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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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