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CS254 Fall 2006Practice Final

To study for the final 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 final: (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) It is comprehensive: There will be two problems from the material up to Midterm 1 (one of these will be from Practice Midterm 1); there will be two problems from the material up to Midterm 2 (one of these will be from Practice Midterm 2); the reamining six problems will be on material since Midterm 2 (one of these will be from the Practice Final).

[Student generated final solutions-PDF]

1. Give a polynomial size circuit family for the language consisting of strings over {0,1} with exactly two 1's. Show your language is logspace uniform.

2. Give a language in P/poly which is not in P.

3. Why does the argument that BPP is contained in P/poly not show that BPP is contained P/n2?

4. Give the definition of what a one-way function is. Try to give an example of a function that could potentially be one-way.

5. Let UNIQUE-SAT be the language consisting of boolean formulas with exactly one satisfying assignment. Why is this class not likely to be complete for UP?

6. Show that if there is a sparse language that is NP-hard then there is a sparse language which is NP-complete.

7. Give an oracle B with respect to which LB is equal to PB

8. Give an oracle B for which there is a sparse language in NPB-PB

9. Give a monotone circuit family for the clique language (it can be of exponential size).

10. What is a sunflower? How is it used in the clique lower bounds from class?