Model(domain elements)and relations amon Truth in first-order logic Complex sentences are made from atomic sentences using conectives Complex sentences Atomic sentences are in the brotherhood relation in the model Under this inte 型le 188 Models for FOL:Lots! Models for FOL:ExampleAtomic sentences Atomic sentence = predicate(term1, . . . ,termn) or term1 = term2 Term = function(term1, . . . ,termn) or constant or variable E.g., Brother(KingJohn, RichardTheLionheart) > (Length(LeftLegOf(Richard)),Length(LeftLegOf(KingJohn))) Chapter 8 7 Complex sentences Complex sentences are made from atomic sentences using connectives ¬S, S1 ∧ S2, S1 ∨ S2, S1 ⇒ S2, S1 ⇔ S2 E.g. Sibling(KingJohn, Richard) ⇒ Sibling(Richard, KingJohn) >(1, 2) ∨ ≤(1, 2) >(1, 2) ∧ ¬>(1, 2) Chapter 8 8 Truth in first-order logic Sentences are true with respect to a model and an interpretation Model contains ≥ 1 objects (domain elements) and relations among them Interpretation specifies referents for constant symbols → objects predicate symbols → relations function symbols → functional relations An atomic sentence predicate(term1, . . . ,termn) is true iff the objects referred to by term1, . . . ,termn are in the relation referred to by predicate Chapter 8 9 Models for FOL: Example R J $ left leg left leg on head brother brother person king person crown Chapter 8 10 Truth example Consider the interpretation in which Richard → Richard the Lionheart John → the evil King John Brother → the brotherhood relation Under this interpretation, Brother(Richard, John) is true just in case Richard the Lionheart and the evil King John are in the brotherhood relation in the model Chapter 8 11 Models for FOL: Lots! Entailment in propositional logic can be computed by enumerating models We can enumerate the FOL models for a given KB vocabulary: For each number of domain elements n from 1 to ∞ For each k-ary predicate Pk in the vocabulary For each possible k-ary relation on n objects For each constant symbol C in the vocabulary For each choice of referent for C from n objects . . . Computing entailment by enumerating FOL models is not easy! Chapter 8 12