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CS254 Fall 2006Practice Midterm 2

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.

[Student generated midterm solutions-PDF]

1. Argue directly that n2 is a proper complexity function.

2. We showed in class that Hf(n) ∉ TIME(⌊n/2⌋). Explain where the factor of 2 comes from.

3. Let G be a graph consisting of 8 vertices arranged in a line. Let one end of the line be x and the other y. Explain in detail what nodes would be considered, what would be stored, etc, if we ran Savitch's algorithm for reachability on this graph.

4. Prove that if SAT were in NL then NL=NP. If you make use of the proposition from class please include its proof.

5. Prove if NL=NP then NL=coNP. You can cite theorems from class for this problem without their proof.

6. Recall a NAND(x,y) gate computes ¬(x ∧ y). Consider the circuit value problem where the only non-input gates are NAND gates. Show this is P-complete.

7. In class we proved the following: Let L be a language. L is in NP iff there is a polynomially decidable and polynomially balanced (by |x|k for some k) relation R, such that L={x | ∃y, |y|≤|x|k ∧ (x,y) ∈ R}. Prove the result still holds if we restrict R to be decidable using logarithmic space.

8. Define what an independent set in a graph is. Then show the problem of determining whether a graph has an independent set of size greater than or equal to k is NP-complete.

9. Show coNP ⊆ PP.

10. Prove all languages in P have polynomial size circuits.