 
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 NPcomplete sets are polynomialtime iso
morphic to each other while according to the encrypted complete set conjecture there is a
poneway function f and an NPcomplete 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 polynomialtime. It is shown that:
1. Relative to reductions computed by oneway logspace DTMs, both the conjectures are
false.
2. Relative to reductions computed by oneway logspace NTMs, the isomorphism conjecture
is true.
3. Relative to reductions computed by oneway, multihead, oblivious logspace DTMs, the
encrypted complete set conjecture is false.
4. Relative to reductions computed by constantscan 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
