CS254 Spring 2017Practice Midterm

Studying for one of my tests does involve some memorization. I believe this is an important skill. Often people waste a lot of time and fail to remember the things they are trying to memorize. Please use a technique that has been shown to work such as the method of loci. Other memorization techniques can be found off the Wiki Page for Moonwalking with Einstein. Given this, to study for the midterm I would suggest you:

• Know how to do (by heart) all the practice problems.
• Go over your notes at least three times. Second and third time try to see how much you can remember from the first time.
• Go over the homework problems.
• Try to create your own problems similar to the ones I have given and solve them.
• Skim the relevant sections from the book.
• 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. Show: (a) `n^5=o(1.5^n)`. (b) `sum_{i=1}^n i^2 = Omega(n^3)`.
2. Suppose we had a variation on a k-tape TM, called a fill TM, which has a special state `q_{fill}`. When a fill TM enters the state `q_{fill}` the contents of its second tape, the tape squares between the start of tape symbol and the current tape head, are all changed to 1 (in 1 step). Explain how a fill TM could be simulated by a usual TM and give an estimate on the slowdown involved.
3. Prove that there exists a function that is uncomputable.
4. Briefly explain how the amortized tape shifting works in our `O(T log T)` Universal TM.
5. Suppose `P=NP`. Show for every `NP`-language `L` and verifier TM `M` for `L`, there is a polynomial-time TM `B` that on input `x in L` outputs a certificate for `x`.
6. Give a p-time reduction from dHAMPATH to TRAVELING SALESMAN.
7. Prove that if `EXP !=NEXP` then `P !=NP`.
8. Give with proof a clause-to-variable threshold above which any 4-SAT instance will almost surely be unsatisfiable.
9. Explain how the DPLL algorithm would find a satisfying assignment for the set of clauses `{{bar{1}, 2, 3, 4}, {bar{2}, 3}, {bar{3}, 4}, {bar{4}, 1}}`.
10. Give an oracle A under which `DQLIN^A != NQLIN^A` where NQLIN and DQLIN are respectively nondeterministic and deterministic quasi-linear time (as used in HW2).