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16
The Isomorphism Conjecture Fails Relative to a Random Oracle
 J. ACM
, 1996
"... Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kc ..."
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Cited by 43 (5 self)
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Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kcreative setsand defined a class of sets (the K k f 's) that are necessarily kcreative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NPcomplete sets. Clearly, the BermanHartmanis and JosephYoung conjectures cannot both be correct. We introduce a family of strong oneway functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NPcomplete sets, as Joseph and Young conjectured, and the BermanHartmanis 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...
Reducing the Complexity of Reductions
 Computational Complexity
, 1997
"... We prove that the BermanHartmanis 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 manyone reductions. Isomorphism: The sets complete for C under AC 0 reductions are all i ..."
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Cited by 33 (14 self)
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We prove that the BermanHartmanis 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 manyone 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 nonuniform and Puniform settings.) To prove the second theorem for Puniform 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 ...
Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem
 Journal of Computer and System Sciences
"... We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions. ..."
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Cited by 31 (12 self)
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We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions.
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 30 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete 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 NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
The Isomorphism Conjecture Holds and Oneway Functions Exist Relative to an Oracle
 Journal of Computer and System Sciences
, 1994
"... In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which t ..."
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Cited by 9 (2 self)
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In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but oneway 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 oneway functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomialtime manytoone reductions and if they are both paddable then they are polynomialtime isomorphic. After surveying all of the thenknown NPcomplete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NPcomplete lan...
An Isomorphism Theorem for Circuit Complexity
, 1996
"... 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 NC¹computable manyone reductions, the sets co ..."
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Cited by 7 (6 self)
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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 NC¹computable 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.
On Pimmunity of nondeterministic complete sets
 In Proceedings of the 10th Annual Conference on Structure in Complexity Theory '95
, 1995
"... We show that every mcomplete set for NEXP as well as ats complement have an infinite subset in P. Thw nnswers an open question first raised in [Ber76]. 1 ..."
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Cited by 6 (0 self)
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We show that every mcomplete set for NEXP as well as ats complement have an infinite subset in P. Thw nnswers an open question first raised in [Ber76]. 1
Every PolynomialTime 1Degree Collapses iff P = PSPACE
, 1996
"... A set A is mreducible (or Karpreducible) to B iff there is a polynomialtime computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1equivalent iff each is mreducible to the other by oneone reductions; ffl pinvertible equivalent iff each is mreducible to the othe ..."
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Cited by 5 (2 self)
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A set A is mreducible (or Karpreducible) to B iff there is a polynomialtime computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1equivalent iff each is mreducible to the other by oneone reductions; ffl pinvertible equivalent iff each is mreducible to the other by oneone, polynomialtime invertible reductions; and ffl pisomorphic iff there is an mreduction from one set to the other that is oneone, onto, and polynomialtime invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1equivalent sets are pisomorphic. (c) Every two pinvertible equivalent sets are pisomorphic. 2 1. Overview If A is mreducible 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...
DSPACE(n) ? = NSPACE(n): A degree theoretic characterization
 in Proc. 10th Structure in Complexity Theory Conference
, 1995
"... It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1 ..."
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Cited by 3 (3 self)
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It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1
The Isomorphism Conjecture for NP
, 2009
"... In this article, we survey the arguments and known results for and against the Isomorphism Conjecture. 1 ..."
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In this article, we survey the arguments and known results for and against the Isomorphism Conjecture. 1