\documentclass[11pt,a4paper]{scrartcl}
\usepackage{algorithm}
\usepackage{algpseudocode}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[margin=2cm,bottom=3cm]{geometry}
\usepackage{hyperref}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{lmodern}
\usepackage{booktabs}
\usepackage{multirow}
\usepackage{graphicx}
\usepackage{float}
\usepackage{multicol}
\usepackage{tikz}
\usepackage{listings}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{subcaption}
\setlength{\parindent}{0cm}
\setlength{\parskip}{0.3em}
\hypersetup{pdfborder={0 0 0}}
%\usepackage[nomessages]{fp} no easter eggs this time
\usepackage{amsmath}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\usepackage{minted}
\usepackage[style=numeric,backend=biber]{biblatex}

% bibliography
\addbibresource{bibliography.bib}

\definecolor{codegreen}{rgb}{0,0.6,0}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}

\lstdefinestyle{mystyle}{
  backgroundcolor=\color{backcolour},   
  commentstyle=\color{codegreen},
  keywordstyle=\color{magenta},
  keywordstyle=[2]{\color{olive}},
  numberstyle=\tiny\color{codegray},
  stringstyle=\color{codepurple},
  basicstyle=\ttfamily\footnotesize,
  breakatwhitespace=false,         
  breaklines=true,                 
  captionpos=b,  
  keepspaces=true,                 
  numbers=left,  
  numbersep=5pt,
  showspaces=false,                
  showstringspaces=false,
  showtabs=false,
  tabsize=2,
  aboveskip=0.8em,
  belowcaptionskip=0.8em
}
\lstset{style=mystyle}

\geometry{left=2cm,right=2cm,top=2cm,bottom=3cm}
\title{
\vspace{-5ex} 
Assignment 4 -- Software Analysis \\\vspace{0.5cm}
\Large Model checking with Spin
\vspace{-1ex} 
}
\author{Claudio Maggioni}
\date{\vspace{-3ex}}

\begin{document}
\maketitle

\section{Introduction}

This assignment consists in using model checking tecniques to verify
the correctness of the algorithm implemented in an existing program. In
particular, a sequential and a multi-threaded implementation of a
array-reversing Java utility class implementation are verified to check
correctness of both reversal procedures, consistency between the results they
produce and for absence of race conditions.

To achieve this I use the Spin model checker \cite{spin} to write an equivalent
finite state automaton implementation of the algorithm using the
\textit{ProMeLa} specification and define linear temporal logic (LTL) properties
to be automatically verified.

This report covers the definition of the model to check and the necessary LTL
properties to verify correctness of the implementation, and additionally
presents a brief analysis on the performance of the automated model checker.

\section{Model Definition}

In this section I define the \textit{ProMeLa} code which implements a FSA model
of the Java implementation. The model I define does not match the exact provided
Java implementation, but aims to replicate the salient algorithmic and
concurrent behaviour of the program.

Due to the way I implement the LTL properties in the following section, I decide
to implement the model as a GNU M4 macro processor \cite{m4} template file.
Therefore, the complete model can be found in the path
\texttt{ReverseModel/reversal.pml.m4} in the assignment repository

\begin{center}
\href{https://gitlab.com/usi-si-teaching/msde/2022-2023/software-analysis/maggioni/assignment-4}{\textit{usi-si-teaching/msde/2022-2023/software-analysis/maggioni/assignment-4}}
\end{center}

on \textit{gitlab.com}.

As suggested by the assignment description, I define some preprocessor constants
to allow for altering some parameters. As mentioned above, I use GNU M4 instead
of the regular \textit{ProMeLa} preprocessor to implement these definitions.
Specifically, I define the following properties:

\begin{description}
\item[N,] which represents the number of parallel threads spawned by the
    parallel reverser;
\item[LENGTH,] which represents the length of the array to reverse;
\item[R,] which represents the upper bound for the random values used to fill
    the array to reverse, the lower bound of them being 0.
\end{description}

The variable values are injected as parameters of the \texttt{m4} command, so no
definition is required in the model code.

Then by using these values the model specification declares the following global
variables:

\begin{minted}{c}
int to_reverse[LENGTH];
int reversed_seq[LENGTH]; 
int reversed_par[LENGTH];
bool done[N + 1];
bool seq_eq_to_parallel = true;
\end{minted}

\texttt{to\_reverse} is the array to reverse, and \texttt{reverse\_seq} and
\texttt{reverse\_par} are respectively where the sequential and parallel
reverser store the reversed array. The \texttt{done} array stores an array of
boolean values: \mintinline{c}{done[0]} stores whether the sequential reverser
has terminated, and each \mintinline{c}{done[i]} for $1 \leq i \leq N$ stores
whether the i-th spawned thread of the parallel reverser has terminated.
Consequently, since threads are joined in order, when 
\mintinline{c}{done[N] == true} the parallel reverser terminates, an effect that
is exploited by the main model body implementation to wait for it. Finally
\texttt{seq\_eq\_to\_parallel} is set to \mintinline{c}{false} when an
incongruence between \texttt{reversed\_seq} and \texttt{reversed\_par} is found
after termination of
both reversers.

The body of the model is structured in the following way: 

\begin{minted}{kotlin}
init {
    { /* array initialization */ }

    /* sequential reverser algorithm */
    run SequentialReverser();
    /* parallel reverser algorithm */
    run ParallelReverser();

    (done[0] == true && done[N] == true);

    { /* congruence check between reversers */ }
}
\end{minted}

Each of the enumerated sections is surrounded by curly braces to emulate the
effect of locally scoped variables in procedures, which do not exist in
\textit{ProMeLa} aside the concurrency emulating \texttt{proctype} construct.

As requested by the assignment, the sequential and parallel reverser are
implemented in a \texttt{proctype} and spawned in parallel in the model. The two
\textit{ProMeLa} processes join before the congruence check thanks to an
``expression'' statement waiting on the termination boolean array to signal that
both reversers have finished doing their job.

The array initialization is carried out as follows:

\begin{minted}{c}
int i;
for (i in to_reverse) {
    int value;
    select(value: 0 .. R);
    to_reverse[i] = value;
    printf("to_reverse[%d]: %d\n", i, value);
}
\end{minted}

As specified above, the array is initialized with values in $[0,R]$.
Specifically, values are generated using a nondeterministic \texttt{select}
statement to allow the model checker to try all possible values efficiently.

The sequential reversed algorithm is implemented with the following code:

\begin{minted}{c}
int k;
for (k: 0 .. (LENGTH - 1)) {
    reversed_seq[LENGTH - k - 1] = to_reverse[k];
    printf("reversed_seq[%d] = to_reverse[%d]\n", LENGTH - k - 1, k);
}
done[0] = true;
\end{minted}

which is a direct translation of the Java implementation to verify.

The sequential reverser is used to implement each thread of the parallel
reverser through the \textit{ThreadedReverser} class. In the model, the class is
translated in a Spin process through the \texttt{proctype} construct with the
following implementation:

\begin{minted}{c}
proctype ThreadedReverser(int from; int to; int n) {
    printf("proc[%d]: started from=%d to=%d\n", n, from, to);
    int k;
    for (k: from .. (to - 1)) {
        printf("reversed_par[%d] = to_reverse[%d]\n", LENGTH - k - 1, k);
        reversed_par[LENGTH - k - 1] = to_reverse[k];
    }
    printf("proc[%d]: ended\n", n);
    done[n] = true;
}
\end{minted}

The implementation is closely related to the sequential one, as it differs only
in \texttt{reversed\_par} being used as the destination array and limiting the
reversal between \texttt{from} and \texttt{to}. The argument \texttt{n} is used
to identify the thread in the \texttt{done} array to store the termination
state. Following the indexing rules of \texttt{done} given earlier, the i-th
spawned thread corresponds to a \texttt{proctype} call with $n = i$, so that at
termination \mintinline{c}{done[i]} is set to \mintinline{c}{true}.

The actual thread-spawning part of the parallel reverser, i.e.\ class
\textit{ParallelReverser} itself, is represented by the following
\textit{ProMeLa} code placed in the \texttt{ParallelReverser proctype}:

\begin{minted}{c}
int n;
int s = LENGTH / N;
for (n: 0 .. (N - 1)) { // fork loop
    int from = n * s;
    int to;
    if
    :: (n == N - 1) -> to = LENGTH;
    :: else -> to = n * s + s;
    fi
    run ThreadedReverser(from, to, n + 1); // fork here
}
for (n: 1 .. N) { // join loop
    (done[n] == true); // join n-th thread here
    printf("[%d] joined\n", n);
}
\end{minted}

Here the values of \texttt{n}, \texttt{s}, \texttt{from} and \texttt{to}
replicate exactly the values used in the Java implementation. The \texttt{n + 1}
parameter identifies maps each \texttt{proctype} invocation to its place in the
invocation order (e.g.\ for $n=0$, \texttt{ThreadedReverser} is called with 
$n+1=1$, since this is the 1st invocation of the process).

The second ``join'' loop waits for each process to complete in order of
invocation, replication the thread joining behaviour of the parallel reverser
implementation in the Java program. Note that the \textit{ProMeLa} statement
\mintinline{c}{(done[n] == true);} will ``wait'' for the value of
\mintinline{c}{done[n]} to be \mintinline{c}{true} before executing the
following statement, thus being an adequate analogy for
\mintinline{java}{Thread.join()}. 

Finally, the congruence check between the arrays produced by both implementation
is implemented with the following code:

\begin{minted}{c}
int i;
for (i: 0 .. (LENGTH - 1)) {
    if
    :: (reversed_seq[i] != reversed_par[i]) -> seq_eq_to_parallel = false;
    fi
}
\end{minted}

Should any matching pair of elements be different,
\texttt{seq\_eq\_to\_parallel} will be set to \mintinline{c}{false}. Note that
this boolean variable is used to implement one of the LTL properties, hence why
it is declared and set to a meaningful value in this block of the model.

\subsection{LTL properties}

In this section I cover the LTL property definitions I included in the model.

\begin{minted}{scala}
ltl seq_eq_parallel {
    [] (seq_eq_to_parallel == true)
}
\end{minted}

This LTL property definition checks that once both reversers have terminated,
the content of the respective reversed arrays they produce is the same. As
discussed in the previous section, this variable can only turn to false during
the execution of the congruence check and only if a pair of array elements of
same index is indeed different. Therefore, if the program is correct, the value
of the variable will always be true.

Note that this property does not ensure
termination of the program, at it relies on the congruence check to eventually
run at the end of the program. To ensure termination, I define the following LTL
property:

\begin{minted}{scala}
ltl termination {
    <> (done[0] == true && done[N] == true)
}
\end{minted}

This mirrors the wait statement introduced in the model code before the
congruence check block, and relies exactly in the same way on the termination
boolean array. Note that the elements of the array can only turn from
\mintinline{c}{false} to \mintinline{c}{true} or not change at all, thus the
property in the ``eventually'' operator is actually always true after it becomes
indeed true (i.e.\ the program cannnot un-terminate according to the model).

I then define other two custom properties showcasing the powers of the
M4 macro processor when compared to the built-in \textit{ProMeLa} one.

\begin{minted}{scala}
// ifelse(LTL, correctness_seq, `
ltl correctness_seq {
    [] (done[0] == true -> (true for(`k', 0, LENGTH-1, ` &&
        r_s[eval(LENGTH - k - 1)] == a[k]')))
}
// ', `')
\end{minted}

This property checks if the array produced by the sequential reverser is indeed
the reverse of the input array. Note that the ``polyglot'' M4 sugar allows for
the property to be arbitrairly unraveled based on the value of \texttt{LENGTH}.
Notice that to simplify the \textit{ProMeLa} source code to compile for long
array lengths, thanks to the \texttt{ifelse} macro the property is omitted by M4
when the property is not actually checked (because long LTL properties actually
make the model fail to parse). Here is an example of the unravelled property for
\mintinline{c}{LENGTH = 10}:

\begin{minted}{scala}
ltl correctness_seq {
    [] (done[0] == true -> (true  &&
        r_s[9] == a[0] &&
        r_s[8] == a[1] &&
        r_s[7] == a[2] &&
        r_s[6] == a[3] &&
        r_s[5] == a[4] &&
        r_s[4] == a[5] &&
        r_s[3] == a[6] &&
        r_s[2] == a[7] &&
        r_s[1] == a[8] &&
        r_s[0] == a[9]))
}
\end{minted}

Note that the use of \mintinline{c}{true} as the first condition after the
implication is simply to allow for \mintinline{c}{&&} to be simply appended at
the start of each condition.

In a similar fashion, I also define a property that is not generally true. The
following LTL formula specifies that when the second thread of the parallel
reverser terminates, the section to be reversed by the first thread is already
reversed.

\begin{minted}{scala}
// ifelse(LTL, correctness_par, `
ltl correctness_par {
    [] (done[2] == true -> (true for(`k', 0, LENGTH / N - 1, ` &&
        r_p[eval(LENGTH - k - 1)] == a[k]')))
}
// ', `')
\end{minted}

This property is clearly not generally true as the first thread may not complete
the reversal process before the second thread terminates due to concurrency.

Here is the counterexample Spin provides to show that the property does not hold
for some executions with $N = 2$, $\text{LENGTH} = 3$ and $R = 2$:

\begin{verbatim}program start
a[0]: 1
a[1]: 0
a[2]: 0
sequential start
parallel start
r_s[2] = a[0]
r_s[1] = a[1]
r_s[0] = a[2]
proc[1]: started from=0 to=1
proc[2]: started from=1 to=3
proc[2]: r_p[1] = a[1]
proc[2]: r_p[0] = a[2]
proc[2]: ended
\end{verbatim}

Indeed, the output shows that the second thread can terminate before the first
does. Here, the first thread (\texttt{proc[1]}) is just started but does not
manage to execute \mintinline{c}{r_p[2] = a[0]} before the second thread
terminates. Indeed, this is a realistic counterexample that could be reproduced
in the Java program.

\section{Model Execution}

To compile and check all LTL properties included in the model execute the
command:

\begin{minted}{shell}
./ReversalModel/run.sh all $N $LENGTH $R
\end{minted}

from the root git directory of the assignment, where \mintinline{shell}{$N} is
the value of the variable $N$, \mintinline{shell}{$LENGTH} is the length of the
array and \mintinline{shell}{$R} is the value of $R$. The parameter \texttt{all}
can be replaced by the name of an LTL property to only check the model for that
property.

\subsection{Execution limits}

To better gauge the efficiency of the model checker algorithm, I run the model
to check the LTL property \texttt{seq\_eq\_parallel} for all possible
combinations of the values of $N \in [2,10]$, $\text{LENGTH} \in [3,10]$ and $R
\in [2,10]$. The script \texttt{ReversalModel/grid-search.sh} implements this
procedure with the following GNU Parallel \cite{tange_2023_7855617} command,
allowing for parallel execution of up to 4 model instances:

\begin{minted}[breaklines]{shell}
parallel --jobs 4 "$SCRIPT_DIR/test-property.sh" ::: "$prop" ::: "$(seq 2 10)" ::: "$(seq 3 10)" ::: "$(seq 2 10)" ::: "$SCRIPT_DIR/$time_out"
\end{minted}

The \texttt{test-property.sh} script times out the model execution after 5
minutes of \texttt{real} execution time due to time constraints. Thus, for each
combination of parameters, I measure the process time (\texttt{user} and
\texttt{sys} time) it takes to verify the LTL property and, if the execution
times out, whether this happens or not. I then summarize the execution times for
successful runs and the frequency of timeouts grouping the execution by each
variable in figure \ref{fig:bigplot}.

It is apparent that even small increases in the array length significantly
hinder the performance of the model checking algorithm and increasee the number
of timeouts. Increasing the number of processes $N$ instead shows a tamer effect
on execution time, and avoids a significantly greater number of timeouts for the
values 8, 9, and 10. Finally, increasing upper bound on the values of the array
($R$) has the least significant effect, albeit being a close second to the
distribution for $N$.

\section{Conclusions}

I find the Spin model checker a relatively straightforward tool to use. The 
\textit{ProMeLa} model implementation was easy to carry out due to syntax
similarities with C, and no tweaks were required after writing it to make the
model work.

The model manages to realistic model program behaviour of the Java reverser
implementations, the moder checker did not flag any surprising behaviour w.r.t.\ 
my understanding of the original code. It is however interesting that the
\texttt{correctness\_par} property was invaidated with such an effective
counterexample, and it shows the value of the tool for analysis of (albeit
simple) concurrent program behaviour.

\begin{figure}
\begin{subfigure}[t]{\linewidth}
\centering
\resizebox{0.85\textwidth}{!}{\input{plots/n.pgf}}
\caption{Variable \texttt{N}}
\end{subfigure}
\begin{subfigure}[t]{\linewidth}
\centering
\resizebox{0.85\textwidth}{!}{\input{plots/length.pgf}}
\caption{Variable \texttt{LENGTH}}
\end{subfigure}
\begin{subfigure}[t]{\linewidth}
\centering
\resizebox{0.85\textwidth}{!}{\input{plots/r.pgf}}
\caption{Variable \texttt{R}}
\end{subfigure}
\caption{Distribution of CPU time and percentage of timeouts (i.e.\
executions with a real execution time greater than 5 minutes, discarded for
sake of time) for different executions of the model checker for different
parameters of \texttt{N}, \texttt{LENGTH} and \texttt{R}.}
\label{fig:bigplot}
\end{figure}

\printbibliography

\end{document}

