Question

Show that if ,PH=PSPACEthen the polynomial time hierarchy has only finitely many distinct levels.

Short answer

The hierarchy in polynomial-size can be extended to only distinct finitely level if a sparse set in NP. Hence, it can be said that if PH=PSPACE, then the polynomial time hierarchy consists levels which are distinct and finite.

Step-by-step solution

Understanding the Polynomial space and time

• The hierarchy in polynomial-time exists between deterministically accepted languages of classes P in polynomial time and deterministically or non-deterministically accepted languages of classPSPACE in polynomial space.
• The minimum three levels of the hierarchy is if a relativization exists. The number of distinct levels, which are used to determine “low” and “high”, in the hierarchy of polynomial-time are, depends upon the perishing of the NP class. Now consider the facts of “low” and “high”, which is explained below:
• If there exist some i for which$\mathbf{\sum }_{\mathbf{i}}^{\mathbf{p}}\left(Y\right)\underline{\mathbf{\subset }}\mathbf{\sum }_{\mathbf{i}}^{\mathbf{p}}$ then a set E in NP is known as “low” and if there exists$\mathbf{\sum }_{\mathbf{i}\mathbf{+}\mathbf{1}}^{\mathbf{p}}\underline{\mathbf{\subset }}\mathbf{\sum }_{\mathbf{i}}^{\mathbf{p}}\left(z\right)$ for some i, then it is known as “high”.

Now checking the polynomial time hierarchy for PH

Suppose the union of the various classes of polynomial time hierarchy is defined as PH.

• There are two principal results exists. First one is based on the fact “either all or none (that is, every sparse set in PH is high or no one is low). Second one is that “No one sets will be extended high or each set of sparse will be extended high”.

• Simply, it can be said that “the hierarchy collapses in polynomial-time are the only way for simultaneous existence of high and low behaviour”.

• A disjoint set can be obtained by combining high sets and low sets. The reason behind it is “the existence of immeasurably many levels extended by the hierarchy in polynomial time and in NP there exist some sets which show neither low nor high.

Therefore, it is understood that the hierarchy in polynomial-size can be extended to only distinct finitely level if a sparse set in NP. So, we can say that the polynomial time hierarchy consists levels which are distinct and finite if PH=PSPACE.