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37
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 40 (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...
Instance Complexity
, 1994
"... We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that ..."
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Cited by 30 (1 self)
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We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that runs in time t, decides x correctly, and makes no mistakes on other strings ("don't know" answers are permitted). We prove that a set A is in P if and only if there exist a polynomial t and a constant c such that ic t (x : A) c for all x
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 29 (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
NP Might Not Be As Easy As Detecting Unique Solutions
, 1998
"... We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the fi ..."
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Cited by 23 (6 self)
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We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the first where P A = UP A 6= NP A = coNP A : ffl The construction gives a much simpler proof than Fenner, Fortnow and Kurtz of a relativized world where all NPcomplete sets are polynomialtime isomorphic. It is the first such computable oracle. ffl Relative to A we have a collapse of \PhiEXP A ` ZPP A ` P A /poly. We also create a different relativized world where there exists a set L in NP that is NP complete under reductions that make one query to L but not under traditional manyone reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide. 1 Introduction Valiant and Vazirani [VV86] show the sur...
Pseudorandom generators and structure of complete degrees
 In 17th Annual IEEE Conference on Computational Complexity
, 2002
"... It is shown that if there exist sets in E that requiresized circuits then sets that are hard for class P, and above, under 11 reductions are also hard under 11, sizeincreasing reductions. Under the assumption of the hardness of solving RSA or Discrete Log problem, it is shown that sets that are h ..."
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Cited by 23 (2 self)
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It is shown that if there exist sets in E that requiresized circuits then sets that are hard for class P, and above, under 11 reductions are also hard under 11, sizeincreasing reductions. Under the assumption of the hardness of solving RSA or Discrete Log problem, it is shown that sets that are hard for class NP, and above, under manyone reductions are also hard under (nonuniform) 11, and sizeincreasing reductions. 1
An Excursion to the Kolmogorov Random Strings
 In Proceedings of the 10th IEEE Structure in Complexity Theory Conference
, 1995
"... We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1) . We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resourcebounded measure ..."
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Cited by 17 (8 self)
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We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1) . We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resourcebounded measure introduced by [17]. From this we conclude that R t is not Turingcomplete for EXP . This contrasts the resource unbounded setting. There R is Turingcomplete for coRE . We show that the class of sets to which R t bounded truthtable reduces, has p 2 measure 0 (therefore, measure 0 in EXP ). This answers an open question of Lutz, giving a natural example of a language that is not weaklycomplete for EXP and that reduces to a measure 0 class in EXP . It follows that the sets that are p btt hard for EXP have p 2 measure 0. 1 Introduction One of the main questions in complexity theory is the relation between complexity classes, such as for example P ; NP , and EXP . It is well known that ...
On the Existence of Hard Sparse Sets under Weak Reductions
, 1996
"... Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace manyone reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more gene ..."
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Cited by 17 (4 self)
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Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace manyone reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more general reducibilities. Our main results are as follows. (1) If there exists a sparse set that is hard for P under bounded truthtable reductions, then P = NC 2 . (2) If there exists a sparse set that is hard for P under randomized logspace reductions with onesided error, then P = Randomized LOGSPACE. (3) If there exists an NPhard sparse set under randomized polynomialtime reductions with onesided error, then NP = RP. (4) If there exists a 2 (log n) O(1) sparse hard set for P under truthtable reductions, then P ` DSPACE[(logn) O(1) ]. As a byproduct of (4), we obtain a uniform O(log 2 n log log n) time parallel algorithm for computing the rank of a 2 log 2 n \Theta n matrix o...
Splittings, Robustness and Structure of Complete Sets
, 1993
"... We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. To address this question for a given hard set A, we construct a sparse set S, ..."
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Cited by 15 (4 self)
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We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. To address this question for a given hard set A, we construct a sparse set S, and ask whether A \Gamma S is still hard. It turns out that for most of the reductions considered and for an arbitrary given sparseness condition, there is a single subexponential time computable set S that meets this condition, such that A \Gamma S is not hard for any A. Not only is this set S subexponential time computable, but a slight modification of the construction can make the complexity of S meet any reasonable superpolynomial function. On the other hand we show that for any polynomialtime computable sparse set S, the set A \Gamma S remains hard. There are other properties than time complexity that make a set `almost' polynomialtime computable. For sparse pselective sets...
Complete Sets and Structure in Subrecursive Classes
 In Proceedings of Logic Colloquium '96
, 1998
"... In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completene ..."
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Cited by 14 (1 self)
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In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...
Structural Properties of Complete Problems for Exponential Time
, 1997
"... The properties and structure of complete sets for exponentialtime classes are surveyed. Strong reductions, those implying manyone completeness, are considered as strengthenings of the usual completeness notions. From the results on strong reductions, immunity properties of complete sets are der ..."
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Cited by 14 (0 self)
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The properties and structure of complete sets for exponentialtime classes are surveyed. Strong reductions, those implying manyone completeness, are considered as strengthenings of the usual completeness notions. From the results on strong reductions, immunity properties of complete sets are derived. Differences are shown between complete sets arising from the various polynomialtime reductions. These include most of the "weak" reduction between p m and p T . Finally we consider complete sets for some other classes such as r.e. sets along with structural properties of these sets. 1 Introduction Complete problems play a central and defining role in complexity theory. They are the canonical sets within a complexity class, are almost always the only sets which arise naturally within a class and as such are the most thoroughly studied. One would like to understand the inherent properties which make a set complete and in particular capture the core of what makes them natural a...