Results 1  10
of
24
On PolynomialTime Bounded TruthTable Reducibility of NP Sets to Sparse Sets
, 1991
"... We prove that if P ≠ NP, then there exists a set in NP that is not polynomial time bounded truthtable reducible (in short, p btt reducible) to any sparse set. In other words, we prove that no sparse p btt hard set exists for NP unless P = NP. By using the technique proving this result, we in ..."
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Cited by 44 (3 self)
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We prove that if P ≠ NP, then there exists a set in NP that is not polynomial time bounded truthtable reducible (in short, p btt reducible) to any sparse set. In other words, we prove that no sparse p btt hard set exists for NP unless P = NP. By using the technique proving this result, we investigate intractability of several number theoretic decision problems, i.e., decision problems defined naturally from number theoretic problems. We show that for these number theoretic decision problems, if it is not in P, then it is not p btt reducible to any sparse set.
PSelective Sets, and Reducing Search to Decision vs. SelfReducibility
, 1993
"... We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to de ..."
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Cited by 39 (9 self)
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We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not selfreducible. Funding for this research was provided by the National Science Foundation under grant CCR9002292. y Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 z Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 x Research performed while visiting the Department of Computer Science, State University of New York at Buffalo, Jan. 1992Dec. 1992. Current address: Department of Computer Science, University of ElectroCommunications, Chofushi, Tokyo 182, Japan.  Department of Computer Science, State University of New York at Buffalo, 226...
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
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...
On Bounded TruthTable, Conjunctive, and Randomized Reductions to Sparse Sets
 IN PROC. 12TH CONFERENCE ON THE FOUNDATIONS OF SOFTWARE TECHNOLOGY & THEORETICAL COMPUTER SCIENCE
, 1992
"... In this paper we study the consequences of the existence of sparse hard sets for different complexity classes under certain types of deterministic, randomized and nondeterministic reductions. We show that if an NPcomplete set is boundedtruthtable reducible to a set that conjunctively reduces to ..."
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Cited by 17 (8 self)
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In this paper we study the consequences of the existence of sparse hard sets for different complexity classes under certain types of deterministic, randomized and nondeterministic reductions. We show that if an NPcomplete set is boundedtruthtable reducible to a set that conjunctively reduces to a sparse set then P = NP. Relatedly, we show that if an NPcomplete set is boundedtruthtable reducible to a set that corp reduces to some set that conjunctively reduces to a sparse set then RP = NP. We also prove similar results under the (apparently) weaker assumption that some solution of the promise problem (1SAT; SAT) reduces via the mentioned reductions to a sparse set. Finally we consider nondeterministic polynomial time manyone reductions to sparse and cosparse sets. We prove that if a coNPcomplete set reduces via a nondeterministic polynomial time manyone reduction to a cosparse set then PH = \Theta p 2 . On the other hand, we show that nondeterministic polynomial ...
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...
How Reductions to Sparse Sets Collapse the Polynomialtime Hierarchy: A Primer
, 1993
"... this paper to give simple proofs, in a uniform format, of the major known (pre1992) results relating how polynomialtime reductions of SAT to sparse sets collapse the polynomialtime hierarchy. To help the reader familiar with basic facts of complexity theory follow the main flow of ideas, while ke ..."
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Cited by 14 (0 self)
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this paper to give simple proofs, in a uniform format, of the major known (pre1992) results relating how polynomialtime reductions of SAT to sparse sets collapse the polynomialtime hierarchy. To help the reader familiar with basic facts of complexity theory follow the main flow of ideas, while keeping the exposition selfcontained, straight forward proofs from elementary complexity theory are relegated to footnotes. We treat polynomialtime Turing reductions (i.e., Cook reductions) in Section 2. Bounded truthtable reductions (and manyone reductions) are treated in Section 3. Sections 2 and 3 may be read independently of each other. Section 4 uses the definitions of Section 3 to give simple proofs of results on conjunctive and disjunctive reductions. A comprehensive discussion of early work on how reductions to sparse sets collapse the polynomialtime hierarchy may be found in [Ma89]. Additional discussions of this topic, as well as extensive bibliographies, may be found in [JY90] and in [Yo90]. 2 PolynomialTime Turing Reductions 2.1 Introduction In [Lo82], Long explicitly used the result (KL1) to prove:
On Sparse Hard Sets for Counting Classes
, 1993
"... In this paper, we study one worddecreasing selfreducible sets which are introduced by Lozano and Toran [21]. These are usual selfreducible sets with the peculiarity that the selfreducibility machine makes at most one query and this is lexicographically smaller than the input. We show rst that for ..."
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Cited by 14 (0 self)
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In this paper, we study one worddecreasing selfreducible sets which are introduced by Lozano and Toran [21]. These are usual selfreducible sets with the peculiarity that the selfreducibility machine makes at most one query and this is lexicographically smaller than the input. We show rst that for all counting classes dened by a predicate on the number of accepting paths there exist complete sets which are one worddecreasing selfreducible. Using this fact we can prove that for any class K chosen from fPP; NP; C=P; MOD 2 P; MOD 3 P; g it holds that (1) if there is a sparse P btt hard set for K then K P, and (2) if there is a sparse SN btt hard set for K then K NP \ coNP. This generalizes the result from [24] to the mentioned complexity classes. 1 Introduction One of the central roles in the study of structural complexity theory resides in nding structural differences or similarities among complexity classes. Since almost every complexity class is dened by us...
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...
Languages that are Easier than their Proofs
, 1991
"... A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduc ..."
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Cited by 13 (7 self)
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A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership. Similarly we present languages where checking is harder than computing membership. Each of the following properties  checkability, randomselfreducibility, reduction from search to decision, and interactive proofs in which the prover's power is limited to deciding membership in the language itself  implies coherence, one of the weakest forms of selfreducibility. Under assumptions about tripleexponential time, we construct incoherent sets in NP....