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.
Stephen A. Fenner. Stuart A. Kurtz. James S. Royer. "Every polynomial-time 1-degree collapses if and only if P = PSPACE." J. Symbolic Logic 69 (3) 713 - 741, September 2004. https://doi.org/10.2178/jsl/1096901763