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By Wolfgang Ertel

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Extra resources for Grundkurs Künstliche Intelligenz

Sample text

Bn – die aktuellen Teilziele – zu beweisen. Nach der Anwendung wird B1 ersetzt durch die neuen Teilziele sUppLex 32 2 Aussagenlogik A1 ∧ . . ∧ Am . Um zu zeigen, dass B1 wahr ist, muss nun gezeigt werden, dass A1 ∧ . . ∧ Am wahr sind. Dieser Prozess setzt sich so lange fort, bis die Liste der Teilziele der aktuellen Klausel (der so genannte goal stack) leer ist. Damit ist ein Widerspruch gefunden. Gibt es zu einem Teilziel ¬Bi keine Klausel mit einem komplement¨ aren Literal Bi als Klauselkopf, so terminiert der Beweis und es kann kein Widerspruch gefunden werden.

N. 5 Die aus der Analysis bekannte Definition der Konvergenz einer Folge (an )n∈N gegen einen Grenzwert a lautet ∀ε > 0 ∃n0 ∈ N ∀n > n0 |an − a| < ε. 4) die offenbar nicht in pr¨anexer Normalform ist. Da die Variablen der inneren Quanandert sich toren ∃n0 und ∀n nicht links vom jeweiligen Quantor vorkommen, ¨ an der Semantik der Formel nichts, wenn wir alle Quantoren nach außen ziehen. Die a ¨quivalente pr¨anexe Normalform ist also ∀ε ∃n0 ∀n (gr(ε, 0) ⇒ (el(n0 , N) ⇒ (gr(n, n0 ) ⇒ gr(ε, abs(minus(a(n), a)))))).

Dass dieses Verfahren auch wirklich wie gew¨ unscht funktioniert, sichert uns der folgende Satz. Damit der Kalk¨ ul auch vollst¨andig wird, ben¨ otigen wir noch eine kleine Erg¨ anzung, wie folgendes einfache Beispiel zeigt. Sei als Wissensbasis die Formel (A ∨ A) gegeben. Um per Resolution zu zeigen, dass daraus (A ∧ A) ableitbar ist, muss gezeigt werden, dass aus (A ∨ A) ∧ (¬A ∨ ¬A) die leere Klausel ableitbar ist. Mit der Resolutionsregel allein ist das nicht m¨ oglich. Mit der Faktorisierung, die es erlaubt, Kopien von Literalen in Klauseln zu l¨ oschen, wird dieses Problem beseitigt.

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