A Direct Proof for the Completeness of SLD-Resolution
نویسنده
چکیده
Lemma 2 Let A be an atom and P |= A. Then A has an implication tree wrt. P . Proof: We construct a model M of P . Let |M| be the set of all terms (with variables) and let fM(~t ) := f(~t ). Let rM(~t ) be true if and only if r(~t ) has an implication tree wrt. P . We claim that M is a model of P . If B is a fact of P then for every substitution (variable assignment) θ the atom Bθ is an implication tree wrt. P and M |= Bθ. If B ← C1 ∧ . . . ∧ Cn is a clause of P (1 ≤ n) and θ is a substitution and M |= C1θ ∧ . . . ∧ Cnθ, then C1θ, . . . , Cnθ have implication trees wrt. P and therefore Bθ and we have M |= Bθ. Since M is a model of P and P |= A, the atom A is true in M and has an implication tree wrt. P . 2
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