The Isomorphism Conjecture Fails Relative to a Random Oracle (1996)
| Venue: | J. ACM |
| Citations: | 41 - 4 self |
BibTeX
@INPROCEEDINGS{Kurtz96theisomorphism,
author = {Stuart A. Kurtz and Stephen R. Mahaney and James S. Royer},
title = {The Isomorphism Conjecture Fails Relative to a Random Oracle},
booktitle = {J. ACM},
year = {1996},
pages = {157--166}
}
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Abstract
Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NP-complete sets---the k-creative sets---and defined a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NP-complete sets. Clearly, the Berman--Hartmanis and Joseph--Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...
