- 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: 30 points Q6: 20 points Q7: 30 points Total: 120 points

**Question 1 (10):***(True/false with justification)*The language {(n,S,k): S is a set of binary strings each of length n, and there exists a boolean circuit C with k or fewer gates such that for any string w of length n, C(w) = 1 if and only if w ∈ S} is Turing decidable.**Question 2 (10):***(True/false with justification)*If G is any context-free language with terminal alphabet {0,1}, there exists a family of boolean circuits {C_{n}: n ≥ 0} such that for any string w of any length n, w ∈ L(G) if and only if C_{n}(w) = 1.**Question 3 (10):***(True/false with justification)*Let EQ_{CFG}be the set of all pairs (G_{1}, G_{2}) of context-free grammars such that L(G_{1}) = L(G_{2}). Then EQ_{CFG}is Turing recognizable.**Question 4 (10):***(True/false with justification)*The language ab^{*}a, with alphabet {a,b}, has a four-state minimal DFA.**Question 5 (20):**Recall that a formula is in k-CNF if it consists of the AND of one or more clauses, where each clause is the AND of exactly k literals. (The literals in a clause need not be distinct -- the clause x_{2}∨ (¬x_{3}) ∨ x_{2}could legally occur in a 3-CNF formula.) The language k-SAT is defined to be the set of k-CNF formulas that are satisfiable.- (a,10) Prove that the language 7-SAT is NP-complete. (Don't forget to show that 7-SAT is in NP.)
- (b,10) Prove that if the language 1-SAT is NP-complete, then NP equals co-NP. (Remember that co-NP is the set of all languages whose complements are in P.)

**Question 6 (20):**A directed graph is said to be**strongly connected**if for any two vertices u and v, there is a directed path from u to v. The**out-degree**of a directed graph is the maximum number of directed edges coming out of any vertex. We define 1-STR-CONN to be the set of directed graphs of out-degree 1 that are strongly connected. Prove that 1-STR-CONN is in the class L = DSPACE(log n).**Question 7 (40):**If N is any kind of nondeterministic machine taking an input string w, define the language Z(N) to be the set of all w such that N eventually accepts on**every**computation path with input w. (If there is any path of N on w that fails to halt, then w is not in Z(N).)- (a,10) Prove that if N is an NFA, then Z(N) is a regular language. (Note that for an NFA, a "path on input w" must read the entire string w.)
- (b,10) Prove that if N is a nondeterministic Turing machine, then Z(N) is Turing recognizable.
- (c,10) Give an example of a nondeterministic Turing machine N such that Z(N) is not Turing decidable. Justify your answer.

Last modified 16 May 2008