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Lower Bounds for the Low Hierarchy
"... this paper. The low hierarchy, as defined in [Sc83], can only be used to classify the complexity of sets in NP. In order to talk about related sets that are not in NP, the extended low hierarchy was introduced in [BBS86]. (The levels of this hierarchy are labeled EL 2 ,EL 2 , A preliminary ver ..."
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Cited by 33 (4 self)
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this paper. The low hierarchy, as defined in [Sc83], can only be used to classify the complexity of sets in NP. In order to talk about related sets that are not in NP, the extended low hierarchy was introduced in [BBS86]. (The levels of this hierarchy are labeled EL 2 ,EL 2 , A preliminary version of this paper was presented at the 16th International Colloquium on Automata, Languages, and Programming [AH89a]
Reductions to Sets of Low Information Content
, 1992
"... this paper was coauthored by K. Wagner) ..."
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The Power of the Middle Bit of a #P Function
, 1995
"... This paper studies the class MP of languages which can be solved in polynomial time with the additional information of one bit from a #P function f . The middle bit of f(x) is shown to be as powerful as any other bit, whereas the O(log n) bits at either end are apparently weaker. The polynomial hie ..."
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Cited by 17 (3 self)
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This paper studies the class MP of languages which can be solved in polynomial time with the additional information of one bit from a #P function f . The middle bit of f(x) is shown to be as powerful as any other bit, whereas the O(log n) bits at either end are apparently weaker. The polynomial hierarchy and the classes Mod k P, k 2, are shown to be low for MP. They are also low for a class we call AmpMP which is defined by abstracting the "amplification" methods of Toda (SIAM J. Comput. 20 (1991), 865877). Consequences of these results for circuit complexity are obtained using the concept of a MidBit gate, which is defined to take binary inputs x 1 ; : : : ; xw and output the blog 2 (w)=2c th bit in the binary representation of the number P w i=1 x i . Every language in ACC can be computed by a family of depth2 deterministic circuits of size 2 (log n) O(1) with a MidBit gate at the root and ANDgates of fanin (log n) O(1) at the leaves. This result improves the known ...
Upper bounds for the Complexity of Sparse and Tally Descriptions
 Mathematical Systems Theory
, 1996
"... We investigate the complexity of computing small descriptions for sets in various reduction classes to sparse sets. For example, we show that if a set A and its complement conjunctively reduce to some sparse set, then they also are conjunctively reducible to a P(A \Phi SAT)printable tally set. As ..."
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Cited by 17 (8 self)
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We investigate the complexity of computing small descriptions for sets in various reduction classes to sparse sets. For example, we show that if a set A and its complement conjunctively reduce to some sparse set, then they also are conjunctively reducible to a P(A \Phi SAT)printable tally set. As a consequence, the class IC[log; poly] of sets with low instance complexity is contained in the EL \Sigma 1 level of the extended low hierarchy. By refining our techniques, we also show that all worddecreasing selfreducible sets in IC[log; poly] are in NP " coNP and therefore low for NP. We derive similar results for sets in R p d (SPARSE)) and R p hd (R p c (SPARSE)), as well as in some nondeterministic reduction classes to sparse sets. Parts of this work have been presented at ISAAC'92, MFCS'93, and CIAC'94 [AKM92b, AKM93, Mu94]. y Work done while visiting Universitat Ulm. Supported in part by an Alexander von Humboldt research fellowship. 1 1 Introduction Sparse sets play...
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 ...
The Structure of Logarithmic Advice Complexity Classes
 Theoretical Computer Science
, 1992
"... A nonuniform class called here FullP/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated b ..."
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Cited by 12 (4 self)
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A nonuniform class called here FullP/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated by the need of a logarithmic advice class closed under polynomialtime deterministic reductions. Several characterizations of FullP/log are shown, formulated in terms of various sorts of tally sets with very small information content. A study of its inner structure is presented, by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets. Partially supported by the E.U. through the ESPRIT Long Term Research Project 20244 (ALCOMIT) and through the HCM Network CHRXCT930415 (COLORET); by the Spanish DGICYT through project PB950787 (KOALA), and by Acciones Integradas HispanoAl...
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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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...
Online Learning and ResourceBounded Dimension: Winnow Yields New Lower Bounds for Hard Sets
"... We establish a relationship between the online mistakebound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Z ..."
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We establish a relationship between the online mistakebound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Zhao (2000), and solves one of Lutz and Mayordomo's "Twelve Problems in ResourceBounded Measure" (1999).
NPHard Sets are Exponentially Dense Unless coNP ⊆ NP/poly
"... We show that hard sets S for NP must have exponential density, for some ɛ> 0 and infinitely many n, unless coNP ⊆ i.e. S=n  ≥ 2nɛ NP/poly and the polynomialtime hierarchy collapses. This result holds for Turing reductions that make n1−ɛ queries. In addition we study the instance complexity of ..."
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Cited by 8 (0 self)
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We show that hard sets S for NP must have exponential density, for some ɛ> 0 and infinitely many n, unless coNP ⊆ i.e. S=n  ≥ 2nɛ NP/poly and the polynomialtime hierarchy collapses. This result holds for Turing reductions that make n1−ɛ queries. In addition we study the instance complexity of NPhard problems and show that hard sets also have an exponential amount of instances that have instance complexity nδ for some δ> 0. This result also holds for Turing reductions that make n1−ɛ queries. 1