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\begin{document}
\fnote{\copyright{} Christian Urban, King's College London, 2017, 2018, 2019, 2020}

\section*{Handout 7 (Compilation)}

The purpose of a compiler is to transform a program a human can read and
write into code the machine can run as fast as possible. The fastest
code would be machine code the CPU can run directly, but it is often
good enough for improving the speed of a program to target a
virtual machine. This produces not the fastest possible code, but code
that is often pretty fast. This way of producing code has the advantage that
the virtual machine takes care of things a compiler would normally need
to take care of (like explicit memory management). 

As a first example in this module we will implement a compiler for the
very simple WHILE-language that we parsed in the last lecture. The
compiler will target the Java Virtual Machine (JVM), but not directly.
Pictorially the compiler will work as follows:

\begin{center}
  \begin{tikzpicture}[scale=1,font=\bf,
                      node/.style={
                      rectangle,rounded corners=3mm,
                      ultra thick,draw=black!50,minimum height=18mm, 
                      minimum width=20mm,
                      top color=white,bottom color=black!20}]

  \node (0) at (-3,0) {};  
  \node (A) at (0,0) [node,text width=1.6cm,text centered] {our compiler};
  \node (B) at (3.5,0) [node,text width=1.6cm,text centered] {Jasmin / Krakatau};
  \node (C) at (7.5,0) [node] {JVM};
 
  \draw [->,line width=2.5mm] (0) -- node [above,pos=0.35] {*.while} (A); 
  \draw [->,line width=2.5mm] (A) -- node [above,pos=0.35] {*.j} (B); 
  \draw [->,line width=2.5mm] (B) -- node [above,pos=0.35] {*.class} (C); 
  \end{tikzpicture}
  \end{center}

\noindent
The input will be WHILE-programs; the output will be assembly files
(with the ending .j). Assembly files essentially contain human-readable
machine code, meaning they are not just bits and bytes, but rather
something you can read and understand---with a bit of practice of
course. An \emph{assembler} will then translate the assembly files into
unreadable class or binary files the JVM can run. Unfortunately, the
Java ecosystem does not come with an assembler which would be handy for
our compiler-endeavour  (unlike Microsoft's  Common Language
Infrastructure for the .Net platform which has an assembler
out-of-the-box). As a substitute we shall therefore use the 3rd-party
programs Jasmin and Krakatau 

\begin{itemize}
  \item \url{http://jasmin.sourceforge.net}
  \item \url{https://github.com/Storyyeller/Krakatau}
\end{itemize}

\noindent
The first is a Java program and the second a program written in Python.
Each of them allow us to generate \emph{assembly} files that are still
readable by humans, as opposed to class-files which are pretty much just
(horrible) zeros and ones. Jasmin (respectively Krakatau) will then take
an assembly file as input and generate the corresponding class file for
us. 

Good about the JVM is that it is a stack-based virtual machine, a fact
which will make it easy to generate code for arithmetic expressions. For
example when compiling the expression $1 + 2$ we need to generate the
following three instructions

\begin{lstlisting}[language=JVMIS,numbers=none]
ldc 1
ldc 2
iadd 
\end{lstlisting}

\noindent The first instruction loads the constant $1$ onto
the stack, the next one loads $2$, the third instruction adds both
numbers together replacing the top two elements of the stack
with the result $3$. For simplicity, we will throughout
consider only integer numbers. Therefore we can
use the JVM instructions \code{iadd}, \code{isub},
\code{imul}, \code{idiv} and so on. The \code{i} stands for
integer instructions in the JVM (alternatives are \code{d} for
doubles, \code{l} for longs and \code{f} for floats).

Recall our grammar for arithmetic expressions (\meta{E} is the
starting symbol):


\begin{plstx}[rhs style=, margin=3cm]
: \meta{E} ::= \meta{T} $+$ \meta{E}
         | \meta{T} $-$ \meta{E}
         | \meta{T}\\
: \meta{T} ::= \meta{F} $*$ \meta{T}
          | \meta{F} $\backslash$ \meta{T}
          | \meta{F}\\
: \meta{F} ::= ( \meta{E} )
          | \meta{Id}
          | \meta{Num}\\
\end{plstx}


\noindent where \meta{Id} stands for variables and \meta{Num}
for numbers. For the moment let us omit variables from arithmetic
expressions. Our parser will take this grammar and given an input
produce abstract syntax trees. For example we will obtain for the
expression $1 + ((2 * 3) + (4 - 3))$ the following tree.

\begin{center}
\begin{tikzpicture}
\Tree [.$+$ [.$1$ ] [.$+$ [.$*$ $2$ $3$ ] [.$-$ $4$ $3$ ]]]
\end{tikzpicture}
\end{center}

\noindent To generate JVM code for this expression, we need to traverse
this tree in \emph{post-order} fashion and emit code for each
node---this traversal in \emph{post-order} fashion will produce code for
a stack-machine (which is what the JVM is). Doing so for the tree above
generates the instructions

\begin{lstlisting}[language=JVMIS,numbers=none]
ldc 1 
ldc 2 
ldc 3 
imul 
ldc 4 
ldc 3 
isub 
iadd 
iadd
\end{lstlisting}

\noindent If we ``run'' these instructions, the result $8$ will be on
top of the stack (I leave this to you to verify; the meaning of each
instruction should be clear). The result being on the top of the stack
will be an important convention we always observe in our compiler. Note,
that a different bracketing of the expression, for example $(1 + (2 *
3)) + (4 - 3)$, produces a different abstract syntax tree and thus also
a different list of instructions. Generating code in this
post-order-traversal fashion is rather easy to implement: it can be done
with the following recursive \textit{compile}-function, which takes the
abstract syntax tree as an argument:

\begin{center}
\begin{tabular}{lcl}
$\textit{compile}(n)$ & $\dn$ & $\pcode{ldc}\; n$\\
$\textit{compile}(a_1 + a_2)$ & $\dn$ &
$\textit{compile}(a_1) \;@\;\textit{compile}(a_2)\;@\; \pcode{iadd}$\\
$\textit{compile}(a_1 - a_2)$ & $\dn$ & 
$\textit{compile}(a_1) \;@\; \textit{compile}(a_2)\;@\; \pcode{isub}$\\
$\textit{compile}(a_1 * a_2)$ & $\dn$ & 
$\textit{compile}(a_1) \;@\; \textit{compile}(a_2)\;@\; \pcode{imul}$\\
$\textit{compile}(a_1 \backslash a_2)$ & $\dn$ & 
$\textit{compile}(a_1) \;@\; \textit{compile}(a_2)\;@\; \pcode{idiv}$\\
\end{tabular}
\end{center}

This is all fine, but our arithmetic expressions can clearly contain
variables and then this code will not be good enough. To fix this we
will represent our variables as the \emph{local variables} in the JVM.
Essentially, local variables are an array or pointers to memory cells,
containing in our case only integers. Looking up a variable can be done
with the instruction

\begin{lstlisting}[language=JVMIS,mathescape,numbers=none]
iload $index$
\end{lstlisting}

\noindent 
which places the content of the local variable $index$ onto 
the stack. Storing the top of the stack into a local variable 
can be done by the instruction

\begin{lstlisting}[language=JVMIS,mathescape,numbers=none]
istore $index$
\end{lstlisting}

\noindent Note that this also pops off the top of the stack. One problem
we have to overcome, however, is that local variables are addressed, not
by identifiers (like \texttt{x}, \texttt{foo} and so on), but by numbers
(starting from $0$). Therefore our compiler needs to maintain a kind of
environment where variables are associated to numbers. This association
needs to be unique: if we muddle up the numbers, then we essentially
confuse variables and the consequence will usually be an erroneous
result. Our extended \textit{compile}-function for arithmetic
expressions will therefore take two arguments: the abstract syntax tree
and an environment, $E$, that maps identifiers to index-numbers.

\begin{center}
\begin{tabular}{lcl}
$\textit{compile}(n, E)$ & $\dn$ & $\pcode{ldc}\;n$\\
$\textit{compile}(a_1 + a_2, E)$ & $\dn$ & 
$\textit{compile}(a_1, E) \;@\;\textit{compile}(a_2, E)\;@\; \pcode{iadd}$\\
$\textit{compile}(a_1 - a_2, E)$ & $\dn$ &
$\textit{compile}(a_1, E) \;@\; \textit{compile}(a_2, E)\;@\; \pcode{isub}$\\
$\textit{compile}(a_1 * a_2, E)$ & $\dn$ &
$\textit{compile}(a_1, E) \;@\; \textit{compile}(a_2, E)\;@\; \pcode{imul}$\\
$\textit{compile}(a_1 \backslash a_2, E)$ & $\dn$ & 
$\textit{compile}(a_1, E) \;@\; \textit{compile}(a_2, E)\;@\; \pcode{idiv}$\\
$\textit{compile}(x, E)$ & $\dn$ & $\pcode{iload}\;E(x)$\\
\end{tabular}
\end{center}

\noindent In the last line we generate the code for variables where
$E(x)$ stands for looking up the environment to which index the variable
$x$ maps to. This is similar to the interpreter we saw earlier in the
module, which also needs an environment: the difference is that the
interpreter maintains a mapping from variables to current values (what
is the currently the value of a variable?), while compilers need a
mapping from variables to memory locations (where can I find the current
value for the variable in memory?).

There is a similar \textit{compile}-function for boolean
expressions, but it includes a ``trick'' to do with
\pcode{if}- and \pcode{while}-statements. To explain the issue
let us first describe the compilation of statements of the
WHILE-language. The clause for \pcode{skip} is trivial, since
we do not have to generate any instruction

\begin{center}
\begin{tabular}{lcl}
$\textit{compile}(\pcode{skip}, E)$ & $\dn$ & $([], E)$\\
\end{tabular}
\end{center}

\noindent whereby $[]$ is the empty list of instructions. Note that
the \textit{compile}-function for statements returns a pair, a
list of instructions (in this case the empty list) and an
environment for variables. The reason for the environment is
that assignments in the WHILE-language might change the
environment---clearly if a variable is used for the first
time, we need to allocate a new index and if it has been used
before, then we need to be able to retrieve the associated index.
This is reflected in the clause for compiling assignments, say
$\textit{x := a}$:

\begin{center}
\begin{tabular}{lcl}
$\textit{compile}(x := a, E)$ & $\dn$ & 
$(\textit{compile}(a, E) \;@\;\pcode{istore}\;index, E')$
\end{tabular}
\end{center}

\noindent We first generate code for the right-hand side of the
assignment (that is the arithmetic expression $a$) and then add an
\pcode{istore}-instruction at the end. By convention running the code
for the arithmetic expression $a$ will leave the result on top of the
stack. After that the \pcode{istore} instruction, the result will be
stored in the index corresponding to the variable $x$. If the variable
$x$ has been used before in the program, we just need to look up what
the index is and return the environment unchanged (that is in this case
$E' = E$). However, if this is the first encounter of the variable $x$
in the program, then we have to augment the environment and assign $x$
with the largest index in $E$ plus one (that is $E' = E(x \mapsto
largest\_index + 1)$). To sum up, for the assignment $x := x + 1$ we
generate the following code

\begin{lstlisting}[language=JVMIS,mathescape,numbers=none]
iload $n_x$
ldc 1
iadd
istore $n_x$
\end{lstlisting}

\noindent 
where $n_x$ is the index (or pointer to the memory) for the variable
$x$. The code for looking-up the index for the variable is as follow:

\begin{center}
\begin{tabular}{lcl}
$index \;=\; E\textit{.getOrElse}(x, |E|)$
\end{tabular}
\end{center}

\noindent
This implements the idea that in case the environment $E$ contains an
index for $x$, we return it. Otherwise we ``create'' a new index by
returning the size $|E|$ of the environment (that will be an index that
is guaranteed not to be used yet). In all this we take advantage of the
JVM which provides us with a potentially limitless supply of places
where we can store values of variables.

A bit more complicated is the generation of code for
\pcode{if}-statements, say

\begin{lstlisting}[mathescape,language={},numbers=none]
if $b$ then $cs_1$ else $cs_2$
\end{lstlisting}

\noindent where $b$ is a boolean expression and where both $cs_{1/2}$
are the statements for each of the \pcode{if}-branches. Let us assume we
already generated code for $b$ and and the two if-branches $cs_{1/2}$.
Then in the true-case the control-flow of the program needs to behave as


\begin{center}
\begin{tikzpicture}[node distance=2mm and 4mm,line cap=round,
 block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm,
               top color=white,bottom color=black!20},
 point/.style={rectangle, inner sep=0mm, minimum size=0mm, fill=red},
 skip loop/.style={black, line width=1mm, to path={-- ++(0,-10mm) -| (\tikztotarget)}}]
\node (A1) [point] {};
\node (b) [block, right=of A1] {code of $b$};
\node (A2) [point, right=of b] {};
\node (cs1) [block, right=of A2] {code of $cs_1$};
\node (A3) [point, right=of cs1] {};
\node (cs2) [block, right=of A3] {code of $cs_2$};
\node (A4) [point, right=of cs2] {};

\draw (A1) edge [->, black, line width=1mm] (b);
\draw (b) edge [->, black, line width=1mm] (cs1);
\draw (cs1) edge [->, black, line width=1mm,shorten >= -0.5mm] (A3);
\draw (A3) edge [->, black, skip loop] (A4);
\node [below=of cs2] {\raisebox{-5mm}{\small{}jump}};
\end{tikzpicture}
\end{center}

\noindent where we start with running the code for $b$; since
we are in the true case we continue with running the code for
$cs_1$. After this however, we must not run the code for
$cs_2$, but always jump to after the last instruction of $cs_2$
(the code for the \pcode{else}-branch). Note that this jump is
unconditional, meaning we always have to jump to the end of
$cs_2$. The corresponding instruction of the JVM is
\pcode{goto}. In case $b$ turns out to be false we need the
control-flow

\begin{center}
\begin{tikzpicture}[node distance=2mm and 4mm,line cap=round,
 block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm,
               top color=white,bottom color=black!20},
 point/.style={rectangle, inner sep=0mm, minimum size=0mm, fill=red},
 skip loop/.style={black, line width=1mm, to path={-- ++(0,-10mm) -| (\tikztotarget)}}]
\node (A1) [point] {};
\node (b) [block, right=of A1] {code of $b$};
\node (A2) [point, right=of b] {};
\node (cs1) [block, right=of A2] {code of $cs_1$};
\node (A3) [point, right=of cs1] {};
\node (cs2) [block, right=of A3] {code of $cs_2$};
\node (A4) [point, right=of cs2] {};

\draw (A1) edge [->, black, line width=1mm] (b);
\draw (b) edge [->, black, line width=1mm,shorten >= -0.5mm] (A2);
\draw (A2) edge [skip loop] (A3);
\draw (A3) edge [->, black, line width=1mm] (cs2);
\draw (cs2) edge [->,black, line width=1mm] (A4);
\node [below=of cs1] {\raisebox{-5mm}{\small{}conditional jump}};
\end{tikzpicture}
\end{center}

\noindent where we now need a conditional jump (if the
if-condition is false) from the end of the code for the 
boolean to the beginning of the instructions $cs_2$. Once we 
are finished with running $cs_2$ we can continue with whatever
code comes after the if-statement.

The \pcode{goto} and the conditional jumps need addresses to
where the jump should go. Since we are generating assembly
code for the JVM, we do not actually have to give (numeric)
addresses, but can just attach (symbolic) labels to our code.
These labels specify a target for a jump. Therefore the labels
need to be unique, as otherwise it would be ambiguous where a
jump should go to. A label, say \pcode{L}, is attached to code
like

\begin{lstlisting}[mathescape,numbers=none]
L:
  $instr_1$
  $instr_2$
    $\vdots$
\end{lstlisting}
 
\noindent where the label needs to be followed by a colon. The task of
the assembler (in our case Jasmin or Krakatau) is to resolve the labels
to actual (numeric) addresses, for example jump 10 instructions forward,
or 20 instructions backwards.
 
Recall the ``trick'' with compiling boolean expressions: the 
\textit{compile}-function for boolean expressions takes three
arguments: an abstract syntax tree, an environment for 
variable indices and also the label, $lab$, to where an conditional 
jump needs to go. The clause for the expression $a_1 = a_2$, 
for example, is as follows:

\begin{center}
\begin{tabular}{lcl}
$\textit{compile}(a_1 = a_2, E, lab)$ & $\dn$\\ 
\multicolumn{3}{l}{$\qquad\textit{compile}(a_1, E) \;@\;\textit{compile}(a_2, E)\;@\; \pcode{if_icmpne}\;lab$}
\end{tabular}
\end{center}

\noindent where we are first generating code for the
subexpressions $a_1$ and $a_2$. This will mean after running
the corresponding code there will be two integers on top of
the stack. If they are equal, we do not have to do anything
(except for popping them off from the stack) and just continue
with the next instructions (see control-flow of ifs above).
However if they are \emph{not} equal, then we need to
(conditionally) jump to the label $lab$. This can be done with
the instruction

\begin{lstlisting}[mathescape,numbers=none,language=JVMIS]
if_icmpne $lab$
\end{lstlisting}

To sum up, the third argument in the compile function for booleans
specifies where to jump, in case the condition is \emph{not} true. I
leave it to you to extend the \textit{compile}-function for the other
boolean expressions. Note that we need to jump whenever the boolean is
\emph{not} true, which means we have to ``negate'' the jump
condition---equals becomes not-equal, less becomes greater-or-equal. 
Other jump instructions for boolean operators are

\begin{center}
\begin{tabular}{l@{\hspace{10mm}}c@{\hspace{10mm}}l}
$\not=$ & $\Rightarrow$ & \pcode{if_icmpeq}\\
$<$ & $\Rightarrow$ & \pcode{if_icmpge}\\
$\le$ & $\Rightarrow$ & \pcode{if_icmpgt}\\
\end{tabular}
\end{center}

\noindent and so on.   If you do not like this design (it can be the
source of some nasty, hard-to-detect errors), you can also change the
layout of the code and first give the code for the else-branch and then
for the if-branch. However in the case of while-loops this
``upside-down-inside-out'' way of generating code still seems the most
convenient.

We are now ready to give the compile function for 
if-statements---remember this function returns for statements a 
pair consisting of the code and an environment:

\begin{center}
\begin{tabular}{lcl}
$\textit{compile}(\pcode{if}\;b\;\pcode{then}\; cs_1\;\pcode{else}\; cs_2, E)$ & $\dn$\\ 
\multicolumn{3}{l}{$\qquad L_\textit{ifelse}\;$ (fresh label)}\\
\multicolumn{3}{l}{$\qquad L_\textit{ifend}\;$ (fresh label)}\\
\multicolumn{3}{l}{$\qquad (is_1, E') = \textit{compile}(cs_1, E)$}\\
\multicolumn{3}{l}{$\qquad (is_2, E'') = \textit{compile}(cs_2, E')$}\\
\multicolumn{3}{l}{$\qquad(\textit{compile}(b, E, L_\textit{ifelse})$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\;is_1$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\; \pcode{goto}\;L_\textit{ifend}$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\;L_\textit{ifelse}:$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\;is_2$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\;L_\textit{ifend}:, E'')$}\\
\end{tabular}
\end{center}

\noindent In the first two lines we generate two fresh labels
for the jump addresses (just before the else-branch and just
after). In the next two lines we generate the instructions for
the two branches, $is_1$ and $is_2$. The final code will
be first the code for $b$ (including the label 
just-before-the-else-branch), then the \pcode{goto} for after
the else-branch, the label $L_\textit{ifesle}$, followed by
the instructions for the else-branch, followed by the 
after-the-else-branch label. Consider for example the 
if-statement:

\begin{lstlisting}[mathescape,numbers=none,language=While]
if 1 = 1 then x := 2 else y := 3
\end{lstlisting}

\noindent 
The generated code is as follows:

\begin{lstlisting}[language=JVMIS,mathescape,numbers=left]
   ldc 1
   ldc 1
   if_icmpne L_ifelse $\quad\tikz[remember picture] \node (C) {\mbox{}};$
   ldc 2
   istore 0
   goto L_ifend $\quad\tikz[remember picture] \node (A) {\mbox{}};$
L_ifelse: $\quad\tikz[remember picture] \node[] (D) {\mbox{}};$
   ldc 3
   istore 1
L_ifend: $\quad\tikz[remember picture] \node[] (B) {\mbox{}};$
\end{lstlisting}

\begin{tikzpicture}[remember picture,overlay]
  \draw[->,very thick] (A) edge [->,to path={-- ++(10mm,0mm) 
           -- ++(0mm,-17.3mm) |- (\tikztotarget)},line width=1mm] (B.east);
  \draw[->,very thick] (C) edge [->,to path={-- ++(10mm,0mm) 
           -- ++(0mm,-17.3mm) |- (\tikztotarget)},line width=1mm] (D.east);
\end{tikzpicture}

\noindent The first three lines correspond to the the boolean
expression $1 = 1$. The jump for when this boolean expression
is false is in Line~3. Lines 4-6 corresponds to the if-branch;
the else-branch is in Lines 8 and 9. Note carefully how the
environment $E$ is threaded through the recursive calls of
\textit{compile}. The function receives an environment $E$,
but it might extend it when compiling the if-branch, yielding
$E'$. This happens for example in the if-statement above
whenever the variable \code{x} has not been used before.
Similarly with the environment $E''$ for the second call to
\textit{compile}. $E''$ is also the environment that needs to
be returned as part of the answer.

The compilation of the while-loops, say 
\pcode{while} $b$ \pcode{do} $cs$, is very similar. In case
the condition is true and we need to do another iteration, 
and the control-flow needs to be as follows

\begin{center}
\begin{tikzpicture}[node distance=2mm and 4mm,line cap=round,
 block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm,
               top color=white,bottom color=black!20},
 point/.style={rectangle, inner sep=0mm, minimum size=0mm, fill=red},
 skip loop/.style={black, line width=1mm, to path={-- ++(0,-10mm) -| (\tikztotarget)}}]
\node (A0) [point, left=of A1] {};
\node (A1) [point] {};
\node (b) [block, right=of A1] {code of $b$};
\node (A2) [point, right=of b] {};
\node (cs1) [block, right=of A2] {code of $cs$};
\node (A3) [point, right=of cs1] {};
\node (A4) [point, right=of A3] {};

\draw (A0) edge [->, black, line width=1mm] (b);
\draw (b) edge [->, black, line width=1mm] (cs1);
\draw (cs1) edge [->, black, line width=1mm,shorten >= -0.5mm] (A3);
\draw (A3) edge [->,skip loop] (A1);
\end{tikzpicture}
\end{center}

\noindent Whereas if the condition is \emph{not} true, we
need to jump out of the loop, which gives the following
control flow.

\begin{center}
\begin{tikzpicture}[node distance=2mm and 4mm,line cap=round,
 block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm,
               top color=white,bottom color=black!20},
 point/.style={rectangle, inner sep=0mm, minimum size=0mm, fill=red},
 skip loop/.style={black, line width=1mm, to path={-- ++(0,-10mm) -| (\tikztotarget)}}]
\node (A0) [point, left=of A1] {};
\node (A1) [point] {};
\node (b) [block, right=of A1] {code of $b$};
\node (A2) [point, right=of b] {};
\node (cs1) [block, right=of A2] {code of $cs$};
\node (A3) [point, right=of cs1] {};
\node (A4) [point, right=of A3] {};

\draw (A0) edge [->, black, line width=1mm] (b);
\draw (b) edge [->, black, line width=1mm,shorten >= -0.5mm] (A2);
\draw (A2) edge [skip loop] (A3);
\draw (A3) edge [->, black, line width=1mm] (A4);
\end{tikzpicture}
\end{center}

\noindent Again we can use the \textit{compile}-function for
boolean expressions to insert the appropriate jump to the
end of the loop (label $L_{wend}$ below).

\begin{center}
\begin{tabular}{lcl}
$\textit{compile}(\pcode{while}\; b\; \pcode{do} \;cs, E)$ & $\dn$\\ 
\multicolumn{3}{l}{$\qquad L_{wbegin}\;$ (fresh label)}\\
\multicolumn{3}{l}{$\qquad L_{wend}\;$ (fresh label)}\\
\multicolumn{3}{l}{$\qquad (is, E') = \textit{compile}(cs_1, E)$}\\
\multicolumn{3}{l}{$\qquad(L_{wbegin}:$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\;\textit{compile}(b, E, L_{wend})$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\;is$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\; \text{goto}\;L_{wbegin}$}\\
\multicolumn{3}{l}{$\qquad\phantom{(}@\;L_{wend}:, E')$}\\
\end{tabular}
\end{center}

\noindent I let you go through how this clause works. As an example
you can consider the while-loop

\begin{lstlisting}[mathescape,numbers=none,language=While]
while x <= 10 do x := x + 1
\end{lstlisting}

\noindent yielding the following code

\begin{lstlisting}[language=JVMIS,mathescape,numbers=left]
L_wbegin: $\quad\tikz[remember picture] \node[] (LB) {\mbox{}};$
   iload 0
   ldc 10
   if_icmpgt L_wend $\quad\tikz[remember picture] \node (LC) {\mbox{}};$
   iload 0
   ldc 1
   iadd
   istore 0
   goto L_wbegin $\quad\tikz[remember picture] \node (LA) {\mbox{}};$
L_wend: $\quad\tikz[remember picture] \node[] (LD) {\mbox{}};$
\end{lstlisting}
 
\begin{tikzpicture}[remember picture,overlay]
  \draw[->,very thick] (LA) edge [->,to path={-- ++(10mm,0mm) 
           -- ++(0mm,17.3mm) |- (\tikztotarget)},line width=1mm] (LB.east);
  \draw[->,very thick] (LC) edge [->,to path={-- ++(10mm,0mm) 
           -- ++(0mm,-17.3mm) |- (\tikztotarget)},line width=1mm] (LD.east);
\end{tikzpicture}

\noindent
As said, I leave it to you to decide whether the code implements
the usual controlflow of while-loops.

Next we need to consider the statement \pcode{write x}, which can be
used to print out the content of a variable. For this we shall use a
Java library function. In order to avoid having to generate a lot of
code for each \pcode{write}-command, we use a separate helper-method and
just call this method with an appropriate argument (which of course
needs to be placed onto the stack). The code of the helper-method is as
follows.

\begin{lstlisting}[language=JVMIS,numbers=left]
.method public static write(I)V 
    .limit locals 1 
    .limit stack 2 
    getstatic java/lang/System/out Ljava/io/PrintStream; 
    iload 0
    invokevirtual java/io/PrintStream/println(I)V 
    return 
.end method
\end{lstlisting}

\noindent The first line marks the beginning of the method,
called \pcode{write}. It takes a single integer argument
indicated by the \pcode{(I)} and returns no result, indicated
by the \pcode{V}. Since the method has only one argument, we
only need a single local variable (Line~2) and a stack with
two cells will be sufficient (Line 3). Line 4 instructs the
JVM to get the value of the mem  \pcode{out} of the class
\pcode{java/lang/System}. It expects the value to be of type
\pcode{java/io/PrintStream}. A reference to this value will be
placed on the stack. Line~5 copies the integer we want to
print out onto the stack. In the next line we call the method
\pcode{println} (from the class \pcode{java/io/PrintStream}).
We want to print out an integer and do not expect anything
back (that is why the type annotation is \pcode{(I)V}). The
\pcode{return}-instruction in the next line changes the
control-flow back to the place from where \pcode{write} was
called. This method needs to be part of a header that is
included in any code we generate. The helper-method
\pcode{write} can be invoked with the two instructions

\begin{lstlisting}[mathescape,language=JVMIS]
iload $E(x)$ 
invokestatic XXX/XXX/write(I)V
\end{lstlisting}

\noindent where we first place the variable to be printed on
top of the stack and then call \pcode{write}. The \pcode{XXX}
need to be replaced by an appropriate class name (this will be
explained shortly).


By generating code for a WHILE-program, we end up with a list
of (JVM assembly) instructions. Unfortunately, there is a bit
more boilerplate code needed before these instructions can be
run. The complete code is shown in Figure~\ref{boiler}. This
boilerplate code is very specific to the JVM. If we target any
other virtual machine or a machine language, then we would
need to change this code. Lines 4 to 8 in Figure~\ref{boiler}
contain a method for object creation in the JVM; this method
is called \emph{before} the \pcode{main}-method in Lines 10 to
17. Interesting are the Lines 11 and 12 where we hardwire that
the stack of our programs will never be larger than 200 and
that the maximum number of variables is also 200. This seem to
be conservative default values that allow is to run some
simple WHILE-programs. In a real compiler, we would of course
need to work harder and find out appropriate values for the
stack and local variables.

\begin{figure}[t]
\begin{lstlisting}[mathescape,language=JVMIS,numbers=left]
.class public XXX.XXX
.super java/lang/Object

.method public static main([Ljava/lang/String;)V
    .limit locals 200
    .limit stack 200

      $\textit{\ldots{}here comes the compiled code\ldots}$

    return
.end method
\end{lstlisting}
\caption{The boilerplate code needed for running generated code.\label{boiler}}
\end{figure}


To sum up, in Figure~\ref{test} is the complete code generated
for the slightly nonsensical program

\begin{lstlisting}[mathescape,language=While]
x := 1 + 2;
write x
\end{lstlisting}

\noindent I let you read the code and make sure the code behaves as
expected. Having this code at our disposal, we need the assembler to
translate the generated code into JVM bytecode (a class file). This
bytecode is then understood by the JVM and can be run by just invoking
the \pcode{java}-program.


\begin{figure}[p]
\lstinputlisting[language=JVMIS,mathescape]{../progs/test-small.j}
\begin{tikzpicture}[remember picture,overlay]
  \draw[|<->|,very thick] (LA.north) -- (LB.south) 
     node[left=0mm,midway] {\footnotesize\texttt{x\,:=\,1\,+\,2}}; 
  \draw[|<->|,very thick] (LC.north) -- (LD.south)
     node[left=0mm,midway] {\footnotesize\texttt{write x}};
\end{tikzpicture}
\caption{The generated code for the test program \texttt{x := 1 + 2; write
x}. This code can be processed by a Java assembler producing a
class-file, which can then be run by the {\tt{}java}-program.\label{test}}
\end{figure}

\subsection*{Arrays}

Maybe a useful addition to the WHILE-language would be arrays. This
would allow us to generate more interesting WHILE-programs by
translating BF*** programs into equivalent WHILE-code. Therefore in this
section let us have a look at how we can support the following three
constructions

\begin{lstlisting}[mathescape,language=While]
new(arr[15000])
x := 3 + arr[3 + y]
arr[42 * n] := ...
\end{lstlisting}

\noindent
The first construct is for creating new arrays. In this instance the
name of the array is \pcode{arr} and it can hold 15000 integers. We do
not support ``dynamic'' arrays, that is the size of our arrays will
always be fixed. The second construct is for referencing an array cell
inside an arithmetic expression---we need to be able to look up the
contents of an array at an index determined by an arithmetic expression.
Similarly in the line below, we need to be able to update the content of
an array at an calculated index.  

For creating a new array we can generate the following three JVM
instructions:

\begin{lstlisting}[mathescape,language=JVMIS]
ldc number 
newarray int 
astore loc_var
\end{lstlisting}

\noindent
First we need to put the size of the array onto the stack. The next
instruction creates the array. In this case the array contains
\texttt{int}s. With the last instruction we can store the array as a
local variable (like the ``simple'' variables from the previous
section). The use of a local variable for each array allows us to have
multiple arrays in a WHILE-program. For looking up an element in an
array we can use the following JVM code

\begin{lstlisting}[mathescape,language=JVMIS]
aload loc_var 
index_aexp 
iaload
\end{lstlisting}

\noindent
The first instruction loads the ``pointer'', or local variable, to the
array onto the stack. Then we have some instructions calculating the
index where we want to look up the array. The idea is that these
instructions will leave a concrete number on the top of the stack, which
will be the index into the array we need. Finally we need to tell the
JVM to load the corresponding element onto the stack. Updating an array
at an index with a value is as follows.

\begin{lstlisting}[mathescape,language=JVMIS]
aload loc_var 
index_aexp 
value_aexp 
iastore
\end{lstlisting}

\noindent
Again the first instruction loads the local variable of
the array onto the stack. Then we have some instructions calculating
the index where we want to update the array. After that come the
instructions for with which value we want to update the array. The last
line contains the instruction for updating the array.

Next we need to modify our grammar rules for our WHILE-language: it
seems best to extend the rule for factors in arithmetic expressions with
a rule for looking up an array.

\begin{plstx}[rhs style=, margin=3cm]
: \meta{E} ::= \meta{T} $+$ \meta{E}
         | \meta{T} $-$ \meta{E}
         | \meta{T}\\
: \meta{T} ::= \meta{F} $*$ \meta{T}
          | \meta{F} $\backslash$ \meta{T}
          | \meta{F}\\
: \meta{F} ::= ( \meta{E} )
          | $\underbrace{\meta{Id}\,[\,\meta{E}\,]}_{new}$
          | \meta{Id}
          | \meta{Num}\\
\end{plstx}

\noindent
There is no problem with left-recursion as the \meta{E} is ``protected''
by an identifier and the brackets. There are two new rules for statements,
one for creating an array and one for array assignment:

\begin{plstx}[rhs style=, margin=2cm, one per line]
: \meta{Stmt} ::=  \ldots
              | \texttt{new}(\meta{Id}\,[\,\meta{Num}\,]) 
              | \meta{Id}\,[\,\meta{E}\,]\,:=\,\meta{E}\\
\end{plstx}

With this in place we can turn back to the idea of creating
WHILE-programs by translating BF programs. This is a relatively easy
task because BF has only eight instructions (we will actually implement
seven because we can omit the read-in instruction from BF). What makes
this translation easy is that BF-loops can be straightforwardly
represented as while-loops. The Scala code for the translation is as
follows:

\begin{lstlisting}[language=Scala,numbers=left]
def instr(c: Char) : String = c match {
  case '>' => "ptr := ptr + 1;"
  case '<' => "ptr := ptr - 1;"
  case '+' => "mem[ptr] := mem [ptr] + 1;"
  case '-' => "mem [ptr] := mem [ptr] - 1;"
  case '.' => "x := mem [ptr]; write x;"
  case '['  => "while (mem [ptr] != 0) do {"
  case ']'  => "skip};"
  case _ => ""
}
\end{lstlisting}

\noindent 
The idea behind the translation is that BF-programs operate on an array,
called here \texttt{mem}. The BP-memory pointer into this array is
represented as the variable \texttt{ptr}. As usual the BF-instructions
\code{>} and \code{<} increase, respectively decrease, \texttt{ptr}. The
instructions \code{+} and \code{-} update a cell in \texttt{mem}. In
Line 6 we need to first assign a \texttt{mem}-cell to an auxiliary variable
since we have not changed our write functions in order to cope with
writing out any array-content directly. Lines 7 and 8 are for
translating BF-loops. Line 8 is interesting in the sense that we need to
generate a \code{skip} instruction just before finishing with the
closing \code{"\}"}. The reason is that we are rather pedantic about
semicolons in our WHILE-grammar: the last command cannot have a
semicolon---adding a \code{skip} works around this snag. Putting all
this together is we can generate WHILE-programs with more than 400
instructions and then run the compiled JVM code for such programs.
\bigskip

\noindent
Hooooray, we can finally run the BF-mandelbrot program on the JVM and it
completes within 20 seconds (after nearly 10 minutes of parsing the
corresponding WHILE-program and generating 270K of a class file). Try
replicating the 20 secs with an interpreter! OK, we now face the
nagging question about what to do next\ldots

\subsection*{Added Value}

As you have probably seen, the compiler writer has a lot of freedom
about how to generate code from what the progarmmer wrote as program.
The only condition is that generated code should behave as expected by
the programmer. Then all is fine\ldots mission accomplished! But
sometimes the compiler writer is expected to go an extra mile, or even
miles. Suppose we are given the following WHILE-program:

\begin{lstlisting}[mathescape,language=While]
new(arr[10]);
arr[14] := 3 + arr[13]
\end{lstlisting}

\noindent 
While admittedly this is a contrived program, and probably not meant to
be like this by any sane programmer, it is supposed to make the
following point: We generate an array of size 10, and then try to access
the non-existing element at index 13 and even updating element with
index 14. Obviously this is baloney. However, our compiler generates
code for this program without any questions asked. We can even run this
code on the JVM\ldots of course the result is an exception trace where
the JVM yells at us for doing naughty things. (This is much better than
C, for example, where such errors are not prevented and as a result
insidious attacks can be mounted against such kind C-programs. I assume
everyone has heard about \emph{Buffer Overflow Attacks}.) 

Imagine we do not want to rely in our compiler on the JVM for producing
an annoying, but safe exception trace, rather we want to handle such
situations ourselves. Lets assume we want to handle them in the
following way: if the programmer access a field out-of-bounds, we just
return the default 0, and if a programmer wants to update an
out-of-bounds filed, we quietly ignore this update.

\end{document}

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