Another Proof That $\mathcal{BPP}\subseteq \mathcal{PH}$ (and More)

Abstract

We provide another proof of the Sipser–Lautemann Theorem by which ${\cal BPP}\subseteq{\cal MA}$ ($\subseteq{\cal PH}$). The current proof is based on strong results regarding the amplification of ${\cal BPP}$, due to Zuckerman (1996). Given these results, the current proof is even simpler than previous ones. Furthermore, extending the proof leads to two results regarding ${\cal MA}$: ${\cal MA}\subseteq{\cal ZPP}^{\cal NP}$ (which seems to be new), and that two-sided error ${\cal MA}$ equals ${\cal MA}$. Finally, we survey the known facts regarding the fragment of the polynomial-time hierarchy that contains ${\cal MA}$.