4.E: Description Logics (Exercises)
- Page ID
- 6418
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
How are DLs typically different from full FOL?
Exercise \(\PageIndex{2}\)
What are the components of a DL knowledge base?
Exercise \(\PageIndex{3}\)
What are (in the context of DLs) the concept and role constructors? You may list them for either \(\mathcal{ALC}\) or \(\mathcal{SROIQ}\).
Exercise \(\PageIndex{4}\)
What distinguishes one DL from another? That is, e.g., \(\mathcal{ALC}\) is different from \(\mathcal{SROIQ}\) and from \(\mathcal{EL}\); what is the commonality of those differences?
Exercise \(\PageIndex{5}\)
Explain in your own words what the following \(\mathcal{ALC}\) reasoning tasks involve and why they are important for reasoning with ontologies:
a. Instance checking.
b. Subsumption checking.
c. Checking for concept satisfiability.
Exercise \(\PageIndex{6}\)
Consider the following TBox \(\mathcal{T}\):
\(\texttt{Vegan}\equiv\texttt{Person}\sqcap\forall\texttt{eats.Plant}\)
\(\texttt{Vegetarian}\equiv\texttt{Person}\sqcap\forall\texttt{eats.(Plant}\sqcup\texttt{Dairy)}\)
We want to know if \(\mathcal{T}\vdash Vegan\sqsubseteq Vegetarian\).
This we convert to a constraint system \(S=\{ (Vegan\sqcap\neg Vegetarian)(a)\}\), which is unfolded (here: complex concepts on the left-hand side are replaced with their properties declared on the right-hand side) into:
\[S=\{ Person\sqcap\forall eats.Plant\sqcap\neg (Person\sqcap\forall eats.(Plant\sqcup Dairy))(a)\}\]
Tasks:
a. Rewrite Eq. 3.4.1 into negation normal form
b. Enter the tableau by applying the rules until either you find a completion or only clashes.
c. \(\mathcal{T}\vdash Vegan\sqsubseteq Vegetarian\)?
- Answer
-
(a) Rewrite (Eq. 3.4.1) into negation normal form:
\(Person\sqcap\forall eats.Plant\sqcap (\neg Person\sqcup\neg\forall eats.(Plant\sqcup Dairy))\)
\(Person\sqcap\forall eats.Plant\sqcap (\neg Person\sqcup\exists\neg eats.(Plant\sqcup Dairy))\)
\(Person\sqcap\forall eats.Plant\sqcap (\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy))\)
So our initial ABox is:
\(S=\{ (Person\sqcap\forall eats.Plant\sqcap (\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy)))(a)\}\)
(b) Enter the tableau by applying the rules until either you find a completion or only clashes.
(\(\sqcap\)-rule): \(\{ Person(a),\forall eats.Plant(a),(\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy))(a)\}\)
(\(\sqcup\)-rule): (i.e., it generates two branches)
(1) \(\{ Person(a),\forall eats.Plant(a),(\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy))(a),\neg Person(a)\}\) \(¡\)clash \(!\)
(2) \(\{ Person(a),\forall eats.Plant(a),(\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy))(a),\exists eats.(\neg Plant\sqcap\neg Dairy)(a)\}\)
\((\exists\)-rule): \(\{ Person(a),\forall eats.Plant(a),(\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy))(a),\exists eats.(\neg Plant\sqcap\neg Dairy)(a), eats(a, b),(\neg Plant\sqcap\neg Dairy)(b)\}\)
(\(\sqcap\)-rule): \(\{ Person(a),\forall eats.Plant(a),(\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy))(a),\exists eats.(\neg Plant\sqcap\neg Dairy)(a), eats(a, b),(\neg Plant\sqcap\neg Dairy)(b),\neg Plant(b),\neg Dairy(b)\}\)
(\(\forall\)-rule): \(\{ Person(a),\forall eats.Plant(a),(\neg Person\sqcup\exists eats.(\neg Plant\sqcap\neg Dairy))(a),\exists eats.(\neg Plant\sqcap\neg Dairy)(a), eats(a, b),(\neg Plant\sqcap\neg Dairy)(b),\neg Plant(b),\neg Dairy(b), Plant(b)\}\)
\(¡\)clash\(!\)
(c) \(\mathcal{T}\vdash Vegan\sqsubseteq Vegetarian\)? yes