COMS W3261 CS Theory
Lecture 19: The Classes P and NP

1. Time Complexity

2. The Classes P and NP

3. Polynomial-Time Reductions

4. NP-Complete Problems

5. Boolean Expressions

6. The Satisfiability Problem

7. Practice Problems

  1. Is the function nlog n polynomial or exponential, where log is to the base 2?
  2. Show that the function 2n2 is O(n2).
  3. Show that the function 2n2 + 3n + 4 is O(n2).
  4. Show that if f(n) and g(n) are polynomial functions, then f(g(n)) is a polynomial function.
  5. If M is a nondeterministic Turing machine of time complexity T(n), then show that M can be simulated by a deterministic Turing machine of time complexity O(2T(n)).
  6. Show that P and NP closed under the following operations:
    1. reversal
    2. union
    3. concatenation
    4. Kleene closure
  7. Show that P is closed under complementation.
  8. List all satisfying truth assignments for x ∧ (yx) ∧ (zy).

8. Reading Assignment



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