Description Logic

Faculty of Computer Science Institute of Theoretical Computer Science, Chair of Automata Theory

Description Logic

Exercise Sheet 5 15th November 2017

Prof. Dr.-Ing. Franz Baader, Dr.-Ing. Stefan Borgwardt

Exercise 5.1 Prove that the negation normal form (NNF) of anALC-conceptCcan be computed in polynomial time and is equivalent toC.

Exercise 5.2 Execute the tableau algorithmconsistent(A)for the normalized ABox

A=(b,a):r, (a,b):r, (a,c):s, (c,b):s, a:∃s.A, b:∀r. (∀s.¬A)t(∃r.B), c:∀s. Bu(∀s.⊥) .

IfAis consistent, draw the model generated by the algorithm.

Exercise 5.3 We consider the concept constructor→(implication) with the following semantics:

(C→D)^I :={x∈_∆^I|x∈C^I impliesx∈ D^I}.

To extendconsistent(A)to this constructor, we propose two alternative new expansion rules:

The deterministic→-rule

Condition: Acontainsa:C→Danda:C, but nota:D Action: A −→ A ∪ {a:D}

The nondeterministic→-rule

Condition: Acontainsa:C→D, but neithera: ˙¬Cnora:D Action: A −→ A ∪ {a:X}for someX∈ {¬˙C,D}

For each rule, determine whether the resulting algorithm is still terminating, sound, and complete.

Exercise 5.4 We consider TBoxesT that only contain the following two kinds of axioms:

• role inclusions of the formrvs, and

• role disjointness constraints of the formdisj(r,s),

whererandsare role names. An interpretationIsatisfies these axioms if

• r^I ⊆s^I, and

• r^I∩s^I =_∅, respectively.

Modify the tableau algorithmconsistent(A)to decide consistency of(T,A), whereAis an ABox and T a TBox that contains only role inclusions and role disjointness constraints. Show that the algorithm remains terminating, sound, and complete.

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