Homework Assignment #4 1. (12 points) Represent the following knowledge base in FOPC. Use the predicates attend(x), fair(t), pass(x,t), prepared(x), smart(x), study(x), wsustudent(x), where arguments x have the domain of all students and arguments t have the domain of all tests. i. Everyone who is smart, studies, and attends class will be prepared. ii. Everyone who is prepared will pass a test if it is fair. iii. Every WSU student is smart. iv. If a test is not fair, no one will pass the test. v. Chris is a WSU student. vi. At least one student passed the 440 exam. vii. Chris attends class. Convert this knowledge base to clausal form. Prove that study(Chris) -> pass(Chris, 440exam) using resolution refutation. Clearly show which sentences were resolved to produce each new sentence. 2. (8 points) Use truth tables to show the following sentences are valid, and thus that the equivalences hold. a. P ^ (Q v R) <=> (P ^ Q) v ( P ^ R) Distributivity of ^ over v b. P v (Q ^ R) <=> (P v Q) ^ ( P v R) Distributivity of v over ^ c. P => Q <=> ~Q => ~P Contraposition d. P => Q <=> ~P v Q e. P <=> Q <=> (P => Q) ^ (Q => P) f. P <=> Q <=> (P ^ Q) v (~Q ^ ~P) g. P ^ ~P <=> False h. P v ~P <=> True 3. (6 points) State whether the following sentences are valid, unsatisfiable, or neither. Verify your decisions where appropriate using the equivalence rules from problem #2 and from Figure 7.11 in [RN] (page 210 of the second edition). a. Smoke => Smoke b. Smoke => Fire c (Smoke => Fire) => (~Smoke => ~Fire) d. Smoke v Fire v ~Fire e. ((Smoke ^ Heat) => Fire) <=> ((Smoke => Fire) v (Heat => Fire)) f. (Smoke => Fire) => ((Smoke ^ Heat) => Fire)