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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...
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 Relative to an Oracle
, 1996
"... We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the BermanHartmanis isomorphism conjecture [BH77] holds relative to any spgeneric oracle, i.e., for any symmetric perfect generic set A, ..."
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Cited by 27 (11 self)
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We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the BermanHartmanis isomorphism conjecture [BH77] holds relative to any spgeneric oracle, i.e., for any symmetric perfect generic set A, all NP^Acomplete sets are polynomialtime isomorphic relative to A. Prior to this work there were no known oracles relative to which the isomorphism conjecture held. As part of our proof that the isomorphism conjecture holds relative to symmetric perfect generic sets we also show that P A = FewP A for any symmetric perfect generic A.
A Short History of Computational Complexity
 IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2002
"... this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quit ..."
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Cited by 18 (1 self)
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this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld
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.
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...
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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Cited by 1 (0 self)
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In this article, we survey the arguments and known results for and against the Isomorphism Conjecture. 1
My Favorite Ten Complexity Theorems of the Past Decade
"... 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. ..."
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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.
The Quasilinear Isomorphism Challenge
"... We show that all the basic facts about the BermanHartmanis Isomorphism Conjecture carry over from polynomial to quasilinear time bounds. However, we show that the connection between uniqueaccepting NTMs and oneway functions, which underlies much subsequent work on the BH conjecture, breaks down ..."
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Cited by 1 (1 self)
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We show that all the basic facts about the BermanHartmanis Isomorphism Conjecture carry over from polynomial to quasilinear time bounds. However, we show that the connection between uniqueaccepting NTMs and oneway functions, which underlies much subsequent work on the BH conjecture, breaks down for quasilinear time under relativizations. Hence we have a whole new ballgame, on a tighter playing field.