One thing nice about space is that it keeps on going ...

—Willem de Kooning

We study complexity classes beyond the class *NP*. First, the polynomial-time hierarchy of complexity classes is introduced based on nondeterministic oracle machines. This hierarchy lies between the class *P* and the class *PSPACE*, and the class *NP* is the first level of the hierarchy. Characterizations of these complexity classes in terms of alternating quantifiers and alternating Turing machines (ATMs) are proven. Finally, we present some natural complete problems for this hierarchy and for complexity classes *PSPACE* and *EXP*.

We have defined in Chapter 2 the notions of polynomial-time Turing reducibility and oracle TMs, and have seen that many optimization problems, when formulated in the search problem form, are solvable in polynomial time relative to a set in *NP*. We now extend this notion to nondeterministic oracle TMs and study problems that are solvable in nondeterministic polynomial time relative to sets in *NP*.

A *nondeterministic (function-)oracle Turing machine* (*oracle NTM*) is an NTM equipped with an additional *query* tape and two additional states: the *query* state and the *answer* state. The computation of an oracle NTM is similar to that of an oracle DTM, except that at each nonquery state an oracle NTM can make a nondeterministic move. We require that the query step of the computation be a deterministic ...

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