On Alternation and the Union Theorem

Under the assumption P= Σ 2 , we prove a new variant of the Union Theorem of McCreight and Meyer for the class Σ 2 . This yields a union function F which is computable in time F (n) for some constant c and satisfies P = DTIME(F ) = Σ2(F ) = Σ p 2 with respect to a subfamily (S̃i) of Σ2-machines. We show that this subfamily does not change the complexity classes P and Σ 2 . Moreover, a padding construction shows that this also implies DTIME(F ) = Σ2(F ). On the other hand, we prove a variant of Gupta’s result who showed that DTIME(t) ( Σ2(t) for time-constructible functions t(n). Our variant of this result holds with respect to the subfamily (S̃i) of Σ2-machines. We show that these two results contradict each other. Hence the assumption P= Σ 2 cannot hold.