Editing NP (complexity) (section)

=== Machine-definition ===
Equivalent to the verifier-based definition is the following characterization: NP is the class of [[decision problem]]s solvable by a [[nondeterministic Turing machine]] that runs in [[polynomial time]]. That is to say, a decision problem <math> \Pi </math> is in NP whenever <math> \Pi </math> is recognized by some polynomial-time nondeterministic Turing machine <math> M </math> with an '''existential acceptance condition''', meaning that <math> w \in \Pi </math> if and only if some computation path of <math> M(w) </math> leads to an accepting state. This definition is equivalent to the verifier-based definition because a nondeterministic Turing machine could solve an NP problem in polynomial time by nondeterministically selecting a certificate and running the verifier on the certificate.  Similarly, if such a machine exists, then a polynomial time verifier can naturally be constructed from it.

In this light, we can define co-NP dually as the class of decision problems recognizable by polynomial-time nondeterministic Turing machines with an existential rejection condition.  Since an existential rejection condition is exactly the same thing as a '''universal acceptance condition''', we can understand the ''NP vs. co-NP'' question as asking whether the existential and universal acceptance conditions have the same expressive power for the class of polynomial-time nondeterministic Turing machines.