Trường Đại Học Bách Khoa Tp.Hồ Chí Minh Khoa Khoa Học và Kỹ Thuật Máy Tính Tutorial 6 PREDICATE LOGIC and STRUCTURED KNOWLEDGE Question 1: Wrong in: ¬gt(5, 2) ∨ ¬succ(y, 2) combines with 6¬gt(x, y) ∨ ¬gt(y, z) ∨ gt(x, z) by the substitution 5/x, y/z This combination is wrong because the variable y in each clause is different to each other. To correct the resolution, we have to change the name of variable y in either first clause or second clause. Question 2: What is wrong with the following argument: • Men are widely distributed over the earth. • Socrates is a man. • Therefore, Socrates is widely distributed over the earth. How should the facts represented by these sentences be represented in logic so that this problem does not arise? Solution The conclusion is false because the property "widely distributed over the earth" is appli- cable over a set (group of men). Not to each individual in the set. To avoid that wrong conclusion, we can represent these facts like this: man(Socrates). widelyDistributed(men,earth) Question 3: Consider the following axioms: 1. All hounds howl at night. 2. Anyone who has any cats will not have any mice. 3. Light sleepers do not have anything which howls at night. 4. John has either a cat or a hound. Use resolution with predicate logic to affirm the following conclusion: If John is a light sleeper, then John does not have any mice. Solution Artificial Intelligence Exercises Page 1/5 Trường Đại Học Bách Khoa Tp.Hồ Chí Minh Khoa Khoa Học và Kỹ Thuật Máy Tính Before going to do resolution, we have to represent these facts in predicate logic: 1. ∀x(HOUND(x) → HOW L(x)) 2. ∀x∀y(HAV E(x, y) ∧ CAT (y) → ¬∃z(HAV E(x, z) ∧ MOUSE(z))) 3. ∀x(LS(x) → ¬∃y(HAV E(x, y) ∧ HOW L(y))) 4. ∃x(HAV E(John, x) ∧ ((CAT (x) ∨ HOUND(x)) ∧ (¬CAT (x) ∨ ¬HOUND(x))) Conclusion: LS(John) → ¬∃z(HAV E(John, z) ∧ MOUSE(z)) Convert to clause form 1.¬HOUN D(x) ∨ HOW L(x) 2.¬HAV E(x, y) ∨ ¬CAT (y) ∨ ¬HAV E(x, z) ∨ ¬MOUSE(z) 3.¬LS(x) ∨ ¬HAV E(x, y) ∨ ¬HOW L(y) 4a.HAV E(John, a) 4b.CAT (a) ∨ HOU ND(a) 4c. ¬CAT (a) ∨ ¬HOUND(a) Negation of conclusion 5a.LS(John) 5b.HAV E(John, b) 5c.MOUSE(b) Resolution 5c b/z 2 6.¬HAV E(x, y) ∨ ¬CAT(y) ∨ ¬HAV E(x, b) 5b John/x 6 7.¬HAV E(John, y) ∨ ¬CAT (y) 4b a/y 7 8.¬HAV E(John, a) ∨ HOU N D(a) 4a 8 9.HOW L(a) 3 a/y 9 10.¬LS(x) ∨ ¬HAV E(x, a) 4a John/x 10 11.¬LS(John) 5a 11 [] Question 4: Prove that each of the following sentences is valid by using existential graph’s rules of inferences: a. P ⇒ (Q ⇒ P ) b. ((Smoke ⇒ F ire) ∨ (Heat ⇒ F ire)) ⇒ ((Smoke ∧ Heat) ⇒ F ire) c. (Q ⇒ ¬P ) ⇒ ((Q ⇒ P ) ⇒ ¬Q) d. ((P ⇒ Q) ∧ (¬P ⇒ R)) ⇒ ((P ∧ Q) ⇒ (¬P ∧ R)) Solution Artificial Intelligence Exercises Page 2/5 Trường Đại Học Bách Khoa Tp.Hồ Chí Minh Khoa Khoa Học và Kỹ Thuật Máy Tính a ( () ) Double negation (P () ) Insertion (P ( (()) ) ) Double negation (P ( ((P)) ) ) Iteration (P ( ( Q (P) ) ) ) Insertion (P ( ( Q (P) ) ) ) Insertion b: ( ( ( (S (F)) ) ( (H (F)) ) ) ((S H (F))) ) ( () ) Double negation ( () ((S H (F))) ) Insertion ( ( ((S H (F))) ) ((S H (F))) ) Iteration ( ( S H (F) ) ((S H (F))) ) Double negation ( ( S H (F) (F) ) ((S H (F))) ) Iteration ( ( S (F) ((H (F))) ) ((S H (F))) ) Double negation ( ( ((S (F))) ((H (F))) ) ((S H (F))) ) Double negation c: ( (Q ((P))) ((Q (P)) ((Q))) ) ( () ) Double negation ( () ((Q (P)) ((Q)))) Insertion ( () ((Q (P)) ((Q))) ((Q (P)) ((Q))) ) Iteration ( () ((Q (P)) Q) ((Q (P)) ((Q))) ) Double negation ( () (((P)) Q) ((Q (P)) ((Q))) ) Deiteration ( () ( P Q) ((Q (P)) ((Q))) ) Double negation ( ( (P Q) ) (P Q) ((Q (P)) ((Q))) ) iteration ( ( (Q ((P))) ) (P Q) ((Q (P)) ((Q))) ) Double negation Artificial Intelligence Exercises Page 3/5 Trường Đại Học Bách Khoa Tp.Hồ Chí Minh Khoa Khoa Học và Kỹ Thuật Máy Tính d: ( (P (Q)) ((P) (R)) (((P Q) ((P) R))) ) ( () ) Double negation ( () ((P)) ((Q)) ((P) (R))) Insertion ( (((P)) ((Q))) ((P)) ((Q)) ((P) (R)) ) Iteration ( (P Q) ((P)) ((Q)) ((R)) ((P) (R)) ) Double negation ( (P Q) ((P)) ((Q) ((P)) ) ((P) (R)) ) Iteration ( (P Q) ((P)) ((Q) P) ((P) (R)) ) Double negation ( (P Q) ((Q) P) ((P) ((P) (R))) ((P) (R)) ) Iteration ( (P Q) ((Q) P) ( (P) R ) ((P) (R)) ) Deiteration + Double negation ( (P (Q)) ((P) (R)) (P Q) ((P) R) ) Just reorder ( (P (Q)) ((P) (R)) (( (P Q) ((P) R) )) ) Double negation Question 5: Convert the following sentence into predicate logic, existential graph (EG) and conceptual graph (CG): If a cat is on a mat, then it is a happy pet. Solution Prediate logic: ∀x∀y : cat(x) ∧ mat(y) ∧ on(x, y) → happyP et(x) ¬(∃x∃y : cat(x) ∧ mat(y) ∧ on(x, y) ∧ ¬happyPet(x) Another possible solution: ∀x∀y∀z((cat(x) ∧ mat(y) ∧ on(x, y)) → (pet(x) ∧ happy(z) ∧ haveattr(x, z))) Artificial Intelligence Exercises Page 4/5 Trường Đại Học Bách Khoa Tp.Hồ Chí Minh Khoa Khoa Học và Kỹ Thuật Máy Tính Artificial Intelligence Exercises Page 5/5 . following argument: • Men are widely distributed over the earth. • Socrates is a man. • Therefore, Socrates is widely distributed over the earth. How should. arise? Solution The conclusion is false because the property "widely distributed over the earth" is appli- cable over a set (group of men). Not