Consider the following logic program Which of the following first order…

Solution: Translate the program rules into first-order sentences and conjoin them.

• Rule: A(x) ← B(x,y), C(y).

  Translation: for every x, if there exists a y such that B(x,y) and C(y), then A(x).

  Formula: (∀x)[(∃y)(B(x,y) ∧ C(y)) ⇒ A(x)]

• Constraint: ← B(x,x).

  Translation: there must be no x with B(x,x).

  Formula: ¬(∃x)B(x,x)

Final equivalent first-order sentence:

(∀x)[(∃y)(B(x,y) ∧ C(y)) ⇒ A(x)] ∧ ¬(∃x)B(x,x)

Notes: use an existential quantifier for the variable y that appears only in the rule body (the body requires some y), and translate denial constraints as a negated existential statement. Do not replace the existential by a universal, and do not change the constraint into an existence claim or into a disjunction with the rule translation.