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Unformatted text preview: Homework 11 CISC 303 Timo Kötzing (tkoe@udel.edu) Friday, May 8.
Friday, May 15. Handed out:
Due Date:
Problem 1. (12 points) (i) Prove informally that L0 = {xy  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
semidecidable.
(iii) Show that L2 = {code(M )  M is TM and ∀x ∈ N : ϕM (x)↑} is not semidecidable.
(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 semidecidable set.
(ii) This is an extra credit problem. Let L be a semidecidable 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 semidecidable 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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This note was uploaded on 02/01/2012 for the course CISC 303 taught by Professor Carberry,m during the Spring '08 term at University of Delaware.
 Spring '08
 Carberry,M

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