Description Logic

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

Description Logic

Exercise Sheet 12 17th January 2018

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

Exercise 12.1 We consider simulations, which are “one-sided” variants of bisimulations. Given interpretationsI₁andI₂, the relationσ⊆_∆^I1×_∆^I2 is asimulationbetweenI₁andI₂if

• wheneverd₁σd₂andd₁ ∈A^I1, thend₂∈ A^I2, for alld₁∈_∆^I1,d₂∈_∆^I2, andA∈C;

• whenever d1 σ d2 and (_d1,d⁰₁) ∈ r^I1, then there exists a d⁰₂ ∈ _∆^I2 such that d⁰₁ σ d⁰₂ and (d₂,d⁰₂)∈r^I2, for alld₁,d⁰₁∈_∆^I1,d₂∈_∆^I2, andr ∈R.

We write(I_1,d₁)^*∼ (I_2,d₂)if there is a simulationσbetweenI₁andI₂such thatd₁σd₂. (a) Show that(I₁,d₁)∼(I₂,d₂)implies(I₁,d₁)^*∼ (I₂,d₂)and(I₂,d₂)^*∼ (I₁,d₁). (b) Is the converse of the implication in (a) also true?

(c) Show that, if (I_1,d₁)^*∼ (I_2,d₂), then for all EL-concepts C it holds that d₁ ∈ C^I1 implies d₂∈C^I2.

(d) Which of the constructors disjunction, negation, or value restriction can be added toELwithout losing the property in (c)?

(e) Show thatALCis more expressive thanEL. (f) Show thatELIis more expressive thanEL.

(g) Can the fact that subsumption inELis decidable in polynomial time, while subsumption inELI is EXPTIME-complete, be used to show thatELIis more expressive thanEL?

Exercise 12.2 Consider the TBox

T ={A₁uA₂v ∃r.B, ∃r⁻.A₂vC, AvA₁uA₂, ∃r.(BuC)vD},

where A,A₁,A2,B,C,D are concept names. Use the classification procedure for ELI to check whether the following subsumption relationships hold w.r.t.T:

(a) AvD (b) ∃r.Av ∃r.D (c) Av ∃r.A

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