1. Hint of final exams jinnjy

2. Outline Hint of final 2007 (1/10/2008)

3. Hint of Problem 1 From ex 4.17, L is RE iff there is a recursive language N exists s.t. L={x | there exists y s.t. is in N} Yes, L is in RE. There exists a language N={ | c is an accepting computation history for M on w}

4. Hint of Problem 2 (a) HALT TM is in RE since we can use universal TM to make the simulation. (b) For every RE language B, there must exist a TM M B s.t. L(M B )=B. We define a mapping f B (w)=. B can be reduced to L u by using f B.

5. Hint of Problem 2 Assume we have R s.t. L(R)=HALT TM, we can reduce L u to HALT TM by construct S s.t. S=L u : S=On input : Run R on If R accepts, simulate M on w until it halts. If M accepts, accepts; otherwise, reject. From all above, every RE language can be reduced to HALT TM.

6. Hint of Problem 3 仔細檢查條件 A 和 B 的邏輯會發現，如果符合條 件 A 的 TM 就不會符合條件 B ，反之亦然 由上 (b), (c) 答案是 No 而 (d) 是 Yes ，因為沒有 TM 可以同時符合條件 A 和 B ， (d) 所提的 langauge 是 empty 所以它的 description 是 trivial property 。 (e) 的答案是 Yes ，因為所有的 TM 不是符合條件 A 就是符合條件 B ，它的 description 是 trivial property

7. Hint of Problem 4 Def 1. A RE language is a language for which there is a TM to accept it. Def 2. A language C is RE iff there is a recursive language D s.t. C={x | there is a y s.t. in D} Proof: Use the answer of 4.17 ，

8. Hint of Problem 5 NP Def 2. NP=the class of languages that can be decided by a polynomial time nondeterminstic TM. NP-Complete: A language L is NP-complete iff 1. L is in NP 2. Every A in NP can be reduced to L in polynomial time. If a language if NP-complete, it can be decided by a polynomial time NTM, it is decidable.