Applying Resolution: unify Pc and Px with MGU[c/x], and then resolveC1withC2to obtain{}.
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Answers and Solutions to Selected Exercises 329
4.4.7 Transformation into a prenex CNF:
¬(∀ y(∀ zQ(y, z)→ P(z))→ ∃ z(P(z)∧ ∀ x(Q(z, y)→ Q(x, z))))
≡ ∀ y(∀ zQ(y, z)→ P(z))∧ ¬∃ z(P(z)∧ ∀ x(Q(z, y)→ Q(x, z)))
≡ ∀ y(¬∀ zQ(y, z)∨ P(z))∧ ∀ z(¬ P(z)∨ ¬∀ x(¬ Q(z, y)∨ Q(x, z)))
≡ ∀ y(∃ z ¬ Q(y, z)∨ P(z))∧ ∀ z(¬ P(z)∨ ∃ x(Q(z, y)∧ ¬ Q(x, z)))
≡ ∀ y1(∃ z1¬ Q(y1, z1)∨ P(z))∧ ∀ z2(¬ P(z2)∨ ∃ x(Q(z2, y)∧ ¬ Q(x, z2)))
≡ ∀ y1∃ z1(¬ Q(y1, z1)∨ P(z))∧ ∀ z2∃ x(¬ P(z2)∨ ( Q(z2, y)∧ ¬ Q(x, z2)))
≡ ∀ y1∃ z1∀ z2∃ x((¬ Q(y1, z1)∨ P(z))∧ (¬ P(z2)∨ Q(z2, y))∧ (¬ P(z2)∨
¬ Q(x, z2))) Skolemization:
∀ y1∀ z2((¬ Q(y1, f(y1))∨ P(z))∧ (¬ P(z2)∨ Q(z2, y))∧ (¬ P(z2)∨
¬ Q(g(y1, z2), z2))) Clausification:
C1 ={¬ Q(y1, f(y1)), P(z)},
C2 ={¬ P(z2), Q(z2, y)},
C3 ={¬ P(z2), ¬ Q(g(y1, z2), z2)}.
Clausal form:{ C1, C2, C3}.
4.4.9 Transformation into a prenex DNF and a prenex CNF:
¬(∀ y(¬∃ zQ(y, z)→ P(z))→ ∃ z((P(z)→ Q(z, y))∧ ¬∃ xR(x, y, z)))
≡ ∀ y(¬∃ zQ(y, z)→ P(z))∧ ∀ z((P(z)∧ ¬ Q(z, y))∨ ∃ xR(x, y, z))
≡ ∀ y(∃ zQ(y, z)∨ P(z))∧ ∀ z((P(z)∧ ¬ Q(z, y))∨ ∃ xR(x, y, z))
≡ ∀ v(∃ wQ(v, w)∨ P(z))∧ ∀ u((P(u)∧ ¬ Q(u, y))∨ ∃ xR(x, y, u))
≡ ∀ v ∃ w ∀ u ∃ x((Q(v, w)∨ P(z))∧ (( P(u)∧ ¬ Q(u, y))∨ R(x, y, u))) ()
≡ ∀ v ∃ w ∀ u ∃ x(((Q(v, w)∨ P(z))∧ ( P(u)∧ ¬ Q(u, y)))∨ (( Q(v, w)∨
P(z))∧ R(x, y, u)))
≡ ∀ v ∃ w ∀ u ∃ x((Q(v, w)∧ P(u)∧ ¬ Q(u, y))∨ ( P(z)∧ P(u)∧ ¬ Q(u, y))∨
(Q(v, w)∧ R(x, y, u))∨ ( P(z)∧ R(x, y, u))) (PDNF)
≡ ∀ v ∃ w ∀ u ∃ x((Q(v, w)∨ P(z))∧ ( P(u)∨ R(x, y, u))∧ (¬ Q(u, y)∨
R(x, y, u)))(PCNF from ()) Skolemization:
∀ v ∀ u((Q(v, f(v))∨ P(z))∧ ( P(u)∨ R(g(v, u), y, u))∧ (¬ Q(u, y)∨
R(g(v, u), y, u))) Clausal form:
{{ Q(v, f(v)), P(z)} , { P(u), R(g(v, u), y, u)} , {¬ Q(u, y), R(g(v, u), y, u)}}
Section 4.5
4.5.2 (a) Transformation of¬(∀ xP(x)→ ∀ yP(y)) into a clausal form:
C1 ={ P(x)} and C2={¬ P(c)}
for some Skolem constantc Applying Resolution: unify P(c) and P(x) with MGU[c/x], and then resolveC1withC2to obtain{}.