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Oblivious Symmetric Alternation
"... We introduce a new class Op2 as a subclass of the symmetric alternation class Sp2. An Op2proof system has the flavor of an S p 2 proof system, but it is more restrictive in nature. Inan Sp 2 proof system, we have two competing provers and a verifier such that for any input,the honest prover has an i ..."
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We introduce a new class Op2 as a subclass of the symmetric alternation class Sp2. An Op2proof system has the flavor of an S p 2 proof system, but it is more restrictive in nature. Inan Sp 2 proof system, we have two competing provers and a verifier such that for any input,the honest prover has an irrefutable certificate. In an Op 2 proof system, we require that the irrefutable certificates depend only on the length of the input, not on the input itself. In other words, the irrefutable proofs are oblivious of the input. For this reason, we call the new class oblivious symmetric alternation. While this might seem slightly contrived, it turns out that this class helps us improve some existing results. For instance, we show that if NP ae P/poly then PH = Op 2, whereas the best known collapse under the same hypothesis was PH = S p 2. We also define classes YOp
Query order in the polynomial hierarchy
 Journal of Universal Computer Science
, 1998
"... Hemaspaandra, Hempel, and Wechsung [HHW] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomia ..."
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Hemaspaandra, Hempel, and Wechsung [HHW] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. P C:D denotes the class of languages computable by a polynomialtime machine that is allowed one query to C followed by one query to D [HHW]. We prove that the levels of the polynomial hierarchy are orderoblivious: P Σp j:Σp k = P Σp k:Σp j. Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes—and thus essentially all standard complexity classes—inherit all orderobliviousness results that hold for P. 1
A Moment of Perfect Clarity I: The Parallel Census Technique
, 2000
"... We discuss the history and uses of the parallel census techniquean elegant tool in the study of certain computational objects having polynomially bounded census functions. A sequel [GH] will discuss advances (including [CNS95] and Glaer [Gla00]), some related to the parallel census technique and ..."
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Cited by 3 (3 self)
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We discuss the history and uses of the parallel census techniquean elegant tool in the study of certain computational objects having polynomially bounded census functions. A sequel [GH] will discuss advances (including [CNS95] and Glaer [Gla00]), some related to the parallel census technique and some due to other approaches, in the complexityclass collapses that follow if NP has sparse hard sets under reductions weaker than (full) truthtable reductions.
Downward Collapse from a Weaker Hypothesis
, 1998
"... Hemaspaandra et al. [1] proved that, for m ? 0 and 0 ! i ! k \Gamma 1: if \Sigma p i \DeltaDIFF m (\Sigma p k ) is closed under complementation, then DIFFm (\Sigma p k ) = coDIFFm (\Sigma p k ). This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allo ..."
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Hemaspaandra et al. [1] proved that, for m ? 0 and 0 ! i ! k \Gamma 1: if \Sigma p i \DeltaDIFF m (\Sigma p k ) is closed under complementation, then DIFFm (\Sigma p k ) = coDIFFm (\Sigma p k ). This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allowing the \Sigma p i to be replaced by any class in its difference hierarchy. We so extend the result by proving that, for s; m ? 0 and 0 ! i ! k \Gamma 1: if DIFF s (\Sigma p i )\DeltaDIFF m (\Sigma p k ) is closed under complementation, then DIFFm (\Sigma p k ) = coDIFFm (\Sigma p k ). 1 Introduction and Preliminaries In complexity theory, countless cases are known in which it can be proven that the collapse of seemingly small classes implies the collapse of classes that before This research was supported in part by grants NSFCCR9322513 and NSFINT9513368 /DAAD315PROfoab, and was done in part during visits to Le Moyne College and to FriedrichSchillerUniversitat Jena. the...