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37
Separability and Oneway Functions
, 2000
"... We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable. ..."
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Cited by 27 (12 self)
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We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable.
Separation of NPcompleteness notions
 SIAM Journal on Computing
, 2001
"... Abstract. We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduc ..."
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Cited by 24 (12 self)
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Abstract. We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduce. Key words. Turing completeness, truthtable completeness, manyone completeness, pselectivity, pgenericity AMS subject classifications. 1. Introduction. Ladner, Lynch, and Selman [LLS75] were the first to compare the strength of polyno), truth), that mialtime reducibilities. They showed, for the common polynomialtime reducibilities, ( ¢ Turing P T ( ¢ table P tt), bounded truthtable ( ¢ P btt), and manyone ( ¢ P m
Hardness hypotheses, derandomization, and circuit complexity
"... 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) ..."
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Cited by 18 (5 self)
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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.
Easy sets and hard certificate schemes
 Acta Informatica
, 1997
"... Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for al ..."
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Cited by 14 (4 self)
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Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. In particular, we provide equivalent characterizations of these classes in terms of relative generalized Kolmogorov complexity, showing that they are robust. We also provide structural conditions—regarding immunity and class collapses—that put upper and lower bounds on the sizes of these two classes. Finally, we provide negative results showing that some of our positive claims are optimal with regard to being relativizable. Our negative results are proven using a novel observation: we show that the classical “wide spacing ” oracle construction technique yields instant nonbiimmunity results. Furthermore, we establish a result that improves upon Baker, Gill, and Solovay’s classical result that NP = P = NP ∩ coNP holds in some relativized world.
Comparing reductions to NPcomplete sets
 Electronic Colloquium on Computational Complexity
, 2006
"... Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondetermin ..."
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Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNPcomplete for NP but not Turingcomplete. (3) Every problem that is manyone complete for NP is complete under lengthincreasing reductions that are computed by polynomialsize circuits. The first item solves one of Lutz and Mayordomo’s “Twelve Problems in ResourceBounded Measure ” (1999). We also show that every manyone complete problem for NE is complete under onetoone, lengthincreasing reductions that are computed by polynomialsize circuits. 1
Proofs, Codes, and PolynomialTime Reducibilities
"... We show how to construct proof systems for NP languages where a deterministic polynomialtime verifier can check membership, given any N (2=3)+ffl bits of an N bit witness of membership. ..."
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Cited by 12 (0 self)
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We show how to construct proof systems for NP languages where a deterministic polynomialtime verifier can check membership, given any N (2=3)+ffl bits of an N bit witness of membership.
Oneway permutations and selfwitnessing languages
 Journal of Computer and System Sciences
, 2003
"... A desirable property of oneway functions is that they be total, onetoone, and onto—in other words, that they be permutations. We prove that oneway permutations exist exactly if PaUPcoUP: This provides the first characterization of the existence of oneway permutations based on a complexityclas ..."
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Cited by 9 (2 self)
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A desirable property of oneway functions is that they be total, onetoone, and onto—in other words, that they be permutations. We prove that oneway permutations exist exactly if PaUPcoUP: This provides the first characterization of the existence of oneway permutations based on a complexityclass separation and shows that their existence is equivalent to a number of previously studied complexitytheoretic hypotheses. We study permutations in the context of witness functions of nondeterministic Turing machines. A language is in PermUP if, relative to some unambiguous, nondeterministic, polynomialtime Turingmachine acceptingthe language, the function mappingeach stringto its unique witness is a permutation of the members of the language. We show that, under standard complexitytheoretic assumptions, PermUP is a strict subset of UP. We study SelfNP, the set of all languages such that, relative to some nondeterministic, polynomialtime Turing machine that accepts the language, the set of all witnesses of strings in the language is identical to the language itself. We show that SATASelfNP; and, under standard complexitytheoretic assumptions, SelfNPaNP:
Properties of NPcomplete sets
 In Proceedings of the 19th IEEE Conference on Computational Complexity
, 2004
"... We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreo ..."
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Cited by 9 (7 self)
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We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p mhard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure oneway permutations to derive consequences for longstanding open questions about whether NPcomplete sets are immune. For example, assuming that pseudorandom generators and secure oneway permutations exist, it follows easily that NPcomplete sets are not pimmune. Assuming only that secure oneway permutations exist, we prove that no NPcomplete set is DTIME(2nɛ)immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomialclose to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turingcomplete sets for NP are not closed under union. Our hypothesis asserts existence of a UPmachine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ timebounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)biimmune sets, then this hypothesis is true.
On NPPartitions over Posets with an Application to Reducing the Set of Solutions of NP Problems
 In Proceedings 25th Symposium on Mathematical Foundations of Computer Science
, 2000
"... . The boolean hierarchy of kpartitions over NP for k 2 was introduced as a generalization of the wellknown boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NPpartiti ..."
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Cited by 9 (3 self)
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. The boolean hierarchy of kpartitions over NP for k 2 was introduced as a generalization of the wellknown boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NPpartitions by considering partition classes which are generated by nite labeled posets. Since we cannot prove it absolutely, we collect evidence for this extended boolean hierarchy to be strict. We give an exhaustive answer to the question of which relativizable inclusions between partition classes can occur depending on the relation between their dening posets. The study of the extended boolean hierarchy is closely related to the issue of whether one can reduce the number of solutions of NP problems. For nite cardinality types, assuming the extended boolean hierarchy of kpartitions over NP is strict, we give a complete characterization when such solution reductions are possible. 1 Introduct...
Graph Isomorphism is Low for ZPP(NP) and other Lowness results
, 2000
"... We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several grouptheoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets th ..."
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Cited by 7 (0 self)
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We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several grouptheoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets that have interactive proof systems with honest provers in P=poly, is also low for ZPP NP . We consider lowness properties of nonuniform function classes, namely, NPMV=poly, NPSV=poly, NPMV t =poly, and NPSV t =poly. Specifically, we show that { Sets whose characteristic functions are in NPSV=poly and that have program checkers (in the sense of Blum and Kannan [8]) are low for AM and ZPP NP . { Sets whose characteristic functions are in NPMV t =poly are low for p 2 .