Homework 9 Due Thursday, Dec' 10

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Homework 9Due Thursday, Dec. 10

• Problem 1. Draw a diagram showing the various
  classes of languages that we have discussed and
  alluded to in terms of which class contains which
  other classes. Indicate (by words if necessary)
  which containments are not known to be strict.
• Note this is an open-ended question, but at a
  minimum include the Kleene hierarchy, the
  polynomial-time hierarchy, PSPACE, NP-complete
  languages, and regular languages

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Problem 2 State the time hierarchy theorem
(consult Sipser). Show the error in the
following argument that NP PProof by
contradiction. Assume NP P. Then SAT is in P.
Therefore there exists a value k such that SAT is
in TIME(nk). Because every language in NP is
polynomial-time reducible to SAT, we know that P
is a subset of TIME(nk). Yet the time hierarchy
theorem tells us that TIME(nk1) contains a
language that is not in TIME(nk). Therefore our
assumption that NP P must be false.
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Problem 3 Define uniform family of circuits
and NC (consult Sipser). Problem 4
Define trapdoor function (consult Sipser).