  Downward translation of equality refers to cases where a collapse of some
pair of complexity classes would induce a collapse of some other pair of
complexity classes that (a priori) one expects are smaller. Recently, the first
downward translation of equality was obtained that applied to the polynomial
hierarchy-in particular, to bounded access to its levels [cs.CC/9910007]. In
this paper, we provide a much broader downward translation that extends not
only that downward translation but also that translation's elegant enhancement
by Buhrman and Fortnow. Our work also sheds light on previous research on the
structure of refined polynomial hierarchies, and strengthens the connection
between the collapse of bounded query hierarchies and the collapse of the
polynomial hierarchy.
