Symmetric Alternation Captures BPP

We introduce the natural class SP 2 containing those languages which may be expressed in terms of two symmetric quanti ers. This class lies between
P 2 and P 2 P 2 and naturally generates a symmetric" hierarchy corresponding to the polynomial-time hierarchy. We demonstrate, using the probabilistic method, new containment theorems for BPP. We show that MA and hence BPP lies within SP 2 , improving the constructions of 10, 8 which show that BPP P 2 P 2 . Symmetric alternation is shown to enjoy two strong structural properties which are used to prove the desired containment results. We o er some evidence that SP 2 6= P 2 P 2 by demonstrating an oracle so that SP;O 2 6= P;O 2 P;O 2 assuming that the machines make only positive" oracle queries.

• Type: Journal
• Year: 1998
• Where: CC
• Authors: Alexander Russell, Ravi Sundaram
• Keywords: CC 1998, Natural Class Sp, Polynomial-time Hierarchy, Symmetric Quanti Ers, System Software