Download Transforming Axioms into Clause Form: Logic and Predicate Calculus and more Assignments Electrical and Electronics Engineering in PDF only on Docsity! A.Show the steps required to put the following axioms into clause form: ∀x∀y[On(x,y) => Above(x,y)] ∀x∀y∀z[(Above(x,y) & Above(y,z)) => Above(x,z)] Eliminate implications ∀x∀y[~On(x,y) V Above(x,y)] Remove ∀ Axiom1: ~On(x,y) V Above(x,y) Eliminate implications ∀x∀y∀z[~(Above(x,y) & Above(y,z)) V Above(x,z)] Move negation to literals ∀x∀y∀z[~Above(x,y) V ~Above(y,z) V Above(x,z)] Remove ∀ Axiom2: ~Above(x,y) V ~Above(y,z) V Above(x,z) B. Put the following axioms in clause form, which says that if x is above y, but not directly on y, there must exist some third block, z, in between. Note that a Skolem function will be needed. ∀x∀y[(Above(x,y) & ~On(x,y)) => ∃z[Above(x, z) & Above(z,y)]] Eliminate implications ∀x∀y[~(Above(x,y) & ~On(x,y)) V ∃z[Above(x, z) & Above(z,y)]] Move negations to literals ∀x∀y[(~Above(x,y) V On(x,y)) V ∃z[Above(x, z) & Above(z,y)]] Eliminate existencial symbols ∀x∀y[(~Above(x,y) V On(x,y)) V [Above(x, f(x,y)) & Above(f(x,y),y)]] Move V’s to literals ∀x∀y[(~Above(x,y) V On(x,y) V Above(x, f(x,y))) &(~Above(x,y) V On(x,y) V Above(f(x,y),y))] Separate &’s ∀x∀y[(~Above(x,y) V On(x,y) V Above(x, f(x,y)))] ∀x∀y[(~Above(x,y) V On(x,y) V Above(f(x,y),y))] Remove ∀ Axiom1: ~Above(x,y) V On(x,y) V Above(x, f(x,y))) Axiom2: ~Above(x,y) V On(x,y) V Above(f(x,y),y)) A more elaborated answer was designed by professor Rozenblit. I recommend to use this one as a study example for your final. Axiom1: JR owns a sports car : Sport_car(c) & Owns (JR,c) Axiom 1a: Sport_car(c) Axiom 1b: Owns(JR,c) Axiom2: Every sports car owner loves races : ∀x [∃y(Sport_car(y) & Owns (x,y)) => loves_races(x)] ∀y~(Sport_car(y)&Owns(x,y)) V loves_races(x) Axiom 2: ~Sport_car(y) V ~Owns(x,y) V loves_races(x) Axiom3: No one who loves races buys a station wagon. ∀x ∀y (loves_races(x) & SW(y)) => ~Buys(x,y) Axiom 3: ~loves_races(x) V ~SW(y) V ~Buys(x,y) Axiom4: Either JR or “Bud” bought the station wagon, which is named Boat Axiom 4: Buys(JR,Boat) V Buys(Bud, Boat) Axiom5: SW(Boat) Negated Theorem ~Buys(Bud, Boat) Did Bud buy the station wagon? Yes he did... Axiom 1a Axiom 2 ~Owns(x,c) V loves_races(x) Axiom 1b loves_races(JR) Axiom 3 ~SW(y)V ~Buys(JR,y) Axiom 5 ~Buys(JR, Boat) Axiom 4 Buys(Bud, Boat) ~Theorem NIL <y,c> <x,JR> <x,JR> <y,Boat>
