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Competing Provers Yield Improved KarpLipton Collapse Results
 Information and Computation
, 2002
"... Via competing provers, we show that if a language A is selfreducible and has polynomialsize circuits then S 2 = S 2 . Building on this, we strengthen the Kamper AFK Theorem, namely, we prove that if NP coNP)/poly then the polynomial hierarchy collapses to S 2 . We also strengthen Yap ..."
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Cited by 15 (2 self)
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Via competing provers, we show that if a language A is selfreducible and has polynomialsize circuits then S 2 = S 2 . Building on this, we strengthen the Kamper AFK Theorem, namely, we prove that if NP coNP)/poly then the polynomial hierarchy collapses to S 2 . We also strengthen Yap's Theorem, namely, we prove that if NP coNP/poly then the polynomial hierarchy collapses to S 2 . Under the same assumptions, the best previously known collapses were to ZPP respectively ([KW98, BCK 94], building on [KL80, AFK89, Kam91, Yap83]). It is known that S 2 [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kamper AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of resultsranging from the study of unique solutions to issues of approximationour results implicitly strengthen all those results.
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 ..."
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Cited by 7 (0 self)
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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 .
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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Cited by 6 (1 self)
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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
Boolean operations, joins, and the extended low hierarchy. Theoretical Computer Science
 Comput. Sci
, 1998
"... We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy [BBS86] than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL2, yet their join is in EL2. That is, in terms of extended lowness, ..."
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Cited by 3 (3 self)
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We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy [BBS86] than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL2, yet their join is in EL2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL2 and prove that EL2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL2 lower bounds for certain notions generalizing Selman’s Pselectivity [Sel79], which may be regarded as an interesting result in its own right. 1
Observations on Measure and Lowness for . . .
 In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science
, 1996
"... Assuming that k 2 and \Delta P k does not have pmeasure 0, it is shown that BP \Delta \Delta P k = \Delta P k . This implies that the following conditions hold if \Delta P 2 does not have pmeasure 0. (i) AM " coAM is low for \Delta P 2 . (Thus BPP and the graph isomorphism problem are low ..."
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Cited by 2 (1 self)
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Assuming that k 2 and \Delta P k does not have pmeasure 0, it is shown that BP \Delta \Delta P k = \Delta P k . This implies that the following conditions hold if \Delta P 2 does not have pmeasure 0. (i) AM " coAM is low for \Delta P 2 . (Thus BPP and the graph isomorphism problem are low for \Delta P 2 .) (ii) If \Delta P 2 6= PH, then NP does not have polynomialsize circuits. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation. 1 Introduction Many widely believed conjectures in computational complexity are "strong" in the sense that they are known to imply that P 6= NP, but are not known to follow from the P 6= NP hypothesis. Recent investigations have shown that a number of these conjectures do follow from the (apparently) stronger hypothesis that NP does not have pmeasure 0. (This hypothesis, written ¯ p (NP) 6= 0, is defined in...
Coding Complexity: The Computational Complexity of Succinct Descriptions
, 1996
"... For a given set of strings, the problem of obtaining a succinct description becomes an important subject of research, related to several areas of theoretical computer science. In structural complexity theory, researchers have developed a reasonable framework for studying the complexity of these prob ..."
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Cited by 1 (1 self)
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For a given set of strings, the problem of obtaining a succinct description becomes an important subject of research, related to several areas of theoretical computer science. In structural complexity theory, researchers have developed a reasonable framework for studying the complexity of these problems. In this paper, we survey how such investigation has proceeded, and explain the current status of our knowledge.
Combining SelfReducibility and Partial Information Algorithms
"... Abstract. A partial information algorithm for a language A computes, for some fixed m, for input words x1,..., xm a set of bitstrings containing χA(x1,..., xm). E.g., pselective, approximable, and easily countable languages are defined by the existence of polynomialtime partial information algorit ..."
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Abstract. A partial information algorithm for a language A computes, for some fixed m, for input words x1,..., xm a set of bitstrings containing χA(x1,..., xm). E.g., pselective, approximable, and easily countable languages are defined by the existence of polynomialtime partial information algorithms of specific type. Selfreducible languages, for different types of selfreductions, form subclasses of PSPACE. For a selfreducible language A, the existence of a partial information algorithm sometimes helps to place A into some subclass of PSPACE. The most prominent known result in this respect is: Pselective languages which are selfreducible are in P [9]. Closely related is the fact that the existence of a partial information algorithm for A simplifies the type of reductions or selfreductions to A. The most prominent known result in this respect is: Turing reductions to easily countable languages simplify to truthtable reductions [8]. We prove new results of this type. We show: 1. Selfreducible languages which are easily 2countable are in P. This partially confirms a conjecture of [8]. 2. Selfreducible languages which are (2m − 1, m)verbose are truthtable selfreducible. This generalizes the result of [9] for pselective languages, which are (m + 1, m)verbose. 3. Selfreducible languages, where the language and its complement are strongly 2membership comparable, are in P. This generalizes the corresponding result for pselective languages of [9]. 4. Disjunctively truthtable selfreducible languages which are 2membership comparable are in UP. Topic: Structural complexity 1
PolynomialTime MultiSelectivity
, 1999
"... We introduce a generalization of Selman’s Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove the f ..."
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We introduce a generalization of Selman’s Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove the first known (and optimal) lower bounds for generalized selectivitylike classes in terms of EL2, the second level of the extended low hierarchy. We study the resulting selectivity hierarchy, denoted by SH, which we prove does not collapse. In particular, we study the internal structure and the properties of SH and completely establish, in terms of incomparability and strict inclusion, the relations between our generalized selectivity classes and Ogihara’s Pmc (polynomialtime membershipcomparable) classes. Although SH is a strictly increasing infinite hierarchy, we show that the core results that hold for the Pselective sets and that prove them structurally simple also hold for SH. In particular, all sets in SH have small circuits; the NP sets in SH are in Low2, the second level of the low hierarchy within NP; and SAT cannot be in SH unless P = NP. Finally, it is known that PSel, the class of Pselective sets, is not closed under union or intersection. We provide an extended selectivity hierarchy that is based on SH and that is large enough to capture those closures of the Pselective sets, and yet, in contrast with the Pmc classes, is refined enough to distinguish them. 1