q3-solutions - 6.045J/18.400J Automata Computability and...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
6.045J/18.400J: Automata, Computability and Complexity Prof. Nancy Lynch Quiz 3- Solutions May 2, 2007 Elena Grigorescu Please write your name in the upper corner of each page. This exam is open book. Problem Points Grade 1 20 2 12 3 8 4 20 5 20 6 20 Total 100 SQ3-1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Problem 1 : True, False, or Unknown (20 points). In each case, say whether the given statement is known to be TRUE, known to be FALSE, or currently not known either way. Full credit will be given for correct answers. If you include justification for your answers, you may obtain partial credit for incorrect answers. 1. True, False, or Unknown: P = NP. Unknown 2. True, False, or Unknown: 3 2 n is O (2 3 n ) . False 3. True, False, or Unknown: There exists a language that is not decidable in time 2 n but is decidable in time 8 n (where n is the length of the input). True ; by the Time Hierarchy Theorem (see section 9.1 of Sipser). 4. True, False, or Unknown: UHAMPATH, the Hamiltonian path problem for undirected graphs, is in coNP. (Here, UHAMPATH= {( G, s, t )| G is an undirected graph with a Hamiltonian path from s to t } ). (coNP is defined to be the set of languages whose complements are in NP.) Unknown ; UHAMPATH is NP-complete. If coNP=NP, then UHAMPATH is in coNP. If coNP negationslash = NP, then UHAMPATH is not in coNP. SQ3-2
Image of page 2
5. True, False, or Unknown: If UHAMPATH P, then P = coNP. True; UHAMPATH is NP-complete. If UHAMPATH P then P = NP = coNP 6. True, False, or Unknown: For every language A , P A negationslash = NP A . False. 7. True, False, or Unknown: If B is decidable in time 2 p ( n ) for some polynomial p , and A p B , then A is decidable in time 2 q ( n ) for some polynomial q . True 8. True, False, or Unknown: Suppose that B is decidable in time 2 p ( n ) by some nondeterministic Tur- ing machine, for some polynomial p , but is not decidable in time 2 q ( n ) by any deterministic Turing machine, for any polynomial q . Then P negationslash = NP . True ; Consider the language C = { x # 2 p ( n ) −| x | | x B } . Notice that C is decidable in polynomial time by a NTM but it is not decidable in polynomial time by any deterministic TM. SQ3-3
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Problem 2 : (12 points) The proof that SAT is NP-complete appears in Sipser’s book, p. 276-282. Part of the main construction involves constructing a formula φ move , which is expressed as the conjunction of formulas saying that 2 × 3 windows of the tableau are “legal”. For each of the following, state whether it represents a legal window. Assume that the underlying polynomial-time NTM is of the form ( Q, Σ , Γ , δ, q 0 , q acc , q rej ) , where Σ = { a, b, c, d } and Γ = { a, b, c, d, ⊔} . 1. Legal or not: a q 1 b a c q 2 , where ( q 2 , c, R ) δ ( q 1 , b ) . Legal. 2. Legal or not: a b c q 2 b c , where ( q 2 , d, R ) δ ( q 1 , a ) for some q 1 .
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern