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10
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...
SemiMembership Algorithms: Some Recent Advances
 SIGACT News
, 1994
"... A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990 ..."
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Cited by 14 (8 self)
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A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semimembership algorithms. 1 Introduction A membership algorithm M for a set A takes as its input any string x and decides whether x 2 A. Informally, a semimembership algorithm M for a set A takes as its input any strings x and y and decides which is "no less likely" to belong to A in the sense that if exactly one of the strings is in A, then M outputs that one string. Semimembership algorithms have been studied in a number of settings. Recursive semimembership algorithms (and the associated semirecursive setsthose sets having recursive semimembership algorithms) were introduced in the 1...
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...
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 .
Reducibility Classes of Pselective Sets
, 1995
"... A set is Pselective [Sel79] if there is a polynomialtime semidecision algorithm for the setan algorithm that given any two strings decides which is \more likely" to be in the set. This paper establishes a strict hierarchy among the various reductions and equivalences to Pselective sets. 1 I ..."
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Cited by 4 (2 self)
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A set is Pselective [Sel79] if there is a polynomialtime semidecision algorithm for the setan algorithm that given any two strings decides which is \more likely" to be in the set. This paper establishes a strict hierarchy among the various reductions and equivalences to Pselective sets. 1 Introduction Given the large number of important problems that do not appear to have easy solutions, researchers have explored more exible approaches to ecient set recognition (or nearrecognition): almost polynomial time [MP79], average polynomial time (see [Gur91]), implicit membership testability [HH91], neartestability [GHJY91], Pcloseness [Sch86,Yes83], Pselectivity [Sel79], and others. One such notion, that of the Pselective sets, has proven useful in many contexts, such as characterizing P [BvHT93] and understanding whether SAT may have unique solutions [HNOSb]. Intuitively, a set is Pselective if there is a 2ary polynomialtime function that chooses which of its inputs is \more...
Logspace Reducibility: Models and Equivalences
 International Journal of Foundations of Computer Science
, 1997
"... We study the relative computational power of logspace reduction models. In particular, we study the relationships between oneway and twoway oracle tapes, resetting of the oracle head, and blanking of the oracle tape. We show that oracle models letting information persist between queries can be qui ..."
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Cited by 2 (0 self)
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We study the relative computational power of logspace reduction models. In particular, we study the relationships between oneway and twoway oracle tapes, resetting of the oracle head, and blanking of the oracle tape. We show that oracle models letting information persist between queries can be quite powerful, even if the information is not readable by the querying machine. We show that logspace f(n)Turing reductions are stronger than polynomialtime f(n)Turing reductions when f(n) = !(log n), and that this is optimal if P = L. 1 Introduction Efficient reductions are a central object of study in computational complexity theory. Polynomialtime reductions have received wide attention, and logspacebounded reductions have also long been studied as a potentially finergrained reducibility than polynomialtime reducibility. But the extent to which logspace reducibilities provably provide a finergrained stratification has remained open. We resolve this with respect to relativizable t...
A Moment of Perfect Clarity II: Consequences of Sparse Sets Hard for NP with Respect to Weak Reductions
 In preparation. Will appear in the SIGACT News Complexity Theory Column
, 2000
"... This paper discusses advances, due to the work of Cai, Naik, and Sivakumar [CNS95] and Glaer [Gla00], in the complexity class collapses that follow if NP has sparse hard sets under reductions weaker than (full) truthtable reductions. 1 Quick Hits Most of this article will be devoted to presenting ..."
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Cited by 2 (2 self)
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This paper discusses advances, due to the work of Cai, Naik, and Sivakumar [CNS95] and Glaer [Gla00], in the complexity class collapses that follow if NP has sparse hard sets under reductions weaker than (full) truthtable reductions. 1 Quick Hits Most of this article will be devoted to presenting the work of Glaer [Gla00]. However, even before presenting the background and definitions for that, let us briefly note some improvements that follow from the work of Cai, Naik, and Sivakumar due to c fl Christian Glaer and Lane A. Hemaspaandra, 2000. Supported in part by grants NSFCCR 9322513 and NSFINT9815095/DAAD315PPPguab, and the Studienstiftung des Deutschen Volkes. Written in part while Lane A. Hemaspaandra was visiting JuliusMaximiliansUniversitat Wurzburg. y Email: glasser@informatik.uniwuerzburg.de. z Email: lane@cs.rochester.edu. 1 the results discussed in the first part of this article [GH00]. (See [GH00] for definitions of the terms and classes used here: U...
Sparse Hard Sets for P
 DIMACS TECHNICAL REPORT
, 1996
"... Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various re ..."
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Cited by 1 (0 self)
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Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for lowlevel complexity classes. In particular, the Hartmanis' sparseness conjectures for P and NL have been resolved. This article overviews the history of sparse hard set problems and exposes some of the recent results.