2. Consider the language that contains the constant symbols

a, b, and c; the predicate symbols p and q; and no function symbols. We have the following knowledge bases built from this language:

KB1 = { p

(a) }

KB2 = { p(X) ← q(X) }

KB3 = { p(X) ← q(X), p(a), q

(b) } .

Now consider possible interpretations for this language of the form I = ⟨D, π, ϕ⟩, where D = {✂,

☎, ✈, ✎}.

(a) How many interpretations with the four domain elements exist for our simple language? Give a brief justification for your answer. [Hint: Consider how many possible assignments ϕ exist for the constant symbols, and consider how many extensions predicates p and q can have to determine how many assignments π exist.] Do not try to enumerate all possible interpretations.

(b) Of the interpretations outlined above, how many are models of KB1? Give a brief justification for your answer.

(c) Of the interpretations outlined above, how many are models of KB2? Give a brief justification for your answer.

(d) Of the interpretations outlined above, how many are models of KB3? Give a brief justification for your answer.