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Hw11 - Homework 11  CISC 303 Timo Kötzing([email protected]

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Unformatted text preview: Homework 11  CISC 303 Timo Kötzing ([email protected]) Friday, May 8. Friday, May 15. Handed out: Due Date: Problem 1. (12 points) (i) Prove informally that L0 = {x|y | x2 + 15 ≤ y 300 − x3 } is decidable. (ii) Prove informally that L1 = {code(M ) | M is TM and there are M TM and x ∈ N : ϕM (x) = code(M )} is semi-decidable. (iii) Show that L2 = {code(M ) | M is TM and ∀x ∈ N : ϕM (x)↑} is not semi-decidable. (4 points) Let F be a nite set. Prove informally (using reduction ) that L = {code(M) | M TM and ∀x ∈ F : ϕM (x)↓} is not decidable. Problem 2. Problem 3. (8 points) (i) Let M be a total TM. Show that L = {x | ∃y ∈ N : ϕM (x, y ) = 1} is a semi-decidable set. (ii) This is an extra credit problem. Let L be a semi-decidable set. Show that there is a total TM M such that L = {x | ∃y ∈ N : ϕM (x, y ) = 1}. (8 points) Think up any number of problems about TMs, decidable languages and/or semi-decidable languages that are not solved in the lecture notes, and solve them. You get points depending on how hard the problems you chose are, on the correctness of your arguments, and on the number of your problems. For full credit (8pts), you need to do roughly as much as for any other 8pts problem. However, you may do as many as you want. The limit of points you may get for this problem is 24pts. If you have trouble nding nice questions, look into the book, or on the Internet (you have to do proofs in our style, though). You may ask me for concrete questions in email, if you cannot think of any yourself, or if your own questions are too hard. However, if you don't ask me for problems, you will receive 2 extra points. You may ask me whether your choice of problems is good (and you can ask me about how many points I'd give you), and still get the 2 extra points (but you don't have to). Problem 4. 1 ...
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