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 (a) `n^3 = o(n!)`, (b) `n^3 = Omega(sum_(i=1)^n i^2)`, (c) `sum_(i=0)^(n-1) 2^i = O(2^n)`
2. Give a TM algorithm for computing palindrome.
3. Show carefully that `n^3` is time constructible.
4. Briefly explain the proof from class on how to simulate a TM with a `k` symbol alphabet using a TM that just has four symbols.
5. Prove the complement of HALT is undecidable.
6. Consider the problem of given a Boolean formula `F` and a truth assignment to its variables determining whether the output of `F` is `1`. Show this problem is in `P`.
7. Briefly explain how the amortized shifting works in our `O(T Log T)` universal TM simulation.
8. Prove that the verifier and nondeterministic time formulations of `NP` define same class.
9. Give a `p`-time reduction from `3SAT` to INDEPENDENT SET.
10. Prove that TAUTOLOGY is `coNP` complete.