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24
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
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 17 (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...
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 16 (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...
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 16 (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...
The Resolution of a Hartmanis Conjecture
, 1995
"... Building on the recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE ..."
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Cited by 13 (4 self)
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Building on the recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. We further prove that if P has a sparse hard set under manyone reductions computable in NC 1 , then P collapses to NC 1 . 1 Introduction A set S is called sparse if there are at most a polynomial number of strings in S up to length n. Sparse sets have been the subject of study in complexity theory for the past 20 years, as they reveal inherent structure and limitations of computation [BH77, HOW92, You92a, You92b]. For instance, it is well known that the class of languages polynomial time Turing reducible (i.e. by Cook reductions) to a sparse set is precisely the class of languages with polynomial size circuits. One major motivation for the study of sparse sets, and various reducib...
Sparse Hard Sets for P Yield SpaceEfficient Algorithms
, 1995
"... In 1978, Hartmanis conjectured that there exist no sparse complete sets for P under logspace manyone reductions. In this paper, in support of the conjecture, it is shown that if P has sparse hard sets under logspace manyone reductions, then P ` DSPACE[log 2 n]. The result is derived from a more ..."
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Cited by 11 (1 self)
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In 1978, Hartmanis conjectured that there exist no sparse complete sets for P under logspace manyone reductions. In this paper, in support of the conjecture, it is shown that if P has sparse hard sets under logspace manyone reductions, then P ` DSPACE[log 2 n]. The result is derived from a more general statement that if P has 2 polylog sparse hard sets under polylogarithmic spacecomputable manyone reductions, then P ` DSPACE[polylog]. 1 Introduction In 1978, Hartmanis conjectured that no Pcomplete sets under logspace manyone reductions can be polynomially sparse; i.e., for any Pcomplete set A, k fx 2 A j jxj ng k cannot be bounded by any polynomial in n [5]. The conjecture is interesting and fascinating. If the conjecture is true, then L 6= P, because L = P implies any nonempty finite set being Pcomplete. So, with expectation that L is different from P, one might believe the validity of the conjecture. Nevertheless, such a reasoning would be fallacious, for, proving thi...
Completeness for Nondeterministic Complexity Classes
, 1991
"... We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomialtime bounded (even logarithmic space bounded) reducibilities turn out to be different for any class ..."
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Cited by 9 (0 self)
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We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomialtime bounded (even logarithmic space bounded) reducibilities turn out to be different for any class containing NE . For space classes the completeness notions under logspace reducibilities can be separated for any class properly containing LOGSPACE . Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes. 1 Introduction Efficient reducibilities and completeness are two of the central concepts of complexity theory. Since the first use of polynomial time bounded Turing reductions by Cook [4] and the introduction of polynomial time bounded manyone reductions by Karp[9], considerable effort has been put in the investigation of properties and the relative strengt...
Resolution of Hartmanis' Conjecture for NLHard Sparse Sets
, 1995
"... We resolve a conjecture of Hartmanis from 1978 about sparse hard sets for nondeterministic logspace (NL). We show that there exists a sparse hard set S for NL under logspace manyone reductions if and only if NL = L (deterministic logspace). ..."
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Cited by 7 (3 self)
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We resolve a conjecture of Hartmanis from 1978 about sparse hard sets for nondeterministic logspace (NL). We show that there exists a sparse hard set S for NL under logspace manyone reductions if and only if NL = L (deterministic logspace).
Sparse Hard Sets for P: Resolution of a Conjecture of Hartmanis
"... Building on a recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. ..."
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Cited by 5 (0 self)
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Building on a recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. We further prove that if P has a sparse hard set under manyone reductions computable in NC 1 , then P collapses to NC 1 .
SpaceEfficient Recognition Of Sparse SelfReducible Languages
, 1994
"... . Mahaney and others have shown that sparse selfreducible sets have timeecient algorithms, and have concluded that it is unlikely that NP has sparse complete sets. Mahaney's work, intuition, and a 1978 conjecture of Hartmanis notwithstanding, nothing has been known about the density of co ..."
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Cited by 5 (3 self)
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. Mahaney and others have shown that sparse selfreducible sets have timeecient algorithms, and have concluded that it is unlikely that NP has sparse complete sets. Mahaney's work, intuition, and a 1978 conjecture of Hartmanis notwithstanding, nothing has been known about the density of complete sets for feasible classes until now. This paper shows that sparse selfreducible sets have spaceecient algorithms, and in many cases, even have timespaceecient algorithms. We conclude that NL, NC k , AC k , LOG(DCFL), LOG(CFL), and P lack complete (or even Turinghard) sets of low density unless implausible complexity class inclusions hold. In particular, if NL (respectively P, k , or NP) has a polylogsparse logspacehard set, then NL SC (respectively P SC, k SC, or PH SC), and if P has subpolynomially sparse logspacehard sets, then P 6= PSPACE. Subject classications. 68Q15, 03D15. 1. Introduction Complete sets are the quintessences of their complexity cla...