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\fnote{\copyright{} Christian Urban, King's College London, 2019, 2023}


% 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. That is pretty much the whole point of the JVM. 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 are some instructions and a chunk of
memory---compiler, or better compiler writers, do something with
them. Consequently, modern compilers need to go the extra mile in
order to generate code that is much easier and faster to process by
actual CPUs, rather than running code on virtual machines that
introduce an additional layer of indirection. To make this all
tractable for this module, we target the \emph{LLVM Intermediate
  Language} (LLVM-IR). In this way we can take advantage of the tools
coming with LLVM.\footnote{\url{http://llvm.org}} For example we do
not have to worry about things like register allocations or peephole
optimisations. By using the LLVM-IR, however, we also have to pay
price in the sense that generating code gets
harder\ldots{}unfor\-tunately nothing comes for free in life.

\subsection*{LLVM and the LLVM-IR}

\noindent LLVM is a beautiful example
that projects from Academia can make a difference in the Real 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 different programming 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 complicated
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. Targetting LLVM-IR will also
allow us to benefit from the modular structure of the LLVM compiler
and we can let the compiler generate code for different CPUs, for
example X86 or ARM.  That means we can be agnostic about where our
code is actually going to run.\footnote{Anybody want to try to run our
  programs on Android phones?}  We can also be somewhat ignorant about
optimising our code and about allocating memory efficiently---the LLVM
tools will take care of all this.

However, what we have to do in order to make LLVM to play ball is to
generate code in \emph{Static Single-Assignment} format (short SSA). 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. Have a look at the
expression $((a + b) * 4) - (3 * (a + b))$ and the corresponding JVM
instructions:

\begin{lstlisting}[language=JVMIS, numbers=none,mathescape]
iload a
iload b
iadd
ldc 4
imul
ldc 3
iload a
iload b
iadd
imul
isub
\end{lstlisting}

\noindent
and try to reorganise the code such that you calculate the expression
$(a + b)$ only once. This requires either quite a bit of
math-understanding from the compiler or you need to add ``copy-and-fetch''
of a result from a local variable.  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 have sequences of very
simple variable assignments where every (tmp)variable is assigned only
once. The assignments need to be simple in the sense that they can be
just be primitive operations like addition, multiplication, jumps,
comparisons, function calls and so on.  Say, we have an expression
$((1 + a) + (foo(3 + 2) + (b * 5)))$, then the corresponding sequence
of assignments in SSA-format are:
 
\begin{lstlisting}[language=LLVMIR,numbers=left]
let tmp0 = add 1 a in
let tmp1 = add 3 2 in  
let tmp2 = call foo(tmp1)  
let tmp3 = mul b 5 in 
let tmp4 = add tmp2 tmp3 in 
let tmp5 = add tmp0 tmp4 in
  return tmp5 
\end{lstlisting}

\noindent where every tmpX-variable is used only once (we could, for
example, not write \texttt{tmp1 = add tmp2 tmp3} in Line 5 even if
this would not change the overall result). At the end we have a
return-instruction wich contains the final result of the
expression. As can be seen the task we have to solve for generating
SSA-code is to take apart compound expressions into its most basic
''particles'' and transfrom them into a sequence of simple assignments
that calculates the desired result. Note that this means we have to
fix the order in which the expression is calculated, like from the
left to right.

There are sophisticated algorithms for imperative languages, like C,
that efficiently transform high-level programs 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
\emph{Continuation-Passing-Style}---basically black programming art or
abracadabra programming. So sit tight.

\subsection*{LLVM-IR}

Before we start, let's however 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 ``local'' 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 (and which we
are going to lift in the final CW).
 
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 are 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 of
booleans.

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. I assume you know what this means,
otherwise just ask.

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 and numbers, but not compound
expressions.

Conveniently, you can use the program \texttt{lli}, which comes with
LLVM, to interpret programs written in the LLVM-IR. Type on the command line

\begin{lstlisting}[language=bash,numbers=none]
lli sqr.ll
\end{lstlisting}

\noindent
and you can see the output of the pragram generated. 
This means 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 of the square function). 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}/\texttt{clang}, whereby \texttt{llc} generates
an object file and \texttt{gcc} (that is actually \texttt{clang} on my Mac) 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 the \texttt{sqr}-function in \texttt{@main} with the argument \texttt{5}
  (Line 20). The
  code for the \texttt{sqr}-function is in Lines 13 -- 16. The main-function
  stores
  the result of sqr in the variable called \texttt{\%1} and then
  prints out the contents of this variable in Line 21. The other
  code is boilerplate for printing out integers.\label{lli}}
\end{figure}   


    
\subsection*{Our Own Intermediate Language}

Let's get back to our compiler: 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}.

\begin{center}
  \begin{tikzpicture}[scale=0.9,font=\bf,
                      node/.style={
                      rectangle,rounded corners=3mm,
                      ultra thick,draw=black!50,minimum height=16mm, 
                      minimum width=20mm,
                      top color=white,bottom color=black!20}]

  %\node (0) at (-3,0) {};  
  \node (A) at (0.0,0) [node,text width=1.6cm,text centered,label=above:{\small\it{}source}] {Fun-Language};
  \node (B) at (3.5,0) [node,text width=1.6cm,text centered,label=above:{\small\it{}stepping stone}] {K-Language};
  \node (C) at (7.0,0) [node,text width=1.6cm,text centered,label=above:{\small\it{}target}] {LLVM-IR};
 
  \draw [->,line width=2.5mm] (A) -- node [above,pos=0.35] {} (B); 
  \draw [->,line width=2.5mm] (B) -- node [above,pos=0.35] {} (C); 
  \end{tikzpicture}
  \end{center}

  \noindent
  To see why we need a stepping stone for generating code in SSA-format,
  considder again arithmetic expressions such as
  $((1 + a) + (3 + (b * 5)))$. They 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
  return 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'' or ``trival'' operation, like addition or multiplication of
a number or a variable.  We are not allowed to have for example a
nested addition or an if-condition on the right-hand side of an
assignment. Such constraints are forced upon us because of how the
SSA-format works in the LLVM-IR. To simplify matters we represent
assignments with two kinds of syntactic entities, namely
\emph{K-values} and \emph{K-expressions}. K-values are all ``things''
that can appear on the right-hand side of an equal. Say we have the
following definition for K-values:

\begin{lstlisting}[language=Scala,numbers=none]
enum KVal {
  case KVar(s: String)
  case KNum(n: Int)
  case KAop(op: String, v1: KVal, v2: KVal)
  case KCall(fname: String, args: List[KVal])
}
\end{lstlisting}

\noindent
where a K-value can be a variable, a number or a ``trivial'' binary
operation, such as addition or multiplication. The two arguments of a
binary operation need to be K-values as well. Finally, we have
function calls, but again each argument of the function call needs to
be a K-value.  One constraint we need to be careful about is that the
arguments of the binary operations and function calls can only be
variables or numbers. For example

\begin{lstlisting}[language=Scala,numbers=none]
KAop("+", KAop("*", KNum(1), KNum(2)), KNum(3))
KCall("foo", List(KAop("+", KNum(4), KNum(5)))  
\end{lstlisting}

\noindent
while perfect instances of K-values according to the types, are not
allowed in the LLVM-IR. To encode this constraint into the type-system
of Scala, however, would make things overly complicated---to satisfy
this constraint is therefore on us. For the moment this will be all
K-values we are considdering. Later on, we will have some more for our
Fun-language.


Our K-expressions will encode the \texttt{let} and the \texttt{return}
from the SSA-format (again for the moment we only consider these two
constructors---later on we will have if-branches as well).

\begin{lstlisting}[language=Scala,numbers=none]
enum KExp {
  case KLet(x: String, v: KVal, e: KExp)
  case KReturn(v: KVal)
}
\end{lstlisting}

\noindent
By having \texttt{KLet} taking first a string (standing for a
tmp-variable) and second a value, we can fulfil the SSA constraint in
assignments ``by con\-struction''---there is no way we could write
anything else than a K-value.  Note that the third argument of a
\texttt{KLet} is again a K-expression, meaning either another
\texttt{KLet} or a \texttt{KReturn}. In this way we can construct a
sequence of computations and indicate what the final result of the
computations is.  According to the SSA-format, we also have to ensure
that all intermediary computations are given (fresh) names---we will
use an (ugly) counter for this.

To sum up, K-values are the atomic operations that can be on the
right-hand side of assignemnts. The K-language is restricted such that
it is easy to generate the SSA-format for the LLVM-IR. What remains to
be done is a translation of our Fun-language into the K-language. The
main difficulty is that we need to order the computation---in the
K-language we only have linear sequences of instructions. Before we
explain this, we have a look at some programs in CPS-style.




\subsection*{CPS-Translations}

CPS stands for \emph{Continuation-Passing-Style}. It is a kind of
programming technique often used in advanced functional programming
and in particular in compilers. Before we delve into the
CPS-translation for our Fun-language compiler, 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 slight variant of the factorial function:

\begin{lstlisting}[language=Scala, numbers=left]
def factC(n: Int, k: Int => Int) : Int = 
  if (n == 0) k(1) 
  else factC(n - 1, x => k(n * x)) 

factC(3, id)
\end{lstlisting}

\noindent
The function is is Lines 1--3. The function call is in Line 5.  We
call this function with a number, in this case 3, and the
identity-function (which returns just its input). The \texttt{k} is
the continuation in this function. The recursive calls are:

\begin{lstlisting}[language=Scala, numbers=none]
factC(2, x => id(3 * x))
factC(1, x => id(3 * (2 * x)))
factC(0, x => id(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 in Line 2). This gives

\begin{lstlisting}[language=Scala, numbers=none]
id(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.  So far, however,
we did not look at this generalisation in earnest.  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=left]
def fibC(n: Int, k: Int => Int) : Int = 
  if (n == 0 || n == 1) k(1) 
  else fibC(n - 1,
             r1 => fibC(n - 2,
               r2 => k(r1 + r2)))
\end{lstlisting}

\noindent
Here the continuation (Lines 4--5) is a nested function essentially wrapping up 
the second recursive call plus the original continuation. 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}

\noindent
The point of this section is that you should be playing around with these
simple definitions and make sure they calculate the expected result.
In the next step, you should understand \emph{how} these functions
calculate the result with the continuations. Now change the initial
continuation which you call the function to

\begin{lstlisting}[language=Scala, numbers=none]
r => { println("The result plus 1 is:") ; r + 1 }
\end{lstlisting}

\noindent
Does this still calculate the expected result?


\section*{Worked Example}

Let us now come back to the CPS-translations for the Fun-language.
Though we will start with a simpler subset containing only numbers,
arithmetic expressions and function calls---no variables for the
moment.  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, we use the CPS-translation
mechanism. What the continuations essentially solve is the following
problem: Given an expressions

\begin{equation}\label{exp}
(1 + 2) * (3 + 4)
\end{equation}  

\noindent
which of the subexpressions should be calculated first? We are going 
arbitrarily to decide that the calculation will be from left to
right. Other languages make different choices---C famously is right to
left. In our case this means we have to tear the expression shown in
\eqref{exp} apart as follows:

\[
(1 + 2) \qquad\text{and}\qquad  \Box * (3 + 4)
\]  

\noindent
The first subexpression can be easily calculated and will give us a result,
but the second one is not really an expression that makes sense. It
will only make sense as the next step in the computation when we
fill-in the result of $1+2$ into the ``hole'' indicated by
$\Box$. Such incomplete expressions can be represented in Scala as
functions

\begin{lstlisting}[language=Scala, numbers=none]
r => r * (3 + 4)
\end{lstlisting}  

\noindent
where \texttt{r} will represent any result that has been computed
earlier. By the way, in lambda-calculus notation we would write
$\lambda r.\, r * (3 + 4)$ for the same function. To sum up, we 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
  return 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.

Lets look at some concrete code. To simplify matters, 
suppose our source language consists just of three kinds of expressions

\begin{lstlisting}[language=Scala,numbers=none]
enum Expr {
    case Num(n: Int)
    case Aop(op: String, e1: Expr, e2: Expr)
    case Call(fname: String, args: List[Expr])
}
\end{lstlisting}

\noindent
This allows us to generate SSA-instructions for expressions like

\[
1 + foo(bar(4 * 7), 3, id(12))  
\]

\noindent
The code of the CPS-translation
that generates these instructions is then of the form

\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. The result of the function is a K-expression, which later
can be compiled into LLVM-IR code.

In case we have numbers, then we can just pass them in the CPS-translation
to the continuations because numbers need not be further teared apart
as they are already primitive. Passing the number to the continuation
means we apply the continuation like

\begin{lstlisting}[language=Scala,numbers=none]
  case Num(i) => k(KNum(i))
\end{lstlisting}

\noindent This would fill in the $\Box$ in a \code{KLet}-expression.
Since \texttt{k} is a function from \texttt{KVar} to \texttt{KExp} we
also optain the correct type for CPS, namely \texttt{KExp}.  There is
no need to create a temporary variable for simple numbers.  More
interesting is the case for arithmetic operations.

\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=0mm]
case Aop(op, e1, e2) => {
  val z = Fresh("tmp")
  CPS(e1)(r1 => 
    CPS(e2)(r2 => KLet(z, KAop(op, r1, r2), k(KVar(z)))))
}
\end{lstlisting}

\noindent
What we essentially have to do in this case is the following: compile
the subexpressions \texttt{e1} and \texttt{e2}. They will produce some
result that is stored in two temporary variables (assuming \texttt{e1} and \texttt{e2} are more
complicated than just numbers). We need to use these two temporary
variables and feed them into a new assignment of the form

\begin{lstlisting}[language=LLVMIR,numbers=none,escapeinside={(*@}{@*)}]
let z = op (*@$\Box_\texttt{r1}$@*) (*@$\Box_\texttt{r2}$@*) in
...
\end{lstlisting}

\noindent
Note that this assignment has two ``holes'', one for the left
subexpression and one for the right subexpression. We can fill both
holes by calling the CPS function twice---this is a bit of the black
art in CPS. We can use the second call of CPS as the continuation
of the first call. The reason is that

\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=0mm]
r1 => CPS(e2)(r2 => KLet(z, KAop(op, r1, r2), k(KVar(z))))
\end{lstlisting}

\noindent is a function from \code{KVal} to \code{KExp}, which we need
as type for the continuation. Once we created the assignment with the
fresh temporary variable \texttt{z}, we need to ``communicate'' that
the result of the computation of the arithmetic expression is stored
in \texttt{z} to the computations that follow. In this way we apply
the continuation \texttt{k} with this new variable (essentially we are
plugging in a hole further down the line).  Hope this makes sense!? If not,
play with the given code yourself.

The last case we need to consider in our small expression language are
function calls. For them remember each argument of the function
call can in SSA-format only be a variable or a number. Here is the
complete code for this case:

\begin{lstlisting}[language=Scala,numbers=left,xleftmargin=0mm]
case Call(fname, args) => {
  def aux(args: List[Expr], vs: List[KVal]): KExp = args match {
       case Nil => {
         val z = Fresh("tmp")
         KLet(z, KCall(fname, vs), k(KVar(z)))
       }
       case a::as => CPS(a)(r => aux(as, vs ::: List(r)))
  }
  aux(args, Nil)  
}
\end{lstlisting}

\noindent
As can be seen, we introduce an auxiliary function that compiles first all
function arguments---remember in our source language we can have calls
such as $foo(3 + 4, g(3))$ where we first have to create temporary
variables (and computations) for each argument. Therefore \texttt{aux}
analyses the argument list from left to right. In case there is an
argument \texttt{a} on the front of the list (the case \texttt{a::as}
in Line 7), we call CPS recursively for the corresponding
subexpression. The temporary variable containing the result for this
argument, we add to the end of the K-values we have already analysed
before. Again very conveniently we can use the recursive call to
\texttt{aux} as the continuation for the computations that
follow. When we reach the end of the argument list (the
\texttt{Nil}-case in Lines 3--6), then we collect all the K-values
\texttt{v1} to \texttt{vn} and call the function in the K-language
like so


\begin{lstlisting}[language=LLVMIR,numbers=none,escapeinside={(*@}{@*)}]
let z = call foo(v1,...,vn) in
...
\end{lstlisting}

\noindent
Again we need to communicate the result of the function call, namely the
fresh temporary variable \texttt{z}, to the rest of the computation. Therefore
we apply the continuation \texttt{k} with this variable.



\begin{figure}[p]\small
  \lstinputlisting[numbers=left,firstline=1,lastline=24]{../progs/fun/simple-cps.sc}
  \hspace{10mm}...
  \lstinputlisting[numbers=left,firstline=32,firstnumber=32,lastline=51]{../progs/fun/simple-cps.sc}
\caption{CPS-translation for a simple expression language.\label{cps}}
\end{figure}

The last question we need to answer in the code (see Figure~\ref{cps})
is how we start the CPS-translation function, or more precisely with
which continuation we should start CPS? It turns out that this initial
continuation will be the last one that is called. What should be the
last step in the computation? Yes, we need to return the temporary
variable where the last result is stored (if it is a simple number,
then we can just return this number). Therefore we call CPS with the
code

\begin{lstlisting}[language=Scala,numbers=none]
def CPSi(e: Expr) : KExp = CPS(e)(KReturn(_))
\end{lstlisting}

\noindent
where we give the function \code{KReturn(_)} as the continuation. With
this we completed the translation of simple expressions into our
K-language. Play around with some more expressions and see how the
CPS-translation generates the correct code. I know this is not easy to
follow code when you see it the first time.  Figure~\ref{absfun} gives
the complete datatypes for the ASTs of the Fun-language and the
K-values and K-expressions for the K-language. The complete
CPS-translation you can find in the code.

\section*{Next Steps}

Having obtained a K-expression, it is relatively straightforward to
generate a valid program for the LLVM-IR---remember the K-language
already enforces the SSA convention of a linear sequence of primitive
instructions involving only unique temporary variables. 
We leave this step to the attentive reader.

What else can we do?  Well it should be relatively easy to apply some
common optimisations to the K-expressions. One optimisations is called
constant folding---for example if we have an expression $3 + 4$ then
we can replace it by just $5$ instead of generating code to compute
$5$. Now this information needs to be propagated to the next
computation step to see whether any further constant foldings are
possible. Another useful technique is common subexpression
elimination, meaning if you have twice a calculation of a function
$foo(a)$, then we want to call it only once and store the result in a
temporary variable that can be used instead of the second, or third,
call to $foo(a)$. Again I leave this to the attentive reader to work
out and implement.


\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 and the K-language.\label{absfun}}
\end{figure}



\noindent
\end{document}


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