On the isomorphism conjecture for weak reducibilities

Abstract

According to the isomorphism conjecture all NP-complete sets are polynomial-time isomorphic to each other while according to the encrypted complete set conjecture there is ap-one way functionfand an NP-complete set A such that A and f(A) are not polynomial-time isomorphic to each other. In this paper, these two conjectures are translated and investigated for reducibilities weaker than polynomial-time. It is shown that: 1. Relative to reductions computed by one-way logspace DTMs, both the conjectures are false. 2. Relative to reductions computed by one-way logspace NTMs, the isomorphism conjecture is true. 3. Relative to reductions computed by one-way, multi-head, oblivious logspace DTMs, the encrypted complete set conjecture is false. 4. Relative to reductions computed by constant-scan logspace DTMs, the encrypted complete set conjecture is true. It is also shown that the complete degrees of NP under the latter two reducibilities coincide.