Every Polynomial-Time 1-Degree Collapses iff P = PSPACE (1996)
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BibTeX
@MISC{Fenner96everypolynomial-time,
author = {Stephen A. Fenner and Stuart A. Kurtz and James S. Royer},
title = {Every Polynomial-Time 1-Degree Collapses iff P = PSPACE},
year = {1996}
}
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Abstract
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 2 A () f(x) 2 B. Two sets are: ffl 1-equivalent iff each is m-reducible to the other by one-one reductions; ffl p-invertible equivalent iff each is m-reducible to the other by one-one, polynomial-time invertible reductions; and ffl 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. 2 1. Overview If A is m-reducible to B, we usually interpret this to mean that A is computationally no more difficult than B, since a procedure for computing B is easily converted into a procedure for computing A of comparable complexity. In fact, this interpretation is supported by muc...
