ctex 本體論作業

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\title{OWL 作業}

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\author{徐威鴻}

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\newpage
\section{Exercise 1}
\subsection{題目}
Exercise 1. Describe - both verbally and formally - the extension of the following
concepts with respect to the interpretation $\mathcal{I}$ defined in Example 16:
\begin{enumerate}[(a)]
  \item $\forall hasSuccessor^{-}.Positive$
  \item $\exists multipleOf.Self$
  \item $\exists multipleOf.\exists hasSuccessor^{-}.\exists hasSuccessor^{-}.\{zero\}$
  \item $\geq 10lessThan^{-}.Prime$
  \item $\neg Prime \sqcap \leq 2multipleOf.\top $
  \item $\exists lessThan.Prime$
  \item $\forall multipleOf. ( \exists hasSuccessor-.\{zero\} \\ \sqcup \exists multipleOf. \exists has Successor^{-}.\exists hasSuccessor^{-}.\{zero\} )$
    \begin{enumerate}[-]
     \item $N_{\mathcal I}=\{zero\}$
     \item $N_C = \{Prime,Positive\}$
     \item $N_R = \{hasSuccessor,lessThan,multipleOf\}$
     \end{enumerate}
\end{enumerate}
Now, we define${\mathcal I}$ as follows: let $\Delta ^ \mathcal{I} = {\mathcal N} = \{0,1,2,...\}$,i.e., the set of all natural numbers including zero. Furthermore, we let $zero^{\mathcal I} = 0$, as well as $Prime^{\mathcal I} = \{n| n is a prime number\}$
and $Positive^{\mathcal I} = \{ n | n > 0\}$. For the roles, we define
\begin{enumerate}[-]
  \item $hasSuccessor^{\mathcal I} = \{<n,n+1>|n \in {\mathcal N}  \}$
  \item $lessThan^{\mathcal I} = \{<n,n'>|n<n',n,n' \ in {\mathcal N}\}$
  \item $multipleOf^{\mathcal I} = \{<n,n'>|\exists k,n = k,n', n,n',k \in {\mathcal N}\}$
\end{enumerate}
\subsection{解法}

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\section{Exercise 2}
\subsection{題目}
Decide whether the following axioms are satisfied by the interpretation ${\mathcal I}$ from Example 16.
\begin{enumerate}[(a)]
  \item $hasSuccessor \sqsubseteq lessThan$
  \item $\exists hasSuccessor^{-}.\exists hasSuccessor^{-}.\{zero\}\sqsubseteq Prime$
  \item $\top \sqsubseteq \forall multipleOf^{-}.\{zero\}$
  \item $Dis(divisileBy,lessThan^{-})$
  \item $multipleOf \circ multipleOf \sqsubseteq multipleOf$
  \item $\top \sqsubseteq \leq 1hasSuccessor.Positive $
  \item $zero \not\approx zero$
  \item $\leq 1mutipleOf^{-}.\top(zero)$
  \item $\top \sqsubseteq \forall lessThan.\exists lessThan.(Prime \sqcap \exists hasSuccessor.\exists hasSuccessor.Prime) $
\end{enumerate}
\subsection{解法}
\newpage
\section{Exercise 11}
\subsection{題目}
Show that the following equivalences are not valid:
\begin{enumerate}[(a)]
  \item $\exists \textbf{r}.(C \sqcap D) \equiv \exists \textbf{r}.C \sqcap \exists \textbf{r}.D$
  \item $C \sqcap (D \sqcup E) \equiv (C \sqcap D) \sqcup E$
  \item $\exists \textbf{r}.\{\textbf{a}\} \sqcap \exists \textbf{r}.\{\textbf{b}\} \equiv \geq 2.\{\textbf{a}, \textbf{b}\}$
  \item $\exists \textbf{r}.\top \sqcap \exists \textbf{s}.\top \equiv \exists \textbf{r}.\exists\textbf{r}^{-}.\exists \textbf{s}.\top .$
\end{enumerate}
\subsection{解法}

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\section{總結}
    \subsection{Description Logics 描述邏輯}
    All DLs are based on a vocabulary [signature] containing individual names [constants], concept names [unary predicates] and role names[binary predicates]. Two specific class names,$\top and \bot$, denote the concept containing all individuals and the empty
concept, respectively.

    Usually, a DL knowledge base [theory] is partitioned into an assertional part, called ABox and a terminological part, which is further subdivided int TBox and RBox.
    \newpage
        \subsubsection{Introduction}
            \begin{enumerate}[-]
                \item ABox:The ABox contains assertional knowledge[ground facts],the notation of which coincides with FOL:there are \textsl{\textbf{concept assertions}} and \textsl{\textbf{role assertions}}
                    \begin{enumerate}[$\circ$]
                      \item concept assertions:C(a)
                      \item role assertions:R(a,b)
                    \end{enumerate}
                \item TBox:TBox contains \textsl{\textbf{universal statements}}. The notation used in DLs does not need variables and is inspired by set theory.
                    \begin{itemize}
                      \item We can specify \textbf{subsumptions}:$ C \sqsubseteq D$
                      \item A specific feature of DLs is that concept names can be combined into complex concepts by \textbf{Boolean operators} :$\exists \textbf{r}.C $
                      \item \textbf{Role inverses} can be used to "traverse" roles backeward:$\exists \textbf{r}.\top \sqsubseteq \forall \textbf{r}^{-}.C$
                      \item \textbf{Cardinality constraints }allow for specifying the number of related instances: $\geq 2.Married.\top$
                      \item By means of \textbf{nominals}, classes can be defined by enumerating their instances: $\exists r.C \sqsubseteq \{...\}$
                    \end{itemize}
                \item RBox:The RBox of DL knowledge base allows for further, role-centric modeling features
                \begin{enumerate}
                  \item \textbf{role inclusion} statements:$\textbf{r} \sqsubseteq \textbf{R}$
                  \item A more general and expressive variant of role inclusions are \textbf{role-chain} axioms: $r_1 \circ r_2 ...\circ r_n \sqsubseteq R$
                \end{enumerate}
            \end{enumerate}
        \newpage
        \subsubsection{Syntax of Description Logics}
        Most of today's mainstream DLs are, in fact, sublanguages of ${\mathcal SRPIQ}$
        \begin{enumerate}[-]
          \item The set $N_I$ of individual names contains all names used to denote singular entity in our domain of interest.
          \item The set $N-C$ of concept names contains names that refer to types, categories, or classes of entities, usually characterized by common properties.
          \item The set $N_R$ of role names contains nameds that denote binary relationships which may hold between individuals of a domain.
        \end{enumerate}
        Having these name sets at hand, we can now turn to the three building blocks of ${\mathcal SROIQ}$ knowledge bases: RBox, TBox and ABox.
        \begin{enumerate}
        \newpage
          \item RBox \{roles | roles.dependencies\}

          A role can be
            \begin{itemize}
                \item a role name \textbf{r}
                \item an inverted role name \textbf{$r^{-}$}
                \item the universal role \textbf{u}
            \end{itemize}

          A \textsl{role inclusion axiom}(RIA) is a statement of the form $$r_1 \circ r_2 \circ ... \circ r_n \sqsubseteq r$$ where $r_1,r_2,...,r_n,r$ are roles
          \begin{itemize}
            \item Given a set of RIAs, roles are divided into \textsl{\textbf{simple}} and\textsl{\textbf{ non-simple}} roles.
            \item Roughly, role are non-simple if they may occur on the rhs of a complex RIA.
            \item More precisely,
            \begin{itemize}
              \item for any RIA \textbf{$r_1 \circ r_2 \circ ...  \circ r_n  \sqsubseteq r$} with $n>1$, \textbf{r} is non-simple
              \item any RIA $s \sqsubseteq r$ with \textbf{s} non-simple, \textbf{r} is non-simple
              \item all other properties are simple
            \end{itemize}
          \end{itemize}
          
          A role disjointness statement has the form $$Dis(s1,s2)$$ where s1 and s2 are simple roles.
          
          \begin{tabular}{|c|c|}            
            \hline
            % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
            Concept expressions & Meaning  \\
            \hline
            $\top$ and $\bot$ & concept expressions \\
            $a_1,..,a_n$ & individual names \\
            $\{a_1,...,a_n\}$ & concept expression \\
            $\lnot C$ and $C \sqcap D$ and $C \sqcup D$ &  concept expressions \\
            $\exists r.C$ and $\forall r.C $ &  concept expressions \\
            $\exists r.Self$ and $\leq ns.C$ and $\geq ns.C $ & concept expressions \\
            \hline            
          \end{tabular}
          
          for C and D concept expressions , for r a role , for s a simple role
          \newpage
          \item TBox \{concepts | concepts.taxonomic dependencies\}
          A general concept inclusion(GCI) has the form $$C \sqsubseteq D$$ where C and D are concept expressions.
          
          A TBox consists of a set of GCIs.
          
          \newpage
          \item ABox \{individuals |  individuals.concept role.memberships\}
          
          \begin{tabular}{|c|c|}
            \hline
            % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
            ABox & Meaning \\
            \hline
            C(a) & concept assertion \\
            r(a,b) & role assertion \\
            $\lnot r(a,b)$ & negated role assertion \\
            $a\approx b$ & equality statement \\
            $a\not\approx b$ & inequality statement \\
            \hline
          \end{tabular}
        \end{enumerate}
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