# Graphol: A Graphical Language for Ontology Modeling Equivalent to OWL 2
## 3 Preliminaries
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Construct** & **Syntax** & **Semantics** \\ \hline Atomic concept & \(A\) & \(A^{I}\subseteq\Delta^{I}_{\mathcal{O}}\) \\ \hline Atomic role & \(p\) & \(P^{I}\subseteq\Delta^{I}_{\mathcal{O}}\times\Delta^{I}_{\mathcal{O}}\) \\ \hline Atomic attribute & \(U\) & \(U^{I}\subseteq\Delta^{I}_{\mathcal{O}}\times\Gamma_{V}\) \\ \hline Atomic value-domain & \(T\) & \(T^{I}\subseteq\Gamma_{V}\) \\ \hline Universal concept & \(\top_{C}\) & \(\Delta^{I}_{\mathcal{O}}\) \\ \hline Universal role & \(\top_{R}\) & \(\Delta^{I}_{\mathcal{O}}\times\Delta^{I}_{\mathcal{O}}\) \\ \hline Universal attribute & \(\top_{A}\) & \(\Delta^{I}_{\mathcal{O}}\times\Gamma_{V}\) \\ \hline Universal value-domain & \(\top_{D}\) & \(\Gamma_{V}\) \\ \hline Empty concept, role attribute, value-domain & \(\bot_{C},\bot_{R},\bot_{A},\bot_{D}\) & \(\emptyset\) \\ \hline Unqualified role existential restriction & \(\exists R\) & \(\{\,o\mid\exists\sigma^{\prime},(o,\sigma^{\prime})\in R^{I}\,\}\) \\ \hline Qualified role existential restriction & \(\exists R.C\) & \(\{\,o\mid\exists\sigma^{\prime},(o,\sigma^{\prime})\in R^{I}\wedge\sigma^{ \prime}\in C^{I}\,\}\) \\ \hline Qualified role universal restriction & \(\forall R.C\) & \(\{\,o\mid\forall\sigma^{\prime},(o,\sigma^{\prime})\in R^{I}\to\sigma^{ \prime}\in C^{I}\,\}\) \\ \hline Qualified maximum cardinality role restriction & \(\leq n\,R.C\) & \(\{\,o\mid\sharp\{\sigma^{\prime}\mid(o,\sigma^{\prime})\in R^{I}\wedge\sigma^{ \prime}\in C^{I}\,\}\leq n\}\) \\ \hline Qualified minimum cardinality role restriction & \(\geq n\,R.C\) & \(\{\,o\mid\sharp\{o^{\prime}\mid(o,\sigma^{\prime})\in R^{I}\wedge\sigma^{ \prime}\in C^{I}\,\}\geq n\}\) \\ \hline Self restriction & \(\exists R.Self\) & \(\{\,o\mid(o,o)\in R^{I}\,\}\) \\ \hline Unqualified attribute existential restriction & \(\exists V\) & \(\{\,o\mid\exists v,(o,v)\in V^{I}\,\}\) \\ \hline Qualified attribute existential restriction & \(\exists V.F\) & \(\{\,o\mid\exists v,(o,v)\in V^{I}\wedge v\in F^{I}\,\}\) \\ \hline Qualified attribute universal restriction & \(\forall V.F\) & \(\{\,o\mid\forall v,(o,v)\in V^{I}\to v\in F^{I}\,\}\) \\ \hline Qualified maximum cardinality attribute restriction & \(\leq n\,V.F\) & \(\{\,o\mid\sharp\{v\mid(o,v)\in V^{I}\wedge v\in F^{I}\,\}\leq n\}\) \\ \hline Qualified minimum cardinality attribute restriction & \(\geq n\,V.F\) & \(\{\,o\mid\sharp\{v\mid(o,v)\in V^{I}\wedge v\in F^{I}\,\}\geq n\}\) \\ \hline One-of (concept) & \(\{c_{1},\ldots,c_{n}\}\) & \(\{c^{I}_{1},\ldots,c^{I}_{n}\}\) \\ \hline One-of (value-domain) & \(\{w_{1},\ldots,w_{n}\}\) & \(\{w_{1},\ldots,w_{n}\}\) \\ \hline Attribute range & \(\exists V^{-}\) & \(\{\,o\mid\exists o,(o,v)\in V^{I}\,\}\) \\ \hline Inverse role & \(p^{-}\) & \(\{\,o,o^{\prime}\mid(o^{\prime},o)\in P^{I}\,\}\) \\ \hline Role chain & \(R\circ R\) & \(\{\,o,o^{\prime}\mid\exists\sigma^{\prime\prime},(o,o^{\prime\prime})\in R^{I}_ {1}\wedge(o^{\prime\prime},o^{\prime})\in R^{I}_{2}\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The DL constructs with their semantics. In the table, \(c_{1}^{l},\ldots c_{n}^{l}\) denote the interpretation of the constants \(c_{1},\ldots c_{n}\) from the alphabet \(\Gamma_{O}\). A DL KB \(\mathcal{K}\) is a pair \(\langle\mathcal{T},\mathcal{A}\rangle\), where \(\mathcal{T}\) is a finite set of intensional assertions, called _TBox_, and \(\mathcal{A}\) is a finite set of extensional assertions, called _ABox_. In this paper we focus on the modeling of intensional knowledge, i.e., the TBox, so we will not detail further the form of the ABox (which is the component of the knowledge base maintaining the data). The TBox assertions we focus on in this work are as follows:  
\[ \begin{array}{ll}C_{1}\sqsubseteq C_{2}&\text{(\emph{concept inclusion});}\\ R_{1}\sqsubseteq R_{1}&\text{(\emph{role inclusion});}\\ V_{1}\sqsubseteq V_{2}&\text{(\emph{attribute inclusion});}\\ F_{1}\sqsubseteq F_{2}&\text{(\emph{value-domain inclusion}).}\\ \end{array}