Description Logic

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

Description Logic

Exercise Sheet 4 8th November 2017

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

Exercise 4.1 Prove or refute the following claim: If anALC-conceptCis satisfiable w.r.t. anALC- TBoxT, then for alln≥1there is a finite modelI_nofT such that|C^In| ≥n.

Does the claim hold if the condition “|C^In| ≥n” is replaced by “|C^In|=n”?

Exercise 4.2 Prove or refute the following claim: Given anALC-conceptCand anALC-TBoxT, ifI is an interpretation andJ its filtration w.r.t.sub(C)∪sub(T)(see Definition 3.14), then the relation ρ={(d,[d])|d∈_∆^I}is a bisimulation betweenI andJ.

Exercise 4.3 We consider bisimulations between an interpretationI and itself, which are called bisimulationsonI. For two elementsd,e∈_∆^I, we writed≈_I eif they are bisimilar, i.e., if there is a bisimulationρonI such thatdρe.

(a) Show that≈_I is an equivalence relation on∆^I. (b) Show that≈_I is a bisimulation onI.

(c) Show that, for finite interpretationsI, the relation≈_I can be computed in time polynomial in the cardinality ofI.

Consider the interpretationJ that is defined like the filtration (Definition 3.14), but with≈_I instead of'.

(d) Show thatρ={(d,[d]_≈I)|d∈_∆^I}is a bisimulation betweenIandJ.

(e) Show that, ifIis a model of anALC-conceptCw.r.t. anALC-TBoxT, then so isJ. (f) Why can we not use the previous result to show the finite model property forALC?

Exercise 4.4 For the following interpretationI, draw the unraveling ofI atd up to depth5, i.e., restricted tod-paths of length at most5(see Definition 3.21):

d A

e A

f r

s r

s

Exercise 4.5 Prove or refute the following claim: IfKis anALC-knowledge base andCanALC- concept description such thatCis satisfiable w.r.t.K, thenChas a tree model w.r.t.K.

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