Karp-reducible, m-reducible, polynomial-time computable function
A set A is m-reducible (or Karp-reducible) to B iff there is a polynomial-time computable function f such that, for all x, x ∈ A <--> f (x) ∈ B. Two sets are: (a) 1-equivalent iff each is m-reducible to the other by one-one reductions; (b) p-invertible equivalent iff each is m-reducible to the other by one-one, polynomial-time invertible reductions; and (c) p-isomorphic iff 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.
Fenner, Stephen A.; Kurtz, Stuart A.; and Royer, James S., "Every Polynomial-Time 1-Degree Collapses iff P = PSPACE" (1996). Electrical Engineering and Computer Science. Paper 51.