Complements of Languages in NP

Osama Awwad

Department of Computer Science

Western Michigan University

April 20, 2023

Complement classes

In general, if C is a complexity class co-C is the complement class, containing

all complements of languages in CL C implies (* - L) co-C(* - L) C implies L co-C

Some classes closed under complement:e.g. 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

NPNP (nondeterministic polynomial-time): (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-NPCo-NP: The set of languages whose complements are in NP

• Does G have nono Hamiltonian cycle?

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

qaccept qreject

x L x L

qacceptqreject

x Lx L

Can we transform this machine:

into this machine?

Is it true that CoNP = 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-NP

NP-Complete and Co-NP

Co-NP NP

P

NPCCo-NPC

NP-Complete and Co-NP

Theorem NP=co-NP if and only if the complement of some NP-complete problem is in NP.

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