1. The Polynomial – Time Hierarchy
Lecture 9 - Theory of Computation By Dexter Kozen
Presented By: JOEL S. GRACIA

2. Web Definition:
In computational complexity theory, the polynomial hierarchy is a hierarchy of complexity classes that generalize the classes P, NP and co-NP to oracle machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic.
Source:

3. The Polynomial – Time Hierarchy
The polynomial-time hierarchy (PH) is a hierarchy of complexity classes lying over P and inside PSPACE. It was first identified by Stockmeyer [117]. It is most easily defined in terms of alternating polynomial-time-bounded TMs, although it was originally defined in terms of oracle Turing machines. The hierarchy is analogous in many ways to the arithmetic hierarchy, which we introduce later in Lecture 35. However, unlike the arithmetic hierarchy, it is not known whether PH is strict.

4. Con.
In this lecture and the next, we define PH in two different ways, in terms of ATMs and oracle TMs, and prove the equivalence of the two definitions. We also give generic ≤log m -complete problems for each level of the hierarchy.

5. Definition of PH in Terms of ATMs
Informally, a ΣK-machine (respectively, ΠK - machine) is an ATM without negations that on any input makes at most k alternations of ∨- and ∧- configurations along any computation path, beginning with ∨ (respectively, ∧).

6. Definition 9.1 A Σk-machine is an ATM such that on any input, every computation path can be divided into contiguous intervals such that (i) in any interval, all configurations are either ∨-configurations or all are ∧-configurations; (ii) there are at most k intervals; and (iii) the first interval consists of ∨-configurations. A ΠK -machine is similar, except we change (iii) to: (iii) the first interval consists of ∧-configurations. A Σ1-machine is just a nondeterministic TM. By convention, Σ0-and Π0 - machines are deterministic TMs.

7. Definition 9.2 The complexity classes and are defined as
{L(M) | M is a polynomial-time-bounded Σk -machine},
{L(M) | M is a polynomial-time-bounded Πk -machine},
Thus =
NP, ,
=co-NP, and =
= P

8. Definition 9.3
Here ∼A denotes the complement of A.
Proof. The first equation is obtained by interchanging ∨- and ∧-states. This gives the dual machine, which accepts the complement of the set accepted by the original machine (provided the machine halts along all computation paths, which it does in this case because it is polynomial time- bounded). The second inclusion is immediate from the definition. The third inclusion follows from the fact that PSPACE = APTIME, in which there are no restrictions on the number of alternations.
It is not known whether any of the inclusions in Lemma 9.3 are strict.

9. Generic Complete Problems
We can define generic problems that are complete for the various levels of the polynomial-time hierarchy. Define
where M is any ATM and is M modified so as to halt on any computation path that tries to alternate between ∨ and ∧ states more than k

10. times, beginning with an ∨, or tries to take more than m steps
times, beginning with an ∨, or tries to take more than m steps. In other words, the computation tree of on any input is essentially the same as that of M, except that it is artificially truncated to depth m and k alternations. Any leaves of the computation tree of
that are not leaves of the computation tree of M are either accept or reject configurations of , according as the corresponding configuration of M is an ∧-configuration or an ∨-configuration, respectively.