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\begin{document}
\chapter{Knowledge Theory --- the Rough Set Approach}
\markboth{Knowledge Theory - the Rough Set Approach}{Knowledge Theory --- the Rough Set Approach}
\markright{Knowledge Theory --- the Rough Set Approach}
\thispagestyle{empty}
We present here a concise introduction, based on literature, to
the notion of Knowledge; our approach is that of Rough Set
Theory as proposed in [17].
\section{Introduction}
Knowledge theory has a long and rich history (cf. [13], [15]).
For understanding, representing, and manipulating knowledge,
there is a variety of opinions and approaches in this area (cf.
[13], [15]).
One can understand knowledge as a body of information about some
parts of reality, which form the domain of interest. But this
definition fails to meet precision standards and on closer
inspection has multiple meaning tending to mean one of several
things depending on the context and the area of interest.
We will give the formal definition of term ``knowledge''
proposed by Pawlak [17] and some of its basic properties.
The concept of knowledge presented here is general enough to cover
various understandings of this concept in the current literature.
We advocate here a rough set concept as a theoretical framework
for discussion about knowledge.
\section{Knowledge and classification}
Of knowledge is assumed here that it is based on the ability to
classify objects, and by objects we mean anything we can think
of. Knowledge in this approach is necessarily connected with the
variety of classification patterns, related to specific parts of
the real or abstract world, called here the universe.
Knowledge here will consist of a family
of various classification patterns of a domain of interest,
which provide explicite facts about reality together with the
reasoning capacity able to deliver implicit facts derivable from
implicit knowledge.
\begin{deff}[17]\hfill\break
Let $U$ be a set of objects we are interested in such that $\#$
$(U)\geq 1$ (the universe). Any subset $X\subseteq U$ of the
universe will be called a {\em concept} or a {\em category} in $U$.
\end{deff}
\begin{deff}[17]\hfill\break
Any family $\{X_{i}\}$ of concepts in $U$ will be referred to as
{\em abstract knowledge} (or in short, {\em knowledge}).
For formal reason we also admit the null set $\emptyset$ as a category.
\end{deff}
\begin{deff}\hfill\break
Any knowledge $\{X_{i}\}^{i=n}_{i=1}$ of a certain universe $U$
such that $X_{i}\subseteq U$, $X_{i}\neq \emptyset$,
$$
X_{i} \cap X_{j} = \emptyset \quad\mbox{for}\quad i\neq j, \; i,j=1,2,\ldots,n,
$$
and $UX_{i}=U$ forms a partition of $U$ and will be called a
{\em classification} of $U$.
Since we usually deal not with a single classification but
with some families of classifications over $U$, we will submit
the following definition.
\end{deff}
\begin{deff}[17]\hfill\break
Any family $\{C_{i}\}$ of classifications over $U$, will be
called a knowledge base over $U$. Thus, knowledge base represents
a variety of basic classification skills (e.g., according to
color, shape and size) of an ``intelligent'' agent or group of
agents (e.g.\ cars, toys).
Now since the concepts of classifications (partitions) and equivalence
relations are mutually interchangeable and relations are easier
to deal with, we will often use equivalence relations. Let us now give
some necessary definitions using equivalence relations.
\end{deff}
\begin{deff}\hfill\break
\vspace*{-4mm}
\begin{enumerate}
\item $U|R = \{[x]\}_{R}: x\in U\}$ is the family of all equivalence
classes of the relation $R$ i.e. $U|R$ is a classification
of $U$ and $[x]_{R}$ (the equivalence class of $x$) is a concept
or category in $R$ containing $x\in U$.
\item Let $U$ be a universe such that \#$(U)\geq 1$. Let $\R =
\{R: R$ is an equivalence relation on $U\}$, then we define a
relational system $K=(U,\R)$ as a knowledge base.
\end{enumerate}
\end{deff}
\begin{proposition}[17]\hfill\break
{\it
If $\P \subseteq \R$, $\P\neq \emptyset$, where $\R$ is a family of
equivalence relations over a non-empty finite universe $U$, then
$\cap_{R\in \P}R$ is an equivalence relation and denoted by $\IND(\P)$,
i.e. $\IND(\P)=\cap_{R\in \P} R$, and we call it an
indiscernibility relation over $\P$.
Moreover,
$$
[x]_{\IND(\P)}= \bigcap_{R\in \P} [x]_{R}.
$$
}
\end{proposition}
\begin{proof}
\begin{itemize}
\item[(i)] Reflexivity: Since $\forall R\in \P \; (x,x)\in R$,
$(x,x)\in \cap_{R\in \P} R$.
\item[(ii)] Symmetry: $(x,y)\in \cap \P$ implies $(x,y)\in R$
for all $R\in \P$, but each $R$ is symmetric so
$(y,x)\in R$ for all $R\in \P$ i.e. $(y,x)\in \cap \P$
\item[(iii)] Transitivity: Let $(x,y)$, $(y,z)\in \cap \P$ which
implies $\forall R\in \P \; (x,y)$, $(y,z)\in R$, but each
$R$ is transitive, therefore $(x,z)\in \cap \P$.
\end{itemize}
\end{proof}
\begin{deff}[17]\hfill\break
\vspace*{-4mm}
\begin{enumerate}
\item The family $U|\IND(\P)$ will be called the $\P$- basic knowledge
about $U$ in $K=(U,\P)$.
\item Equivalence classes of $\IND(\P)$ are called {\em basic concepts
(categories)} of knowledge $\P$ in particular if $Q\in \R$,
then $Q$ will be called a $Q$-{\em elementary knowledge} (about
$U$ in $K$), and equivalence classes of $Q$ are referred to
as $Q$-{\em elementary concepts} of knowledge $\R$.
\item We define the minimal set of all equivalence relations
defined in $K$ to be:
$$
\IND(K)= \{\IND(\P):\emptyset \neq \P\subseteq \R\}.
$$
\end{enumerate}
\end{deff}
\hbox{}
\begin{example}
\vspace*{-4mm}
$$
\mbox{Let}\;\; U=\{x_{1},x_{2},x_{3}, x_{4},x_{5}\}
$$
where $x_{i}$ is a student in University at Warsaw,
for each $i=1,\ldots,5$.
These students have different sex, nationality and specialty.
Suppose $U$ is classified according to sex, nationality and specialty
for example as shown below:
\[
\begin{array}{l}
\left.
\begin{array}{ll}
x_{1}, x_{3}, x_{5} \;\;\mbox{\em female}\\
x_{2}, x_{4} \;\; \mbox{\em male}\\
\end{array}
\right\}
\mbox{according to sex}\\
\\
\left.
\begin{array}{ll}
x_{1}, x_{3} \;\;\mbox{\em Polish}\\
x_{2}, x_{4} \;\; \mbox{\em Libyan}\\
x_{5} \;\; \mbox{\em German}\\
\end{array}
\right\}
\mbox{according to nationality}\\
\\
\left.
\begin{array}{ll}
x_{1}, x_{2} \;\;\mbox{\em Mathematics}\\
x_{3} \;\; \mbox{\em Physics}\\
x_{4}, x_{5} \;\; \mbox{\em Chemistry}\\
\end{array}
\right\}
\mbox{according to specialty}\\
\end{array}
\]
By these three classifications we defined three equivalence
relations $R_{1}, R_{2}, R_{3}$ respectively having the following
equivalence classes:
\begin{eqnarray*}
U|R_{1} &=& \{\{x_{1}, x_{3}, x_{5}\}, \{x_{2}, x_{4}\}\}\\
U|R_{2} &=& \{\{x_{1}, x_{3}\}, \{x_{2}, x_{4}\}, \{x_{5}\}\\
U|R_{3} &=& \{\{x_{1}, x_{2}\}, \{x_{3}\}, \{x_{4}, x_{5}\}\}
\end{eqnarray*}
These are elementary concepts in the knowledge base $K=(U$,
$\{R_{1}$, $R_{2}$, $R_{3}\})$. Examples of basic concepts:
$$
\{x_{1}, x_{3}, x_{5}\} \cap \{x_{1}, x_{3}\} = \{x_{1}, x_{3}\}
$$
this set is $\{R_{1}, R_{2}\}$-basic concept {\em female} and
{\em Polish}.
The set
$$
\{x_{1},x_{3},x_{5}\}\cap \{x_{1},x_{3}\}\cap \{x_{1},x_{2}\}= \{x_{1}\}
$$
is $\{R_{1}, R_{2}, R_{3}\}$-basic category {\em female, Polish} and
{\em Mathematics}
\end{example}
\begin{deff}[17]\hfill\break
\vspace*{-4mm}
\begin{enumerate}
\item Let $K=(U,\P)$ and $K'=(U,Q)$ be two knowledge bases. We
will say that $K$ and $K'$ ($\P$ and $Q$) are equivalent, denoted
$K\simeq K'$, $(\P\simeq Q)$, if $\IND(\P)=\IND(Q)$ or
$U|_{\P}=U|_{Q}$.
\item Let $K=(U,\P)$ and $K'=(U,Q)$ be two knowledge bases. If
$\IND(\P)\subset \IND(Q)$,
we say that knowledge $\P$
(knowledge base $K$) is finer than knowledge $Q$,
(knowledge base $K'$) or $Q$ is coarser than $\P$.
We will also say that if $\P$ is finer than $Q$, then $Q$ is a
specialization of $Q$ and $Q$ is a generalization of $\P$.
\end{enumerate}
\end{deff}
\begin{example}\hfill\break
Let $U= \{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\}$, let
\begin{eqnarray*}
R_{1} &=& \{(x_{1}, x_{1}), (x_{2},x_{2}), (x_{3},x_{3}), (x_{4},x_{4}), (x_{5},x_{5}),\\
& & (x_{1},x_{2}), (x_{2},x_{1}), (x_{3},x_{4}),
(x_{4},x_{3})\}\\
R_{2} &=& \{(x_{1}, x_{1}), (x_{2},x_{2}), (x_{3},x_{3}),
(x_{4},x_{4}), (x_{5},x_{5}), (x_{1},x_{3}),\\
& & (x_{3},x_{1}), (x_{2},x_{4}), (x_{4},x_{2}),
(x_{2},x_{3}), (x_{3},x_{2}), (x_{4},x_{5}), (x_{5}, x_{4})\}\\
R_{3} &=& \{(x_{1}, x_{1}), (x_{2},x_{2}), (x_{3},x_{3}),
(x_{4},x_{4}), (x_{5},x_{5}),\\
& & (x_{1},x_{5}), (x_{5},x_{1}), (x_{1},x_{2}),
(x_{2},x_{1}), (x_{5},x_{2}), (x_{2},x_{5})\}\\
R_{4} &=& \{(x_{1}, x_{1}), (x_{2},x_{2}), (x_{3},x_{3}),
(x_{4},x_{4}), (x_{5},x_{5}),\\
& & (x_{1},x_{3}), (x_{3},x_{1}), (x_{2},x_{5}),
(x_{5},x_{2})\}\\
R_{5} &=& \{(x_{1}, x_{1}), (x_{2},x_{2}), (x_{3},x_{3}), (x_{4},x_{4})\\
& & (x_{5},x_{5}), (x_{2},x_{3}), (x_{3},x_{2})\}\\
R_{6} &=& \{(x_{1}, x_{1}), (x_{2},x_{2}), (x_{3},x_{3}), (x_{4},x_{4}),
(x_{5},x_{5}), (x_{1},x_{2}),\\
& & (x_{2},x_{1}), (x_{1},x_{3}), (x_{3},x_{1}), (x_{1},x_{4}),
(x_{4},x_{1}), (x_{2},x_{3}), (x_{3}, x_{2}),\\
& & (x_{2}, x_{4}), (x_{4},x_{2}), (x_{3},
x_{4}), (x_{4}, x_{3})\}\\
R_{7} &=& \{(x_{1}, x_{1}), (x_{2},x_{2}), (x_{3},x_{3}), (x_{4},x_{4}),
(x_{5},x_{5}), (x_{1},x_{4}),\\
& & (x_{4},x_{1}), (x_{2},x_{5}),
(x_{5},x_{2}), (x_{2},x_{3}), (x_{3},x_{2}), (x_{5},x_{3}),
(x_{3}, x_{5})\}
\end{eqnarray*}
\begin{eqnarray*}
U/R_{1} & =& \{\{ x_{1},x_{2}\}, \{x_{3}, x_{4}\}, \{x_{5}\}\}\\
U/R_{2} & =& \{\{ x_{1},x_{3}\}, \{x_{2}, x_{4}, x_{5}\}\}\\
U/R_{3} & =& \{\{ x_{1},x_{2}, x_{5}\}, \{x_{3}\}, \{x_{4}\}\}\\
U/R_{4} & =& \{\{ x_{1},x_{3}\}, \{x_{2}, x_{5}\}, \{x_{4}\}\}\\
U/R_{5} &=& \{\{ x_{1}\}, \{x_{2}, x_{3}\}, \{x_{4}\}, \{x_{5}\}\}\\
U/R_{6} &=& \{\{ x_{1},x_{2}, x_{3}, x_{4}\}, \{x_{5}\}\}\\
U/R_{7} &=& \{\{ x_{1},x_{4}\}, \{x_{2}, x_{5}\}, \{x_{3}\}\}
\end{eqnarray*}
Now take
$\P=\{R_{1}, R_{2}, R_{3}, R_{4}\}$ and $Q = \{R_{5}, R_{6}, R_{7}\}$
\begin{eqnarray*}
\IND(\P) &=& R_{1}\cap R_{2}\cap R_{3}\\
&=& \{(x_{1},x_{1}), (x_{2},x_{2}), (x_{3},x_{3}), (x_{4},x_{4}),
(x_{5}, x_{5})\},\\
\IND(Q) &=& R_{4}\cap R_{5}\cap R_{6}\cap R_{7}\\
&=& \{(x_{1},x_{1}), (x_{2},x_{2}), (x_{3},x_{3}), (x_{4},x_{4}),
(x_{5}, x_{5}), (x_{2},x_{3}), (x_{3},x_{2})\},
\end{eqnarray*}
It is clear that $\IND(\P) \subset \IND(Q)$.
\end{example}
Therefore $\P$ is a specialization of $Q$ or $Q$ is a
generalization of $\P$.
\end{document}