Polynomial hierarchy

In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH.
Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) which ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level.
Definitions.
There are multiple equivalent definitions of the classes of the polynomial hierarchy.
Oracle definition.
For the oracle definition of the polynomial hierarchy, define
where P is the set of decision problems solvable in polynomial time. Then for i ≥ 0 define
where formula_5 is the set of decision problems solvable in polynomial time by a Turing machine augmented by an oracle for some complete problem in class A; the classes formula_6 and formula_7 are defined analogously. For example, formula_8, and formula_9 is the class of problems solvable in polynomial time with an oracle for some NP-complete problem.
Quantified boolean formulae definition.
For the existential/universal definition of the polynomial hierarchy, let be a language (i.e. a decision problem, a subset of {0,1}*), let be a polynomial, and define
where formula_11 is some standard encoding of the pair of binary strings "x" and "w" as a single binary string. "L" represents a set of ordered pairs of strings, where the first string "x" is a member of formula_12, and the second string "w" is a "short" (formula_13) witness testifying that "x" is a member of formula_12. In other words, formula_15 if and only if there exists a short witness "w" such that formula_16. Similarly, define
Note that De Morgan's laws hold: formula_18 and formula_19, where "L"c is the complement of "L".
Let be a class of languages. Extend these operators to work on whole classes of languages by the definition
Again, De Morgan's laws hold: formula_22 and formula_23, where formula_24.
The classes NP and co-NP can be defined as formula_25, and formula_26, where P is the class of all feasibly (polynomial-time) decidable languages. The polynomial hierarchy can be defined recursively as
Note that formula_30, and formula_31.
This definition reflects the close connection between the polynomial hierarchy and the arithmetical hierarchy, where R and RE play roles analogous to P and NP, respectively. The analytic hierarchy is also defined in a similar way to give a hierarchy of subsets of the real numbers.
Alternating Turing machines definition.
An alternating Turing machine is a non-deterministic Turing machine with non-final states partitioned into existential and universal states. It is eventually accepting from its current configuration if: it is in an existential state and can transition into some eventually accepting configuration; or, it is in a universal state and every transition is into some eventually accepting configuration; or, it is in an accepting state.
We define formula_32 to be the class of languages accepted by an alternating Turing machine in polynomial time such that the initial state is an existential state and every path the machine can take swaps at most "k" – 1 times between existential and universal states. We define formula_33 similarly, except that the initial state is a universal state.
If we omit the requirement of at most "k" – 1 swaps between the existential and universal states, so that we only require that our alternating Turing machine runs in polynomial time, then we have the definition of the class AP, which is equal to PSPACE.
Relations between classes in the polynomial hierarchy.
The union of all classes in the polynomial hierarchy is the complexity class PH.
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 formula_37, or if any formula_38, then the hierarchy "collapses to level k": for all formula_39, formula_40. In particular, we have the following implications involving unsolved problems:
Relationships to other classes.
The polynomial hierarchy is an analogue (at much lower complexity) of the exponential hierarchy and arithmetical hierarchy.
It is known that PH is contained within PSPACE, but it is not known whether the two classes are equal. One useful reformulation of this problem is that PH = PSPACE if and only if second-order logic over finite structures gains no additional power from the addition of a transitive closure operator.
If the polynomial hierarchy has any complete problems, then it has only finitely many distinct levels. Since there are PSPACE-complete problems, we know that if PSPACE = PH, then the polynomial hierarchy must collapse, since a PSPACE-complete problem would be a formula_42-complete problem for some "k".
Each class in the polynomial hierarchy contains formula_43-complete problems (problems complete under polynomial-time many-one reductions). Furthermore, each class in the polynomial hierarchy is "closed under formula_43-reductions": meaning that for a class in the hierarchy and a language formula_45, if formula_46, then formula_47 as well. These two facts together imply that if formula_48 is a complete problem for formula_49, then formula_50, and formula_51. For instance, formula_52. In other words, if a language is defined based on some oracle in , then we can assume that it is defined based on a complete problem for . Complete problems therefore act as "representatives" of the class for which they are complete.
The Sipser–Lautemann theorem states that the class BPP is contained in the second level of the polynomial hierarchy.
Kannan's theorem states that for any "k", formula_53 is not contained in SIZE(nk).
Toda's theorem states that the polynomial hierarchy is contained in P#P.