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.

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Received: 16 March 1995

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Russell, A., Sundaram, R. Symmetric alternation captures BPP. Comput. complex. 7, 152–162 (1998).

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  • Key words. Alternation; complexity classes; symmetric quantifiers; randomness.