Symmetric alternation captures BPP

We introduce the natural class ${\bf\,S}^P_2$ containing those languages that may be expressed in terms of two symmetric quantifiers. This class lies between $\Delta^P_2$ and $\Sigma^P_2\,\cap\,\Pi^P_2$ and naturally generates a “symmetric” hierarchy corresponding to the polynomial-time hierarchy. We demonstrate, using the probabilistic method, new containment the theorems for BPP. We show that MA (and hence BPP) lies within ${\bf\,S}^P_2$ , orems for BPP. We show that MA (and hence BPP) lies within ${\bf\,S}^P_2$ , improving the constructions of Sipser and Lautemann which show that ${\bf BPP}\subseteq\,\Sigma^P_2\,\cap\,\Pi^P_2$ . Symmetric alternation is shown to enjoy two strong structural properties which are used to prove the desired containment results. We offer some evidence that ${\bf S}^P_2\,\neq\,\Sigma^P_2\,\cap\,\Pi^P_2$ by constructing an oracle O such that ${\bf S}^{P,O}_2\,\neq\,\Sigma^{P,O}_2\,\cap\,\Pi^{P,O}_2$ , assuming that the machines make only “positive” oracle queries.