Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 353

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Logic as a tool  a guide to formal logical reasoning ( PDFDrive ) 353

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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 constant c Applying Resolution: unify P (c) and P (x) with MGU[c/x], and then resolve C1 with C2 to obtain {}

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