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\begin{document}+ −
\fnote{\copyright{} Christian Urban, King's College London, 2017, 2018, 2019}+ −
+ −
\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. It will generate code for the Java Virtual+ −
Machine (JVM). 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 use in this module+ −
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 post-order fashion and emit code for+ −
each node---this traversal in post-order fashion will produce+ −
code for a stack-machine (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 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}+ −
+ −
However, our arithmetic expressions can also contain+ −
variables. We will represent them as \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, 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 an interpreter,+ −
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 and then add an \pcode{istore}-instruction at+ −
the end. By convention the result of the arithmetic expression+ −
$a$ will be on top of the stack. After 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+ −
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 to be not 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. Lets assume we+ −
already generated code for $b$ and $cs_{1/2}$. Then in the true-case the+ −
control-flow of the program needs to behave as + −
+ −
\begin{center}+ −
\begin{tikzpicture}[node distance=2mm and 4mm,+ −
block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm},+ −
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] (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 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,+ −
block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm},+ −
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] (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 a label is indicated by a colon. The task of the+ −
assmbler (in our case Jasmin or Krakatau) is to resolve the labels+ −
to actual 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}+ −
+ −
\noindent 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. 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. 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,+ −
block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm},+ −
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] (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,+ −
block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm},+ −
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] (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+ −
I leave it to you to read the code and follow its controlflow.+ −
+ −
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 need to 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 argument (which 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 field \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).+ −
+ −
+ −
\begin{figure}[t]+ −
\begin{lstlisting}[mathescape,language=JVMIS,numbers=left]+ −
.class public XXX.XXX+ −
.super java/lang/Object+ −
+ −
.method public <init>()V+ −
aload_0+ −
invokenonvirtual java/lang/Object/<init>()V+ −
return+ −
.end method+ −
+ −
.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{Boilerplate code needed for running generated code.\label{boiler}}+ −
\end{figure}+ −
+ −
+ −
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.+ −
+ −
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]{../progs/test-small.j}+ −
\caption{Generated code for a test program. This code can be + −
processed by an Java assembler producing a class-file, which+ −
can 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 let us generate more interesting While-programs by translating+ −
BF*** programs into equivalent While-code. So in this section lets 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. The+ −
second 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 dimension of the array onto the stack. The next+ −
instruction creates the array. With the last 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'' to the array onto the stack.+ −
Then we have some instructions corresponding to the index where we want+ −
to look up the array. The idea is that these instructions will leave a+ −
concrete number on 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 ``pointer'' to the array onto the+ −
stack. Then we have some instructions corresponding to the index where+ −
we want to update the array. After that come the instructions for with+ −
what 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 only has eight instructions (we will actually only have seven+ −
because we can omit the read-in instruction from BF). But also translating+ −
BF-loops is going to be easy since they straightforwardly can be + −
represented by 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 '+' => "field[ptr] := field[ptr] + 1;"+ −
case '-' => "field[ptr] := field[ptr] - 1;"+ −
case '.' => "x := field[ptr]; write x;"+ −
case '[' => "while (field[ptr] != 0) do {"+ −
case ']' => "skip};"+ −
case _ => ""+ −
}+ −
\end{lstlisting}+ −
+ −
\noindent + −
The idea behind the translation is that BF-programs operate on an array,+ −
called \texttt{field}. The BP-memory pointer into this array is+ −
represented as the variable \texttt{ptr}. The BF-instructions \code{>}+ −
and \code{<} increase, respectively decrease, \texttt{ptr}. The+ −
instructions \code{+} and \code{-} update a cell in \texttt{field}. In+ −
Line 6 we need to first assign a field-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.+ −
+ −
+ −
\end{document}+ −
+ −
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