Relations Between Classes in The Polynomial Hierarchy
The definitions imply the relations:
Unlike the arithmetic and analytic hierarchies, whose inclusions are known to be proper, it is an open question whether any of these inclusions are proper, though it is widely believed that they all are. If any, or if any, then the hierarchy collapses to level k: for all, . In particular, if P = NP, then the hierarchy collapses completely.
The union of all classes in the polynomial hierarchy is the complexity class PH.
Read more about this topic: Polynomial Hierarchy
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