\documentclass{article}

\usepackage{../style}

\usepackage{../langs}

\usepackage{../graphics}

\usepackage{../grammar}

\usepackage{multicol}

\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}

\begin{document}

\fnote{\copyright{} Christian Urban, King's College London, 2014}

%% why are shuttle flights so good with software

%%http://www.fastcompany.com/28121/they-write-right-stuff

\section*{Handout 9 (Static Analysis)}

If we want to improve the safety and security of our programs,

we need a more principled approach to programming. Testing is

good, but as Edsger Dijkstra famously wrote: 

\begin{quote}\it 

``Program testing can be a very effective way to show the

\underline{\smash{presence}} of bugs, but it is hopelessly

inadequate for showing their \underline{\smash{absence}}.''

\end{quote}

\noindent While such a more principled approach has been the

subject of intense study for a long, long time, only in the

past few years some impressive results have been achieved. One

is the complete formalisation and (mathematical) verification

of a microkernel operating system called seL4.

\begin{center}

\url{http://sel4.systems}

\end{center}

\noindent In 2011 this work was included in the MIT Technology

Review in the annual list of the world’s ten most important

emerging

technologies.\footnote{\url{http://www2.technologyreview.com/tr10/?year=2011}}

While this work is impressive, its technical details are too

enormous for an explanation here. Therefore let us look at

something much simpler, namely finding out properties about

programs using \emph{static analysis}.

Static analysis is a technique that checks properties of a

program without actually running the program. This should

raise alarm bells with you---because almost all interesting

properties about programs  are equivalent to the halting

problem, which we know is undecidable. For example estimating

the memory consumption of programs is in general undecidable,

just like the halting problem. Static analysis circumvents

this undecidability-problem by essentially allowing answers

\emph{yes} and \emph{no}, but also \emph{don't know}. With

this ``trick'' even the halting problem becomes

decidable\ldots{}for example we could always say \emph{don't

know}. Of course this would be silly. The point is that we

should be striving for a method that answers as often as

possible either \emph{yes} or \emph{no}---just in cases when

it is too difficult we fall back on the

\emph{don't-know}-answer. This might sound all like abstract

nonsense. Therefore let us look at a concrete example.

\subsubsection*{A Simple, Idealised Programming Language}

Our starting point is a small, idealised programming language.

It is idealised because we cut several corners in comparison

with real programming languages. The language we will study

contains, amongst other things, variables holding integers.

Using static analysis, we want to find out what the sign of

these integers (positive or negative) will be when the program

runs. This sign-analysis seems like a very simple problem. But

even such simple problems, if approached naively, are in

general undecidable, just like Turing's halting problem. I let

you think why?

Is sign-analysis of variables an interesting problem? Well,

yes---if a compiler can find out that for example a variable

will never be negative and this variable is used as an index

for an array, then the compiler does not need to generate code

for an underflow-check. Remember some languages are immune to

buffer-overflow attacks, but they need to add underflow and

overflow checks everywhere. According to John Regher, an

expert in the field of compilers, overflow checks can cause

5-10\% slowdown, in some languages even 100\% for tight

loops.\footnote{\url{http://blog.regehr.org/archives/1154}} If

the compiler can omit the underflow check, for example, then

this can potentially drastically speed up the generated code. 

What do programs in our simple programming language look like?

The following grammar gives a first specification:

\begin{multicols}{2}

\begin{plstx}[rhs style=,one per line,left margin=9mm]

: \meta{Stmt} ::= \meta{label} \texttt{:}

                | \meta{var} \texttt{:=} \meta{Exp}

                | \texttt{jmp?} \meta{Exp} \meta{label}

                | \texttt{goto} \meta{label}\\

: \meta{Prog} ::= \meta{Stmt} \ldots{} \meta{Stmt}\\

\end{plstx}

\columnbreak

\begin{plstx}[rhs style=,one per line]
: \meta{Exp} ::= \meta{Exp} \texttt{+} \meta{Exp}

               | \meta{Exp} \texttt{*} \meta{Exp}

               | \meta{Exp} \texttt{=} \meta{Exp} 

               | \meta{num}

               | \meta{var}\\

\end{plstx}

\end{multicols}

\noindent I assume you are familiar with such

grammars.\footnote{\url{http://en.wikipedia.org/wiki/Backus–Naur_Form}}

There are three main syntactic categories: \emph{statments}

and \emph{expressions} as well as \emph{programs}, which are

sequences of statements. Statements are either labels,

variable assignments, conditional jumps (\pcode{jmp?}) and

unconditional jumps (\pcode{goto}). Labels are just strings,

which can be used as the target of a jump. We assume that in

every program the labels are unique---if there is a clash,

then we do not know where to jump to. The conditional jumps

and variable assignments involve (arithmetic) expressions.

Expressions are either numbers, variables or compound

expressions built up from \pcode{+}, \pcode{*} and \emph{=}

(for simplicity reasons we do not consider any other

operations). We assume we have negative and positive numbers,

\ldots \pcode{-2}, \pcode{-1}, \pcode{0}, \pcode{1},

\pcode{2}\ldots{} An example program that calculates the

factorial of 5 is in our programming language as follows:

\begin{lstlisting}[language={},xleftmargin=10mm]

      a := 1

      n := 5 

top:  

      jmp? n = 0 done 

      a := a * n 

      n := n + -1 

      goto top 

done:

\end{lstlisting}

\noindent As can be seen each line of the program contains a

statement. In the first two lines we assign values to the

variables \pcode{a} and \pcode{n}. In line 4 we test whether

\pcode{n} is zero, in which case we jump to the end of the

program marked with the label \pcode{done}. If \pcode{n} is

not zero, we multiply the content of \pcode{a} by \pcode{n},

decrease \pcode{n} by one and jump back to the beginning of

the loop, marked with the label \pcode{top}. Another program

in our language is shown in Figure~\ref{fib}. I let you think

what it calculates.

\begin{figure}[t]

\begin{lstlisting}[numbers=none,

                   language={},xleftmargin=10mm]

      n := 6

      m1 := 0

      m2 := 1

loop: 

      jmp? n = 0 done

      tmp := m2

      m2 := m1 + m2

      m1 := tmp

      n := n + -1

      goto top

done:

\end{lstlisting}

\caption{A mystery program in our idealised programming language.

Try to find out what it calculates! \label{fib}}

\end{figure}

Even if our language is rather small, it is still Turing

complete---meaning quite powerful. However, discussing this

fact in more detail would lead us too far astray. Clearly, our

programming is rather low-level and not very comfortable for

writing programs. It is inspired by real machine code, which

is the code that is executed by a CPU. So a more interesting

question is what is missing in comparison with real machine

code? Well, not much\ldots{}in principle. Real machine code,

of course, contains many more arithmetic instructions (not

just addition and multiplication) and many more conditional

jumps. We could add these to our language if we wanted, but

complexity is really beside the point here. Furthermore, real

machine code has many instructions for manipulating memory. We

do not have this at all. This is actually a more serious

simplification because we assume numbers to be arbitrary small

or large, which is not the case with real machine code. In

real machine code, basic number formats have a range and might

over-flow or under-flow this range. Also the number of

variables in our programs is potentially unlimited, while

memory in an actual computer, of course, is always limited. To

sum up, our language might look ridiculously simple, but it is

not too far removed from practically relevant issues.

\subsubsection*{An Interpreter}

Designing a language is like playing god: you can say what

names for variables you allow; what programs should look like;

most importantly you can decide what each part of the program

should mean and do. While our language is quite simple and the

meaning of statements, for example, is rather straightforward,

there are still places where we need to make real choices. For

example consider the conditional jumps, say the one in the

factorial program: 

\begin{center}

\code{jmp? n = 0 done}

\end{center}

\noindent How should they work? We could introduce Booleans

(\pcode{true} and \pcode{false}) and then jump only when the

condition is \pcode{true}. However, since we have numbers in

our language anyway, why not just encoding \pcode{true} as

one, and \pcode{false} as zero? In this way we can dispense

with the additional concept of Booleans.

I hope the above discussion makes it already clear we need to

be a bit more careful with our programs. Below we shall

describe an interpreter for our programming language, which

specifies exactly how programs are supposed to be

run\ldots{}at least we will specify this for all \emph{good}

programs. By good programs I mean where all variables are

initialised, for example. Our interpreter will just crash if

it cannot find out the value for a variable when it is not

initialised. Also, we will assume that labels in good programs

are unique, otherwise our programs will calculate ``garbage''.

First we will pre-process our programs. This will simplify the

definition of our interpreter later on. By pre-processing our

programs we will transform programs into \emph{snippets}. A

snippet is a label and all the code that comes after the

label. This essentially means a snippet is a \emph{map} from

labels to code.\footnote{Be sure you know what maps are. In a

programming context they are often represented as association

list where some data is associated with a key.} 

Given that programs are sequences (or lists) of statements, we

can easily calculate the snippets by just traversing this

sequence and recursively generating the map. Suppose a program

is of the general form

\begin{center}

$stmt_1\;stmt_2\; \ldots\; stmt_n$

\end{center}

\noindent The idea is to go through this sequence of

statements one by one and check whether they are a label. If

yes, we add the label and the remaining statements to our map.

If no, we just continue with the next statement. To come up

with a recursive definition for generating snippets, let us

write $[]$ for the program that does not contain any

statement. Consider the following definition:

\begin{center}

\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

$\textit{snippets}([])$ & $\dn$ & $\varnothing$\\

$\textit{snippets}(stmt\;\; rest)$ & $\dn$ &

$\begin{cases}

   \textit{snippets}(rest)[label := rest] & \text{if}\;stmt = \textit{label:}\\

   \textit{snippets}(rest)                & \text{otherwise}

\end{cases}$   

\end{tabular}

\end{center}

\noindent In the first clause we just return the empty map for

the program that does not contain any statement. In the second

clause, we have to distinguish the case where the first

statement is a label or not. As said before, if not, then we

just ``throw away'' the label and recursively calculate the

snippets for the rest of the program (the otherwise clause).

If yes, then we do the same, but also update the map so that

$label$ now points to the rest of the statements (the if

clause). This looks all realtively straightforward, but there

is one small problem we need to overcome: our two programs

shown so far have no label as \emph{entry point}---that is

where the execution is supposed to start. We usually assume

that the first statement will be run first. To make this the

default, it is convenient if we add to all our programs a

default label, say \pcode{""} (the empty string). With this we

can define our pre-processing of programs as follows

\begin{center}

$\textit{preproc}(prog) \dn \textit{snippets}(\pcode{"":}\;\; prog)$ 

\end{center} 

\noindent Let us see how this pans out in practice. If we

pre-process the factorial program shown earlier, we obtain the 

following map:

\begin{center}

\small

\lstset{numbers=none,

        language={},

        xleftmargin=0mm,

        aboveskip=0.5mm,

        belowskip=0.5mm,

        frame=single,

        framerule=0mm,

        framesep=0mm}

\begin{tikzpicture}[node distance=0mm]

\node (A1) [draw]{\pcode{""}};

\node (B1) [right=of A1] {$\mapsto$};

\node (C1) [right=of B1,anchor=north west] {

\begin{minipage}{3.5cm}

\begin{lstlisting}[language={},xleftmargin=0mm]

  a := 1

  n := 5 

top:  

  jmp? n = 0 done 

  a := a * n 

  n := n + -1 

  goto top 

done:

\end{lstlisting}

\end{minipage}};

\node (A2) [right=of C1.north east,draw] {\pcode{top}};

\node (B2) [right=of A2] {$\mapsto$};

\node (C2) [right=of B2, anchor=north west] {

\begin{minipage}{3.5cm}

\begin{lstlisting}[language={},xleftmargin=0mm]

  jmp? n = 0 done 

  a := a * n 

  n := n + -1 

  goto top 

done:

\end{lstlisting} 

\end{minipage}}; 

\node (A3) [right=of C2.north east,draw] {\pcode{done}};

\node (B3) [right=of A3] {$\mapsto$};

\node (C3) [right=of B3] {$[]$};

\end{tikzpicture}

\end{center}

\noindent I highlighted the \emph{keys} in this map. Since

there are three labels in the factorial program (remember we

added \pcode{""}), there are three keys. When running the

factorial program and encountering a jump, then we only have

to consult this snippets-map in order to find out what the

next statements should be.

We should now be in the position to define how a program

should be run. In the context of interpreters, this

``running'' of programs is often called \emph{evaluation}. Let

us start with the definition of how expressions are evaluated.

A first attempt might be the following recursive function:

\begin{center}

\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

$\textit{eval\_exp}(\texttt{n})$ & $\dn$ & $n$

  \qquad\text{if}\; \texttt{n}\; \text{is a number like} 

  \ldots \pcode{-2}, \pcode{-1}, \pcode{0}, \pcode{1},

\pcode{2}\ldots{}\\

$\textit{eval\_exp}(\texttt{e}_\texttt{1} \,\texttt{+}\, 

                    \texttt{e}_\texttt{2})$ & 

  $\dn$ & $\textit{eval\_exp}(\texttt{e}_\texttt{1}) + 

           \textit{eval\_exp}(\texttt{e}_\texttt{2})$\\

$\textit{eval\_exp}(\texttt{e}_\texttt{1} \,\texttt{*}\, 

                    \texttt{e}_\texttt{2})$ & 

  $\dn$ & $\textit{eval\_exp}(\texttt{e}_\texttt{1}) * 

           \textit{eval\_exp}(\texttt{e}_\texttt{2})$\\

$\textit{eval\_exp}(\texttt{e}_\texttt{1} \,\texttt{=}\, 

                    \texttt{e}_\texttt{2})$ & 

  $\dn$ & $\begin{cases}

  1 & \text{if}\;\textit{eval\_exp}(\texttt{e}_\texttt{1}) =

                 \textit{eval\_exp}(\texttt{e}_\texttt{2})\\

  0 & \text{otherwise}

  \end{cases}$          

\end{tabular}

\end{center}

\noindent While this should look all rather intuitive`, still

be very careful. There is a subtlety which can be easily

overlooked: The function \textit{eval\_exp} takes an

expression of our programming language as input and returns a

number as output. Therefore whenever we encounter a number in

our program, we just return this number---this is defined in

the first clause above. Whenever we encounter an addition,

well then we first evaluate the left-hand side

$\texttt{e}_\texttt{1}$ of the addition (this will give a

number), then evaluate the right-hand side

$\texttt{e}_\texttt{2}$ (this gives another number), and

finally add both numbers together. Here is the subtlety: on

the left-hand side of the $\dn$ we have a \texttt{+} (in the

teletype font) which is the symbol for addition in our

programming language. On the right-hand side we have $+$ which

stands for the arithmetic operation from ``mathematics'' of

adding two numbers. These are rather different concepts---one

is a symbol (which we made up), and the other a mathematical

operation. When we will have a look at an actual

implementation of our interpreter, the mathematical operation

will be the function for addition from the programming

language in which we \underline{\smash{implement}} our

interpreter. While the \texttt{+} is just a symbol that is

used in our programming language. Clearly we have to use a

symbol that is a good mnemonic for addition, otherwise we will

confuse the programmers working with our language. Therefore

we use $\texttt{+}$. A similar choice is made for times in the

third clause and equality in the fourth clause. 

Remember I wrote at the beginning of this section about being

god when designing a programming language. You can see this

here: we need to give meaning to symbols. At the moment

however, we are a poor fallible god. Look again at the grammar

of our programming language and our definition. Clearly, an

expression can contain variables. So far we have ignored them.

What should our interpreter do with variables? They might

change during the evaluation of a program. For example the

variable \pcode{n} in the factorial program counts down from 5

down to 0. How can we improve our definition above to give also

an answer whenever our interpreter encounters a variable in an

expression? The solution is to add an \emph{environment},

written $env$, as an additional input argument to our

\textit{eval\_exp} function.

\begin{center}

\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

$\textit{eval\_exp}(\texttt{n}, env)$ & $\dn$ & $n$

  \qquad\text{if}\; \texttt{n}\; \text{is a number like} 

  \ldots \pcode{-2}, \pcode{-1}, \pcode{0}, \pcode{1},

\pcode{2}\ldots{}\\

$\textit{eval\_exp}(\texttt{e}_\texttt{1} \,\texttt{+}\, 

                    \texttt{e}_\texttt{2}, env)$ & 

  $\dn$ & $\textit{eval\_exp}(\texttt{e}_\texttt{1}, env) + 

           \textit{eval\_exp}(\texttt{e}_\texttt{2}, env)$\\

$\textit{eval\_exp}(\texttt{e}_\texttt{1} \,\texttt{*}\, 

                    \texttt{e}_\texttt{2}, env)$ & 

  $\dn$ & $\textit{eval\_exp}(\texttt{e}_\texttt{1}, env) * 

           \textit{eval\_exp}(\texttt{e}_\texttt{2}, env)$\\

$\textit{eval\_exp}(\texttt{e}_\texttt{1} \,\texttt{=}\, 

                    \texttt{e}_\texttt{2}, env)$ & 

  $\dn$ & $\begin{cases}

  1 & \text{if}\;\textit{eval\_exp}(\texttt{e}_\texttt{1}, env) =

                 \textit{eval\_exp}(\texttt{e}_\texttt{2}, env)\\

  0 & \text{otherwise}

  \end{cases}$\\

$\textit{eval\_exp}(\texttt{x}, env)$ & $\dn$ & $env(x)$      

\end{tabular}

\end{center}

\noindent This environment $env$ also acts like a map: it

associates variables with their current values. For example

after evaluating the first two lines in our factorial

program, such an environment might look as follows

\begin{center}

\begin{tabular}{ll}

$\fbox{\texttt{a}} \mapsto \texttt{1}$ &

$\fbox{\texttt{n}} \mapsto \texttt{5}$

\end{tabular}

\end{center}

\noindent Again I highlighted the keys. In the clause for

variables, we can therefore consult this environment and

return whatever value is currently stored for this variable.

This is written $env(x)$---meaning we query this map with $x$

we obtain the corresponding number. You might ask what happens

if an environment does not contain any value for, say, the

variable $x$? Well, then our interpreter just ``crashes'', or

more precisely will raise an exception. In this case we have a

``bad'' program that tried to use a variable before it was

initialised. The programmer should not have done this. In a

real programming language, we would of course try a bit harder

and for example give an error at compile time, or design our

language in such a way that this can never happen. With the

second version of \textit{eval\_exp} we completed our

definition for evaluating expressions.

Next comes the evaluation function for statements. We define

this function in such a way that we recursively evaluate a

whole sequence of statements. Assume a program $p$ (you want

to evaluate) and its pre-processed snippets $sn$. Then we can

define:

\begin{center}

\begin{tabular}{@{}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}

$\textit{eval\_stmts}([], env)$ & $\dn$ & $env$\\

$\textit{eval\_stmts}(\texttt{label:}\;rest, env)$ &

  $\dn$ & $\textit{eval\_stmts}(rest, env)$ \\ 

$\textit{eval\_stmts}(\texttt{x\,:=\,e}\;rest, env)$ & 

  $\dn$ & $\textit{eval\_stmts}(rest, 

           env[x := \textit{eval\_exp}(\texttt{e}, env)])$\\  

$\textit{eval\_stmts}(\texttt{goto\,lbl}\;rest, env)$ 

 & $\dn$ & $\textit{eval\_stmts}(sn(\texttt{lbl}), env)$\\

$\textit{eval\_stmts}(\texttt{jmp?\,e\,lbl}\;rest, env)$ 

 & $\dn$ & $\begin{cases}\begin{array}{@{}l@{\hspace{-12mm}}r@{}}

 \textit{eval\_stmts}(sn(\texttt{lbl}), env)\\