CS254 Fall 2011Practice Midterm

To study for the midterm I would suggest you: (1) Know how to do (by heart) all the practice problems. (2) Go over your notes at least three times. Second and third time try to see how much you can remember from the first time. (3) Go over the homework problems. (4) Try to create your own problems similar to the ones I have given and solve them. (5) Skim the relevant sections from the book. (6) If you want to study in groups, at this point you are ready to quiz each other. The practice midterm is below. Here are some facts about the actual midterm: (a) It is closed book, closed notes. Nothing will be permitted on your desk except your pen (pencil) and test. (b) You should bring photo ID. (c) There will be more than one version of the test. Each version will be of comparable difficulty. (d) One problem (less typos) on the actual test will be from the practice test.

1. Prove `NTIME(n) stackrel(subset)(ne) co-NTIME(n^(1.5))`.
2. State and prove the nondeterministic time hierarchy theory.
3. Use closure properties of `NP` and the space hierarchy theorem to show that `SPACE(n) ne NP`.
4. Explain why `SAT_H` for the `H` we used in the proof of Ladner's Theorem is neither in `P` nor in `NP`.
5. State and prove Savitch's Theorem.
6. Prove `PATH` is in `NL`.
7. Assuming `PATH` is in `coNL` prove `NL=coNL`.
8. Prove `Sigma_i^p = cup_c Sigma_i TIME(n^c)`.
9. `TISP(n^(14), n^(2)) subseteq Sigma_2TIME(n^9)`.
10. Prove `NP^(SAT) = \Sigma_2^p`.