Notes on Assignment 3 Foundations of Artificial Intelligence

2 English to FOPC (Similar to Assignment 3 - Problem 2) Not all students take both History and Biology ~  x student(x) => (takes(x, hist ) /\ takes(x, bio ))  x student(x) /\ ~(takes(x, hist ) /\ takes(x, bio )) or alternatively Only one student failed History  x student(x) /\ fails(x, hist ) /\ (  y student(y) /\ fails(y, hist ) => x = y

Foundations of Artificial Intelligence 3 Forward Chaining Proof – Example (Similar to Assingment 3 - Problem 1) KB1: (A \/ B) => E KB2: ~D \/ C KB3: D /\ C => A KB4: D Give a derivation (proof) for the sentence E. 1.DKB4 2.CStep 1, KB2, using Resolution rule 3.D /\ CStep 1, Step 2, using AND-Introduction 4.AStep 3, KB3, using Modus Ponens 5.A \/ BStep 4, using OR-Introduction (introducing B) 6.EStep 5, KB1, using Modus Ponens

Foundations of Artificial Intelligence 4 Similar to Assignment 3 - Problem 1 (resolution-refutation)  Can we do this using resolution-refutation?  First need to convert everything into clause form  Next we need to negate the sentence to be proved and convert to clause form: KB1: (A \/ B) => E KB2: ~D \/ C KB3: D /\ C => A KB4: D KB1.1: ~A \/ E KB1.2: ~B \/ E KB2: ~D \/ C KB3: ~D \/ ~C \/ A KB4: D ~E E Initial subgoal for resolution

Foundations of Artificial Intelligence 5 Similar to Assignment 3 - Problem 1 (resolution-refutation) ~E Subgoal Resolve with KB3: ~D \/ ~C \/ A ~A KB1.1: ~A \/ E ~D \/ ~C KB4: D ~C KB1.1: ~A \/ E KB1.2: ~B \/ E KB2: ~D \/ C KB3: ~D \/ ~C \/ A KB4: D KB2: ~D \/ C ~D KB4: D X

Foundations of Artificial Intelligence 6 Resolution Example (Assignment 3 - Problem 3)  John likes all kinds of food.  Apples are food.  Chicken is food.  Anything anyone eats and isn't killed by is food.  Bill eats peanuts and is still alive.  Anyone who is killed by anything is not alive.  Sue eats everything Bill eats.   x food(x) => likes( john, x)  food( apple )  food( chicken )   x  y eats(x, y) /\ ~killed-by(x, y) => food(y)  eats( bill, peanuts ) /\ alive( bill )   x  y killed-by(x, y) => ~alive(x)   x eats( bill, x) => eats( sue, x)

Foundations of Artificial Intelligence 7 Resolution Example (Assignment 2 - Problem 5)   x food(x) => likes( john, x)  food( apple )  food( chicken )   x  y eats(x, y) /\ ~killed-by(x, y) => food(y)  eats( bill, peanuts ) /\ alive( bill )   x  y killed-by(x, y) => ~alive(x)   x eats( bill, x) => eats( sue, x) 1. ~food(x) \/ likes( john, x) 2. food( apple ) 3. food( chicken ) 4. ~eats(x, y) \/ killed-by(x, y) \/ food(y) 5a. eats( bill, peanuts ) 5b. alive( bill ) 6. ~killed-by(x,y) \/ ~alive(x) 7. ~eats( bill, x) \/ eats( sue, x) 1. ~food(x) \/ likes( john, x) 2. food( apple ) 3. food( chicken ) 4. ~eats(x, y) \/ killed-by(x, y) \/ food(y) 5a. eats( bill, peanuts ) 5b. alive( bill ) 6. ~killed-by(x,y) \/ ~alive(x) 7. ~eats( bill, x) \/ eats( sue, x) Clausal Form

Foundations of Artificial Intelligence 8 Resolution Example (Assignment 2 - Problem 5) Prove that john likes peanuts : Prove that john likes peanuts : 1. ~food(x) \/ likes( john, x) 2. food( apple ) 3. food( chicken ) 4. ~eats(x, y) \/ killed-by(x, y) \/ food(y) 5a. eats( bill, peanuts ) 5b. alive( bill ) 6. ~killed-by(x,y) \/ ~alive(x) 7. ~eats( bill, x) \/ eats( sue, x) 1. ~food(x) \/ likes( john, x) 2. food( apple ) 3. food( chicken ) 4. ~eats(x, y) \/ killed-by(x, y) \/ food(y) 5a. eats( bill, peanuts ) 5b. alive( bill ) 6. ~killed-by(x,y) \/ ~alive(x) 7. ~eats( bill, x) \/ eats( sue, x)

Foundations of Artificial Intelligence 9 Resolution Example (Assignment 3 -Problem 3) What food does sue eat? ( i.e.,  x food(x) /\ eats( sue, x)? )  x food(x) /\ eats( sue, x)  x food(x) /\ eats( sue, x))  food(x) \/ ~eats( sue, x)) negate Convert to clause form This can be resolved with candidate clauses 2, 3, and 4 This can be resolved with candidate clauses 2, 3, and 4 1. ~food(x) \/ likes( john, x) 2. food( apple ) 3. food( chicken ) 4. ~eats(x, y) \/ killed-by(x, y) \/ food(y) 5a. eats( bill, peanuts ) 5b. alive( bill ) 6. ~killed-by(x,y) \/ ~alive(x) 7. ~eats( bill, x) \/ eats( sue, x) 1. ~food(x) \/ likes( john, x) 2. food( apple ) 3. food( chicken ) 4. ~eats(x, y) \/ killed-by(x, y) \/ food(y) 5a. eats( bill, peanuts ) 5b. alive( bill ) 6. ~killed-by(x,y) \/ ~alive(x) 7. ~eats( bill, x) \/ eats( sue, x)

Foundations of Artificial Intelligence 10 English to FOPC (Similar to Assignment 3 - Problem 2) Not all students take both History and Biology ~  x student(x) => (takes(x, hist ) /\ takes(x, bio ))  x student(x) /\ ~(takes(x, hist ) /\ takes(x, bio )) or alternatively Only one student failed History  x student(x) /\ fails(x, hist ) /\ (  y student(y) /\ fails(y, hist ) => x = y