Every polynomial-time 1-degree collapses if and only if P = PSPACE

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 ∈ A if and only if f(x) ∈ B. Two sets are: • 1-equivalent if and only if each is m-reducible to the other by one-one reductions; • p-invertible equivalent if and only if each is m-reducible to the other by one-one, polynomial-time invertible reductions; and • p-isomorphic if and only if there is an m-reduction from one set to the other that is one-one, onto, and polynomial-time invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1-equivalent sets are p-isomorphic. (c) Every two p-invertible equivalent sets are p-isomorphic.