Faculty of Computer Science Institute of Theoretical Computer Science, Chair of Automata Theory
Description Logic
Winter Semester 2017/18Exercise Sheet 7 28th November 2017
Prof. Dr.-Ing. Franz Baader, Dr.-Ing. Stefan Borgwardt
Exercise 7.1 We consider another form of blocking, where an individual can be blocked by an individual that is not necessarily an ancestor:anywhere blocking. Instead of the ancestor relation, it uses the age of an individual to determine the blocking relation.
Theageof an individuala, denoted byage(a), is defined as0for individuals that occur in the input ABoxAand asnfor a new individual that was generated by thenth application of the∃-rule.
LetA0 be an ABox obtained by applying the tableau rules ofconsistent(T,A)for general TBoxes. A tree individualbisanywhere blocked by an individualainA0 if
• conA0(b)⊆conA0(a),
• age(a)<age(b), and
• ais not blocked.
As before, the descendants ofbare then also considered blocked.
Prove that the tableau algorithm with anywhere blocking is a decision procedure for consistency of ALC-knowledge bases with general TBoxes.
Exercise 7.2 LetK= (T,A0)be anALC-knowledge base, whereT is a general TBox. Aprecom- pletionofKis a clash-free ABoxAthat is obtained fromKby exhaustively applying all expansion rules except the∃-rule.
Show thatKis consistent iff there is a precompletionAofKsuch that, for all individual namesa occurring inA, the conceptCAa := l
a:C∈A
Cis satisfiable w.r.t.T.
Exercise 7.3 Let C be an ALC-concept. We denote by #C the number of occurrences of the constructorst,u,∃, and∀withinC. The multisetM(C)contains, for each occurrence of a subconcept of the form¬DinC, the number#D.
Use this representation to prove that exhaustively applying the following transformation rules to an ALC-concept always terminates, regardless of the order of rule applications:
¬(CuD) ¬¬¬Ct ¬¬¬D
¬(CtD) ¬¬¬Cu ¬¬¬D
¬¬C C
¬(∃r.C) ∀r.¬C
¬(∀r.C) ∃r.¬C
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