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17
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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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...
On the Structure of Low Sets
 PROC. 10TH STRUCTURE IN COMPLEXITY THEORY CONFERENCE, IEEE
, 1995
"... Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected ..."
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Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomialsize circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.
Locating P/poly Optimally in the Extended Low Hierarchy
, 1993
"... The low hierarchy within NP and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes. We relocate P/poly from the third \Sigmalevel EL P;\Sigma 3 (Balc'azar et al., 1986) to the third \Thetalevel EL P;\Theta 3 of the extended low hier ..."
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The low hierarchy within NP and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes. We relocate P/poly from the third \Sigmalevel EL P;\Sigma 3 (Balc'azar et al., 1986) to the third \Thetalevel EL P;\Theta 3 of the extended low hierarchy. The location of P=poly in EL P;\Theta 3 is optimal since, as shown by Allender and Hemachandra (1992), there exist sparse sets that are not contained in the next lower level EL P;\Sigma 2 . As a consequence of our result, all NP sets in P=poly are relocated from the third \Sigmalevel L P;\Sigma 3 (Ko and Schoning, 1985) to the third \Thetalevel L P;\Theta 3 of the low hierarchy.
Monotonous and Randomized Reductions to Sparse Sets
, 1994
"... An oracle machine is called monotonous, if after a negative answer the machine does not ask further queries to the oracle. For example, one truthtable, conjunctive, and Hausdorff reducibilities are monotonous. We study the consequences of the existence of sparse hard sets for different complexity cl ..."
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Cited by 4 (2 self)
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An oracle machine is called monotonous, if after a negative answer the machine does not ask further queries to the oracle. For example, one truthtable, conjunctive, and Hausdorff reducibilities are monotonous. We study the consequences of the existence of sparse hard sets for different complexity classes under monotonous and randomized reductions. We prove tradeoffs between the randomized time complexity of NP sets that reduce to a set B via such reductions and the density of B as well as the number of queries made by the monotonous reduction. As a consequence, bounded Turing hard sets for NP are not corp reducible to a sparse set unless 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.
Sparse Sets, Approximable Sets, and Parallel Queries to NP
 Proc. Sixteenth Symposium on Theoretical Aspects of Computing (STACS '99), LNCS 1563
, 1999
"... We show that if an NPcomplete set or a coNPcomplete set is polynomialtime disjunctive truthtable reducible to a sparse set then FP NP jj = FP NP [log]. With a similar argument we show also that if SAT is O(log n)approximable then FP NP jj = FP NP [log]. Since FP NP jj = FP NP [lo ..."
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We show that if an NPcomplete set or a coNPcomplete set is polynomialtime disjunctive truthtable reducible to a sparse set then FP NP jj = FP NP [log]. With a similar argument we show also that if SAT is O(log n)approximable then FP NP jj = FP NP [log]. Since FP NP jj = FP NP [log] implies that SAT is O(logn)approximable [BFT97], it follows from our result that the two hypotheses are equivalent. We also show that if an NPcomplete set or a coNPcomplete set is disjunctively reducible to a sparse set of polylogarithmic density then P = NP. 1 Introduction The study of the existence of sparse hard sets for complexity classes has occupied complexity theorists for over two decades. The first results in this area were motivated by the BermanHartmanis isomorphism conjecture [BH77] and by the study of connections between uniform and nonuniform complexity classes [KL80]. The focus shifted to proving, for various reducibilities 1 (whose strengths lie between the many...
AverageCase Intractability vs. WorstCase Intractability
 IN THE 23RD INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE
, 1998
"... We use the assumption that all sets in NP (or other levels of the polynomialtime hierarchy) have efficient averagecase algorithms to derive collapse consequences for MA, AM, and various subclasses of P/poly. As a further consequence we show for C 2 fP(PP);PSPACEg that C is not tractable in the a ..."
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We use the assumption that all sets in NP (or other levels of the polynomialtime hierarchy) have efficient averagecase algorithms to derive collapse consequences for MA, AM, and various subclasses of P/poly. As a further consequence we show for C 2 fP(PP);PSPACEg that C is not tractable in the averagecase unless C = P.
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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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...
On reductions to sets that avoid EXPSPACE
, 1995
"... A set B is called EXPSPACEavoiding, if every subset of B in EXPSPACE is sparse. Sparse sets and sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACEavoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACEavoiding sets, we show th ..."
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A set B is called EXPSPACEavoiding, if every subset of B in EXPSPACE is sparse. Sparse sets and sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACEavoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACEavoiding sets, we show that if the reducibility used has a property, called rangeconstructibility, then A must also reduce to a sparse set under the same reducibility. Keywords: Computational Complexity, Reducibilities, Sparse Sets. 1 Introduction The study of reductions to low information content sets has received much attention in structural complexity theory research in recent years. There is a series of results showing that complexity classes containing intractable problems cannot be reduced to sets of low information content unless there is an unlikely collapse of complexity classes. The class of sparse sets [8, 10, 11] is an example of a wellstudied class of low information content sets. A research trend ...
Circuit Expressions of Low Kolmogorov Complexity
 In preparation
, 1999
"... We study circuit expressions of logarithmic and polylogarithmic polynomialtime Kolmogorov complexity, focusing on their complexitytheoretic characterizations and learnability properties. They provide a nontrivial circuitlike characterization for a natural nonuniform complexity class that lacked ..."
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We study circuit expressions of logarithmic and polylogarithmic polynomialtime Kolmogorov complexity, focusing on their complexitytheoretic characterizations and learnability properties. They provide a nontrivial circuitlike characterization for a natural nonuniform complexity class that lacked it up to now. We show that circuit expressions of this kind can be learned with membership queries in polynomial time if and only if every NEpredicate is Esolvable. Thus they are learnable given that the learner is allowed the extra use of an oracle in NP. The precise way of accessing the oracle is shown to be optimal under relativization. We present a precise characterization of the subclass defined by Kolmogoroveasy circuit expressions that can be constructed from membership queries in polynomial time, with some consequences for the structure of reduction and equivalence classes of tally sets of very low density. Preliminary, sometimes weaker versions of the results in this paper were...
unknown title
"... (Extended abstracty) V. Arvindz J. K"oblerx M. Mundhenkk Abstract A set B is called EXPSPACEavoiding, if every subset of B in EXPSPACE is sparse. For example, sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACEavoiding. Investigating the complexity of set ..."
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(Extended abstracty) V. Arvindz J. K&quot;oblerx M. Mundhenkk Abstract A set B is called EXPSPACEavoiding, if every subset of B in EXPSPACE is sparse. For example, sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACEavoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACEavoiding sets, we show that if the reducibility used has a property, called rangeconstructibility, then A must also reduce to a sparse set under the same reducibility.