# Practice for Final Exam

### Directions:

• Answer the problems on the exam pages.
• There are seven problems, for 120 total points. Probable scale is somewhere around A=105, C=70.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first four problems are true/false, with five points for the correct boolean answer and up to five for a convincing justification (proof, counterexample, quotation of result from lecture, etc.)

```  Q1: 10 points
Q2: 10 points
Q3: 10 points
Q4: 10 points
Q5: 20 points
Q6: 20 points
Q7: 40 points
Total: 120 points
```

• Question 1 (10): (True/false with justification) Every language in the class L (or DSPACE(log n)) is context-free.

• Question 2 (10): (True/false with justification) The language PATH = {(G,s,t): G is a directed graph and G has a path from s to t} has polynomial-size circuits.

• Question 3 (10): (True/false with justification) If A and B are two languages where A ≤m B and B is NP-complete, then A is Turing decidable.

• Question 4 (10): (True/false with justification) If X is a language that has polynomial-size circuits, i.e., for every n there is a circuit Cn that decides membership in X for inputs of size n, and size(Cn) = nO(1), then X must be decidable.

• Question 5 (20): Let ELOGSPACE be the language {M: M is a Turing machine that uses O(log n) space on inputs of size n such that L(M) is not empty. Prove that ELOGSPACE is not Turing decidable.

• Question 6 (20): Recall that ECFG is the language {G: G is a context-free grammar and L(G) is not empty}. Prove that ECFG is Turing decidable. (Note: This is a result proved in Sipser.)

• Question 7 (40): We define an instance of the box problem to be a natural number n (with n ≥ 3), a finite alphabet Σ, an assignment of letters in Σ to all the border cells of an n by n box of cells, and a set of rules indicating exactly which 3 by 3 subboxes are allowed. (Thus a set of rules is a subset of the set Σ9 indicating the legal 3 by 3 boxes.) A box problem instance is called feasible if there exists an assignment of letters in Σ to all the n2 cells such that border cells are assigned as indicated and all the 3 by 3 subboxes are legal. The language BOX is {b: b is a feasible box problem instance}.

• (a,10) Prove that for any fixed n, the set of (strings denoting) feasible n by n boxes is a regular language.
• (b,10) Prove that given a set of rules and an assignment of letters to all the cells of an n by n box, we can check whether the assignment follows the rules using O(log n) space.
• (c,20) Either prove that BOX is in P or prove that BOX is NP-complete. (Extra credit for proving both.)