Download PDFOpen PDF in browserCurrent versionPH = PSPACEEasyChair Preprint no. 7481, version 26 pages•Date: October 3, 2022AbstractIn 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. The last can be done locally for each leaf/ branch/ iteration (keep in mind the algebraic normal form (ANF)), i.e. in polynomial time, since all arguments are specified. Relativization is defeated due to the wellknown fact: PH = PSPACE iff secondorder logic over finite structures gains no additional power from the addition of a transitive closure operator. Boolean algebra is finite. The exchanging quantifiers is possible only for finite possibilities of 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. Keyphrases: BQP, computational complexity, polynomial hierarchy, PSPACE, QBFs, Quantified Boolean Formula
