\documentclass{article}

\usepackage{../style}

\usepackage{../langs}

\usepackage{../grammar}

\usepackage{../graphics}

\begin{document}

\section*{Handout 8 (A Functional Language)}

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 enough to improve the speed of a

program by just targeting a virtual machine. This produces not

the fastest possible code, but code that is fast enough and

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 we will implement a compiler for the very

simple While-language. It will generate code for the Java

Virtual Machine (JVM). This is a stack-based virtual machine,

a fact which will make it easy to generate code for arithmetic

expressions. For example for generating code for the

expression $1 + 2$ we need to generate the following three

instructions

\begin{lstlisting}[numbers=none]

ldc 1

ldc 2

iadd 

\end{lstlisting}

\noindent The first instruction loads the constant $1$ onto

the stack, the next one $2$, 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 and results. 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 ($E$ 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 for

the expression $1 + ((2 * 3) + (4 - 3))$ it will produce the

following tree.

\begin{center}

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

\end{center}

\noindent To generate 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}[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 a convention we always

observe in our compiler, that is the results of arithmetic

expressions will always be on top of the stack. Note, that a

different bracketing of the expression, for example $(1 + (2 *

3)) + (4 - 3)$, produces a different abstract syntax tree and

thus potentially also a different list of instructions.

Generating code in this 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}[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}[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 the

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.

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 $[]$ 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, we need to be able to retrieve the associated index.

This is reflected in the clause for compiling assignments:

\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 $E' = E(x \mapsto largest\_index + 1)$). 

That means for the assignment $x := x + 1$ we generate the

following code

\begin{lstlisting}[mathescape,numbers=none]

iload $n_x$

ldc 1

iadd

istore $n_x$

\end{lstlisting}

\noindent 

where $n_x$ is the index for the variable $x$.

More complicated is the code for \pcode{if}-statments, 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 the $cs_{1/2}$

are the statements for each \pcode{if}-branch. 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 be

\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. 

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]

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 way of generating code still seems

the most convenient.

We are now ready to give the compile function for 

if-statments---remember this function returns for staments 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={}]

if 1 = 1 then x := 2 else y := 3

\end{lstlisting}

\noindent 

The generated code is as follows:

\begin{lstlisting}[mathescape,language={}]

   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 x <= 10 do x := x + 1

\end{lstlisting}

\noindent yielding the following code

\begin{lstlisting}[mathescape,language={}]

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}

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