Every polynomial-time 1-degree collapses iff P=PSPACE

Abstract

A set A is m-reducible (or Karp-reducible) to B if and only if there is a polynomial-time computable function f such that for all x, x in A if and only if f(x) in B. Two sets are 1-equivalent if each is m-reducible to the other by one-one reductions; p-invertible equivalent iff each is m-reducible to the other by one-one, polynomial-time invertible reductions; and p-isomorphic iff there is an m-reduction from one set to the other that is one-one, onto, and polynomial-time invertible. It is proved that the following statements are equivalent: (1) P=PSPACE. (2) Every two 1-equivalent sets are p-isomorphic. (3) Every two p-invertible equivalent sets are p-isomorphic. >