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...

This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

We show that all sets that arecomplete for NP under non-uniform AC are isomorphic under non-uniform AC -computable isomorphisms. Furthermore, these sets remain NP-complete even under non-uniform NC reductions.

We prove that the Berman-Hartmanis isomorphism conjecture is true under AC 0 reductions. More generally, we show three theorems that hold for any complexity class C closed under (uniform) TC 0 -computable many-one reductions. Isomorphism: The sets complete for C under AC 0 reductions are all isomorphic under isomorphisms computable and invertible by AC 0 circuits of depth three. Gap: The sets that are complete for C under AC 0 and NC 0 reducibility coincide. Stop Gap: The sets that are complete for C under AC 0 [mod 2] and AC 0 reducibility do not coincide. (These theorems hold both in the non-uniform and P-uniform settings.) To prove the second theorem for P-uniform settings, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.) 1 Introduction The notion of complete sets in complexity classes provides one of ...

In this paper we demonstrate an oracle relative to which there are one-way functions but every paddable 1-li-degree collapses to an isomorphism type, thus yielding a relativized failure of the Joseph-Young Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but one-way functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of one-way functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomial-time many-to-one reductions and if they are both paddable then they are polynomial-time isomorphic. After surveying all of the then-known NP-complete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NP-complete lan...

We show that all sets complete for NC¹ under AC 0 reductions are isomorphic under AC 0 -computable isomorphisms. Although our proof does not generalize directly to other complexity classes, we do show that, for all complexity classes C closed under NC¹-computable manyone reductions, the sets complete for C under NC 0 reductions are all isomorphic under AC 0 -computable isomorphisms. Our result showing that the complete degree for NC¹ collapses to an isomorphism type follows from a theorem showing that in NC¹, the complete degrees for AC 0 and NC 0 reducibility coincide.

A set A is m-reducible (or Karp-reducible) to B iff there is a polynomial-time computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1-equivalent iff each is m-reducible to the other by one-one reductions; ffl p-invertible equivalent iff each is m-reducible to the other by one-one, polynomial-time invertible reductions; and ffl p-isomorphic iff there is an m-reduction from one set to the other that is one-one, onto, and polynomial-time invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1-equivalent sets are p-isomorphic. (c) Every two p-invertible equivalent sets are p-isomorphic. 2 1. Overview If A is m-reducible to B, we usually interpret this to mean that A is computationally no more difficult than B, since a procedure for computing B is easily converted into a procedure for computing A of comparable complexity. In fact, this interpretation is supported by muc...

We show that every m-complete set for NEXP as well as ats complement have an infinite subset in P. Thw nnswers an open question first raised in [Ber76]. 1

It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a non-trivial ≤1−NL m-degree that coincides with a ≤1−L m-degree. 3. For every class C closed under log-lin reductions, the ≤1−NL m coincides with the ≤1−L m-complete degree of C. 1

We review the past ten years in computational complexity theory by focusing on ten theorems that the author enjoyed the most. We use each of the theorems as a springboard to discuss work done in various areas of complexity theory.