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A bit more text

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Théophile Bastian 2017-08-19 18:03:20 +02:00
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3 changed files with 151 additions and 4 deletions
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72
common/math.sty Normal file
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@ -0,0 +1,72 @@
\usepackage{stmaryrd}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{fancybox}
% Intervalle discret.
\newcommand{\discrIv}[1]{\llbracket #1 \rrbracket}
% ensembliste
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\card}[1]{\left\vert{} #1 \right\vert}
\newcommand{\abs}[1]{\left\vert{} #1 \right\vert}
\newcommand{\interior}[1]{\left({#1}\right)^\circ}
\newcommand{\floor}[1]{\left\lfloor{} #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil{} #1 \right\rceil}
% Abréviations courrantes
\newcommand{\ie}{\textit{ie.}}
\newcommand{\eg}{\textit{eg.}}
\newcommand{\wrt}{\textit{wrt.}}
% Matrices
\newcommand{\transp}{\top}
% Notations à polices étranges
\newcommand{\domain}{\mathcal{D}}
\newcommand{\bigO}{\mathcal{O}}
\newcommand{\calA}{\mathcal{A}}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\calG}{\mathcal{G}}
\newcommand{\calV}{\mathcal{V}}
\newcommand{\calT}{\mathcal{T}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\risk}{\mathcal{R}}
\newcommand{\vect}[1]{\overrightarrow{#1}}
% Ensembles
\newcommand{\realset}{\mathbb{R}}
\newcommand{\natset}{\mathbb{N}}
\newcommand{\relset}{\mathbb{Z}}
\newcommand{\funcspace}{\mathcal{F}}
% Probas
\newcommand{\prob}{\mathbb{P}}
\newcommand{\probP}[1]{\mathbb{P}\left(#1\right)}
\newcommand{\expec}{\mathbb{E}}
\newcommand{\expecP}[1]{\mathbb{E}\left[#1\right]}
\newcommand{\variance}{\mathbb{V}}
\newcommand{\ber}{\mathcal{B}er}
\newcommand{\bin}{\mathcal{B}in}
\newcommand{\poi}{\mathcal{P}oi}
% Suppression des points
\newcommand{\ibar}{\overline{\imath}}
\newcommand{\jbar}{\overline{\jmath}}
% Fonctions
%\newcommand{\functiondef}[4]{\left\lbrace \begin{tabular}{r l} #1 & \rightarrow #2 \\ #3 & \mapsto #4\end{tabular} \right.}
\newcommand{\functiondef}[4]{\begin{cases}
#1 & \to #2 \\
#3 & \mapsto #4
\end{cases}}
\newcommand{\argmin}{\operatorname{argmin}}
% Preuve par équivalence - puces
\newcommand{\impliesbullet}{\ovalbox{$\implies$}}
\newcommand{\impliedbybullet}{\ovalbox{$\impliedby$}}

18
common/refs.bib
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@ -24,3 +24,21 @@
organization={IEEE}
}
@inproceedings{babai2016graph,
title={Graph isomorphism in quasipolynomial time},
author={Babai, L{\'a}szl{\'o}},
booktitle={Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing},
pages={684--697},
year={2016},
organization={ACM}
}
@inproceedings{cook1971complexity,
title={The complexity of theorem-proving procedures},
author={Cook, Stephen A},
booktitle={Proceedings of the third annual ACM symposium on Theory of computing},
pages={151--158},
year={1971},
organization={ACM}
}

65
report/report.tex
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@ -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}