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

  • Stephen A. Fenner (a1), Stuart A. Kurtz (a2) and James S. Royer (a3)

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, xA 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.

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References

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

  • Stephen A. Fenner (a1), Stuart A. Kurtz (a2) and James S. Royer (a3)

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