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Complements of Languages in NP. Osama Awwad Department of Computer Science Western Michigan University April 3, 2014. Complement classes. In general, if C is a complexity class co-C is the complement class, containing all complements of languages in C L  C implies (* - L)  co-C

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complements of languages in np

Complements of Languages in NP

Osama Awwad

Department of Computer Science

Western Michigan University

April 3, 2014

complement classes
Complement classes
  • In general, if C is a complexity class
  • co-Cis the complement class, containing all complements of languages in C
    • L  C implies (* - L)  co-C
    • (* - L) C implies L  co-C
  • Some classes closed under complement:
    • e.g. co-P = P
co p p
Co-P = P
  • Given a polynomial-time TM M for L, we can modify M to accept the complement of L as follows:
  • Make each accepting state of M a nonaccepting state from which there are no moves. Thus, if M accepts, the new TM (L) will halt without accepting.
  • Create a new state q, which is the only accepting state in the new TM. For each state-symbol combination that has no move, the new TM enters state q, whereupon it accepts and halts.
co np
Co-NP
  • NP (nondeterministic polynomial-time): class of problems for which, if the answer is yes, then there's a polynomial-size proof of that fact that you can check in polynomial time.
  • Co-NP: The set of languages whose complements are in NP
example
Example
  • Input: graph G=(V,E)
  • Does G have a Hamiltonian cycle?
          • Hamiltonian cycle: cycle visiting each vertex exactly once
  • If G has a Hamiltonian cycle then let us take a promised Hamiltonian cycle – we can check if this is indeed one in O(n) time!
  • Hamiltonian cycle is in NP
  • Does G have no Hamiltonian cycle?
slide6

Is it true that CoNP = NP?

x  L

Can we transform this machine:

x  L

x  L

x  L

qaccept

qreject

into this machine?

qaccept

qreject

np complete and co np
NP-Complete and Co-NP
  • Every language in P has its complement also in P, and therefore in NP.
  • We believe that none of the NP-complete problems have their complements in NP.

no NP-complete problem is in Co-NP.

  • We believe the complements of NP-complete problems, which are by definition in Co-NP, are not in NP.
np complete and co np9
NP-Complete and Co-NP
  • Theorem NP=co-NP if and only if the complement of some NP-complete problem is in NP.
proof
Proof
  • (Only if): Should NP and Co-NP be the same
    • Every NP-complete problem L, being in NP, is also in co-NP
    • The complement of a problem in Co-NP is in NP  the complement of L is in NP
  • (If)
    • Let P be an NP-complete problem
    • Suppose Pc is in NP
    • For each L in NP, there is a polynomial-time reduction of L to P
    • Lc is polynomial reducible to Pc

_ Lc is in NP