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-isumorphic 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.