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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 ..."
Abstract

Cited by 41 (4 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...
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
Easy sets and hard certificate schemes
 Acta Informatica
, 1997
"... Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for al ..."
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Cited by 16 (4 self)
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Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. In particular, we provide equivalent characterizations of these classes in terms of relative generalized Kolmogorov complexity, showing that they are robust. We also provide structural conditions—regarding immunity and class collapses—that put upper and lower bounds on the sizes of these two classes. Finally, we provide negative results showing that some of our positive claims are optimal with regard to being relativizable. Our negative results are proven using a novel observation: we show that the classical “wide spacing ” oracle construction technique yields instant nonbiimmunity results. Furthermore, we establish a result that improves upon Baker, Gill, and Solovay’s classical result that NP = P = NP ∩ coNP holds in some relativized world.
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...
PSelectivity: Intersections and Indices (Extended Abstract)
, 1994
"... ) Lane A. Hemaspaandra and Zhigen Jiang y Department of Computer Science University of Rochester Rochester, NY 14627 Abstract The Pselective sets [Sel79] are those sets for which there is a polynomialtime algorithm that, given any two strings, determines which is "more likely" to belon ..."
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) Lane A. Hemaspaandra and Zhigen Jiang y Department of Computer Science University of Rochester Rochester, NY 14627 Abstract The Pselective sets [Sel79] are those sets for which there is a polynomialtime algorithm that, given any two strings, determines which is "more likely" to belong to the set: if either of the strings is in the set, the algorithm chooses one that is in the set. We prove that, for each k, the kary Boolean connectives under which the Pselective sets are closed are exactly those that are either completely degenerate or almostcompletely degenerate. We determine the complexity of the index set of the r.e. Pselective sets\Sigma 0 3 complete. Keywords: Computational Complexity Theory, Theory of Computation Supported in part by grants CCR8957604 and NSFINT9116781/JSPSENGR207, and a postdoctoral fellowship from the government of the People's Republic of China. y Permanent address: Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Be...
A note on the Isomorphism Conjecture and oneway functions
"... omputable and the subscript m stands for manyone. There is also an mreduction from TSP to SAT so TSP p m SAT. So, in some sense, these two problems are very similar: If we can solve one efficiently, we can solve the other efficiently as well. But we are interested in a more stringent kind of s ..."
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omputable and the subscript m stands for manyone. There is also an mreduction from TSP to SAT so TSP p m SAT. So, in some sense, these two problems are very similar: If we can solve one efficiently, we can solve the other efficiently as well. But we are interested in a more stringent kind of similarity: What if the NPcomplete languages are isomorphic? An isomorphism between languages is a polynomialtime computable reduction that is onetoone and polynomialtime invertible. A reduction f is polynomialtime invertible if there is a polynomialtime computable function g such that g(f(x)) = x. Berman and Hartmanis [BH77] were the first to consider whether the NPcomplete languages are isomorphic to one another. They proved the following
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
DiplomInformatiker und DiplomMathematiker
"... This dissertation presents the lion’s share of the research I did since I started working at the Technical University of Berlin at the end of 1999. Unfortunately, since I failed to focus my research on a single subject during the last three years, the results of several interesting research projects ..."
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This dissertation presents the lion’s share of the research I did since I started working at the Technical University of Berlin at the end of 1999. Unfortunately, since I failed to focus my research on a single subject during the last three years, the results of several interesting research projects could not be included. If I had included them, the title would have had to be ‘On a Bunch of Interesting, but Unrelated Theorems of Theoretical Computer Science’. When I went to my advisor Dirk Siefkes at the beginning of 2002, we pondered on which papers would be fit to form the basis of a dissertation. There were two alternatives: either two papers on a new concept, namely enumerability by finite automata; or different technical reports on reducibility to selective languages, proververifier protocols, and reachability problems. Reducibility classes of selective languages would have been more ‘en vogue ’ and the results applicable in standard complexity theory and in practice (like a constant parallel time reachability algorithm for tournaments). In the end, the mathematical beauty of results that are presented in the following won over. I have presented the central theorems of this dissertation at two conferences in 2002, namely at the 19th Symposium on Theoretical Aspects of
Received Revised
"... This paper proves that “promise classes ” are so fragilely structured that they do not robustly (i.e. with respect to all oracles) possess Turinghard sets even in classes far larger than themselves. In particular, this paper shows that FewP does not robustly possess Turing hard sets for UP ∩ coUP an ..."
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This paper proves that “promise classes ” are so fragilely structured that they do not robustly (i.e. with respect to all oracles) possess Turinghard sets even in classes far larger than themselves. In particular, this paper shows that FewP does not robustly possess Turing hard sets for UP ∩ coUP and IP ∩ coIP does not robustly possess Turing hard sets for ZPP. It follows that ZPP, R, coR, UP∩coUP, UP, FewP∩coFewP, FewP, and IP ∩ coIP do not robustly possess Turing complete sets. This both resolves open questions of whether promise classes lacking robust downward closure under Turing reductions (e.g., R, UP, FewP) might robustly have Turing complete sets, and extends the range of classes known not to robustly contain manyone complete sets. Keywords: Structural complexity theory; Polynomialtime reductions; Complete languages. ∗Some of these results were reported at Logic at TVER ’92—Symposium on Logical Foundations of Computer Science.
On The Topological Size of pmComplete Degrees
"... All polynomial manyone degrees are shown to be of second Baire category in the superset topology when witness functions are allowed to run in 2 log h n time, for any h. Any improvement of this result for the complete pmdegrees of RE, EXP or NP implies P<F NaN> 6= NP. 1 Introduction U ..."
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All polynomial manyone degrees are shown to be of second Baire category in the superset topology when witness functions are allowed to run in 2 log h n time, for any h. Any improvement of this result for the complete pmdegrees of RE, EXP or NP implies P<F NaN> 6= NP. 1 Introduction Understanding the properties of complete degrees is a central theme of research in complexity theory. Questions like ffl Are complete sets pcreative for P ? ffl Are all complete sets paddable ? Are they selfreducible ? ffl Do all complete sets contain infinite polynomial time sets ? ffl Which are the collapsing complete degrees, i.e., degrees consisting of only one pisomorphism type ? have direct, decisive implications on the relation between the most important complexity classes. Kurtz, Mahaney and Royer have written an excellent survey of these topics [KMR90]. In this work we pursue the study of the topological size of some interesting classes of sets begun in [Zim93], focusing this time on c...