The Polynomial‐Time Hierarchy and Polynomial Space

This chapter introduces the polynomial-time hierarchy of complexity classes 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. The chapter presents some natural complete problems for this hierarchy and for complexity classes PSPACE and EXP. The chapter extends polynomial-time Turing Reducibility to nondeterministic oracle TMs and study problems that are solvable in nondeterministic polynomial time relative to sets in NP. 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. The time complexity of a set-oracle NTM is also defined similar to that of a set-oracle DTM.