% !TEX program = xelatex

\documentclass{article}

\usepackage{../style}

\usepackage{../langs}

\usepackage{../graphics}

\usepackage{../grammar}

\begin{document}

\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. The problem is that ``real'' CPUs, although supporting stack

operations, are not really designed to be \emph{stack machines}.  The

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 LLVM Intermediate

Language. 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

\noindent LLVM 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 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

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),

because that is what the LLVM-IR expects from us. A reason why LLVM

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.

comparisons, function calls and so on.  Say, we have an expression

\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}

example, not write \texttt{tmp1 = add tmp2 tmp3} in Line 5 even if this

There are sophisticated algorithms for imperative languages, like C,

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

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

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. 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 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

  (Line 20). The

  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

assignments with two kinds of syntactic entities, namely

\emph{K-values} and \emph{K-expressions}. K-values are all ``things''

\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 point we need to be careful, however, is that the

constraint is therefore on us. For our Fun-language, we will later on

consider some further K-values.

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 in \texttt{KLet} taking first a string (standing for an

intermediate variable) and second a value, we can fulfil the SSA

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 return a final result.  Of

course 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

be done is a translation of our Fun-language into the K-language. The

\subsection*{CPS-Translations}

technique often used in advanced functional programming. Before we delve

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

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.

So far 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=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}

\noindent

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. 

\section*{Worked Example}

essentially solve is the following problem: Given an expressions

\begin{equation}\label{exp}

(1 + 2) * (3 + 4)

\end{equation}  

\noindent

\[

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

\]  

\noindent

\begin{lstlisting}[language=Scala, numbers=none]

r => r * (3 + 4)

\end{lstlisting}  

\noindent

where \texttt{r} is any result that has been computed earlier. By the

$\lambda r.\, r * (3 + 4)$ for the same function. To sum up, we use

functions (``continuations'') to represent what is coming next in a