CS254 Fall 2011Practice Final

To study for the final 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 final is below. Here are some facts about the actual final: (a) It is comprehensive (b) It is closed book, closed notes. Nothing will be permitted on your desk except your pen (pencil) and test. (c) You should bring photo ID. (d) There will be more than one version of the test. Each version will be of comparable difficulty. (e) It is 10 problems, 6 problems will be on material since the midterm, four problems will come from the topics covered prior to the midterm. (f) Two problems will be exactly (less typos) off of the practice final, and one will be off of practice midterm

1. Suppose to AND, OR, and NOT, we added XOR. Explain how this would affect (if at all) the bounds given by Shannon's Theorem.
2. Give an `NC` circuit family whose member `n`th member, `C_n`, computes the parity of its `n` input bits.
3. Consider the language `{langle a, k, langle a_1, ... a_n rangle rangle | mbox(a is the kth smallest ) a_i}`. Show this is in `BPP`.
4. State Markov's Theorem and prove it.
5. Give the randomized p-time algorithm for 2SAT and explain how its running time is determined.
6. State Chernoff bounds.
7. Prove `BPP` is in `Sigma_2^p`.
8. Show `GNI` is in `IP`.
9. State the Set Lower Bound Protocol.
10. Explain how the Razborov Smolensky approximating polynomial for an `AC^0(3)` circuit is constructed.