Knowledge reduction of dynamic covering decision information systems with immigration of more objects
Guangming Lang \thanks{Corresponding author.\quad Tel./fax: +86 731 85258787
langguangming1984@126.com \newline\mbox{
1504.00136.tex

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\title{Knowledge reduction of dynamic covering decision information systems with immigration of more objects}

\author
{Guangming Lang
\thanks{Corresponding author.\quad Tel./fax: +86 731 85258787,
langguangming1984@126.com
\newline\mbox{}\hspace{0.55cm}
%E-mail address: langguangming1984@126.com(G.M.Lang), cmjlong@163.com
%(M.J.Cai), math-yangtian@126.com
. }\hspace{1cm}
\\
\small { School of Mathematics and Computer Science, Changsha University of Science and Technology}\\
\small {Changsha, Hunan 410114, P.R. China}\\
%\small {$^{b}$ College of Mathematics and Econometrics, Hunan University}\\
%\small {Changsha, Hunan 410082, P.R. China}\\
%\small {$^{c}$ College of Science, Central South University of Forestry and Technology}\\
%\small {Changsha, Hunan 410004, P.R. China
}
\date{}

\begin{document}
\maketitle \baselineskip=17pt
\begin{center}
\begin{quote}
{{\bf Abstract.}
In practical situations, it is of interest to investigate computing
approximations of sets as an important step of knowledge reduction
of dynamic covering decision information systems. In this paper, we present
incremental approaches to computing the type-1 and type-2
characteristic matrices of dynamic coverings whose cardinalities increase with immigration of more objects. We also present the incremental algorithms of computing the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. 

{\bf Keywords:} Boolean matrice; Characteristic matrice; Dynamic
covering approximation space; Dynamic covering information system; Rough set
\\}
\end{quote}
\end{center}
\renewcommand{\thesection}{\arabic{section}}

\section{Introduction}

Covering-based rough set theory\cite{Zakowski1}, as a powerful
mathematical tool for studying covering approximation spaces, has
attracted a lot of attention of researchers in various fields of
sciences. Especially, various kinds of approximation operators have
been proposed for covering approximation spaces. Recently, Wang et
al.\cite{Wang1} transformed the computation of approximations of a
set into products of the characteristic matrices and the
characteristic function of the set. However, it paid little
attention to approaches to calculating the characteristic matrices.
In practice, the covering approximation space varies with time due
to the characteristics of data collection, and the non-incremental
approach to constructing the characteristic matrices is often very
costly or even intractable in dynamic covering approximation spaces.
It is necessary to present effective approaches to computing
characteristic matrices of dynamic coverings.

To the best of our knowledge,
researchers\cite{Chen1,Chen3,Li12,Luo1,Luo2, Zhang1,Zhang4} have
focused on computing approximations of sets. For instance, Chen et
al.\cite{Chen1,Chen3} constructed approximations of sets when
coarsening or refining attribute values. Li et al.\cite{Li12}
computed approximations in dominance-based rough sets approach under
the variation of attribute set. Luo et al.\cite{Luo1,Luo2} studied
dynamic maintenance of approximations in set-valued ordered decision
systems under the attribute generalization and the variation of
object set. Zhang et al.\cite{Zhang1,Zhang4} updated rough set
approximations based on relation matrices and investigated
neighborhood rough sets for dynamic data mining. These works
demonstrate that incremental approaches are effective and efficient
for computing approximations of sets. It motivates us to apply an
incremental updating scheme to conduct approximations of sets by
using characteristic matrices in dynamic covering approximation
spaces, which will provide an effective approach to computing
approximations of sets from the view of matrices.

The purpose of this paper is to compute approximations of sets by
using incremental approaches in dynamic covering approximation
spaces. First, we present incremental approaches to computing the
type-1 and type-2 characteristic matrices in dynamic covering
approximation spaces. We mainly focus on the situation: the
variation of elements in coverings when adding and deleting objects.
Furthermore, we provide incremental algorithms for constructing the
second and sixth lower and upper approximations of sets based on the
type-1 and type-2 characteristic matrices, respectively. We compare
computation complexities of the incremental algorithms with those of
non-incremental algorithms. Several examples are employed to
illustrate that calculating approximations of sets is simplified
greatly by utilizing the proposed approach.

The rest of this paper is organized as follows: Section 2 briefly
reviews the basic concepts of covering-based rough set theory. In
Section 3, we introduce incremental approaches to computing the
type-1 and type-2 characteristic matrices of dynamic coverings whose cardinalities increase with immigration of more objects. In Section 4, we present incremental algorithms of calculating the
second and sixth lower and upper approximations of sets by using the
type-1 and type-2 characteristic matrices, respectively. We conclude the paper in Section 5.

\section{Preliminaries}

In this section, we  briefly review some concepts of covering-based rough
sets.

\begin{definition}\cite{Zakowski1}
Let $U$ be a finite universe of discourse, and $\mathscr{C}$ a
family of subsets of $U$. Then $\mathscr{C}$ is called a covering of
$U$ if none of elements of $\mathscr{C}$ is empty and
$\bigcup\{C|C\in \mathscr{C}\}=U$.
\end{definition}

If $\mathscr{C}$ is a covering of $U$, then $(U,\mathscr{C})$ is referred to as a covering approximation
space.

\begin{definition}\cite{Wang1}
Let $U=\{x_{1},x_{2},...,x_{n}\}$ be a finite universe, and
$\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$ a covering of $U$. For any
$X\subseteq U$, the second, fifth and sixth upper and lower
approximations of $X$ with respect to $\mathscr{C}$
are defined as

$(1)$ $SH_{\mathscr{C}}(X)=\bigcup\{C\in\mathscr{C}|C\cap X\neq
\emptyset\}$, $SL_{\mathscr{C}}(X)=[SH_{\mathscr{C}}(X^{c})]^{c}$;

$(2)$ $IH_{\mathscr{C}}(X)=\bigcup\{N(x)|N(x)\cap X\neq \emptyset,x\in U\}$,
$IL_{\mathscr{C}}(X)=\bigcup\{N(x)|N(x)\subseteq X,x\in U\}$;

$(3)$ $XH_{\mathscr{C}}(X)=\{x\in U|N(x)\cap X\neq \emptyset\}$,
$XL_{\mathscr{C}}(X)=\{x\in U|N(x)\subseteq X\}$,
 where $N(x)=\bigcap\{C_{i}|x\in C_{i}\in \mathscr{C}\}$.
\end{definition}

For simplicity, we omit $\mathscr{C}$ in the following
description of approximation operators.



\begin{definition}\cite{Wang1}
Let $U=\{x_{1},x_{2},...,x_{n}\}$ be a finite universe,
$\mathscr{C}=\{C_{1}, C_{2}, ..., C_{m}\}$ a family of subsets of $U$,
and $M_{\mathscr{C}}=(a_{ij})_{n\times m}$, where $a_{ij}=\left\{
\begin{array}{ccc}
1,&{\rm}& x_{i}\in C_{j};\\
0,&{\rm}& x_{i}\notin C_{j}.
\end{array}
\right. $ Then $M_{\mathscr{C}}$ is called a matrice representation
of $\mathscr{C}$.
\end{definition}

Accordingly, we have the characteristic function $\mathcal
{X}_{X} =\left[\begin{array}{cccccc}
  a_{1}&a_{2}&.&.&. & a_{n} \\
  \end{array}
\right]^{T}$ for $X\subseteq U$, where $a_{i}=\left\{
\begin{array}{ccc}
1,&{\rm}& x_{i}\in X;\\
0,&{\rm}& x_{i}\notin X.
\end{array}
\right. $.

\begin{definition}\cite{Wang1}
Let $\mathscr{C}$ be a covering of the
universe $U$, $A=(a_{ij})_{n\times m}$ and $B=(b_{ij})_{m\times p}$ Boolean
matrices, $A\odot B=(c_{ij})_{n\times p}$, where
$c_{ij}=\bigwedge^{m}_{k=1}(b_{kj}-a_{ik}+1).$ Then

$(1)$ $\Gamma(\mathscr{C})=M_{\mathscr{C}}\cdot
M_{\mathscr{C}}^{T}=(d_{ij})_{n\times n}$ is called the type-1 characteristic matrice
of $\mathscr{C}$, where $d_{ij}=\bigvee^{m}_{k=1}(a_{ik}\cdot
a_{jk})$, and $M_{\mathscr{C}}\cdot M_{\mathscr{C}}^{T}$ is the
boolean product of $M_{\mathscr{C}}$ and its transpose $
M_{\mathscr{C}}^{T}$;

$(2)$ $\prod(\mathscr{C})=M_{\mathscr{C}}\odot M_{\mathscr{C}}^{T}$ is referred to as the type-2 characteristic matrice of
$\mathscr{C}$.
\end{definition}

Wang et al. axiomatized two important types of covering
approximation operators equivalently by using the type-1 and  type-2 characteristic matrice of
$\mathscr{C}$.

\begin{definition}\cite{Wang1}
Let $U=\{x_{1},x_{2},...,x_{n}\}$ be a finite universe,
$\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$ a covering of $U$, and
$\mathcal {X}_{X}$ the characteristic function of $X$ in $U$. Then

$(1)$ $\mathcal {X}_{SH(X)}=\Gamma(\mathscr{C})\cdot \mathcal
{X}_{X}$, $\mathcal {X}_{SL(X)}=\Gamma(\mathscr{C})\odot \mathcal
{X}_{X}$; $(2)$ $\mathcal {X}_{XH(X)}=\prod(\mathscr{C})\cdot
\mathcal {X}_{X}$, $\mathcal {X}_{XL(X)}=\prod(\mathscr{C})\odot
\mathcal {X}_{X}$.
\end{definition}


\section{Update approximations of sets
with immigration of more objects}

In this section, we introduce incremental approaches to computing
the second and sixth lower and upper approximation of sets with immigration of more objects.

\begin{definition}
Let $(U,\mathscr{C})$ and $(U^{+},\mathscr{C}^{+})$ be covering
approximation spaces, where $U=\{x_{1},x_{2},...,x_{n}\}$,
$U^{+}=U\cup \{x_{n+1},x_{n+2},...,x_{n+t}\}(t\geq 2)$, $\mathscr{C}=\{C_{1},C_{2},...,C_{m}\}$,
$\mathscr{C}^{+}=\{C^{+}_{1},C^{+}_{2},...,C^{+}_{m},C^{+}_{m+1},C^{+}_{m+2},...,C^{+}_{m+l}\}(l\geq 2)$,
where $C^{+}_{i}=C_{i}\cup \Delta C_{i}$ or $C_{i}$ $(1\leq i\leq
m)$, $\Delta C_{i}\subseteq \{x_{n+1},x_{n+2},...,x_{n+t}\}$, and $\{x_{n+1},x_{n+2},...,x_{n+t}\}\subseteq \{C^{+}_{m+j}|1\leq j\leq l\}$. Then $(U^{+},\mathscr{C}^{+})$ is
called a dynamic covering approximation space.
\end{definition}

By Definition 3.1, we refer $\mathscr{C}^{+}$ to as a dynamic covering.
Although there are several types of coverings when adding
objects, we only discuss this type of dynamic coverings for simplicity in this work.

In what follows, we discuss how to construct
$\Gamma(\mathscr{C}^{+})$ based on $\Gamma(\mathscr{C})$. For
convenience, we denote $M_{\mathscr{C}}=(a_{ij})_{n\times m}$,
$M_{\mathscr{C}^{+}}=(a_{ij})_{(n+t)\times (m+l)}$,
$\Gamma(\mathscr{C})=(b_{ij})_{n\times n}$ and
$\Gamma(\mathscr{C}^{+})=(c_{ij})_{(n+t)\times (n+t)}$.

\begin{theorem}
Let $(U^{+},\mathscr{C}^{+})$ be a dynamic covering approximation space
of $(U,\mathscr{C})$, $\Gamma(\mathscr{C})$ and
$\Gamma(\mathscr{C}^{+})$ the type-1 characteristic matrices of
$\mathscr{C}$ and $\mathscr{C}^{+}$, respectively. Then
 \begin{eqnarray*}
 \Gamma(\mathscr{C}^{+})=\left[
  \begin{array}{cc}
    \Gamma(\mathscr{C}) & 0\\
    0 & 0\\
  \end{array}
\right]\bigvee \left[
  \begin{array}{cc}
    \triangle_{1}(\Gamma(\mathscr{C})) & (\triangle_{2}(\Gamma(\mathscr{C})))^{T}\\
    \triangle_{2}(\Gamma(\mathscr{C})) & \triangle_{3}(\Gamma(\mathscr{C}))\\
  \end{array}
\right],
\end{eqnarray*} where
  \begin{eqnarray*}
  \triangle_{1}(\Gamma(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right]^{T}\cdot \left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right];
 \\
\triangle_{2}(\Gamma(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\cdot \left[\begin{array}{cccccc}
  a_{11}&a_{12}&.&.&. & a_{1(m+l)} \\
  a_{21}&a_{22}&.&.&. & a_{2(m+l)} \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  a_{n1}&a_{n2}&.&.&. & a_{n(m+l)} \\
  \end{array}
\right]^{T};\\
 \triangle_{3}(\Gamma(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\cdot \left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T}.
\end{eqnarray*}
\end{theorem}

\noindent\textbf{Proof.} By Definition 3.1, we get
$\Gamma(\mathscr{C})$ and $\Gamma(\mathscr{C}^{+})$ as follows:
\begin{eqnarray*}
\Gamma(\mathscr{C})&=&M_{\mathscr{C}}\cdot
M_{\mathscr{C}}^{T}\\&=&\left[
  \begin{array}{cccccc}
    a_{11} & a_{12} & . & . & . & a_{1m} \\
    a_{21} & a_{22} & . & . & . & a_{2m} \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    a_{n1} & a_{n2} & . & . & . & a_{nm} \\
  \end{array}
\right] \cdot \left[
  \begin{array}{cccccc}
    a_{11} & a_{12} & . & . & . & a_{1m} \\
    a_{21} & a_{22} & . & . & . & a_{2m} \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    a_{n1} & a_{n2} & . & . & . & a_{nm} \\
  \end{array}
\right]^{T} \\&=&\left[
  \begin{array}{cccccc}
    b_{11} & b_{12} & . & . & . & b_{1n} \\
    b_{21} & b_{22} & . & . & . & b_{2n} \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    b_{n1} & b_{n2} & . & . & . & b_{nn} \\
  \end{array}
\right];\end{eqnarray*}
\begin{eqnarray*}
\Gamma(\mathscr{C}^{+})&=&M_{\mathscr{C}^{+}}\cdot
M_{\mathscr{C}^{+}}^{T}\\&=&\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
    a_{21} & a_{22} & . & . & . & a_{2m}& a_{2(m+1)} & . & . & . & a_{2(m+l)}\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    a_{n1} & a_{n2} & . & . & . & a_{nm}& a_{n(m+1)} & . & . & . & a_{n(m+l)}\\
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)} & . & . & . & a_{(n+1)(m+l)}\\
    . & . & . & . & . & .& . & . & . & . & .\\
    . & . & . & . & . & .&. & . & . & . & .\\
    . & . & . & . & . & .& . & . & . & . & .\\
    a_{(n+t)1} & a_{(n+t)2} & . & . & . & a_{(n+t)m}& a_{(n+t)(m+1)} & . & . & . & a_{(n+t)(m+l)}\\
  \end{array}
\right]\\&&\cdot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
    a_{21} & a_{22} & . & . & . & a_{2m}& a_{2(m+1)} & . & . & . & a_{2(m+l)}\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    a_{n1} & a_{n2} & . & . & . & a_{nm}& a_{n(m+1)} & . & . & . & a_{n(m+l)}\\
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)} & . & . & . & a_{(n+1)(m+l)}\\
    . & . & . & . & . & .& . & . & . & . & .\\
    . & . & . & . & . & .&. & . & . & . & .\\
    . & . & . & . & . & .& . & . & . & . & .\\
    a_{(n+t)1} & a_{(n+t)2} & . & . & . & a_{(n+t)m}& a_{(n+t)(m+1)} & . & . & . & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T} \\
&=&\left[
  \begin{array}{ccccccccccccccc}
    c_{11} & c_{12} & . & . & . & c_{1n}& c_{1(n+1)}& . & . & . &c_{1(n+t)}\\
    c_{21} & c_{22} & . & . & . & c_{2n}& c_{2(n+1)}& . & . & . &c_{2(n+t)}\\
    . & . & . & . & . & . & . & . & . & . &.\\
    . & . & . & . & . & . & . & . & . & . &.\\
    . & . & . & . & . & . & . & . & . & . &.\\
    c_{n1} & c_{n2} & . & . & .& c_{nn}& c_{n(n+1)} &. & . & . &c_{n(n+t)}\\
    c_{(n+1)1} & c_{(n+1)2} & . & . & . & c_{(n+1)n}& c_{(n+1)(n+1)}& . & . & . &c_{(n+1)(n+t)}\\
    . & . & . & . & . & .& .& . & . & . &.\\
    . & . & . & . & . & .& .& . & . & . &.\\
    . & . & . & . & . & .& .& . & . & . &.\\
    c_{(n+t)1} & c_{(n+t)2} & . & . & . & c_{(n+t)n}& c_{(n+t)(n+1)}& . & . & . &c_{(n+t)(n+t)}\\
  \end{array}
\right];
\end{eqnarray*}

In the sense of the type-1 characteristic matrice of
$\mathscr{C}^{+}$, we have
\begin{eqnarray*}
c_{11}&=&\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\\&&\cdot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T} \\
&=&\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}\\
  \end{array}
\right]\cdot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}\\
  \end{array}
\right]^{T}\\&&\vee\left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\cdot \left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T}\\&=&b_{11}\vee\left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\cdot \left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T};\\
c_{(n+1)1}&=&\left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]\\&&\cdot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T}\\&=&0\vee \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T};
\end{eqnarray*}
\begin{eqnarray*}
c_{(n+1)(n+1)}&=&\left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]\\&&\cdot \left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]^{T}\\&=&0\vee\left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]\\&&\cdot \left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]^{T}.
\end{eqnarray*}

Since
$c_{11}\in\triangle_{1}(\Gamma(\mathscr{C}))$,
$c_{(n+1)1}\in\triangle_{2}(\Gamma(\mathscr{C}))$ and $c_{(n+1)(n+1)}\in\triangle_{3}(\Gamma(\mathscr{C}))$, we can compute other elements of $\triangle_{1}(\Gamma(\mathscr{C}))$,
$\triangle_{2}(\Gamma(\mathscr{C}))$ and $\triangle_{3}(\Gamma(\mathscr{C}))$ similarly.
Thus,
to obtain $\Gamma(\mathscr{C}^{+})$, we only need to compute
$\triangle_{1}(\Gamma(\mathscr{C}))$,
$\triangle_{2}(\Gamma(\mathscr{C}))$ and $\triangle_{3}(\Gamma(\mathscr{C}))$
on the basis of
$\Gamma(\mathscr{C})$ as follows:
  \begin{eqnarray*}
  \triangle_{1}(\Gamma(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right]^{T}\cdot \left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right];
 \\
\triangle_{2}(\Gamma(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\cdot \left[\begin{array}{cccccc}
  a_{11}&a_{12}&.&.&. & a_{1(m+l)} \\
  a_{21}&a_{22}&.&.&. & a_{2(m+l)} \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  a_{n1}&a_{n2}&.&.&. & a_{n(m+l)} \\
  \end{array}
\right]^{T};
\\
 \triangle_{3}(\Gamma(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\cdot \left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T}.
\end{eqnarray*}

Therefore, we have
\begin{eqnarray*}
 \Gamma(\mathscr{C}^{+})=\left[
  \begin{array}{cc}
    \Gamma(\mathscr{C}) & 0\\
    0 & 0\\
  \end{array}
\right]\bigvee \left[
  \begin{array}{cc}
    \triangle_{1}(\Gamma(\mathscr{C})) & (\triangle_{2}(\Gamma(\mathscr{C})))^{T}\\
    \triangle_{2}(\Gamma(\mathscr{C})) & \triangle_{3}(\Gamma(\mathscr{C}))\\
  \end{array}
\right].
\end{eqnarray*}


\begin{example}
Let $U=\{x_{1},x_{2},x_{3},x_{4}\}$, $U^{+}=U\cup\{x_{5},x_{6}\}$,
$\mathscr{C}=\{C_{1},C_{2},C_{3}\}$,
$\mathscr{C}^{+}=\{C^{+}_{1},C^{+}_{2},C^{+}_{3},C^{+}_{4},C^{+}_{5}\}$, where
$C_{1}=\{x_{1},x_{4}\}$, $C_{2}=\{x_{1},x_{2},x_{4}\}$,
$C_{3}=\{x_{3},x_{4}\}$, $C^{+}_{1}=\{x_{1},x_{4},x_{5}\}$,
$C^{+}_{2}=\{x_{1},x_{2},x_{4},x_{5}\}$,
$C^{+}_{3}=\{x_{3},x_{4}\}$, $C^{+}_{4}=\{x_{3},x_{5},x_{6}\}$,$C^{+}_{5}=\{x_{1},x_{6}\}$, and
$X=\{x_{3},x_{4},x_{5}\}$. By Definition 3.1, we first have that
\begin{eqnarray*}
\Gamma(\mathscr{C})&=&M_{\mathscr{C}}\cdot
M_{\mathscr{C}}^{T}=\left[
\begin{array}{cccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1 \\
\end{array}
\right] \cdot \left[
\begin{array}{cccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1 \\
\end{array}
\right]^{T}  =\left[
\begin{array}{cccc}
1 & 1 & 0 & 1 \\
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}
\right].
\end{eqnarray*}

Second, by Theorem 3.2, we get that
\begin{eqnarray*}
\triangle_{1}(\Gamma(\mathscr{C}))&=&\left[
\begin{array}{ccccc}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right]^{T}\cdot \left[
\begin{array}{cccccc}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right]=\left[
\begin{array}{ccccc}
1 & 0 & 0 & 0  \\
0 & 0 & 0 & 0  \\
0 & 0 & 1 & 0  \\
0 & 0 & 0 & 0  \\
\end{array}
\right];
\end{eqnarray*}
\begin{eqnarray*} \triangle_{2}(\Gamma(\mathscr{C}))&=&\left[
\begin{array}{ccccc}
1 & 1 & 0 & 1& 0\\
0 & 0 & 1 & 0& 1\\
\end{array}
\right]\cdot \left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 & 1\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 1 & 0\\
1 & 1 & 1 & 0 & 0\\
\end{array}
\right]^{T}=\left[
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 \\
\end{array}
\right];\\
\triangle_{3}(\Gamma(\mathscr{C}))&=&\left[
\begin{array}{ccccc}
1 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 1\\
\end{array}
\right]\cdot \left[
\begin{array}{ccccc}
1 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 1\\
\end{array}
\right]^{T}=\left[
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right].
\end{eqnarray*}

Thus, we obtain that
\begin{eqnarray*}
 \Gamma(\mathscr{C}^{+})&=&(c_{ij})_{66}\\&=&\left[
  \begin{array}{cc}
    \Gamma(\mathscr{C}) & 0\\
    0 & 0\\
  \end{array}
\right]\bigvee \left[
  \begin{array}{cc}
    \triangle_{1}(\Gamma(\mathscr{C}) & (\triangle_{2}(\Gamma(\mathscr{C})))^{T}\\
    \triangle_{2}(\Gamma(\mathscr{C})) & \triangle_{3}(\Gamma(\mathscr{C}))\\
  \end{array}
\right]\\ &=&\left[
\begin{array}{cccccc}
1 & 1 & 0 & 1 & 0& 0\\
1 & 1 & 0 & 1 & 0& 0\\
0 & 0 & 1 & 1 & 0& 0\\
1 & 1 & 1 & 1 & 0& 0\\
0 & 0 & 0 & 0 & 0& 0\\
0 & 0 & 0 & 0 & 0& 0\\
\end{array}
\right]\bigvee \left[
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 1& 1\\
0 & 0 & 0 & 0 & 1& 0\\
0 & 0 & 1 & 0 & 1& 1\\
0 & 0 & 0 & 0 & 1& 1\\
1 & 1 & 1 & 1 & 1& 0\\
1 & 0 & 1 & 1 & 0& 1\\
\end{array}
\right]\\&=& \left[
\begin{array}{cccccc}
1 & 1 & 0 & 1 & 1& 1\\
1 & 1 & 0 & 1 & 1& 0\\
0 & 0 & 1 & 1 & 1& 1\\
1 & 1 & 1 & 1 & 1& 1\\
1 & 1 & 1 & 1 & 1& 0\\
1 & 0 & 1 & 1 & 0& 1\\
\end{array}
\right].
\end{eqnarray*}
By Definition 2.5, we have that
\begin{eqnarray*}
\mathcal {X}_{SH(X)}&=&\Gamma(\mathscr{C}^{+})\cdot \mathcal {X}_{X}
=\left[
\begin{array}{cccccc}
1 & 1 & 0 & 1 & 1& 1\\
1 & 1 & 0 & 1 & 1& 0\\
0 & 0 & 1 & 1 & 1& 1\\
1 & 1 & 1 & 1 & 1& 1\\
1 & 1 & 1 & 1 & 1& 0\\
1 & 0 & 1 & 1 & 0& 1\\
\end{array}
\right]\cdot\left[
\begin{array}{c}
0 \\
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right] =\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & 1\\
\end{array}
\right]^{T};
\\
 \mathcal {X}_{SL(X)}&=&\Gamma(\mathscr{C}^{+})\odot
\mathcal {X}_{X}=\left[
\begin{array}{cccccc}
1 & 1 & 0 & 1 & 1& 1\\
1 & 1 & 0 & 1 & 1& 0\\
0 & 0 & 1 & 1 & 1& 1\\
1 & 1 & 1 & 1 & 1& 1\\
1 & 1 & 1 & 1 & 1& 0\\
1 & 0 & 1 & 1 & 0& 1\\
\end{array}
\right]\odot \left[
\begin{array}{c}
0 \\
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right]=\left[
\begin{array}{ccccccc}
0 & 0 & 1 & 0 & 0& 0\\
\end{array}
\right]^{T}.
\end{eqnarray*}

Therefore, $SH(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\}$ and
$SL(X)=\{x_{3}\}$.
\end{example}

In Example 3.3, we only need to calculate elements in
$\triangle_{1}(\Gamma(\mathscr{C}))$,
$\triangle_{2}(\Gamma(\mathscr{C}))$ and $\triangle_{3}(\Gamma(\mathscr{C}))$ by Theorem 3.2.
Thereby, the incremental algorithm is effective to compute the
second lower and upper approximations of sets.

In practical situations, there exists a need to construct the type-2
characteristic matrices of dynamic coverings for computing the sixth
lower and upper approximations of sets. Subsequently, we construct
$\prod(\mathscr{C}^{+})$ based on $\prod(\mathscr{C})$. For
convenience, we denote $\prod(\mathscr{C})=(d_{ij})_{n\times n}$ and
$\prod(\mathscr{C}^{+})=(e_{ij})_{(n+t)\times (n+t)}$.

\begin{theorem}
Let $(U^{+},\mathscr{C}^{+})$ be a dynamic covering approximation space
of $(U,\mathscr{C})$, $\prod(\mathscr{C})$ and
$\prod(\mathscr{C}^{+})$ the type-2 characteristic matrices of
$\mathscr{C}$ and $\mathscr{C}^{+}$, respectively. Then
\begin{eqnarray*}
 \prod(\mathscr{C}^{+})=\left[
  \begin{array}{cc}
   \prod(\mathscr{C}) & 1\\
    1 & 1\\
  \end{array}
\right]\bigwedge \left[
  \begin{array}{cc}
    \triangle_{1}(\prod(\mathscr{C})) & \triangle_{3}(\prod(\mathscr{C}))\\
    \triangle_{2}(\prod(\mathscr{C})) &\triangle_{4}(\prod(\mathscr{C}))\\
  \end{array}
\right], \end{eqnarray*}
where
  \begin{eqnarray*}
  \triangle_{1}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right]^{T}\odot \left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right];
 \\
\triangle_{2}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\odot \left[\begin{array}{cccccc}
  a_{11}&a_{12}&.&.&. & a_{1(m+l)} \\
  a_{21}&a_{22}&.&.&. & a_{2(m+l)} \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  a_{n1}&a_{n2}&.&.&. & a_{n(m+l)} \\
  \end{array}
\right]^{T};\\
\\
\triangle_{3}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{11}&a_{12}&.&.&. & a_{1(m+l)}\\
  a_{21}&a_{22}&.&.&. & a_{2(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{n1}&a_{n2}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right]\odot \left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T};
\\
 \triangle_{4}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\odot \left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T}.
\end{eqnarray*}
\end{theorem}

\noindent\textbf{Proof.} By Definition 3.1, we have
$\prod(\mathscr{C})$ and $\prod(\mathscr{C}^{+})$ as follows:
\begin{eqnarray*}
\prod(\mathscr{C})&=&M_{\mathscr{C}}\odot M_{\mathscr{C}}^{T}\\&=&\left[
  \begin{array}{cccccc}
    a_{11} & a_{12} & . & . & . & a_{1m} \\
    a_{21} & a_{22} & . & . & . & a_{2m} \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    a_{n1} & a_{n2} & . & . & . & a_{nm} \\
  \end{array}
\right] \odot \left[
  \begin{array}{cccccc}
    a_{11} & a_{12} & . & . & . & a_{1m} \\
    a_{21} & a_{22} & . & . & . & a_{2m} \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    a_{n1} & a_{n2} & . & . & . & a_{nm} \\
  \end{array}
\right]^{T} \\&=&\left[
  \begin{array}{cccccc}
    d_{11} & d_{12} & . & . & . & d_{1n} \\
    d_{21} & d_{22} & . & . & . & d_{2n} \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    . & . & . & . & . & . \\
    d_{n1} & d_{n2} & . & . & . & d_{nn} \\
  \end{array}
\right];
\end{eqnarray*}\begin{eqnarray*}
\prod(\mathscr{C}^{+})&=&M_{\mathscr{C}^{+}}\odot
M_{\mathscr{C}^{+}}^{T}\\&=&\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
    a_{21} & a_{22} & . & . & . & a_{2m}& a_{2(m+1)} & . & . & . & a_{2(m+l)}\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    a_{n1} & a_{n2} & . & . & . & a_{nm}& a_{n(m+1)} & . & . & . & a_{n(m+l)}\\
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)} & . & . & . & a_{(n+1)(m+l)}\\
    . & . & . & . & . & .& . & . & . & . & .\\
    . & . & . & . & . & .&. & . & . & . & .\\
    . & . & . & . & . & .& . & . & . & . & .\\
    a_{(n+t)1} & a_{(n+t)2} & . & . & . & a_{(n+t)m}& a_{(n+t)(m+1)} & . & . & . & a_{(n+t)(m+l)}\\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
    a_{21} & a_{22} & . & . & . & a_{2m}& a_{2(m+1)} & . & . & . & a_{2(m+l)}\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    . & . & . & . & . & . & . & . & . & . & .\\
    a_{n1} & a_{n2} & . & . & . & a_{nm}& a_{n(m+1)} & . & . & . & a_{n(m+l)}\\
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)} & . & . & . & a_{(n+1)(m+l)}\\
    . & . & . & . & . & .& . & . & . & . & .\\
    . & . & . & . & . & .&. & . & . & . & .\\
    . & . & . & . & . & .& . & . & . & . & .\\
    a_{(n+t)1} & a_{(n+t)2} & . & . & . & a_{(n+t)m}& a_{(n+t)(m+1)} & . & . & . & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T} \\ \end{eqnarray*} \begin{eqnarray*}
&=&\left[
  \begin{array}{ccccccccccccccc}
    e_{11} & e_{12} & . & . & . & e_{1n}& e_{1(n+1)}& . & . & . &e_{1(n+t)}\\
    e_{21} & e_{22} & . & . & . & e_{2n}& e_{2(n+1)}& . & . & . &e_{1(n+t)}\\
    . & . & . & . & . & . & . & . & . & . &.\\
    . & . & . & . & . & . & . & . & . & . &.\\
    . & . & . & . & . & . & . & . & . & . &.\\
    e_{n1} & e_{n2} & . & . & .& e_{nn}& e_{n(n+1)} &. & . & . &e_{1(n+t)}\\
    e_{(n+1)1} & e_{(n+1)2} & . & . & . & e_{(n+1)n}& e_{(n+1)(n+1)}& . & . & . &e_{1(n+1)}\\
    . & . & . & . & . & .& .& . & . & . &.\\
    . & . & . & . & . & .& .& . & . & . &.\\
    . & . & . & . & . & .& .& . & . & . &.\\
    e_{(n+t)1} & e_{(n+t)2} & . & . & . & e_{(n+t)n}& e_{(n+t)(n+1)}& . & . & . &e_{(n+t)(n+t)}\\
  \end{array}
\right].
\end{eqnarray*}

In the sense of the type-2 characteristic matrice of
$\mathscr{C}^{+}$, we have
\begin{eqnarray*}
e_{11}&=&\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T} \\
&=&\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}\\
  \end{array}
\right]\odot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}\\
  \end{array}
\right]^{T}\\&&\wedge\left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\odot \left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T}\\&=&d_{11}\wedge\left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\odot \left[
  \begin{array}{ccccc}
     a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T};\\
e_{(n+1)1}&=&\left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T}\\&=&1\wedge\left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]^{T};\end{eqnarray*}\begin{eqnarray*}
e_{1(n+1)}&=&\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]^{T}\\&=&1\wedge\left[
  \begin{array}{ccccccccccc}
    a_{11} & a_{12} & . & . & . & a_{1m}& a_{1(m+1)}& . & . & . & a_{1(m+l)} \\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]^{T};\\
e_{(n+1)(n+1)}&=&\left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]^{T}\\&=&1\wedge\left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]\\&&\odot \left[
  \begin{array}{ccccccccccc}
    a_{(n+1)1} & a_{(n+1)2} & . & . & . & a_{(n+1)m}& a_{(n+1)(m+1)}& . & . & . & a_{(n+1)(m+l)} \\
  \end{array}
\right]^{T}.
\end{eqnarray*}

Since
$e_{11}\in\triangle_{1}(\prod(\mathscr{C}))$,
$e_{(n+1)1}\in\triangle_{2}(\prod(\mathscr{C}))$,
$e_{1(n+1)}\in\triangle_{3}(\prod(\mathscr{C}))$ and $e_{(n+1)(n+1)}\in\triangle_{3}(\prod(\mathscr{C}))$, we can compute other elements of $\triangle_{1}(\prod(\mathscr{C}))$,
$\triangle_{2}(\prod(\mathscr{C}))$, $\triangle_{3}(\prod(\mathscr{C}))$ and $\triangle_{4}(\prod(\mathscr{C}))$ similarly.
Thus,
to compute $\prod(\mathscr{C}^{+})$ on the basis of
$\prod(\mathscr{C})$, we only need to compute
$\triangle_{1}(\prod(\mathscr{C}))$,
$\triangle_{2}(\prod(\mathscr{C}))$, $\triangle_{3}(\prod(\mathscr{C}))$
and $\triangle_{4}(\prod(\mathscr{C}))$ as follows:
   \begin{eqnarray*}
  \triangle_{1}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right]^{T}\odot \left[\begin{array}{cccccc}
  a_{1(m+1)}&a_{2(m+1)}&.&.&. & a_{n(m+1)}\\
  a_{1(m+2)}&a_{2(m+2)}&.&.&. & a_{n(m+2)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{1(m+l)}&a_{2(m+l)}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right];
 \\
\triangle_{2}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\odot \left[\begin{array}{cccccc}
  a_{11}&a_{12}&.&.&. & a_{1(m+l)} \\
  a_{21}&a_{22}&.&.&. & a_{2(m+l)} \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  .&.&.&.&. & . \\
  a_{n1}&a_{n2}&.&.&. & a_{n(m+l)} \\
  \end{array}
\right]^{T};\\
\\
\triangle_{3}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{11}&a_{12}&.&.&. & a_{1(m+l)}\\
  a_{21}&a_{22}&.&.&. & a_{2(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{n1}&a_{n2}&.&.&. & a_{n(m+l)}\\
  \end{array}
\right]\odot \left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T};\\
 \triangle_{4}(\prod(\mathscr{C}))&=&\left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]\odot \left[\begin{array}{cccccc}
  a_{(n+1)1}&a_{(n+1)2}&.&.&. & a_{(n+1)(m+l)}\\
  a_{(n+2)1}&a_{(n+2)2}&.&.&. & a_{(n+2)(m+l)}\\
  .&.&.&.&. & .\\
  .&.&.&.&. & .\\
  .&.&.&.&. &.\\
  a_{(n+t)1}&a_{(n+t)2}&.&.&. & a_{(n+t)(m+l)}\\
  \end{array}
\right]^{T}.
\end{eqnarray*}

Therefore, we have
\begin{eqnarray*}
 \prod(\mathscr{C}^{+})=\left[
  \begin{array}{cc}
    \prod(\mathscr{C}) & 1\\
    1 & 1\\
  \end{array}
\right]\bigwedge \left[
  \begin{array}{cc}
    \triangle_{1}(\prod(\mathscr{C})) & \triangle_{3}(\prod(\mathscr{C}))\\
    \triangle_{2}(\prod(\mathscr{C})) & \triangle_{4}(\prod(\mathscr{C}))\\
  \end{array}
\right].
\end{eqnarray*}


The following example illustrates that how to compute the sixth
lower and upper approximations of set by using the incremental
algorithm.
\begin{example} (Continuation of Example 3.3)
We obtain that
\begin{eqnarray*}
\prod(\mathscr{C})&=&M_{\mathscr{C}}\odot M_{\mathscr{C}}^{T}
=\left[
\begin{array}{cccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1 \\
\end{array}
\right] \odot \left[
\begin{array}{cccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1 \\
\end{array}
\right]^{T}=\left[
\begin{array}{cccc}
1 & 0 & 0 & 1 \\
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 \\
\end{array}
\right].
\end{eqnarray*}

By Theorem 3.4, we have that
\begin{eqnarray*}
\triangle_{1}(\prod(\mathscr{C}))&=&\left[
\begin{array}{ccccc}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right]^{T}\odot \left[
\begin{array}{cccccc}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}
\right]=\left[
\begin{array}{ccccc}
1 & 0 & 0 & 0  \\
1 & 1 & 1 & 1  \\
0 & 0 & 1 & 0  \\
1 & 1 & 1 & 1  \\
\end{array}
\right];\\ \triangle_{2}(\prod(\mathscr{C}))&=&\left[
\begin{array}{ccccc}
1 & 1 & 0 & 1& 0\\
0 & 0 & 1 & 0& 1\\
\end{array}
\right]\odot \left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 & 1\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 1 & 0\\
1 & 1 & 1 & 0 & 0\\
\end{array}
\right]^{T}=\left[
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right];\\
\triangle_{3}(\prod(\mathscr{C}))&=&\left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 & 1\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 1 & 0\\
1 & 1 & 1 & 0 & 0\\
\end{array}
\right]^{T}\odot\left[
\begin{array}{ccccc}
1 & 1 & 0 & 1& 0\\
0 & 0 & 1 & 0& 1\\
\end{array}
\right] =\left[
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right];\\
\triangle_{4}(\prod(\mathscr{C}))&=&\left[
\begin{array}{ccccc}
1 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 1\\
\end{array}
\right]\odot \left[
\begin{array}{ccccc}
1 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 1\\
\end{array}
\right]^{T}=\left[
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right].
\end{eqnarray*}
\end{example}

Thus, we have
\begin{eqnarray*}
\prod(\mathscr{C}^{+})&=&\left[
  \begin{array}{cc}
    \prod(\mathscr{C}) & 1\\
    1 & 1\\
  \end{array}
\right]\bigwedge\left[
  \begin{array}{cc}
    \triangle_{1}(\prod(\mathscr{C})) & \triangle_{3}(\prod(\mathscr{C}))\\
    \triangle_{2}(\prod(\mathscr{C})) & \triangle_{4}(\prod(\mathscr{C}))\\
  \end{array}
\right]=\left[
\begin{array}{ccccccc}
1  &   0   &  0   &  0  &   0&   0\\
1  &   1   &  0   &  1  &   1&   0\\
0  &   0   &  1   &  0  &   0&   0\\
0  &   0   &  0   &  1  &   0&   0\\
0  &   0   &  0   &  0  &   1&   0\\
0  &   0   &  0   &  0  &   0&   1\\
\end{array}
\right].
\end{eqnarray*}

By Definition 2.5, we obtain
\begin{eqnarray*}
\mathcal {X}_{XH(X)}&=&\prod(\mathscr{C}^{+})\cdot \mathcal {X}_{X}
=\left[
\begin{array}{ccccccc}
1  &   0   &  0   &  0  &   0&   0\\
1  &   1   &  0   &  1  &   1&   0\\
0  &   0   &  1   &  0  &   0&   0\\
0  &   0   &  0   &  1  &   0&   0\\
0  &   0   &  0   &  0  &   1&   0\\
0  &   0   &  0   &  0  &   0&   1\\
\end{array}
\right] \cdot \left[
\begin{array}{c}
0 \\
0 \\
1 \\
1 \\
1 \\
0\\
\end{array}
\right] =\left[
\begin{array}{cccccc}
0 & 1 & 1 & 1 & 1 & 0 \\
\end{array}
\right]^{T};\\
 \mathcal {X}_{XL(X)}&=&\prod(\mathscr{C}^{+})\odot
\mathcal {X}_{X}=\left[
\begin{array}{ccccccc}
1  &   0   &  0   &  0  &   0&   0\\
1  &   1   &  0   &  1  &   1&   0\\
0  &   0   &  1   &  0  &   0&   0\\
0  &   0   &  0   &  1  &   0&   0\\
0  &   0   &  0   &  0  &   1&   0\\
0  &   0   &  0   &  0  &   0&   1\\
\end{array}
\right] \odot \left[
\begin{array}{c}
0 \\
0 \\
1 \\
1 \\
1 \\
0\\
\end{array}
\right]=\left[
\begin{array}{ccccccc}
0 & 0 & 1 & 1 & 1 & 0 \\
\end{array}
\right]^{T}.
\end{eqnarray*}

Therefore, $XH(X)=\{x_{2},x_{3},x_{4},x_{5}\}$ and
$XL(X)=\{x_{3},x_{4},x_{5}\}$.

In Example 3.5, we need to compute all elements in
$\prod(\mathscr{C}^{+})$ for constructing approximations of sets by
Definition 3.1. By Theorem 3.4, we only need to calculate elements in
$\triangle_{1}(\prod(\mathscr{C}))$,
$\triangle_{2}(\prod(\mathscr{C}))$, $\triangle_{3}(\prod(\mathscr{C}))$
and $\triangle_{4}(\prod(\mathscr{C}))$. Thereby, the incremental algorithm is more effective
to compute approximations of sets.

\section{Non-incremental and incremental algorithms for computing the second and sixth lower and upper approximations of sets}

In this section, we show non-incremental and incremental algorithms of computing the second lower and
upper approximations of sets.

\begin{algorithm}(Non-incremental algorithm of computing $SH_{\mathscr{C}^{+}}(X^{+})$ and $SL_{\mathscr{C}^{+}}(X^{+})\bf{(NCS)}$)

Step 1: Input $(U^{+},\mathscr{C}^{+})$ and $X^{+}\subseteq
U^{+}$;

Step 2: Construct $M_{\mathscr{C}^{+}}$ and
$\Gamma(\mathscr{C}^{+})=M_{\mathscr{C}^{+}}\cdot M^{T}_{\mathscr{C}^{+}};$

Step 3: Compute $\mathcal {X}_{SH_{\mathscr{C}^{+}}(X^{+})}=\Gamma(\mathscr{C}^{+})\cdot \mathcal
{X}_{X^{+}}$ and $\mathcal {X}_{SL_{\mathscr{C}^{+}}(X^{+})}=\Gamma(\mathscr{C}^{+})\odot \mathcal
{X}_{X^{+}}$;

Step 4: Output $SH_{\mathscr{C}^{+}}(X^{+})$ and $SL_{\mathscr{C}^{+}}(X^{+})$.
\end{algorithm}

\begin{algorithm}(Incremental algorithm of computing $SH_{\mathscr{C}^{+}}(X^{+})$ and $SL_{\mathscr{C}^{+}}(X^{+})\bf{(ICS)}$)

Step 1: Input $(U,\mathscr{C})$, $(U^{+},\mathscr{C}^{+})$ and $X\subseteq
U^{+}$;

Step 2: Calculate $\Gamma(\mathscr{C})=M_{\mathscr{C}}\cdot
M^{T}_{\mathscr{C}}$, where $ M_{\mathscr{C}}=(a_{ij})_{n\times m};
$

Step 3: Compute $\triangle_{1}(\Gamma(\mathscr{C}))$ and
$\triangle_{2}(\Gamma(\mathscr{C}))$  and  $\triangle_{3}(\Gamma(\mathscr{C}))$;

Step 4: Construct $\Gamma(\mathscr{C}^{+})$, where
\begin{eqnarray*}
 \Gamma(\mathscr{C}^{+})&=&(c_{ij})_{(n+1)(n+1)}=\left[
  \begin{array}{cc}
    \Gamma(\mathscr{C}) & 0\\
    0 & 0\\
  \end{array}
\right]\bigvee \left[
  \begin{array}{cc}
    \triangle_{1}(\Gamma(\mathscr{C})) & (\triangle_{2}(\Gamma(\mathscr{C})))^{T}\\
    \triangle_{2}(\Gamma(\mathscr{C})) & \triangle_{3}(\Gamma(\mathscr{C}))\\
  \end{array}
\right];
\end{eqnarray*}

Step 5:  Obtain $\mathcal {X}_{SH(X)}$ and $\mathcal
{X}_{SL(X)}$, where
\begin{eqnarray*}
\mathcal {X}_{SH(X)}&=&\Gamma(\mathscr{C}^{+})\cdot \mathcal
{X}_{X}; \mathcal {X}_{SL(X)}=\Gamma(\mathscr{C}^{+})\odot \mathcal
{X}_{X}.
\end{eqnarray*}
\end{algorithm}

Subsequently, we present non-incremental and incremental algorithms of computing the sixth lower and
upper approximations of sets.

\begin{algorithm}(Non-incremental algorithm of computing $XH_{\mathscr{C}^{+}}(X^{+})$ and $XL_{\mathscr{C}^{+}}(X^{+})\bf{(NCX)}$)

Step 1: Input $(U^{+},\mathscr{C}^{+})$ and $X^{+}\subseteq
U^{+}$;

Step 2: Construct $M_{\mathscr{C}^{+}}$ and
$\prod(\mathscr{C}^{+})=M_{\mathscr{C}^{+}}\cdot M^{T}_{\mathscr{C}^{+}};$

Step 3: Compute $\mathcal {X}_{XH_{\mathscr{C}^{+}}(X^{+})}=\prod(\mathscr{C}^{+})\cdot \mathcal
{X}_{X^{+}}$ and $\mathcal {X}_{XL_{\mathscr{C}^{+}}(X^{+})}=\prod(\mathscr{C}^{+})\odot \mathcal
{X}_{X^{+}}$;

Step 4: Output $XH_{\mathscr{C}^{+}}(X^{+})$ and $XL_{\mathscr{C}^{+}}(X^{+})$.
\end{algorithm}


\begin{algorithm}(Incremental algorithm of computing $XH_{\mathscr{C}^{+}}(X^{+})$ and $XL_{\mathscr{C}^{+}}(X^{+})\bf{(ICX)}$)

Step 1: Input $(U,\mathscr{C})$, $(U^{+},\mathscr{C}^{+})$ and $X\subseteq
U^{+}$;

Step 2: Construct $\prod(\mathscr{C})$, where
$\prod(\mathscr{C})=M_{\mathscr{C}}\odot M^{T}_{\mathscr{C}};$

Step 3: Compute $\triangle_{1}(\prod(\mathscr{C}))$ and
$\triangle_{2}(\prod(\mathscr{C}))$, $\triangle_{3}(\prod(\mathscr{C}))$ and $\triangle_{4}(\prod(\mathscr{C}))$;

Step 4: Calculate $\prod(\mathscr{C}^{+})$, where
 $\prod(\mathscr{C}^{+})=\left[
  \begin{array}{cc}
    \prod(\mathscr{C}) & 1\\
    1 & 1\\
  \end{array}
\right]\bigwedge\left[
  \begin{array}{cc}
    \triangle_{1}(\prod(\mathscr{C})) & \triangle_{3}(\prod(\mathscr{C}))\\
    \triangle_{2}(\prod(\mathscr{C})) & \triangle_{4}(\prod(\mathscr{C}))\\
  \end{array}
\right];$

Step 5:  Get $\mathcal {X}_{XH(X)}$ and $\mathcal {X}_{XL(X)}$,
where
\begin{eqnarray*}
\mathcal {X}_{XH(X)}&=&\prod(\mathscr{C}^{+})\cdot \mathcal {X}_{X};
\mathcal {X}_{XL(X)}=\prod(\mathscr{C}^{+})\odot \mathcal {X}_{X}.
\end{eqnarray*}
\end{algorithm}


\section{Conclusions}

In this paper, we have provided effective approaches to constructing
approximations of concepts in dynamic covering approximation spaces.
Concretely, we have constructed type-1 and type-2 characteristic
matrices of coverings with the incremental approaches. Incremental
algorithms have been presented for computing the second and sixth
lower and upper approximations of sets. Several examples have been
employed to illustrate that computing approximations of sets could
be reduced greatly by using the incremental approaches.

In the future, we will propose more effective approaches to
constructing the type-1 and type-2 characteristic matrices of
coverings. Additionally, we will focus on the development of
effective approaches for knowledge discovery in dynamic covering
approximation spaces.




\section*{ Acknowledgments}

We would like to thank the anonymous reviewers very much for their
professional comments and valuable suggestions. This work is
supported by the National Natural Science Foundation of China (NO.
11201490,11371130,11401052,11401195), the Scientific Research Fund of Hunan
Provincial Education Department(No.14C0049).

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C. Luo, T.R. Li, H.M. Chen, Dynamic maintenance of approximations in
set-valued ordered decision systems under the attribute
generalization, Information Sciences 257 (2014) 210-228.\vskip.10in

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C. Luo, T.R. Li, H.M. Chen, D. Liu, Incremental approaches for
updating approximations in set-valued ordered information systems,
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C. Luo, T.R. Li, H.M. Chen, L.X. Lu, Fast algorithms for computing rough approximations in
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Information Sciences (2014) http://dx.doi.org/10.1016/j.ins.2014.12.029.



%\bibitem{Ma1}
%L.W. Ma, On some types of neighborhood-related covering rough sets,
%International Journal of Approximate Reasoning 53(6) (2012)
%901-911.\vskip.10in
%
%\bibitem{Mordeson1}
%J. Mordeson, Rough set theory applied to (fuzzy) ideal theory, Fuzzy
%Sets and Systems 121(2) (2001) 315-324.\vskip.10in

\bibitem{Ozawa}
S. Ozawa, S. Pang, N. Kasabov, Incremental Learning of chunk data
for online pattern classification systems, IEEE Transactions on
Neural Networks 19(6) (2008) 1061-1074.\vskip.10in

%\bibitem{Pawlak1}
%Z. Pawlak, Rough sets, International Journal of Computer and
%Information Sciences 11(5) (1982) 341-356. \vskip.10in

\bibitem{Pawlak5}
Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data,
Kluwer Academic Publishers, Dordrecht, MA, 1991.

%\bibitem{Pawlak2}
%Z. Pawlak, A. Skowron, Rudiments of rough sets, Information Sciences
%177 (2007) 3-27.\vskip.10in
%
%\bibitem{Pawlak3}
%Z. Pawlak, A. Skowron, Rough sets: some extensions, Information
%Sciences 177 (2007) 28-40.\vskip.10in
%
%\bibitem{Pawlak4}
%Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning, Information
%Sciences 177 (2007) 41-73.\vskip.10in

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Y.H. Qian, J.Y. Liang, D.Y. Li, F. Wang, N.N. Ma, Approximation
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M.W. Shao, Y. Leung, W.Z. Wu, Rule acquisition and complexity
reduction in formal decision contexts, International Journal of
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N. Shan, W. Ziarko, Data-based acquisition and incremental
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\bibitem{Shu1}
W.H. Shu, H. Shen, Updating attribute reduction in incomplete
decision systems with the variation of attribute set, International
Journal of Approximate Reasoning 55(3) (2013) 867-884.\vskip.10in

\bibitem{Shu}
W.H. Shu, H. Shen, Incremental feature selection based on rough set
in dynamic incomplete data, Pattern Recognition 47(12) (2014)
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A. Skowron, C. Rauszer, The discernibility matrices and functions in
information systems, in: R. Slowin' ski (Ed.), Intelligent Decision
Support, Handbook of Applications and Advances of the Rough Sets
Theory, Kluwer, Dordrecht, 1992.\vskip.10in

%\bibitem{Tsang1}
%Eric C.C. Tsang, D. G. Chen, D. S. Yeung, Approximations and reducts
%with covering generalized rough sets, Computers and Mathematics with
%Applications 56(1) (2008) 279-289.\vskip.10in

\bibitem{Wang5}
C.Z. Wang, M.W. Shao, B.Q. Sun, Q.H. Hu, An improved attribute reduction scheme with covering based rough sets, Applied Soft Computing 26 (2015) 235-243.\vskip.10in

\bibitem{Wang6}
C.Z. Wang, D.G. Chen, B.Q. Sun, Q.H. Hu, Communication between information systems with covering based
rough sets, Information Sciences 216 (2012) 17-33.\vskip.10in

\bibitem{Wang7}
C.Z. Wang, Q. He, D.G. Chen, Q.H. Hu, A novel method for attribute reduction of covering decision
systems, Information Sciences 254 (2014) 181-196.\vskip.10in

\bibitem{Wang8}
C.Z. Wang, D.G. Chen, C. Wu, Q.H. Hu, Data compression with homomorphism in covering information systems, International Journal of Approximate Reasoning 52 (2011) 519-525.\vskip.10in



\bibitem{Wang3}
F. Wang, J.Y. Liang, C.Y. Dang, Attribute reduction for dynamic data
sets, Applied Soft Computing 13 (2013) 676-689. \vskip.10in

\bibitem{Wang}
F. Wang, J.Y. Liang, Y.H. Qian, Attribute reduction: A dimension
incremental strategy, Knowledge-Based Systems 39 (2013) 95-108.
\vskip.10in

\bibitem{Wang4}
J. Wang, J. Wang, Reduction algorithms based on discernibility
matrix: the ordered attributes method, Journal of Computer Science
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\bibitem{Wang1}
S.P. Wang, W. Zhu, Q.H. Zhu, F. Min, Characteristic matrix of
covering and its application to boolean matrice decomposition and
axiomatization, Information Sciences 263(1) (2014)
186-197.\vskip.10in

\bibitem{Wojna}
A. Wojna, Constraint Based Incremental Learning of Classification
Rules, Lecture Notes in Computer Science 2005 (2001)
428-435.\vskip.10in

%\bibitem{Wu2}
%W.Z. Wu, M. Zhang, H.Z. Li, J.S. Mi, Knowledge reduction in random
%information systems via Dempster-Shafer theory of evidence,
%Information Sciences 174 (2005) 143-164.\vskip.10in

\bibitem{Wu1}
W.Z. Wu, W.X. Zhang, Neighborhood operator systems and
approximations, Information Sciences 144(1) (2002)
201-217.\vskip.10in

%\bibitem{Xiang1}
%X.Q. Xiang, J.Z. Zhou, C.S. Li, Q.Q. Li, Z.M. Luo, Fault diagnosis
%based on Walsh transform and rough sets, Mechanical Systems and
%Signal Processing 23(4) (2009) 1313-1326.\vskip.10in

\bibitem{Xu1}
W.H. Xu, W.X. Zhang, Measuring roughness of generalized rough sets
induced by a covering, Fuzzy Sets and Systems 158(22) (2007)
2443-2455.\vskip.10in

\bibitem{Yao1}
Y.Y. Yao, Relational interpretations of neighborhood operators and
rough set approximation operators, Information Sciences 111(1)
(1998) 239-259.\vskip.10in

%\bibitem{Yao2}
%Y.Y. Yao, Constructive and algebraic methods of theory of rough
%sets, Information Sciences 109(1) (1998) 21-47.\vskip.10in

\bibitem{Yao3}
Y.Y. Yao, Y. Zhao, Discernibility matrix simplification for
constructing attribute reducts, Information Sciences 179 (2009)
867-882.\vskip.10in


\bibitem{Yang3}
T. Yang, Q.G. Li, Reduction about approximation spaces of covering
generalized rough sets, International Journal of Approximate
Reasoning 51(3) (2010) 335-345.\vskip.10in

\bibitem{Yang2}
X.B. Yang, M. Zhang, H.L. Dou, J. Y. Yang, Neighborhood systems-based rough sets
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X.B. Yang, Y. Qi, H.L. Yu, X.N. Song, J.Y. Yang, Updating
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structures, Knowledge-Based Systems 64 (2014) 59-69.\vskip.10in

%\bibitem{Yun1}
%Z.Q. Yun, X. Ge, X.L. Bai, Axiomatization and conditions for
%neighborhoods in a covering to form a partition, Information
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W. Zakowski, Approximations in the space $(u, \pi)$, Demonstratio
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J.B. Zhang, T.R. Li, D. Ruan, D. Liu, Rough sets based matric
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P. Zhu, Covering rough sets based on neighborhoods: an approach
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W. Zhu, Relationship between generalized rough sets based on binary
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%P. Zhu, Q.Y. Wen, Entropy and co-entropy of a covering approximation
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\end{thebibliography}
\end{document}
\endinput

\\Assume $\textsf{M}=(P,Q,M)$ with two weak morphisms $\textsf{F}$ and $\textsf{G}$ is a weak
bisimulation for $\textsf{K}$ and $\textsf{L}$, if $P$ is a
measurable subset of $X\times A$, $Q$ is a measurable subset of
$Y\times B$, and $\textsf{F}, \textsf{G}$ are weak projections, then
we call $\textsf{M}$ is a 2-weak bisimulation for $\textsf{K}$ and
$\textsf{L}$.

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{Luo1 C. Luo, T.R. Li, H.M. Chen, Dynamic maintenance of approximations in set-valued ordered decision systems under the attribute generalization, Information Sciences 257 (2014) 210-228.\vskip.10in
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{Luo2 C. Luo, T.R. Li, H.M. Chen, D. Liu, Incremental approaches for updating approximations in set-valued ordered information systems, Knowledge-Based Systems 50 (2013) 218-233.\vskip.10in
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{Luo3 C. Luo, T.R. Li, H.M. Chen, L.X. Lu, Fast algorithms for computing rough approximations in 4 set-valued decision systems while updating criteria values, Information Sciences (2014) http://dx.doi.org/10.1016/j.ins.2014.12.029. %
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{Ma1 %L.W. Ma, On some types of neighborhood-related covering rough sets, %International Journal of Approximate Reasoning 53(6) (2012) %901-911.\vskip.10in % %
1504.00136.extracted_bib
{Mordeson1 %J. Mordeson, Rough set theory applied to (fuzzy) ideal theory, Fuzzy %Sets and Systems 121(2) (2001) 315-324.\vskip.10in
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{Ozawa S. Ozawa, S. Pang, N. Kasabov, Incremental Learning of chunk data for online pattern classification systems, IEEE Transactions on Neural Networks 19(6) (2008) 1061-1074.\vskip.10in %
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{Pawlak1 %Z. Pawlak, Rough sets, International Journal of Computer and %Information Sciences 11(5) (1982) 341-356. \vskip.10in
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{Pawlak5 Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, Dordrecht, MA, 1991. %
1504.00136.extracted_bib
{Pawlak2 %Z. Pawlak, A. Skowron, Rudiments of rough sets, Information Sciences %177 (2007) 3-27.\vskip.10in % %
1504.00136.extracted_bib
{Pawlak3 %Z. Pawlak, A. Skowron, Rough sets: some extensions, Information %Sciences 177 (2007) 28-40.\vskip.10in % %
1504.00136.extracted_bib
{Pawlak4 %Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning, Information %Sciences 177 (2007) 41-73.\vskip.10in
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{Qian1 Y.H. Qian, J.Y. Liang, D.Y. Li, F. Wang, N.N. Ma, Approximation reduction in inconsistent incomplete decision tables, Knowledge-Based Systems 23(5) (2010) 427-433.\vskip.10in
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{Shao M.W. Shao, Y. Leung, W.Z. Wu, Rule acquisition and complexity reduction in formal decision contexts, International Journal of Approximate Reasoning 55 (2014) 259-274.\vskip.10in
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{Shan1 N. Shan, W. Ziarko, Data-based acquisition and incremental modification of classification rules, Computation Intelligence 11(2) (1995) 357-370.\vskip.10in
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{Shu1 W.H. Shu, H. Shen, Updating attribute reduction in incomplete decision systems with the variation of attribute set, International Journal of Approximate Reasoning 55(3) (2013) 867-884.\vskip.10in
1504.00136.extracted_bib
{Shu W.H. Shu, H. Shen, Incremental feature selection based on rough set in dynamic incomplete data, Pattern Recognition 47(12) (2014) 3890-3906.\vskip.10in
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{Skowron1 A. Skowron, C. Rauszer, The discernibility matrices and functions in information systems, in: R. Slowin' ski (Ed.), Intelligent Decision Support, Handbook of Applications and Advances of the Rough Sets Theory, Kluwer, Dordrecht, 1992.\vskip.10in %
1504.00136.extracted_bib
{Tsang1 %Eric C.C. Tsang, D. G. Chen, D. S. Yeung, Approximations and reducts %with covering generalized rough sets, Computers and Mathematics with %Applications 56(1) (2008) 279-289.\vskip.10in
1504.00136.extracted_bib
{Wang5 C.Z. Wang, M.W. Shao, B.Q. Sun, Q.H. Hu, An improved attribute reduction scheme with covering based rough sets, Applied Soft Computing 26 (2015) 235-243.\vskip.10in
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{Wang6 C.Z. Wang, D.G. Chen, B.Q. Sun, Q.H. Hu, Communication between information systems with covering based rough sets, Information Sciences 216 (2012) 17-33.\vskip.10in
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{Wang7 C.Z. Wang, Q. He, D.G. Chen, Q.H. Hu, A novel method for attribute reduction of covering decision systems, Information Sciences 254 (2014) 181-196.\vskip.10in
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{Wang8 C.Z. Wang, D.G. Chen, C. Wu, Q.H. Hu, Data compression with homomorphism in covering information systems, International Journal of Approximate Reasoning 52 (2011) 519-525.\vskip.10in
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{Wang3 F. Wang, J.Y. Liang, C.Y. Dang, Attribute reduction for dynamic data sets, Applied Soft Computing 13 (2013) 676-689. \vskip.10in
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{Wang F. Wang, J.Y. Liang, Y.H. Qian, Attribute reduction: A dimension incremental strategy, Knowledge-Based Systems 39 (2013) 95-108. \vskip.10in
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{Wang4 J. Wang, J. Wang, Reduction algorithms based on discernibility matrix: the ordered attributes method, Journal of Computer Science and Technology 16 (2001) 489-504.\vskip.10in
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{Wang1 S.P. Wang, W. Zhu, Q.H. Zhu, F. Min, Characteristic matrix of covering and its application to boolean matrice decomposition and axiomatization, Information Sciences 263(1) (2014) 186-197.\vskip.10in
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{Wojna A. Wojna, Constraint Based Incremental Learning of Classification Rules, Lecture Notes in Computer Science 2005 (2001) 428-435.\vskip.10in %
1504.00136.extracted_bib
{Wu2 %W.Z. Wu, M. Zhang, H.Z. Li, J.S. Mi, Knowledge reduction in random %information systems via Dempster-Shafer theory of evidence, %Information Sciences 174 (2005) 143-164.\vskip.10in
1504.00136.extracted_bib
{Wu1 W.Z. Wu, W.X. Zhang, Neighborhood operator systems and approximations, Information Sciences 144(1) (2002) 201-217.\vskip.10in %
1504.00136.extracted_bib
{Xiang1 %X.Q. Xiang, J.Z. Zhou, C.S. Li, Q.Q. Li, Z.M. Luo, Fault diagnosis %based on Walsh transform and rough sets, Mechanical Systems and %Signal Processing 23(4) (2009) 1313-1326.\vskip.10in
1504.00136.extracted_bib
{Xu1 W.H. Xu, W.X. Zhang, Measuring roughness of generalized rough sets induced by a covering, Fuzzy Sets and Systems 158(22) (2007) 2443-2455.\vskip.10in
1504.00136.extracted_bib
{Yao1 Y.Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111(1) (1998) 239-259.\vskip.10in %
1504.00136.extracted_bib
{Yao2 %Y.Y. Yao, Constructive and algebraic methods of theory of rough %sets, Information Sciences 109(1) (1998) 21-47.\vskip.10in
1504.00136.extracted_bib
{Yao3 Y.Y. Yao, Y. Zhao, Discernibility matrix simplification for constructing attribute reducts, Information Sciences 179 (2009) 867-882.\vskip.10in
1504.00136.extracted_bib
{Yang3 T. Yang, Q.G. Li, Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximate Reasoning 51(3) (2010) 335-345.\vskip.10in
1504.00136.extracted_bib
{Yang2 X.B. Yang, M. Zhang, H.L. Dou, J. Y. Yang, Neighborhood systems-based rough sets in incomplete information system, Knowledge-Based Systems 24(6) (2011) 858-867.\vskip.10in
1504.00136.extracted_bib
{Yang X.B. Yang, Y. Qi, H.L. Yu, X.N. Song, J.Y. Yang, Updating multigranulation rough approximations with increasing of granular structures, Knowledge-Based Systems 64 (2014) 59-69.\vskip.10in %
1504.00136.extracted_bib
{Yun1 %Z.Q. Yun, X. Ge, X.L. Bai, Axiomatization and conditions for %neighborhoods in a covering to form a partition, Information %Sciences 181(9) (2011) 1735-546.\vskip.10in
1504.00136.extracted_bib
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