Artificial Intelligence: Knowledge Representation and Reasoning

Week 8 Assessment 1 – Answers Glossary: PL = propositional logic, FOL = first order logic, CNF = conjunctive normal form, DNF = disjunctive normal form, KB = knowledge base, SLD = selected literals, linear pattern over definite clauses 1. Which of the following is a description of an FOL formula in clause form? (a,d,f) a. It has only universal quantifiers and should be in CNF form. b. It has only existential quantifiers and should be in CNF form. c. Mix of universal quantifiers and existential quantifiers and should be in CNF form. d. The only logical operators it has are AND, OR and NOT. e. Only universal quantifiers and should be in DNF form. f. The negation operator is only applied to atomic formulas.

2. An inference in backward chaining and forward chaining is, respectively, from (c) a. a set of facts to a new fact, a goal to a set of subgoals. b. a set of subgoals to a goal, a set of facts to new fact. c. a goal to a set of subgoals, a set of facts to a new fact d. None of the above.

3. Which of the following statements concerning Propositional Logic (PL) and First Order Logic (FOL) are true (c,f) a. PL is undecidable, FOL is decidable. b. PL is semi-decidable, FOL is undecidable. c. PL is decidable, FOL is semi-decidable. d. Both PL and FOL are undecidable. e. PL and FOL are decidable. f. Both PL and FOL are sound and complete. g. PL is complete but FOL is incomplete.

4. Which of the following are true of Resolution Refutation Method. (a,c,e,f,g) a. It is a proof procedure in which only the resolution rule is used. b. It employs only the implication and negation operators. c. It employs only the and, or, and negation operators. d. It is used to show that a given proof is not sound. e. It is an approach to proof by contradiction. f. It can used to show that a goal formula is entailed by a KB. g. Derivation of the null clause means that the input KB is inconsistent.

5. FOL with Resolution Refutation is (b) a. complete but not sound. b. both complete and sound. c. sound but not complete. d. neither sound nor complete.

6. Given an inconsistent KB, resolution refutation using the answer predicate (c) a. terminates when only the null clause remains. b. terminates when neither the null clause nor the answer predicate remains c. terminates when only the answer predicate remains d. may not terminate.

7. Convert to Clause form the sentence (a) ∀x [Uncle(x) ≡ [∃z (isBrotherOf(x,z) ∧ ∃y Child(z,y)] ]

a. 1.¬Uncle(?x1) ∨ isBrotherOf(?x1,sk1(?x1)) 2. ¬Uncle(?x2) ∨ Child(sk1(?x2),sk2(?x2)) 3. ¬isBrotherOf(?x3,?z2) ∨ ¬Child(?z2,?y1) ∨ Uncle(?x3)

b. 1.¬Uncle(?x) ∨ isBrotherOf(?x,sk1(?x)) 2. ¬Uncle(?x) ∨ Child(?z1,sk2(?x)) 3. ¬isBrotherOf(?x,?z) ∨ ¬Child(?z,?y) ∨ Uncle(?x)

c. 1.Uncle(?x1) ∨ isBrotherOf(?x1,sk1)) 2. Uncle(?x2) ∨ Child(?z1,sk2)) 3. isBrotherOf(?x3,?z2) ∨ Child(?z2,?y1) ∨ Uncle(?x3)

d. 1.¬Uncle(?x1) ∧ isBrotherOf(?x1,sk1(?x1)) 2. ¬Uncle(?x2) ∧ Child(?z1,sk2(?x2)) 3. ¬isBrotherOf(?x3,?z2) ∧ ¬Child(?z2,?y1) ∧ Uncle(?x3)

8. Which of the following is/are true for FOL? (c, e) a. Resolution Refutation, Forward Chaining and Backward Chaining are complete. b. Forward Chaining and Backward Chaining are complete, Resolution Refutation is not. c. Resolution Refutation is complete, Forward Chaining and Backward Chaining are not. d. Forward Chaining and Resolution Refutaion are complete, Backward Chaining is not. e. Resolution Refutation is complete, Resolution in general is not.

9. Which axiom(s) of equality is/are used to derive 5=f(3) from f(3)=5 (b) a. Only reflexivity. b. Only symmetry. c. Only transitivity. d. Only substitution for functions. e. Reflexivity and symmetry. f. Symmetry and substtitution for functions.

10. What is the motivation behind the paramodulation rule? (a) a. It is a shortcut to perform substitutions avoiding the use of equality axioms. b. It is a conflict resolution strategy for resolving two long clauses. c. Make sure that the derivation is sound. d. None of the above.

11. How does the “set of support” strategy choose clauses in the resolution refutation method? (b) a. It prefers longer clauses to shorter clauses. b. It prefers a descendant of the negated goal as one of the clauses to be resolved. c. It prefers that both the clauses are derived from the given input set of clauses. d. It prefers smaller clauses to larger clauses for resolution.

12. A Horn clause is (c) a. a clause with only one positive literal. b. a clause with only one negative literal. c. a clause with at most one positive literal. d. a clause with at most one negative literal. e. a clause with at least one positive and one negative literal.

13. A positive definite Horn clause (a Prolog statement) (a,b,c,d) a. has exactly one positive literal. By definition. b. has at least one positive literal. If (a) is true then (b) is true. c. has at most one positive literal. If (a) is true then (c) is true. d. is an implication statement of the kind (A ∧ B ∧ C) ⊃ D. e. is an implication statement of the kind A ⊃ (B ∧ C ∧ D).

14. A negative Horn clause (a Prolog goal) (a,d) a. has at least one negative literal. b. has exactly one negative literal. c. can have zero negative literals. (false: must have at least one, and only negative literals) d. can have zero positive literals. (true by definition, has to have zero positive literals) e. has exactly one positive literal and more than one negative literals. f. none of the above.

15. What can one say about the two input clauses in a resolution step with Horn clauses? (a,c,f) a. Both can be positive definite clauses. True. For example (¬C ∨ D) and (¬B ∨ C) b. One must be a positive definite clause and the other must be a negative clause. Because (a) is true (b) must be false. c. One must be a positive definite clause and the other can be anything. d. Both can be negative clauses. False. Because you cannot resolve two negative clauses. e. It is not possible that both are positive. False (because option (a) is true) f. It is not possible that both are negative.

16. What can one say about the two input clauses in a SLD resolution step with Horn clauses? (a,e,f) a. Both can be positive definite clauses. True. If, for example, the goal is a negative literal, then its negation is a positive literal, which can resolve with another positive clause. For example, From {¬D, (¬C ∨ D), (¬B ∨ C)} derive ¬B Observe that the statement would be true for SLD resolution with positive definite Horn clauses (as in Prolog). b. One must be a positive definite clause and the other must be a negative clause. False, since (a) is true. c. One must be a positive definite clause and the other can be anything. True. Look at the derivation for the example in explanation for (a). d. Both can be negative clauses. False. Because you cannot resolve two negative clauses. e. It is not possible that both are positive. False because (a) is true. f. It is not possible that both are negative. True. Because you cannot resolve two negative clauses.

17. Given a set of clauses that are inconsistent, which of the following is/are true? (a,b,e,g) a. A null clause can always be derived using general resolution. b. If the input is in Horn clause form then a null clause can always be derived by general resolution. c. A null clause can be derived by general resolution only if the input is in Horn clause form. Counter example: From {(¬A ∧ B ∧ C), A, ¬B} derive C d. A null clause can always be derived using SLD resolution. Counter example: From: {(A ∨ ¬B), (¬A ∨ B), (A ∨ B)} derive (A ∧ B) e. If the input is in Horn clause form then a null clause can always be derived by SLD resolution. f. A null clause can be derived by SLD resolution only if the input is in Horn clause form. Counter example: From {(¬A ∧ B ∧ C), A, ¬B} derive C g. If a null clause can be derived using SLD resolution that it cen be derived by general resolution. h. If a null clause can be derived using general resolution, then it can be derived using SLD resolution. Counter example: From: {(A ∨ ¬B), (¬A ∨ B), (A ∨ B)} derive (A ∧ B)