Paper 2022/929

PH = PSPACE

Abstract

In this paper it is shown that PSPACE is equal to 4th level in the polynomial hierarchy. A lot of important consequences are also deduced. True quantified Boolean formula is indeed 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 is to skolemize, and then use additional formulas from the second 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" in the paper 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. Boolean algebra is finite. 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 Polynomial Hierarchy is infinite relative to a random oracle with probability 1.