Results 1  10
of
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. ..."
Abstract

Cited by 26 (12 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 25 (12 self)
 Add to MetaCart
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
 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 ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
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].
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 ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
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:
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 ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
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 .
Reducing the Number of Solutions of NP Functions
, 2000
"... We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functi ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses.