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32
New Lowness Results for ZPP^NP and other Complexity Classes
, 2000
"... We show that the class AM\coAM is low for ZPP . As a consequence, it follows that Graph Isomorphism and several grouptheoretic problems are low for ZPP . We also ..."
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We show that the class AM\coAM is low for ZPP . As a consequence, it follows that Graph Isomorphism and several grouptheoretic problems are low for ZPP . We also
A Tight Relationship between Generic Oracles and Type2 Complexity Theory
, 1997
"... We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct. ..."
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We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct.
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...
Complicated Complementations
 In Proceedings 14th IEE Conference on Computational Complexity
, 1998
"... Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs that use complicated combinatorial arguments. In this paper we use Kolmogorov complexity for oracle construction. We obtain separation results that are much stronger than separations obtained previously eve ..."
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Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs that use complicated combinatorial arguments. In this paper we use Kolmogorov complexity for oracle construction. We obtain separation results that are much stronger than separations obtained previously even with the use of very complicated combinatorial arguments. Moreover the use of Kolmogorov arguments almost trivializes the construction itself. In particular we construct relativized worlds where: 1. NP " CoNP = 2 P=poly. 2. NP has a set that is both simple and NP " CoNPimmune. 3. CoNP has a set that is both simple and NP " CoNPimmune. 4. \Pi p 2 has a set that is both simple and \Pi p 2 " \Sigma p 2 immune. 1 Introduction Some complexity classes are closed under complementation and some are not. Obviously all deterministic complexity classes are closed under complementation, but for most nondeterministic complexity classes the closure remains a big open question. When results in ...
Oracles are subtle but not malicious
 In Proc. IEEE Conference on Computational Complexity
, 2006
"... Theoretical computer scientists have been debating the role of oracles since the 1970’s. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an ..."
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Theoretical computer scientists have been debating the role of oracles since the 1970’s. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linearsized circuits, by proving a new lower bound for perceptrons and lowdegree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first nonrelativizing separation of “traditional ” complexity classes, as opposed to interactive proof classes such as MIP and MAEXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size n k for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size n k with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPP NP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPP NP   and even BPP NP   have linearsize circuits. On the other hand, we also show that the NP queries could be parallelized if P = NP. Thus, classes such as ZPP NP inhabit a “twilight zone, ” where we need to distinguish between relativizing and blackbox techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem. 1
Complexity analysis and variational inference for interpretationbased probabilistic description logics
 IN CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE
, 2009
"... This paper presents complexity analysis and variational methods for inference in probabilistic description logics featuring Boolean operators, quantification, qualified number restrictions, nominals, inverse roles and role hierarchies. Inference is shown to be PEXPcomplete, and variational methods ..."
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This paper presents complexity analysis and variational methods for inference in probabilistic description logics featuring Boolean operators, quantification, qualified number restrictions, nominals, inverse roles and role hierarchies. Inference is shown to be PEXPcomplete, and variational methods are designed so as to exploit logical inference whenever possible.
Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds
"... Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argum ..."
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Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argument can prove that a problem requires circuits of superpolynomial size, even for some very restricted classes of circuits (under reasonable cryptographic assumptions). This barrier is so daunting, that some researchers have decided to focus their attentions elsewhere. Yet the goal of proving circuit lower bounds is of such importance, that some in the community have proposed concrete strategies for surmounting the obstacle. This lecture will discuss some of these strategies, and will dwell at length on a recent approach proposed by Michal Koucky and the author.
Derandomizing ArthurMerlin games and approximate counting implies exponentialsize lower bounds
 In Proceedings of the IEEE Conference on Computational Complexity
, 2010
"... Abstract. We show that if ArthurMerlin protocols can be derandomized, then there is a language computable in deterministic exponentialtime with access to an NP oracle, that requires circuits of exponential size. More formally, if every promise problem in prAM, the class of promise problems that hav ..."
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Abstract. We show that if ArthurMerlin protocols can be derandomized, then there is a language computable in deterministic exponentialtime with access to an NP oracle, that requires circuits of exponential size. More formally, if every promise problem in prAM, the class of promise problems that have ArthurMerlin protocols, can be computed by a deterministic polynomialtime algorithm with access to an NP oracle then there is a language in ENP that requires circuits of size Ω(2n /n). The lower bound in the conclusion of our theorem suffices to construct pseudorandom generators with exponential stretch. We also show that the same conclusion holds if the following two related problems can be computed in polynomial time with access to an NPoracle: (i) approximately counting the number of accepted inputs of a circuit, up to multiplicative factors; and (ii) recognizing an approximate lower bound on the number of accepted inputs of a circuit, up to multiplicative factors.
Generic Separations and Leaf Languages
 Mathematical Logic Quaterly
, 2001
"... In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P (polynomial time) and PSPACE (polynomial space). It was shown that the separability of two complexity classes can be ..."
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In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P (polynomial time) and PSPACE (polynomial space). It was shown that the separability of two complexity classes can be reduced to a combinatorial property of the corresponding dening leaf languages. In the present paper, it is shown that every separation obtained in this way holds for every generic oracle in the sense of Blum and Impagliazzo. We obtain several consequences of this result, regarding, e.g., simultaneous separations and universal oracles, resourcebounded genericity, and type2 complexity. Keywords: computational and structural complexity, leaf language, oracle separation, generic oracle, type2 complexity theory. 1
Some Results on Derandomization
 Theory of Computing Systems
"... We show several results about derandomization including 1. If NP is easy on average then ecient pseudorandom generators exist and P = BPP. 2. If NP is easy on average then given an NP machine M we can easily on average nd accepting computations of M(x) when it accepts. 3. For any A in EXP, if NEXP ..."
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We show several results about derandomization including 1. If NP is easy on average then ecient pseudorandom generators exist and P = BPP. 2. If NP is easy on average then given an NP machine M we can easily on average nd accepting computations of M(x) when it accepts. 3. For any A in EXP, if NEXP A is in P A =poly then NEXP A = EXP A . 4. If A is p k complete then NEXP A = EXP = MA A . 1