, identity)
\end{lstlisting}

\noindent
This function is called with the number, in this case 3, and the
identity-function (which returns just its input). The recursive
calls are:

\begin{lstlisting}[language=Scala, numbers=none]
factC(2, x => identity(3 * x))
factC(1, x => identity(3 * (2 * x)))
factC(0, x => identity(3 * (2 * (1 * x))))
\end{lstlisting}

\noindent
Having reached 0, we get out of the recursion and apply 1 to the
continuation (see if-branch above). This gives

\begin{lstlisting}[language=Scala, numbers=none]
identity(3 * (2 * (1 * 1)))
= 3 * (2 * (1 * 1))
= 6
\end{lstlisting}

\noindent
which is the expected result. If this looks somewhat familiar to you,
than this is because functions with continuations can be
seen as a kind of generalisation of tail-recursive functions. Anyway
notice how the continuations is ``stacked up'' during the recursion and
then ``unrolled'' when we apply 1 to the continuation. Interestingly, we
can do something similar to the Fibonacci function where in the traditional
version we have two recursive calls. Consider the following function

\begin{lstlisting}[language=Scala, numbers=none]
def fibC(n: Int, ret: Int => Int) : Int = 
  if (n == 0 || n == 1) ret(1) 
  else fibC(n - 1,
             r1 => fibC(n - 2,
               r2 => ret(r1 + r2)))
\end{lstlisting}

\noindent
Here the continuation is a nested function essentially wrapping up 
the second recursive call. Let us check how the recursion unfolds
when called with 3 and the identity function:

\begin{lstlisting}[language=Scala, numbers=none]
fibC(3, id)
fibC(2, r1 => fibC(1, r2 => id(r1 + r2)))
fibC(1, r1 => 
   fibC(0, r2 => fibC(1, r2a => id((r1 + r2) + r2a))))
fibC(0, r2 => fibC(1, r2a => id((1 + r2) + r2a)))
fibC(1, r2a => id((1 + 1) + r2a))
id((1 + 1) + 1)
(1 + 1) + 1
3
\end{lstlisting}

Let us now come back to the CPS-translations for the Fun language. The
main difficulty of generating instructions in SSA format is that large
compound expressions need to be broken up into smaller pieces and
intermediate results need to be chained into later instructions. To do
this conveniently, CPS-translations have been developed. They use
functions (``continuations'') to represent what is coming next in a
sequence of instructions. In our case, continuations are functions of type
\code{KVal} to \code{KExp}. They can be seen as a sequence of
\code{KLet}s where there is a ``hole'' that needs to be filled. Consider
for example

\begin{lstlisting}[language=LLVMIR,numbers=left,escapeinside={(*@}{@*)}]
let tmp0 = add 1 a in   
let tmp1 = mul (*@$\Box$@*) 5 in 
let tmp2 = add 3 tmp1 in 
let tmp3 = add tmp0 tmp2 in
  tmp3 
\end{lstlisting}

\noindent
where in the second line is a $\Box$ which still expects a \code{KVal}
to be filled in before it becomes a ``proper'' \code{KExp}. When we 
apply an argument to the continuation (remember they are functions)
we essentially fill something into the corresponding hole. The code
of the CPS-translation is 

\begin{lstlisting}[language=Scala,numbers=none]
def CPS(e: Exp)(k: KVal => KExp) : KExp = 
  e match { ... }
\end{lstlisting}

\noindent 
where \code{k} is the continuation and \code{e} is the expression 
to be compiled. In case we have numbers or variables, we can just
apply the continuation like 

\begin{center}
\code{k(KNum(n))} \qquad \code{k(KVar(x))}
\end{center}

\noindent This would just fill in the $\Box$ in a \code{KLet}-expression.
More interesting is the case for an arithmetic expression.

\begin{lstlisting}[language=Scala,numbers=none]
case Aop(o, e1, e2) => {
    val z = Fresh("tmp")
    CPS(e1)(y1 => 
      CPS(e2)(y2 => KLet(z, Kop(o, y1, y2), k(KVar(z)))))
}
\end{lstlisting}

\noindent
For more such rules, have a look to the code of the fun-llvm
compiler.

\noindent
\end{document}


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