On Alternation and the Union Theorem

Mathias Hauptmann

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arXiv · 1602.04781

On Alternation and the Union Theorem

Mathias Hauptmann

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Abstract

Under the assumption $P=\Sigma_2^p$, we prove a new variant of the Union Theorem of McCreight and Meyer for the class $\Sigma_2^p$. This yields a union function $F$ which is computable in time $F(n)^c$ for some constant $c$ and satisfies $P=DTIME(F)=\Sigma_2(F)=\Sigma_2^p$ with respect to a subfamily $(\tilde{S}_i)$ of $\Sigma_2$-machines. We show that this subfamily does not change the complexity classes $P$ and $\Sigma_2^p$. Moreover, a padding construction shows that this also implies $DTIME(F^c)=\Sigma_2(F^c)$. On the other hand, we prove a variant of Gupta's result who showed that $DTIME(t)\subsetneq\Sigma_2(t)$ for time-constructible functions $t(n)$. Our variant of this result holds with respect to the subfamily $(\tilde{S}_i)$ of $\Sigma_2$-machines. We show that these two results contradict each other. Hence the assumption $P=\Sigma_2^p$ cannot hold.

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Mathias Hauptmann. 2016-02-15. On Alternation and the Union Theorem. https://arxiv.org/abs/1602.04781

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