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Table 17 Syntax and semantics of description logic [18]

From: Big knowledge-based semantic correlation for detecting slow and low-level advanced persistent threats

Syntax Description Semantics
CD Concepts or Classes \(C^{{\mathcal {I}}},D^{{\mathcal {I}}}\subseteq \Theta\)
RS Relations or Properties \(R^{{\mathcal {I}}},S^{{\mathcal {I}}}\subseteq \Theta \times \Theta\)
\(C\sqsubseteq D\) Class subsumption \(C^{{\mathcal {I}}}\subseteq D^{{\mathcal {I}}}\)
\(R\sqsubseteq S\) Property subsumption \(R^{{\mathcal {I}}}\subseteq S^{{\mathcal {I}}}\)
\(C\sqcup D\) Union of classes \((C\sqcup D)^{\mathcal {I}}=C^{{\mathcal {I}}}\cup D^{\mathcal {{\mathcal {I}}}}\)
\(C \sqcap D\) Intersection of classes \((C \sqcap D)^{{\mathcal {I}}}=C^{{\mathcal {I}}} \cap D^{{\mathcal {I}}}\)
\(\lnot C\) Complement \((\lnot C)^{\mathcal {I}}=\Theta - C^{{\mathcal {I}}}\)
\(\exists R.C\) Existential quantification \({(\exists R.C)}^{\mathcal {I}}=\{a \in \Theta | \exists b, \langle a, b \rangle \in R^{\mathcal {I}} \wedge b \in C^{\mathcal {I}}\}\)
\(\forall R.C\) Value restriction \({(\forall R.C)}^{\mathcal {I}}=\{a \in \Theta | \forall b, \langle a, b \rangle \in R^{\mathcal {I}} \rightarrow b \in C^{{\mathcal {I}}}\}\)
\(\top\) Top concept or Thing \(\top ^{{\mathcal {I}}}=\Theta\)
\(\bot\) Bottom concept or Nothing \(\bot ^{{\mathcal {I}}}=\varnothing\)
  1. - The model is \({\mathcal {M}}=\langle \Theta ,.^ {\mathcal {I}} \rangle\), where
  2. \(~~~\Theta\) is the domain of objects, and
  3. \(~~~{\mathcal {I}}\) is an interpretation function