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Summary: On the Isomorphism Conjecture for Weak Reducibilities
Manindra Agrawal
School of Mathematics, SPIC Science Foundation
Madras 600 017, India
email : manindra@ssf.ernet.in
Abstract
According to the isomorphism conjecture all NP-complete sets are polynomial-time iso-
morphic to each other while according to the encrypted complete set conjecture there is a
p-one-way function f and 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 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 com-
plete set conjecture is true.
It is also shown that the complete degrees for NP under the latter two reducibilities coin-
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