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\chapter{Main explanation?}
\label{cha:incremental}

\carlos{Review if we want to call nodes ``Enter'' and ``Exit'' or ``Start'' and ``End'' (I'd prefer the first one).}
\sergio{Enter o Entry?}

\section{First definition of the SDG}
\label{sec:first-def-sdg}

The system dependence graph (SDG) is a method for program slicing that was first
proposed by Horwitz, Reps and Blinkey \cite{HorwitzRB88}. It builds upon the
existing control flow graph (CFG), defining dependencies between vertices of the
CFG, and building a program dependence graph (PDG), which represents them. The
system dependence graph (SDG) is then built from the assembly of the different
PDGs (each representing a method of the program), linking each method call to
its corresponding definition. Because each graph is built from the previous one,
new constructs can be added with to the CFG, without the need to alter the
algorithm that converts CFG to PDG and then to SDG. The only modification
possible is the redefinition of a dependency or the addition of new kinds of
dependence.

The language covered by the initial proposal was a simple one, featuring
procedures with modifiable parameters and basic instructions, including calls to
procedures, variable assignments, arithmetic and logic operators and conditional
instructions (branches and loops): the basic features of an imperative
programming language. The control flow graph was as simple as the programs
themselves, with each graph representing one procedure. The instructions of the
program are represented as vertices of the graph and are split into two
categories: statements, which have no effect on the control flow (assignments,
procedure calls) and predicates, whose execution may lead to one of multiple
---though traditionally two--- paths (conditional instructions). Statements are
connected sequentially to the next instruction. Predicates have two outgoing
edges, each connected to the first statement that should be executed, according
to the result of evaluating the conditional expression in the guard of the
predicate.

\begin{definition}[Control Flow Graph \carlos{add original citation}]
	A \emph{control flow graph} $G$ of a program $P$ is a directed graph, represented as a tuple $\langle N, E \rangle$, where $N$ is a set of nodes, composed of a method's statements plus two special nodes, ``Start'' and ``End''; and $E$ is a set of edges of the form $e = \left(n_1, n_2\right) | n_1, n_2 \in N$. Most algorithms to generate the SDG mandate the ``Start'' node to be the only source and ``End'' to be the only sink in the graph. \carlos{Is it necessary to define source and sink in the context of a graph?}.

	Edges are created according to the possible execution paths that exist; each statement is connected to any statement that may immediately follow it. Formally, an edge $e = (n_1, n_2)$ exists if and only if there exists an execution of the program where $n_2$ is executed immediately after $n_1$. In general, expressions are not evaluated; so an \texttt{if} instruction has two outgoing edges even if the condition is always true or false, e.g. \texttt{1 == 0}.
\end{definition}

To build the PDG and then the SDG, there are two dependencies based directly on the CFG's structure: data and control dependence.

\begin{definition}[Postdominance \carlos{add original citation?}]
	Vertex $b$ \textit{postdominates} vertex $a$ if and only if $b$ is on every path from $a$ to the  ``End'' vertex.
\end{definition}

\begin{definition}[Control dependency \carlos{add original citation}]
	\label{def:ctrl-dep}
	Vertex $b$ is \textit{control dependent} on vertex $a$ ($a \ctrldep b$) if and only if $b$ postdominates one but not all of $a$'s successors. It follows that a vertex with only one successor cannot be the source of control dependence.
\end{definition}

\begin{definition}[Data dependency \carlos{add original citation}]
	Vertex $b$ is \textit{data dependent} on vertex $a$ ($a \datadep b$) if and only if $a$ may define a variable $x$, $b$ may use $x$ and there exists a \carlos{could it be ``an''??} $x$-definition free path from $a$ to $b$.
	
	Data dependency was originally defined as flow dependency, and split into loop and non--loop related dependencies, but that distinction is no longer useful to compute program slices.
	It should be noted that variable definitions and uses can be computed for each statement independently, analyzing the procedures called by it if necessary. The variables used and defined by a procedure call are those used and defined by its body.
\end{definition}

With the data and control dependencies, the PDG may be built by replacing the
edges from the CFG by data and control dependence edges. The first tends to be
represented as a thin solid line, and the latter as a thick solid line. In the
examples, data dependencies will be thin solid red lines.

\begin{definition}[Program dependence graph]
	The \textsl{program dependence graph} (PDG) is a directed graph (and originally a tree) represented by three elements: a set of nodes $N$, a set of control edges $E_c$ and a set of data edges $E_d$.

	The set of nodes corresponds to the set of nodes of the CFG, excluding the ``End'' node.
	
	Both sets of edges are built as follows. There is a control edge between two nodes $n_1$ and $n_2$ if and only if $n_1 \ctrldep n_2$, and a data edge between $n_1$ and $n_2$ if and only if $n_1 \datadep n_2$. Additionally, if a node $n$ does not have any incoming control edges, it has a ``default'' control edge $e = (\textnormal{Start},n)$; so that ``Start'' is the only source node of the graph.

	Note: the most common graphical representation is a tree--like structure based on the control edges, and nodes sorted left to right according to their position on the original program. Data edges do not affect the structure, so that the graph is easily readable.
\end{definition}

Finally, the SDG is built from the combination of all the PDGs that compose the
program.

\begin{definition}[System dependence graph]
	The \textsl{system dependence graph} (SDG) is a directed graph that represents the control and data dependencies of a whole program. It has three kinds of edges: control, data and function call. The graph is built combining multiple PDGs, with the ``Start'' nodes labeled after the function they begin. There exists one function call edge between each node containing one or more calls and each of the ``Start'' node of the method called. In a programming language where the function call is ambiguous (e.g. with pointers or polymorphism), there exists one edge leading to every possible function called.
\end{definition}

\begin{example}[Creation of a SDG from a simple program]
	Given the program shown below (left), the control flow graphs for both methods are shown on the right: \\
	\begin{minipage}{0.2\linewidth}
		\begin{lstlisting}
proc main() {
	a = 10;
	b = 20;
	f(a, b);
}

proc f(x, y) {
	while (x > y) {
		x = x - 1;
	}
	print(x);
}
		\end{lstlisting}
	\end{minipage}
	\begin{minipage}{0.79\linewidth}
		\includegraphics[width=0.6\linewidth]{img/cfgsimple}
	\end{minipage}

	Then, control and data dependencies are computed, arranging the nodes in the PDG. Finally, the two graphs are connected with summary edges to create the SDG:

	\begin{center}
	\includegraphics[width=0.8\linewidth]{img/sdgsimple}
	\end{center}
\end{example}

\subsubsection{Function calls and data dependencies}

\carlos{Vocabulary: when is appropriate the use of method, function and procedure????}

In the original definition of the SDG, there was special handling of data dependencies when calling functions, as it was considered that parameters were passed by value, and global variables did not exist. \carlos{Name and cite paper that introduced it} solves this issue by splitting function calls and function into multiple nodes. This proposal solved everything related to parameter passing: by value, by reference, complex variables such as structs or objects and return values.

To such end, the following modifications are made to the different graphs:

\begin{description}
	\item[CFG.] In each CFG, global variables read or modified and parameters are added to the label of the ``Start'' node in assignments of the form $par = par_{in}$ for each parameter and $x = x_{in}$ for global variables. Similarly, global variables and parameters modified are added to the label of the ``End'' node as $x_{out} = x$. The parameters are only passed back if the value set by the called method can be read by the callee. Finally, in method calls the same values must be packed and unpacked: each statement containing a function called is relabeled to contain input (of the form $par_{in} = \textnormal{exp}$ for parameters or $x_{in} = x$ for global variables) and output (always of the form $x = x_{out}$).
	\item[PDG.] Each node modified in the CFG is split into multiple nodes: the original label is the main node and each assignment is represented as a new node, which is control--dependent on the main one. Visually, input is placed on the left and output on the right; with parameters sorted accordingly.
	\item[SDG.] Three kinds of edges are introduced: parameter input (param--in), parameter output (param--out) and summary edges. Parameter input edges are placed between each method call's input node and the corresponding method definition input node. Parameter output edges are placed between each method definition's output node and the corresponding method call output node. Summary edges are placed between the input and output nodes of a method call, according to the dependencies inside the method definition: if there is a path from an input node to an output node, that shows a dependence and a summary method is placed in all method calls between those two nodes.

	Note: parameter input and output edges are separated because the traversal algorithm traverses them only sometimes (the output edges are excluded in the first pass and the input edges in the second).
\end{description}

\begin{example}[Variable packing and unpacking]
	Let it be a function $f(x, y)$ with two integer parameters, and a call $f(a + b, c)$, with parameters passed by reference if possible. The label of the method call node in the CFG would be ``\texttt{x\_in = a + b, y\_in = c, f(a + b, c), c = y\_out}''; method $f$ would have \texttt{x = x\_in, y = y\_in} in the ``Start'' node and \texttt{y\_out = y} in the ``End'' node. The relevant section of the SDG would be:
	\begin{center}
	\includegraphics[width=0.5\linewidth]{img/parameter-passing}
	\end{center}
\end{example}

\section{Unconditional control flow}

Even though the initial definition of the SDG was useful to compute slices, the
language covered was not enough for the typical language of the 1980's, which
included (in one form or another) unconditional control flow.  Therefore, one of
the first additions contributed to the algorithm to build system dependence
graphs was the inclusion of unconditional jumps, such as ``break'',
``continue'', ``goto'' and ``return'' statements (or any other equivalent).  A
naive representation would be to treat them the same as any other statement, but
with the outgoing edge landing in the corresponding instruction (outside the
loop, at the loop condition, at the method's end, etc.).  An alternative
approach is to represent the instruction as an edge, not a vertex, connecting
the previous statement with the next to be executed.  Both of these approaches
fail to generate a control dependence from the unconditional jump, as the
definition of control dependence (see Definition~\ref{def:ctrl-dep}) requires a
vertex to have more than one successor for it to be possible to be a source of
control dependence.  From here, there stem two approaches: the first would be to
redefine control dependency, in order to reflect the real effect of these
instructions ---as some authors~\cite{DanBHHKL11} have tried to do--- and the
second would be to alter the creation of the SDG to ``create'' those
dependencies, which is the most widely--used solution \added{\cite{pending1,pending2}}.

The most popular approach was proposed by Ball and Horwitz\cite{BalH93}, and
represents unconditional jumps as a \textsl{pseudo--predicate}.  The true edge
would lead to the next instruction to be executed, and the false edge would be
non-executable or \textit{dummy} edges, connected to the instruction that would
be executed were the unconditional jump a \textit{nop}\josep{esta frase no se entiende}.  The consequence of this
solution is that every instruction placed after the unconditional jump is
control dependent on the jump, as can be seen in Figure~\ref{fig:break-graphs}.
In the example \josep{con "the example" te refieres a la figura? Es importante distinguir entre figuras y ejemplos. Lo que este texto te está pidiendo a gritos es que crees un entorno ejemplo, con su identificador. Ese ejemplo muestre y explique la figura, y desde aqui cites el ejemplo, no la figura.}, when slicing with respect to variable $a$ on line 5, every
statement would be included, save for ``print(a)''.  Line 4 is not strictly
necessary in this example ---in the context of weak slicing---, but is included
nonetheless.  In the original paper\josep{cual? citalo entre parentesis si es un recordatorio}, the transformation is proved to be
complete, but not correct, as for some examples, the slice includes more
unconditional jumps that would be strictly necessary, even for weak slicing. \josep{yo incluiria uno de esos ejemplos aqui}
Ball and Horwitz theorize that a more correct approach would be possible, if it
were \added{not}\deleted{n't} for the limitation of slices to be a subset of statements of the
program, in the same order as in the original.

\begin{figure}
\centering
\begin{minipage}{0.3\linewidth}
	\begin{lstlisting}
static void f() {
	int a = 1;
	while (a > 0) {
		if (a > 10) break;
		a++;
	}
	System.out.println(a);
}
	\end{lstlisting}
\end{minipage}
\begin{minipage}{0.6\linewidth}
	\includegraphics[width=0.4\linewidth]{img/breakcfg}
	\includegraphics[width=0.59\linewidth]{img/breakpdg}
\end{minipage}
\caption{A program with unconditional control flow, its CFG (center) and PDG(right).}
\label{fig:break-graphs}
\end{figure}

\section{Exceptions}

As seen in section~\ref{sec:intro-exception}, exception handling in Java adds
two constructs: the \texttt{throw} and the \texttt{try-catch} statements. The
first one resembles an unconditional control flow statement, with an unknown (on
compile time) destination. The exception will be caught by a \texttt{catch} of
the corresponding type or a supertype ---if it exists. Otherwise, it will crash
the corresponding thread (or in single-threaded programs, stop the Java Virtual
Machine). The second stops the exceptional control flow conditionally, based on
the dynamic typing of the exception thrown. Both introduce challenges that must
be solved.

\subsection{\texttt{throw} statement}

The \texttt{throw} statement represents two elements at the same time: an
unconditional jump and an erroneous exit from its method. The first one has
been extensively covered and solved, but the second \added{one} requires a small addition
to the CFG: instead of having a single ``End'' node, it will be split in two
---normal and error exit---, though the ``End'' cannot be removed, as a restriction
of most slicing algorithms is that the CFG have only one sink node. Therefore all
nodes that connected to the ``End'' will now lead to ``Normal exit'', all throw
statements' true outgoing edges will connect to the ``Error exit'', and both exit
nodes will converge on the ``End'' node.

\texttt{throw} statements in Java take a single value, a subtype of \texttt{Throwable},
and that value is used to stop the propagation of the exception; which can be handled
as a returned value. This treatment of \texttt{throw} statements only modifies the
structure of the CFG, without altering any other algorithm, nor the basic definitions
for control and data dependencies, making it very easy to incorporate to any existing
slicing software solution that follows the general model described.

\begin{example}[CFG of an uncaught \texttt{throw} statement] \ \\
	\begin{minipage}{0.69\linewidth}
	\begin{lstlisting}
	double f(int x) {
		if (x < 0)
			throw new RuntimeException()
		return Math.sqrt(x)
	}
	\end{lstlisting}
	By analyzing the CFG, we can see that both exits are control dependent on the \texttt{throw}
	statement; data dependencies present no special case in this example.
	\end{minipage}
	\begin{minipage}{0.3\linewidth}
		\includegraphics[width=\linewidth]{img/throw-example-cfg}
	\end{minipage}
\end{example}

\subsection{\texttt{try-catch} statement}

The \texttt{try-catch-finally} statement is the only way to stop an exception once \added{it is}\deleted{it's} thrown,
filtering by type, or otherwise letting it propagate further up the call stack. On top of that,
\texttt{finally} helps guarantee consistency, executing in any case (even when an exception is
left uncaught, the program returns or an exception occurs in a \texttt{catch} block). The main
problem with this construct is that \texttt{catch} blocks are not always necessary, but their
absence may make the compilation fail ---because a \texttt{try} block has no \texttt{catch} or
\texttt{finally} block---, or modify the execution in unexpected ways that are not always accounted
for in slicing software.

For the \texttt{try} block, it is normally represented as a pseudo--predicate, connected to the
first statement inside it and to the end of the first instruction after the whole \texttt{try-catch-finally}
construct. Inside the \texttt{try} there can be four distinct sources of exceptions:

\begin{description}
	\item[Method calls.] If an exception is thrown inside a method and it is not caught, it will
		surface inside the \texttt{try} block. As \textit{checked} exceptions must be declared
		explicitly, method declarations may be consulted to see if a method call may or may not
		throw any exceptions. On this front, polymorphism and inheritance present no problem, as
		inherited methods may not modify the signature ---which includes the exceptions that may
		be thrown. If \textit{unchecked} exceptions are also considered, all method calls shall
		be included, as any can trigger at the very least a \texttt{StackOverflowException}.
	\item[\texttt{throw} statements.] The least common, but most simple, as it is treated as a
		\texttt{throw} inside a method.
	\item[Implicit unchecked exceptions.] If \textit{unchecked} exceptions are considered, many
		common expressions may throw an exception, with the most common ones being trying to call
		a method or accessing a field of a \texttt{null} object (\texttt{NullPointerException}),
		accessing an invalid index on an array (\texttt{ArrayIndexOutOfBoundsException}), dividing
		an integer by 0 (\texttt{ArithmeticException}), trying to cast to an incompatible type
		(\texttt{ClassCastException}) and many others. On top of that, the user may create new
		types that inherit from \texttt{RuntimeException}, but those may only be explicitly thrown.
		Their inclusion in program slicing and therefore in the method's CFG generates extra
		dependencies that make the slices produced bigger.
	\item[\added{Errors}\deleted{Erorrs}.] May be generated at any point in the execution of the program, but they normally
		signal a situation from which it may be impossible to recover, such as an internal JVM error.
		In general, most programs do not consider these to be ``catch-able''.
\end{description}

All exception sources are treated in a similar fashion: the statement that may throw an exception
is treated as a predicate, with the true edge connected to the next instruction were the statement
to execute without raising exceptions; and the false edge connected to the \texttt{catch} node.

\carlos{CATCH Representation doesn't matter, it is similar to a switch but checking against types.
	The difference exists where there exists the possibility of not catching the exception;
	which is semantically possible to define. When a \texttt{catch (Throwable e)} is declared,
	it is impossible for the exception to exit the method; therefore the control dependency must
	be redefined.}

The filter for exceptions in Java's \texttt{catch} blocks is a type (or multiple types since
Java 8), with a class that encompasses all possible exceptions (\texttt{Throwable}), which acts
as a catch--all.
In the literature there exist two alternatives to represent \texttt{catch}: one mimics a static
switch statement, placing all the \texttt{catch} block headers at the same height, all pending
from the exception-throwing exception and the other mimics a dynamic switch or a chain of \texttt{if}
statements. The option chosen affects how control dependencies should be computed, as the different
structures generate different control dependencies by default.

\begin{description}
	\item[Switch representation.] There exists no relation between different \texttt{catch} blocks,
		each exception--throwing statement is connected through an edge labeled false to each
		of the \texttt{catch} blocks that could be entered. Each \texttt{catch} block is a
		pseudo--statement, with its true edge connected to the end of the \texttt{try} and the 
		As an example, a \texttt{1 / 0} expression may be connected to \texttt{ArithmeticException},
		\texttt{RuntimeException}, \texttt{Exception} or \texttt{Throwable}.
		If any exception may not be caught, there exists a connection to the ``Error exit'' of the method.
	\item[If-else representation.] Each exception--throwing statement is connected to the first
		\texttt{catch} block. Each \texttt{catch} block is represented as a predicate, with the true
		edge connected to the first statement inside the \texttt{catch} block, and the false edge
		to the next \texttt{catch} block, until the last one. The last one will be a pseudo--predicate
		connected to the first statement after the \texttt{try} if it is a catch--all type or to the
		``Error exit'' if it \added{is not}\deleted{isn't}.
\end{description}

\begin{example}[Catches.]\ \\
	\begin{minipage}{0.49\linewidth}
		\begin{lstlisting}
			try {
				f();
			} catch (CheckedException e) {
			} catch (UncheckedException e) {
			} catch (Throwable e) {
			}
		\end{lstlisting}
	\end{minipage}
	\begin{minipage}{0.49\linewidth}
		\carlos{missing figures with 4 alternatives: if-else (with catch--all and without) and switch (same two)}\josep{Definitely!!!}
%		\includegraphics[0.5\linewidth]{img/catch1}
%		\includegraphics[0.5\linewidth]{img/catch2}
%		\includegraphics[0.5\linewidth]{img/catch3}
%		\includegraphics[0.5\linewidth]{img/catch4}
	\end{minipage}
\end{example}

Regardless of the approach, when there exists a catch--all block, there is no dependency generated
from the \texttt{catch}, as all of them will lead to the next instruction. However, this means that
if no data is outputted from the \texttt{try} or \texttt{catch} block, the catches will not be picked
up by the slicing algorithm, which may alter the results unexpectedly. If this problem arises, the
simple and obvious solution would be to add artificial edges to force the inclusion of all \texttt{catch}
blocks, which adds instructions to the slice ---lowering its score when evaluating against benchmarks---
but are completely innocuous as they just stop the exception, without running any extra instruction.

Another alternative exists, though, but slows down the process of creating a slice from a SDG.
The \texttt{catch} block is only strictly needed if an exception that it catches may be thrown and
an instruction after the \texttt{try-catch} block should be executed; in any other case the \texttt{catch}
block is irrelevant and should not be included. However, this change requires analyzing the inclusion
of \texttt{catch} blocks after the two--pass algorithm has completed, slowing it down. In any case, each
approach trades time for accuracy and vice--versa, but the trade--off is small enough to be negligible.

Regarding \textit{unchecked} exceptions, an extra layer of analysis should be performed to tag statements
with the possible exceptions they may throw. On top of that, methods must be analyzed and tagged
accordingly. The worst case is that of inaccessible methods, which may throw any \texttt{RuntimeException},
but with the source code unavailable, they must be marked as capable of throwing it. This results on
a graph where each instruction is dependent on the proper execution of the previous statement; save
for simple statements that may not generate exceptions. The trade--off here is between completeness and
correctness, with the inclusion of \textit{unchecked} exceptions increasing both the completeness and the
slice size, reducing correctness. A possible solution would be to only consider user--generated exceptions
or assume that library methods may never throw an unchecked exception. A new slicing variation that
annotates methods or limits the unchecked exceptions to be considered.

Regarding the \texttt{finally} block, most approaches treat it properly; representing it twice: once
for the case where there is no active exception and another one for the case where it executes with
an exception active. An exception could also be thrown here, but that would be represented normally.

% vim: set noexpandtab:ts=2:sw=2:wrap