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#
# Make the propositional logic diagram a paper
#
paper: paper.tex fzd_paper.tex
#latex paper.tex
#dvipdf paper pdflatex cannot use eps ffs
pdflatex paper.tex
okular paper.pdf
# Remove the need for referncing graphics in subdirectories
#
fzd_paper.tex: fzd.tex paper.tex
cat fzd.tex | sed 's/fzd\///' > fzd_paper.tex
#component_failure_modes_definition_paper.tex

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\abstract{
This paper discusses a two stage algorithm designed to greatly
reduce the number of Area compare operations required to determine which zones are `available' in an Euler
diagram.
This algorithm will aid in the quick resolution of complex drawn
Euler diagrams where the available zones on the diagram must be known.
An Euler diagram of $N$ contours has a possible $2^N$ zones.
A `brute force' search for available zones (determining for availability of all possible $2^N$
zones) involves $N.2^N$ Area Compare operations.
The first stage of the algorithm identifies $M$ number of unique groups of contours that are isolated
w.r.t. zone production. Thus each identified group of $nn$ contours
has a maximum of $2^{nn}$ zones. This reduces the exponential overhead of the $N.2^N$ order.
In fact we reduce the number of stages to search from a $N.2^N$ order
to approximately $N^2 + M.2^{nn}$.
The next stage reduces the number of searches required within the isolated
groups, (thus reducing the $M.2^{nn}$ terms) by traverssing a graph
of the relationships between the contours.
}
\section{Introduction : Euler Diagram and Zones Available for use}
Euler diagrams consist of closed curves in the plane which are used to represent sets.
The spaitial ralationship between the curves defines the set theoretic relationships.
The spaitial relationship between the curves defines the set theoretic relationships.
\begin{itemize}
\item Intersection - if the curves overlap
\item Sub-set - if a curve is enclosed by another
\item disjoint - if the curves are separate
\end{itemize}
A zone is defined as a region of the plane where a set of curves will enclose it
and another set is disjoint from it.
\paragraph{Defining a Zone as two sets of Contours}
%A zone is defined as a region of the plane where a set of curves will enclose it
%and another set is disjoint from it.
A zone can be defined by the sets that enclose the region on the plane defining the zone (the $A$ set), and the sets
that are disjoint from it (the $B$ sets).
\paragraph{Difference between a Venn and an Euler Diagram}
A Venn diagram is an Euler diagram where all possible zones are present.
Thus if we have a Venn diagram with $N$ countours it will have $2^N$
Thus if a Venn diagram has $N$ countours it will have $2^N$
zones.
%
An Euler diagram, does not have to make all possible zones available.
@ -23,6 +48,7 @@ not only combinations of contours but also the number of available zones.
In Constraint Diagram and PLD's Euler diagrams are used
and objects are placed upon the available zones.
%
\paragraph{Importance of determining available zones}
When performing logical reasoning on euler diagrams,
it is important to note which available zones do not have objects associated with them in order to flag the unhandled cases in the diagram.
These could represent cases where the user has left them undefined, or considers them to
@ -45,6 +71,8 @@ an enclosing contour.
\par
\paragraph{definition of `available'}
For a zone to be available for use it must
\begin{itemize}
@ -64,18 +92,26 @@ $ Z_{n} $
$ B_{n} $.
\subsubsection{Testing a Zone for Existance}
\begin{figure}[h]
\centering
\includegraphics[width=400pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction1.jpg}
% exampleareasubtraction1.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
\caption{Simple Euler Diagram}
\label{fig:ex1}
\end{figure}
\subsection{Testing a Zone for Existance}
The Java Area, Shape and Polygon classes, provide functions to
intersect, subtract and `exclusive or' Areas on the plane. They are thus
very useful in testing objects drawn under a Java environment.
%
To determine if a zone exists we can apply the intersection
functions to all the $A_{N}$ contours. If any Area remains
functions to all the $A_{N}$ contours. If any $|Area|$ remains
the zone exists. To check that the zone is available, we must
also ensure that it is not covered over by a contour in the $ B_{N}$ set.
also ensure that it is not covered over/obscured by a contour in the $ B_{N}$ set.
\subsubsection{Testing for Obscuration}
\subsection{Testing for Obscuration}
Firstly apply intersection to all the $A_{n}$ contours in the Zone.
We are then left with the area of intersection only.
%
@ -86,7 +122,7 @@ In other words, a zone obscured by other contours is one that forms the area of
of the set $ A_{n} $ and then having all the areas from set $ B_{n} $ subtracted from it
no surface area left.
%
Firstly let us define the meaning of availability in concrete area termsi by means of an example diagram.
Firstly let us define the meaning of availability in concrete area terms by means of an example diagram.
In figure \ref{fig:ex1}, there is an Euler diagram with the following zones available.
@ -116,14 +152,6 @@ In figure \ref{fig:ex1}, there is an Euler diagram with the following zones avai
Note $B \cap C$ and $ C $ are not available in this diagram because it is impossible to place objects on them.
Objects could be placed on $ B \cap C \cap D $ and $ B \cap D $ however.
\begin{figure}[h]
\centering
\includegraphics[width=200pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction1.jpg}
% exampleareasubtraction1.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
\caption{Simple Euler Diagram}
\label{fig:ex1}
\end{figure}
%
% \begin{figure}
% \vskip 4.2cm
@ -181,7 +209,7 @@ In figure \ref{fig:ex1}, there is an Euler diagram with the following zones avai
%\clearpage
\subsubsection{Formal expression of Area Operations}
\subsection{Formal expression of Area Operations}
This section deals with operations on the concrete diagrams using Areas.
An intersection for instance therefore represents an intersection of the Areas
@ -228,7 +256,7 @@ the expression is true.
%\clearpage
\subsubsection{Testing for zone Availability}
\subsection{Testing for zone Availability}
Firstly that the intersection exists in the concrete diagram.
\equation
@ -377,15 +405,15 @@ The algorithm for fast finding of available zones depends upon defining three ne
\item {Belonging to a Pure Intersection Chain}
\end{itemize}
\subsubsection { Pure Intersection }
\subsection { Pure Intersection }
A pair of contours are said to have 'pure intersection' if the contours overlap.
\subsubsection{Enclosure}
\subsection{Enclosure}
A countour is said to be enclosed if it fits completely within another contour.
\subsubsection { Pure Intersection Chains }
\subsection { Pure Intersection Chains }
\begin{figure}[h]
\centering
@ -442,7 +470,7 @@ Because {\em A,B,C,D} all share pure intersection connections they all belong to
NB: Three or more linked pure intersections contitute a 'pure intersection chain'.
\subsubsection { Determining the Pure Intersection and Enclosure Relationships }
\subsection { Determining the Pure Intersection and Enclosure Relationships }
\label{detpe}
By applying java Area searches for enclosure and intersection on each
contour against all others a collection of pure intersection relationships
@ -475,7 +503,7 @@ This forms a list of relationship pairs from the cross product of all the contou
% \label{fig:picwaie}}
% \end{figure}
\subsubsection {Determining Enclosure }
\subsection {Determining Enclosure }
When a contour completely encloses another contour, it has an enclosing relationship.
See figure \ref{fig:piee}
@ -485,6 +513,7 @@ To determine the enclosure relationships for the diagram,
This again, forms a list of relationship pairs from cross product of all the contours.
\equation
%\label{crossprodsingle}
\begin{array}{l}
@ -496,13 +525,22 @@ This again, forms a list of relationship pairs from cross product of all the co
\endequation
\section{The Pure Intersection Chain}
A pure intersection chain is isolated in terms of zone production.
If the number of contours in the chain is $nn$ then the maximum
number of zones within the chain can be $2^{nn}$.
If contours enclose the chain, the $A$ set defining the zones will
have the contours that enclose it, but the same number of
zones will be produced by examining the pure instersection chains with Area Comparisons.
\subsection { Rules that can be derived from the three relationships }
\subsubsection { Rule 1: Simple Zone Creation }
\subsection { Rule 1: Simple Zone Creation }
Any contour not belonging to a pure intersection chain, will create a zone containing itself, and any enclosing contours.
\subsubsection { Rule 2: All Pure Intersection chains and enclosures can be represented on a directed graph }
\subsection { Rule 2: All Pure Intersection chains and enclosures can be represented on a directed graph }
By displaying pure intersection relations and enclosure relations in different colours
both can be represented on the same coloured directed graph (CDG). I have chosen blue for pure
@ -543,18 +581,18 @@ Figure \ref{fig:pig1} Shows a CDG for the diagram in figure \ref{fig:pic}. Note
% \end{figure}
\subsubsection { Graph Traversal }
\subsection { Graph Traversal }
By traversing the graphs and applying tests for implicit enclosure within a pure intersection chain
from each contour belonging to it, and applying any enclosure relations all possible
zone combinations are revealed.
A path may not loop, i.e. it cannot branch to a contour all ready examined in the path.
\subsubsection{ Rule 3: Traversal Reduction : Avoiding Repeated Area checking }
\subsection{ Rule 3: Traversal Reduction : Avoiding Repeated Area checking }
As each potential zone is discovered and checked it is temporarily stored,
and if re-discovered on a new path, is not subjected to Area testing.
\subsubsection { Rule 4: Pure Intersection Pair Zone Creation }
\subsection { Rule 4: Pure Intersection Pair Zone Creation }
If any pure intersection exists, a potential zone exists. This zone
will intersect with any contour which has an enclosure relationship with
@ -575,7 +613,7 @@ Note : In figure \ref{fig:picwaie} note that contours \em{D} and \em{E} are in
\subsubsection { Rule 5: Multiple Zone Creation }
\subsection { Rule 5: Multiple Zone Creation }
A circular reference (often described as a circuit \cite{gtl} \cite{alggraph}) containing more than one pair of pure intersections

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\documentclass[a4paper,10pt]{article}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{tikz}
\usepackage{amsfonts,amsmath,amsthm}
\usepackage{algorithm}
\usepackage{algorithmic}
%\input{../style}
%\newtheorem{definition}{Definition:}
\begin{document}
\pagestyle{fancy}
%\outerhead{{\small\bf Symptom Extraction Process}}
%\innerfoot{{\small\bf R.P. Clark } }
% numbers at outer edges
\pagenumbering{arabic} % Arabic page numbers hereafter
\author{R.P.Clark}
\title{Fast Zone discrimination Extraction Process}
\maketitle
\input{fzd_paper}
\bibliographystyle{plain}
\bibliography{../vmgbibliography,../mybib}
\today
\end{document}