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  1. 1. Wissensrepr¨sentation a Prof. Dr. Nebel, Dr. W¨lfl o Universit¨t Freiburg a M. Helmert, M. Ragni Institut f¨r Informatik u WS 2005/2006 ¨ Ubungsblatt 9 — L¨sungen o Aufgabe 9.1 (Realisierung) Betrachten Sie die Beispiel-TBox und -ABox aus der Vorlesung (Kapitel 11, Folie 3) und die zugeh¨rige Klassifikationshierarchie (Kapitel 11, Folie 18). Die o Realisierung (Folie 23) eines Objektes a in der ABox besteht aus den am meisten spezialisierten Konzepten C der TBox, f¨r die a : C gilt. u (a) Bestimmen Sie die Realisierung von DIANA, ELIZABETH, CHARLES, EDWARD, ANDREW und WILLIAM (nur Ergebnisse, keine Beweise). L¨sung: o • DIANA: Mother-without-daughter • ELIZABETH: Mother-with-many-children • CHARLES: Man • EDWARD: Man • ANDREW: Man • WILLIAM: Male Anmerkung: Aus der ABox geht zwar hervor, dass CHARLES ein Kind hat, nicht aber, dass dieses menschlich ist. Daher k¨nnen wir nicht folgern, dass o er ein Vater ist. Somit folgt auch nicht, dass ELIZABETH eine Großmutter ist. (b) Um die Realisierung eines Objektes algorithmisch zu bestimmen, muss eine Menge von Aussagen der Art a : C folgt aus der Wissensbasis“ und ” a : C folgt nicht aus der Wissensbasis“ bewiesen werden. Geben Sie eine ” minimale Menge solcher Aussagen an, mit denen Ihr Ergebnis aus Teil (a) bewiesen werden k¨nnte. (Sie d¨rfen dabei die Klassifikationshierarchie als o u bereits bewiesen ansehen.) L¨sung: o Wir schreiben T , A |= a : C um zu notieren, dass aus der Wissensbasis (TBox und ABox) die Aussage a : C folgt und T , A |= a : C um zu notieren, dass sie nicht folgt. Dann m¨ssten wir beweisen: u • F¨r a = DIANA: u T , A |= a : Mother-without-daughter, T , A |= a : Mother-with-many-children, T , A |= a : Grandmother, T , A |= a : Male 1
  2. 2. • F¨r a = ELIZABETH: u T , A |= a : Mother-with-many-children, T , A |= a : Mother-without-daughter, T , A |= a : Grandmother, T , A |= a : Male • F¨r a ∈ {CHARLES, EDWARD, ANDREW}: u T , A |= a : Man, T , A |= a : Female, T , A |= a : Parent • F¨r a = WILLIAM: u T , A |= a : Male, T , A |= a : Female, T , A |= a : Living Entity Aufgabe 9.2 (Beschreibungslogik ALC) Ein Fleischfresser ist ein Lebewesen, das andere Lebewesen isst. Ein Vegetarier ist ein Lebewesen, das keine anderen Lebewesen isst. Ein Meta-Vegetarier ist ein Lebewesen, das keine Fleischfresser isst. (a) Geben Sie eine TBox in der Beschreibungslogik ALC an, die die drei Kon- zepte Carnivore, Vegetarian und MetaVegetarian definiert. Normalisie- ren und entfalten Sie die TBox und ubersetzen Sie sie in Negationsnor- ¨ malform (falls n¨tig). o L¨sung: o Die Konzepte sind wie folgt definiert: . Carnivore = Being ∃eats.Being . Vegetarian = Being ¬∃eats.Being . MetaVegetarian = Being ¬∃eats.Carnivore Alle Konzepte sind bereits normalisiert, die ersten beiden Konzepte sind bereits entfaltet. Wir entfalten das dritte Konzept: . Carnivore = Being ∃eats.Being . Vegetarian = Being ¬∃eats.Being . MetaVegetarian = Being ¬∃eats.(Being ∃eats.Being) Das erste Konzept ist bereits in Negationsnormalform, die anderen beiden noch nicht. Wir stellen die Normalform her: . Carnivore = Being ∃eats.Being . Vegetarian = Being ∀eats.¬Being . MetaVegetarian = Being ∀eats.(¬Being ∀eats.¬Being) (b) F¨hren Sie die folgenden Aussagen (bezogen auf die TBox in Teil (a)) u auf Erf¨llbarkeitsaussagen zur¨ck und beweisen oder widerlegen Sie sie u u mithilfe des Tableau-Algorithmus f¨r ALC: u 2
  3. 3. • Vegetarian MetaVegetarian L¨sung: o Im Folgenden k¨rzen wir das Konzept Being mit B und die Rolle u eats mit e ab. Die Aussage ist genau dann wahr, wenn das Konzept Vegetarian ¬MetaVegetarian in der gegebenen TBox unerf¨llbar ist. Entfaltet u ergibt sich das Konzept (B ∀e.¬B) ¬(B ∀e.(¬B ∀e.¬B)), in Negationsnormalform also (B ∀e.¬B) (¬B ∃e.(B ∃e.B)). Der Tableau-Algorithmus ergibt: x : (B ∀e.¬B) (¬B ∃e.(B ∃e.B)) x:B ∀e.¬B x : ¬B ∃e.(B ∃e.B) x:B x : ∀e.¬B x : ¬B x : ∃e.(B ∃e.B) xey y:B ∃e.B y : ¬B y:B Das Constraint-System ist nicht erf¨llbar, die Aussage Vegetarian u MetaVegetarian ist also wahr. • MetaVegetarian Vegetarian L¨sung: o Die Aussage gilt genau dann, wenn MetaVegetarian ¬Vegetarian in der gegebenen TBox unerf¨llbar ist. Entfaltet ergibt sich das Kon- u zept (B ∀e.(¬B ∀e.¬B)) ¬(B ∀e.¬B), in Negationsnormalform also (B ∀e.(¬B ∀e.¬B)) (¬B ∃e.B). Der Tableau-Algorithmus ergibt: 3
  4. 4. x : (B ∀e.(¬B ∀e.¬B)) (¬B ∃e.B) x:B ∀e.(¬B ∀e.¬B) x : ¬B ∃e.B x:B x : ∀e.(¬B ∀e.¬B) x : ¬B x : ∃e.B xey y:B y : ¬B ∀e.¬B) y : ¬B y : ∀e.¬B Das Constraint-System ist erf¨llbar, die Aussage MetaVegetarian u Vegetarian ist also falsch. Aus dem Tableau ergibt sich das Gegen- beispiel {x : Being, y : Being, (x, y) : eats}, f¨r das zwar gilt u x : MetaVegetarian, aber nicht x : Vegetarian. • MetaVegetarian Carnivore L¨sung: o Die Aussage gilt genau dann, wenn MetaVegetarian ¬Carnivore in der gegebenen TBox unerf¨llbar ist. Entfaltet ergibt sich das Konzept u (B ∀e.(¬B ∀e.¬B)) ¬(B ∃e.B), in Negationsnormalform also (B ∀e.(¬B ∀e.¬B)) (¬B ∀e.¬B). Der Tableau-Algorithmus ergibt: x : (B ∀e.(¬B ∀e.¬B)) (¬B ∀e.¬B) x:B ∀e.(¬B ∀e.¬B) x : ¬B ∀e.¬B x:B x : ∀e.(¬B ∀e.¬B) x : ¬B x : ∀e.¬B Das Constraint-System ist erf¨llbar, die Aussage MetaVegetarian u Carnivore ist also falsch. Aus dem Tableau ergibt sich das Gegenbei- spiel {x : Being}, f¨r das zwar gilt x : MetaVegetarian, aber nicht u x : Carnivore. 4

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