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11
Hardness hypotheses, derandomization, and circuit complexity
 In Proceedings of the 24th Conference on Foundations of Software Technology and Theoretical Computer Science
, 2004
"... Abstract We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have pmeasure 0.* The pseudoNP hypothesis: there is an NP language that can be distinguished from anyDT ..."
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Cited by 18 (5 self)
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Abstract We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have pmeasure 0.* The pseudoNP hypothesis: there is an NP language that can be distinguished from anyDTIME(2 nffl) language by an NP refuter. * The NPmachine hypothesis: there is an NP machine accepting 0 * for which no 2n ffltime machine can find infinitely many accepting computations. We show that the NPmachine hypothesis is implied by each of the first two. Previously, norelationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuitsize lower bounds that are known to followfrom the first two hypotheses also follow from the NPmachine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizesAM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): * The measure hypothesis: NP does not have pmeasure 0 [24].
Redundancy in complete sets
 In Proceedings 23nd Symposium on Theoretical Aspects of Computer Science
, 2006
"... We show that a set is mautoreducible if and only if it is mmitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al. [11], complete sets for all of the following complexity classes are mmitotic: NP, coNP, ..."
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Cited by 6 (6 self)
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We show that a set is mautoreducible if and only if it is mmitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al. [11], complete sets for all of the following complexity classes are mmitotic: NP, coNP, ⊕P, PSPACE, and NEXP, as well as all levels of PH, MODPH, and the Boolean hierarchy over NP. In the cases of NP, PSPACE, NEXP, and PH, this at once answers several wellstudied open questions. These results tell us that complete sets share a redundancy that was not known before. We disprove the equivalence between autoreducibility and mitoticity for all polynomialtimebounded reducibilities between 3ttreducibility and Turingreducibility: There exists a sparse set in EXP that is polynomialtime 3ttautoreducible, but not weakly polynomialtime Tmitotic. In particular, polynomialtime Tautoreducibility does not imply polynomialtime weak Tmitoticity, which solves an open question by Buhrman and Torenvliet. We generalize autoreducibility to define polyautoreducibility and give evidence that NPcomplete sets are polyautoreducible. 1
Autoreducibility, mitoticity and immunity
 Mathematical Foundations of Computer Science: Thirtieth International Symposium, MFCS 2005
, 2005
"... We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are we ..."
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Cited by 6 (4 self)
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We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are weakly manyone mitotic. • PSPACEcomplete sets are weakly Turingmitotic. • If oneway permutations and quick pseudorandom generators exist, then NPcomplete languages are mmitotic. • If there is a tally language in NP ∩ coNP − P, then, for every ɛ> 0, NPcomplete sets are not 2 n(1+ɛ)immune. These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets. 1
Robustness of PSPACEcomplete sets
, 2008
"... We study the robustness of complete languages in PSPACE and prove that they are robust against Pselective sparse sets. Earlier similar results are known for EXPcomplete sets [3] and NPcomplete sets [7]. ..."
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Cited by 1 (1 self)
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We study the robustness of complete languages in PSPACE and prove that they are robust against Pselective sparse sets. Earlier similar results are known for EXPcomplete sets [3] and NPcomplete sets [7].
The Informational Content of Canonical Disjoint NPPairs
, 2007
"... We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: Where does the implication ..."
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Cited by 1 (0 self)
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We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: Where does the implication [can(f) ≤ pp m can(g) ⇒ f ≤s g] hold, and where does it fail? Q2: Where can we find proof systems of different strengths, but equivalent canonical pairs? Q3: What do (non)equivalent canonical pairs tell about the corresponding proof systems? Q4: Is every NPpair (A, B), where A is NPcomplete, strongly manyone equivalent to the canonical pair of some proof system? In short, we show that Q1 and Q2 can be answered with ‘everywhere’, which generalizes previous results by Pudlák and Beyersdorff. Regarding Q3, inequivalent canonical pairs tell that the proof systems are not “very similar”, while equivalent, Pinseparable canonical pairs tell that they are not “very different”. We can relate Q4 to the open problem in structural complexity that asks whether unions of disjoint NPcomplete sets are NPcomplete. This demonstrates a new connection between proof systems, disjoint NPpairs, and unions of disjoint NPcomplete sets. 1
Unions of Disjoint NPComplete Sets
"... Abstract. We study the following question: if A and B are disjoint NPcomplete sets, then is A ∪ B NPcomplete? We provide necessary and sufficient conditions under which the union of disjoint NPcomplete sets remain complete. 1 ..."
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Abstract. We study the following question: if A and B are disjoint NPcomplete sets, then is A ∪ B NPcomplete? We provide necessary and sufficient conditions under which the union of disjoint NPcomplete sets remain complete. 1
Hardness Hypotheses, Derandomization, and Circuit Complexity
"... Abstract We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: ffl The measure hypothesis: NP does not have pmeasure 0. ffl The pseudoNP hypothesis: there is an NP language that can be distinguished from ..."
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Abstract We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: ffl The measure hypothesis: NP does not have pmeasure 0. ffl The pseudoNP hypothesis: there is an NP language that can be distinguished from anyDTIME(2nffl) language by an NP refuter. ffl The NPmachine hypothesis: there is an NP machine accepting 0\Lambda for which no 2nffltimemachine can find infinitely many accepting computations. We show that the NPmachine hypothesis is implied by each of the first two. Previously, norelationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuitsize lower bounds that are known to followfrom the first two hypotheses also follow from the NPmachine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizesAM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): ffl The measure hypothesis: NP does not have pmeasure 0 [20]. ffl The pseudoNP hypothesis: NP has a language that can be distinguished from any DTIME(2nffl)
unknown title
, 2005
"... We disprove the equivalence between autoreducibility and mitoticity for all polynomialtimebounded reducibilities between 3ttreducibility and Turingreducibility: There exists a sparse set in EXP that is polynomialtime 3ttautoreducible, but not weakly polynomialtime Tmitotic. In particular, p ..."
Abstract
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We disprove the equivalence between autoreducibility and mitoticity for all polynomialtimebounded reducibilities between 3ttreducibility and Turingreducibility: There exists a sparse set in EXP that is polynomialtime 3ttautoreducible, but not weakly polynomialtime Tmitotic. In particular, polynomialtime Tautoreducibility does not imply polynomialtime weak Tmitoticity, which solves an open question by Buhrman and Torenvliet.
Abstract
"... We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: • The measure hypothesis: NP does not have pmeasure 0. • The pseudoNP hypothesis: there is an NP language that can be distinguished from any DTIME(2nǫ) ..."
Abstract
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We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: • The measure hypothesis: NP does not have pmeasure 0. • The pseudoNP hypothesis: there is an NP language that can be distinguished from any DTIME(2nǫ) language by an NP refuter. • The NPmachine hypothesis: there is an NP machine accepting 0 ∗ for which no 2nǫtime machine can find infinitely many accepting computations. We show that the NPmachine hypothesis is implied by each of the first two. Previously, no relationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuitsize lower bounds that are known to follow from the first two hypotheses also follow from the NPmachine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizes AM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1