Description Logic

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

Description Logic

Exercise Sheet 8 6th December 2017

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

Exercise 8.1 Recall the tableau algorithm forALCN with blocking, and the example that shows non-termination when using the≤-rulewithout pruning. At each step of the algorithm, write down the multisetµ(A)for the current ABoxA, and explain why the termination proof fails in this case.

Exercise 8.2 Prove that the tableau algorithm forALCN is sound and complete.

Exercise 8.3 We extend the tableau algorithm fromALCN toALCQby modifying the≥-rule and the≤-rule as follows:

The≥-rule

Condition: Acontainsa:(≥n r.C), but there are nondistinct individualsb₁, . . . ,b_n with{(a,b_i):r,b_i:C|1≤i≤n} ⊆ A, andais not blocked

Action: A −→ A ∪ {(a,d_i):r,d_i:C |1 ≤ i≤ n} ∪ {d_i 6= d_j |1 ≤ i< j≤ n}, whered₁, . . . ,d_nare new individual names

The≤-rule

Condition: Acontainsa:(≤n r.C), and there aren+1distinct individualsb₀, . . . ,b_n with{(a,b_i):r,b_i:C|0≤i≤n} ⊆ A

Action: A −→prune(A,b_j)[b_j 7→b_i]∪ {b_i =b_j}fori6=jsuch that, ifb_jis a root individual, then so isb_i

For the knowledge base CvE ,

a:(≤1r.(DuE)), (a,b):r, b:CuD, (a,c):r, c:DuE, c:¬C , determine whether it is consistent, and whether the proposed algorithm detects this.

Exercise 8.4 LetT be an acyclic TBox in NNF, and letT^vbe obtained fromT by replacing each definitionA≡ C∈ T withAvC. Prove that every concept name A₀is satisfiable w.r.t.T iff it is satisfiable w.r.t.T^v. Does this also hold for the acyclic TBox{A≡Cu ¬B, B≡P, C≡P}?

Exercise 8.5 Use theALC-Worldsalgorithm to decide satisfiability of the concept nameA₀w.r.t. the following simple TBox:

{A₀≡ A₁uA₂, A₁≡ ∃r.A₃, A₃ ≡P, A2≡ A₄uA5, A₄≡ ∃r.A6, A6 ≡Q, A₅≡ A₇uA₈, A₇≡ ∀r.A₄, A₈ ≡ ∀r.A₉, A₉≡ ∀r.A₁₀, A₁₀≡ ¬P}

Draw the recursion tree of a successful run and of an unsuccessful run. Does the algorithm return a positive or a negative result on this input?

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