Summary: The Isomorphism Conjecture for NP
December 19, 2009
In this article, we survey the arguments and known results for and against the Isomorphism
The Isomorphism Conjecture for the class NP states that all polynomial-time many-one complete
sets for NP are polynomial-time isomorphic to each other. It was made by Berman and Hart-
manis 1, inspired in part by a corresponding result in computability theory for computably
enumerable sets , and in part by the observation that all the existing NP-complete sets known
at the time were indeed polynomial-time isomorphic to each other. This conjecture has attracted
a lot of attention because it predicts a very strong structure of the class of NP-complete sets, one
of the fundamental classes in complexity theory.
After an initial period in which it was believed to be true, Joseph and Young  raised serious
doubts against the conjecture based on the notion of one-way functions. This was followed by
investigation of the conjecture in relativized worlds [33, 46, 27] which, on the whole, also suggested
that the conjecture may be false. However, disproving the conjecture using one-way functions, or
proving it, remained very hard (either implies P = NP). Hence research progressed in three distinct
directions from here.