Practice Midterm Exam 2 - Computability and Complexity | COMPSCI 172, Exams of Computer Science

Material Type: Exam; Class: Computability and Complexity; Subject: Computer Science; University: University of California - Berkeley; Term: Fall 2009;

Typology: Exams

2010/2011

Uploaded on 05/23/2011

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U.C. Berkeley CS172: Automata, Computability and Complexity Practice Midterm II
Professor Luca Trevisan November 4, 2009
Practice Midterm II
1. Define the language
L:= { hMi: for all strings x,M(x) halts within |x|2steps }.
Show that Lis not recognizable but Lis recognizable.
2. Define the language
L:= { hx, yi:K(x)> K(y)}.
Show that Lis not recognizable.
3. Prove that the class of NP -complete languages is not closed under union and intersection.
That is
Show that there are languages A,Bwhich are NP complete but such at ABis not
N P -complete.
Show that there are languages A,Bwhich are NP -complete but such that ABis not
N P -complete.
[Hint: recall that and Σcannot be NP-complete]
4. Define the problem
CLI QU E 1
2:= { hG, ki:Gis an undirected graph with a clique of size at least |V|/2 }.
Show that this problem is NP-complete.
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U.C. Berkeley — CS172: Automata, Computability and Complexity Practice Midterm II Professor Luca Trevisan November 4, 2009

Practice Midterm II

  1. Define the language L := { 〈M 〉 : for all strings x, M (x) halts within |x|^2 steps }. Show that L is not recognizable but L is recognizable.
  2. Define the language L := { 〈x, y〉 : K(x) > K(y) }. Show that L is not recognizable.
  3. Prove that the class of N P -complete languages is not closed under union and intersection. That is - Show that there are languages A, B which are N P complete but such at A ∪ B is not N P -complete. - Show that there are languages A, B which are N P -complete but such that A ∩ B is not N P -complete. [Hint: recall that ∅ and Σ∗^ cannot be NP-complete]
  4. Define the problem CLIQU E 12 := { 〈G, k〉 : G is an undirected graph with a clique of size at least |V |/2 }. Show that this problem is NP-complete.