Results 1  10
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12
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
The Isomorphism Conjecture Fails Relative to a Random Oracle
 J. ACM
, 1996
"... Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kc ..."
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Cited by 40 (4 self)
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Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kcreative setsand defined a class of sets (the K k f 's) that are necessarily kcreative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NPcomplete sets. Clearly, the BermanHartmanis and JosephYoung conjectures cannot both be correct. We introduce a family of strong oneway functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NPcomplete sets, as Joseph and Young conjectured, and the BermanHartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...
The Isomorphism Conjecture Holds Relative to an Oracle
, 1996
"... We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the BermanHartmanis isomorphism conjecture [BH77] holds relative to any spgeneric oracle, i.e., for any symmetric perfect generic set A, ..."
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Cited by 26 (11 self)
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We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the BermanHartmanis isomorphism conjecture [BH77] holds relative to any spgeneric oracle, i.e., for any symmetric perfect generic set A, all NP^Acomplete sets are polynomialtime isomorphic relative to A. Prior to this work there were no known oracles relative to which the isomorphism conjecture held. As part of our proof that the isomorphism conjecture holds relative to symmetric perfect generic sets we also show that P A = FewP A for any symmetric perfect generic A.
The Isomorphism Conjecture Holds and Oneway Functions Exist Relative to an Oracle
 Journal of Computer and System Sciences
, 1994
"... In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which t ..."
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Cited by 9 (2 self)
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In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but oneway functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of oneway functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomialtime manytoone reductions and if they are both paddable then they are polynomialtime isomorphic. After surveying all of the thenknown NPcomplete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NPcomplete lan...
UP and the Low and High Hierarchies: A Relativized Separation
 Mathematical Systems Theory
, 1992
"... The low and high hierarchies within NP were introduced by Schoning in order to classify sets in NP. It is not known whether the low and high hierarchies include all sets in NP. In this paper, using the circuit lower bound techniques of Hastad and Ko, we construct an oracle set relative to which UP c ..."
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Cited by 7 (0 self)
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The low and high hierarchies within NP were introduced by Schoning in order to classify sets in NP. It is not known whether the low and high hierarchies include all sets in NP. In this paper, using the circuit lower bound techniques of Hastad and Ko, we construct an oracle set relative to which UP contains a language that is not in any level of the low hierarchy and such that no language in UP is in any level of the high hierarchy. Thus, in order to prove that UP contains languages that are in the high hierarchy or that UP is contained in the low hierarchy, one needs to use nonrelativized proof techniques. Since it is known that UP A is low for PP A for all sets A, our result also shows that the interaction between UP and PP is crucial for the lowness of UP for PP; changing the base class to any level of the polynomialtime hierarchy destroys the lowness of UP. 1 Introduction The low and high hierarchies in NP were first introduced by Schoning in order to analyze the internal s...
Two Oracles that Force a Big Crunch
, 1999
"... The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new ty ..."
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Cited by 5 (1 self)
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The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new type of injury argument that we call \resource bounded injury." In the special case of the construction of this oracle, a tree method can be used to transform unbounded search into exponentially bounded, hence recursive, search. This transformation of the construction can be interleaved with another construction so that relative to the new combined oracle also P = UP = NP\coNP. This leads to the curious situation where LOW(NP) = P, but LOW(P NP ) = NEXP, and the complete p m degree for P NP collapses to a single pisomorphism type. 1 Introduction In 1978, Seiferas, Fischer and Meyer [SFM78] showed a very strong separation theorem for nondeterministic time: For time constru...
WitnessIsomorphic Reductions and Local Search
 Complexity, Logic, and Recursion Theory
, 1997
"... We study witnessisomorphic reductions, a type of structurepreserving reduction between NP decision problems. We completely determine the relative power of the different models of witnessisomorphic reduction, and we show that witnessisomorphic reductions can be used in a uniform approach to the loc ..."
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Cited by 5 (2 self)
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We study witnessisomorphic reductions, a type of structurepreserving reduction between NP decision problems. We completely determine the relative power of the different models of witnessisomorphic reduction, and we show that witnessisomorphic reductions can be used in a uniform approach to the local search problem. 1 Introduction The "natural" NP complete decision problems are very much alike. They not only are of the same complexity, but also are in the same polynomialtime isomorphism degree [BH77], and the reductions/isomorphisms between many of these problems are parsimonious [Sim75]. One would expect that such a tight connection between NPcomplete problems "of interest" would lead to an integrated approach when dealing with the closely related optimization problems. This however is not common practice in operations research. Indeed, typically, for each individual NP optimization problem new techniques and heuristics are invented. It seems that though existing reductions show a ...
Enforcing and defying associativity, commutativity, totality, and strong noninvertibility for oneway functions in complexity theory
 In ICTCS
, 2005
"... Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist oneway functions (i.e., ptime computable, honest, ptime noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve th ..."
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Cited by 2 (2 self)
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Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist oneway functions (i.e., ptime computable, honest, ptime noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve the sufficient condition to P = NP. More generally, in this paper we completely characterize which types of oneway functions stand or fall together with (plain) oneway functions—equivalently, stand or fall together with P = NP. We look at the four attributes used in Rabi and Sherman’s seminal work on algebraic properties of oneway functions (see [RS97,RS93]) and subsequent papers—strongness (of noninvertibility), totality, commutativity, and associativity—and for each attribute, we allow it to be required to hold, required to fail, or “don’t care. ” In this categorization there are 3 4 = 81 potential types of oneway functions. We prove that each of these 81 featureladen types stand or fall together with the existence of (plain) oneway functions. Key words: computational complexity, complexitytheoretic oneway functions, associativity, 1.1
My Favorite Ten Complexity Theorems of the Past Decade
"... We review the past ten years in computational complexity theory by focusing on ten theorems that the author enjoyed the most. We use each of the theorems as a springboard to discuss work done in various areas of complexity theory. ..."
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Cited by 1 (0 self)
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We review the past ten years in computational complexity theory by focusing on ten theorems that the author enjoyed the most. We use each of the theorems as a springboard to discuss work done in various areas of complexity theory.
Beyond P^NP = NEXP
"... . Buhrman and Torenvliet created an oracle relative to which P NP = NEXP and thus P NP = P NEXP . Their proof uses a delicate finite injury argument that leads to a nonrecursive oracle. We simplify their proof removing the injury to create a recursive oracle making P NP = NEXP. In addition, ..."
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. Buhrman and Torenvliet created an oracle relative to which P NP = NEXP and thus P NP = P NEXP . Their proof uses a delicate finite injury argument that leads to a nonrecursive oracle. We simplify their proof removing the injury to create a recursive oracle making P NP = NEXP. In addition, in our construction we can make P = UP = NP " coNP. This leads to the curious situation where LOW(NP) = P but LOW(P NP ) = NEXP, and the complete p m  degree for P NP collapses to a pisomorphism type. 1 Introduction In 1978, Seiferas, Fischer and Meyer [SFM78] showed a very strong separation theorem for nondeterministic time: For time constructible t 1 (n) and t 2 (n), if t 1 (n + 1) = o(t 2 (n)) then NTIME(t 1 (n)) does not contain NTIME(t 2 (n)). Thus we have a huge gap between nondeterministic polynomial time (NP) and nondeterministic exponential time (NEXP). We would also expect then a separation between P NP and P NEXP . Indeed, we have some evidence for that direction:...