  In this paper we show that PSPACE is equal to 4th level in the polynomial
hierarchy. We also deduce a lot of important consequences.
  True quantified Boolean formula is a generalisation of the Boolean
Satisfiability Problem, where determining of interpretation that satisfies a
given Boolean formula is replaced by existence of Boolean functions that makes
a given QBF to be tautology. Such functions are called the Skolem functions.
  The essential idea of the proof is to show that for any quantified Boolean
formula $\phi$ we can obtain a formula $\phi'$ which is in the 4th level of the
polynomial hierarchy, no more than polynomial in the size of a given $\phi$,
such that the truth of $\phi$ can be determined from the truth of $\phi'$. The
idea is to skolemize, and then use additional formulas from the 2nd level of
the polynomial hierarchy inside the skolemized prefix to enforce that the
skolem variables indeed depend only on the universally quantified variables
they are supposed to. However, some dependence is lost when the quantification
is reversed. It is called "XOR issue" because the functional dependence can be
expressed by means of an XOR formula. Thus, it is needed to locate these XORs,
but there is no need to locate all chains with XORs: any chain includes a XOR
of only two variables. The last can be done locally in each iteration (keep in
mind the algebraic normal form (ANF)), when all arguments are specified, i.e.
as a polynomial subroutine.
  Relativization is defeated due to the well-known fact: PH = PSPACE iff
second-order logic over finite structures gains no additional power from the
addition of a transitive closure operator. The exchange is possible due to
finite possibilities for arguments. So, the theorems with oracles are not
applicable since a random oracle is an arbitrary set. And that's why PH is
infinite relative to a random oracle with probability 1.
