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14
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.
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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Cited by 13 (4 self)
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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
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.
Immunity and Simplicity for Exact Counting and Other Counting Classes
 R.A.I.R.O. Theoretical Informatics and Applications, 33(2):159–176, March/April
, 1998
"... Ko [Ko90] and Bruschi [Bru92] showed that in some relativized world, PSPACE (in fact, \PhiP) contains a set that is immune to the polynomial hierarchy (PH). In this paper, we study and settle the question of (relativized) separations with immunity for PH and the counting classes PP, C=P, and \PhiP i ..."
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Cited by 5 (1 self)
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Ko [Ko90] and Bruschi [Bru92] showed that in some relativized world, PSPACE (in fact, \PhiP) contains a set that is immune to the polynomial hierarchy (PH). In this paper, we study and settle the question of (relativized) separations with immunity for PH and the counting classes PP, C=P, and \PhiP in all possible pairwise combinations. Our main result is that there is an oracle A relative to which C=P contains a set that is immune to BPP \PhiP . In particular, this C=P A set is immune to PH A and \PhiP A . Strengthening results of Tor'an [Tor91] and Green [Gre91], we also show that, in suitable relativizations, NP contains a C=Pimmune set, and \PhiP contains a PP PH immune set. This implies the existence of a C=P B simple set for some oracle B, which extends results of Balc'azar et al. [Bal85,BR88] and provides the first example of a simple set in a class not known to be contained in PH. Our proof technique requires a circuit lower bound for "exact counting" that is der...
Tally NP Sets and Easy Census Functions
, 1998
"... We study the question of whether every P set has an easy (i.e., polynomialtime computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P 1 ` FP, where #P 1 is the class of functions that count the witnesses for tally NP sets ..."
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Cited by 2 (1 self)
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We study the question of whether every P set has an easy (i.e., polynomialtime computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P 1 ` FP, where #P 1 is the class of functions that count the witnesses for tally NP sets. We prove that every #P PH 1 function can be computed in FP #P #P 1 1 . Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does. We show that the assumption #P 1 ` FP implies P = BPP and PH ` MOD k P for each k 2, which provides further evidence that not all sets in P have an easy census function. We also relate a set's property of having an easy census function to other wellstudied properties of sets, such as rankability and scalability (the closure of the rankable sets under Pisomorphisms). Finally, we prove that it is no more likely that the census function of any set in P can be approximated (more precisely, can be n ff e...
DistributionallyHard Languages
 In COCOON 99, Lecture Notes in Computer Science 1627
, 2000
"... Cai and Selman [CS99] defined a modification of Levin's notion of average polynomial time and proved, for every Pbiimmune language L and every polynomialtime computable distribution with infinite support, that L is not recognizable in polynomial time on the average. We call such languag ..."
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Cai and Selman [CS99] defined a modification of Levin's notion of average polynomial time and proved, for every Pbiimmune language L and every polynomialtime computable distribution with infinite support, that L is not recognizable in polynomial time on the average. We call such languages distributionallyhard. Pavan and Selman [PS00] proved that there exist distributionallyhard sets that are not Pbiimmune if and only P contains Pprintableimmune sets. We extend this characterizion to include assertions about several traditional questions about immunity, about finding witnesses for NPmachines, and about existence of oneway functions. Similarly, we address the question of whether NP contains sets that are distributionally hard. Several of our results are implications for which we cannot prove whether or not their converse holds. In nearly all such cases we provide oracles relative to which the converse fails.