A bit more text

report/report.tex

@@ -18,6 +18,7 @@
 \usepackage{my_listings}
 \usepackage{my_hyperref}
 \usepackage{../common/internship}
+\usepackage{../common/math}
 
 \bibliography{../common/refs}
 
@@ -95,9 +96,11 @@ functions for this task.
 
 \todo{Rename this section}
 
+\subsection{Circuit description}
+
 \begin{figure}[!h]
     \begin{align*}
-        \textbf{Integer constant } n, m, \ldots \qquad& \\
+        \textbf{Integer constant } n, m, p, q, \ldots \qquad& \\
         \\
         \textbf{Wire } in0, out0, ctl0, \ldots \qquad& \\
         \\
@@ -109,9 +112,9 @@ functions for this task.
                 &\textit{(three-state gate)} \\
             \vert~&\text{comb} (\evec{in0}{n}, \evec{out0}{m}, \evec{e}{m})
                 &\textit{(combinatorial gate)} \\
-            \vert~&\text{assert} (\evec{in0}{n}, \evec{e}{m)}
+            \vert~&\text{assert} (\evec{in0}{n}, \evec{e}{m})
                 &\textit{(assertion gate)} \\
-            \vert~&\text{group} (\evec{c}{n})
+            \vert~&\text{group} (\evec{in0}{n}, \evec{out0}{m}, \evec{c}{p})
                 &\textit{(circuit hierarchical group)} \\
         \\
         \textbf{Binary operator } \otimes ::=~
@@ -147,10 +150,64 @@ functions for this task.
     \caption{AST of circuits used}\label{fig:ast}
 \end{figure}
 
+The circuits on which \emph{isomatch} is working are described, and internally
+represented, by the AST in Figure~\ref{fig:ast}.
+
+The most important thing in the description of circuits here, is that those
+circuits are organized as a hierarchy of \emph{circuit groups}. This hierarchy
+can be seen as the construction of a circuit by assembling smaller integrated
+circuits (ICs), themselves built the same way, etc. A group is composed of
+sub-circuits, input pins and output pins. Each level can of course contain
+``leaf'' gates, like \textit{and} or \textit{delay} gates. This is important,
+because it allows the program to work on smaller areas the circuit (\eg{}
+loading in memory only a part of the circuit, etc.).
+
+\subsection{Sought efficiency}
+
+The goal of \textit{isomatch} is to be applied to large circuits on-the-fly,
+during their conception. Those circuits can (and will probably) be as large as
+a full processor, and the software will be operated by a human, working on
+their circuit. Thus, \textit{isomatch} must be as fast as possible, since
+matching operation will be executed often, and often multiple times in a row.
+It must then remain fast enough for the human not to lose too much time, and
+eventually lose patience.
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{General approach}
 
-\todo{}
+More precisely, the problems that \emph{isomatch} must solve are the following.
+
+\begin{enumerate}
+    \item\label{prob:equal} Given two circuit groups, are they structurally
+        equivalent? That is, are they the same circuit, arranged in a different
+        way, with possibly different names, etc.?
+
+    \item\label{prob:match} Given two circuits, \emph{needle} and
+        \emph{haystack}, find every (non-overlapping) occurrence of \emph{needle} in
+        \emph{haystack}. An occurrence is a set $S$ of sub-circuits of
+        \emph{haystack} such that there is a one-to-one mapping of structurally
+        equivalent circuits of $S$ with circuits of \emph{needle}, and those
+        circuits are connected the same way in both circuits.
+\end{enumerate}
+
+Both problems are hard. The first one is an instance of graph isomorphism, as
+the actual question is whether there exists a one-to-one mapping between
+sub-circuits of the two groups, such that every mapped circuit is equal to the
+other (either directly if it is a leaf gate, or recursively with the same
+procedure); and whether this mapping respects connections (edges) between those
+circuits. Graph isomorphism is known to be in NP (given a permutation of the
+first graph, it is polynomial to check whether the first is equal to the second
+\wrt{} the permutation), but not known to be in either P or NP-complete. Thus,
+since Babai's work on graph isomorphism~\cite{babai2016graph} is only of
+theoretical interest, the known algorithms remain in worst-case exponential
+time, and require ad-hoc heuristics for specific kind of graphs to get maximum
+efficiency.
+
+The second one is an instance of subgraph isomorphism problem, which is known
+to be NP-complete~\cite{cook1971complexity}. Even though a few algorithms
+(discussed later) are known to be efficient in most cases for this problem, it
+is nevertheless necessary to implement them the right way, and with the right
+heuristics, to get the desired efficiency for the given problem.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Signatures}