Relativized worlds with an infinite hierarchy

We introduce the “Book Trick” as a method for using results about random oracles to create relativized worlds with an infinite polynomial-time hierarchy. We use it to show, for example, a single relativized world where • • one can find satisfying assignments to satisfiable formulae with only nonadaptive queries to an NP oracle, and • • the counting class SPP strictly contains the polynomial-time hierarchy, and • • the polynomial-time hierarchy is infinite.