The Boolean hierarchy and the polynomial hierarchy: a closer connection

It is shown that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH/sub 3/(k), where BH/sub 3/(k) is the kth level of the Boolean hierarchy over Sigma /sub 2//sup p/. This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. This results also points to some previously unexplored connections between the Boolean and query hierarchies of Delta /sub 2//sup p/ and Delta /sub 3//sup p/.<
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