Exercises

Exercise 1: Reduction between Reasoning Tasks We show that A v B if and only if A u ¬B is not satisfiable. If A v B, then in any model I of the TBox, it holds that the AI ⊆ BI . Hence AI ∩ (∆I \BI) = ∅. As (A u ¬B)I = AI ∩ (∆I \ BI), the interpretation of A u ¬B is empty in any model of the TBox, hence A u ¬B is not satisfiable. Conversely, if A is not a subconcept of B there exists a model I of the TBox in which there exists e ∈ AI such that e 6∈ BI . Hence, e ∈ ¬BI , and by definition of conjunction, e ∈ (A ∧ ¬B)I . We have exhibited a model in which (A ∧ ¬B) has a non empty interpretation, and A ∧ ¬B is thus satisfiable. Thus, in order to decide whether A is a subconcept of B, one can check whether A∧¬B is satisfiable. As satisfiability in EL is trivial (every concept is satisfiable) and subsumption is not, there cannot be a reduction from subsumption to satisfiability.