EECS 492 Midterm 2 Review Questions
Part 1
1. How many models are there for 𝐴 ⇔ 𝐵 ⇔ 𝐶?
2. How do you define a sound inference and a complete inference? What about a sound/complete
proof procedure?
3. What is model checking? What is its time and space complexity?
4. Is model checking sound and complete for propositional logic? Why?
5. Be familiar with inference rules: modus ponens, and-elimination, biconditional-elimination,
contrapositive, adjunction, addition and so on.
6. How does DeMorgan’s rule work?
7. What is monotonicity? Why is it important?
8. How does unit/full resolution rule work? Is it sound/complete?
9. Convert the following sentence to CNF 𝑝 ∧ 𝑞 ∨ 𝑟 ∧ 𝑠 ∨ ~𝑞 ∧ 𝑝 ∨ 𝑡
10. Prove the following through refutation 𝑃 ∨ 𝑄 ∧ ~𝑃 ∨ 𝑅
11. Prove by refutation resolution:
∀𝑥, 𝑦 𝑆𝑖𝑏𝑙𝑖𝑛𝑔𝑂𝑓 𝑥, 𝑦 ∧ 𝑀𝑎𝑙𝑒 𝑦 ⇒ 𝐵𝑟𝑜𝑡𝑕𝑒𝑟𝑂𝑓 𝑥, 𝑦
∀𝑥, 𝑦 𝑆𝑖𝑏𝑙𝑖𝑛𝑔𝑂𝑓 𝑥, 𝑦 ∧ ~𝑀𝑎𝑙𝑒 𝑦 ⇒ 𝑆𝑖𝑠𝑡𝑒𝑟𝑂𝑓 𝑥, 𝑦
∀𝑥, 𝑦 𝑆𝑖𝑏𝑙𝑖𝑛𝑔𝑂𝑓 𝑥, 𝑦 ⇒ 𝑆𝑖𝑏𝑙𝑖𝑛𝑔𝑂𝑓 𝑦, 𝑥
𝑀𝑎𝑙𝑒 𝐽𝑜𝑕𝑛
~𝑀𝑎𝑙𝑒 𝑀𝑎𝑟𝑦
𝑆𝑖𝑏𝑙𝑖𝑛𝑔𝑂𝑓 𝐽𝑜𝑕𝑛, 𝑀𝑎𝑟𝑦
Prove: 𝐵𝑟𝑜𝑡𝑕𝑒𝑟𝑂𝑓(𝑀𝑎𝑟𝑦, 𝐽𝑜𝑕𝑛)
12. Prove by forward-chaining:
𝐵𝑢𝑓𝑓𝑎𝑙𝑜 𝑥 ∧ 𝑃𝑖𝑔 𝑦 ⇒ 𝐹𝑎𝑠𝑡𝑒𝑟 𝑥, 𝑦
𝑃𝑖𝑔 𝑦 ∧ 𝑆𝑙𝑢𝑔 𝑧 ⇒ 𝐹𝑎𝑠𝑡𝑒𝑟 𝑦, 𝑧
𝐹𝑎𝑠𝑡𝑒𝑟 𝑥, 𝑦 ∧ 𝐹𝑎𝑠𝑡𝑒𝑟 𝑦, 𝑧 ⇒ 𝐹𝑎𝑠𝑡𝑒𝑟 𝑥, 𝑧
𝐵𝑢𝑓𝑓𝑎𝑙𝑜 𝐵𝑜𝑏
𝑃𝑖𝑔 𝑃𝑎𝑡
𝑆𝑙𝑢𝑔 𝑆𝑡𝑒𝑣𝑒
Prove: 𝐹𝑎𝑠𝑡𝑒𝑟(𝐵𝑜𝑏, 𝑆𝑡𝑒𝑣𝑒)
⇒ 𝑄∨𝑅
13. Prove by backward-chaining:
𝑃𝑖𝑔 𝑦 ∧ 𝑆𝑙𝑢𝑔 𝑧 ⇒ 𝐹𝑎𝑠𝑡𝑒𝑟 𝑦, 𝑧
𝑆𝑙𝑖𝑚𝑦 𝑧 ∧ 𝐶𝑟𝑒𝑒𝑝𝑠 𝑧 ⇒ 𝑆𝑙𝑢𝑔 𝑧
𝑃𝑖𝑔 𝑃𝑎𝑡 ∧ 𝑆𝑙𝑖𝑚𝑦 𝑆𝑡𝑒𝑣𝑒 ∧ 𝐶𝑟𝑒𝑒𝑝𝑠 𝑆𝑡𝑒𝑣𝑒
Prove: 𝐹𝑎𝑠𝑡𝑒𝑟(𝑃𝑎𝑡, 𝑆𝑡𝑒𝑣𝑒)
14. Translate numbers like zero, exactly one/two, at least one/two, at most one/two, only … is, all
but, some, only some, only A’s are B’s, no A’s are B’s, time, place-related English sentences into
first order logic.
15. Unify
𝑃 𝐴, 𝐵, 𝐵 , 𝑃 𝑥, 𝑦, 𝑧
𝑃 𝐴, 𝐵 , 𝑃 𝑥, 𝑥
𝑃 𝑄 𝑎 , 𝑏 , 𝑃(𝑏, 𝑎)
16. Use progressive and regressive planning to solve Task 4 in PS3.