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

Published online by Cambridge University Press:  12 March 2014

Stephen A. Fenner ,
Stuart A. Kurtz  and
James S. Royer
Stephen A. Fenner
Affiliation:
Dept. of Computer Science, University of South Carolina, Columbia, SC 29208, USA, E-mail: [email protected]
Stuart A. Kurtz
Affiliation:
Dept. of Computer Science, University of Chicago, 1100 E. 58TH ST., Chicago, IL 60637-1581, USA, E-mail: [email protected]
James S. Royer
Affiliation:
Dept. of Elec. Eng. and Computer Science, Syracuse University, Syracuse, NY 13244, USA, E-mail: [email protected]
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Abstract.

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 3 , September 2004 , pp. 713 - 741
Copyright
Copyright © Association for Symbolic Logic 2004

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