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22
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
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Cited by 71 (23 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
New Collapse Consequences Of NP Having Small Circuits
, 1995
"... . We show that if a selfreducible set has polynomialsize circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomialtime hierarchy PH to ZPP(NP) under the assumption that NP has polynomialsize circuits. This improves on the wellknown ..."
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Cited by 59 (7 self)
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. We show that if a selfreducible set has polynomialsize circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomialtime hierarchy PH to ZPP(NP) under the assumption that NP has polynomialsize circuits. This improves on the wellknown result of Karp, Lipton, and Sipser (1980) stating a collapse of PH to its second level \Sigma P 2 under the same assumption. As a further consequence, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C=P have polynomialsize circuits. Finally, we investigate the circuitsize complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))size circuits. Key words. polynomialsize circuits, advice classes, lowness, randomized computation AMS subject classifications. 03D10, 03D15, 68Q10, 68Q15 1. Introduction. The question of whether intractable sets ca...
Threshold Computation and Cryptographic Security
 SIAM JOURNAL ON COMPUTING
, 1995
"... Threshold machines are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. ..."
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Cited by 35 (6 self)
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Threshold machines are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. In 1975, Simon proved that for unboundederror polynomialtime machines these two notions yield the same class, PP. Perhaps because Simon's result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored. In this paper, we compare the boundederror probabilistic class BPP with the analogous threshold class, BPP path , and, more generally, we study the structural properties of BPP path . We prove that BPP path contains both NP BPP and P NP[log] , and that BPP path is contained in P \Sigma p 2 [log] , BPP NP , and PP. We conclude that, unless the polynomial hierarchy collapses, boundederror threshold computation is strictly more powerful than boundederror probabilistic computation. We also consider the natural notion of secure access to a database: an adversary who watches the queries should gain no information about the input other than perhaps its length. We show, for both BPP and BPP path , that if there is any database for which this formalization of security differs from the security given by oblivious database access, then P 6= PSPACE. It follows that if any set lacking small circuits can be securely accepted, then P 6= PSPACE.
Turing Machines With Few Accepting Computations And Low Sets For PP
, 1992
"... this paper we study two different ways to restrict the power of NP: We consider languages accepted by nondeterministic polynomial time machines with a small number of accepting paths in case of acceptance, and also investigate subclasses of NP that are low for complexity classes not known to be in t ..."
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Cited by 31 (5 self)
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this paper we study two different ways to restrict the power of NP: We consider languages accepted by nondeterministic polynomial time machines with a small number of accepting paths in case of acceptance, and also investigate subclasses of NP that are low for complexity classes not known to be in the polynomial time hierarchy. The first complexity class defined following the idea of bounding the number of accepting paths was Valiant's class UP (unique P) [Va76] of languages accepted by nondeterministic Turing machines that have exactly one accepting computation path for strings in the language, and none for strings not in the language. This class plays an important role in the areas of oneway functions and cryptography, for example in [GrSe84] it is shown that P6=UP if and only if oneway functions exist. The class UP can be generalized in a natural way by allowing a polynomial number of accepting paths. This gives rise to the class FewP defined by Allender [Al85] in connection with the notion of Pprintable sets. We study complexity classes defined by such pathrestricted nondeterministic polynomial time machines, and show results that exploit the fact that the machines for these classes have a bounded number of accepting computation paths. We will not only consider these subclasses of NP, namely UP and FewP, but also the class Few, an extension of FewP defined by Cai and Hemachandra [CaHe89], in which the accepting mechanism of the machine is more flexible. 1 The three classes UP, FewP and Few are all defined in terms of nondeterministic machines with a bounded number of accepting paths for every input string, but for the last two classes this number is not known beforehand, and can range over a space of polynomial size. We show in Section 3 that a polynomial numb...
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 30 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
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
An Observation on Probability versus Randomness with Applications to Complexity Classes
 MATHEMATICAL SYSTEMS THEORY
, 1993
"... Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of MartinLof. This result is used to derive separation properties of algorithmically random oracles and to gi ..."
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Cited by 18 (7 self)
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Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of MartinLof. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classes P, BPP, AM, and PH in terms of reducibility to such oracles. These characterizations lead to results like: P = NP if and only if there exists an algorithmically random set that is P btt hard for NP.
Unambiguous Computation: Boolean Hierarchies and Sparse TuringComplete Sets
, 1994
"... This paper studies, for UP, two topics that have been intensely studied for NP: Boolean hierarchies and the consequences of the existence of sparse Turingcomplete sets. Unfortunately, as is often the case, the results for NP draw on special properties of NP that do not seem to carry over straightfor ..."
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Cited by 17 (13 self)
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This paper studies, for UP, two topics that have been intensely studied for NP: Boolean hierarchies and the consequences of the existence of sparse Turingcomplete sets. Unfortunately, as is often the case, the results for NP draw on special properties of NP that do not seem to carry over straightforwardly to UP. For example, it is known for NP (and more generally for any class containing \Sigma and ; and closed under union and intersection) that the symmetric difference hierarchy, the Boolean hierarchy, and the Boolean closure all are equal. We prove that closure under union is not needed for this claim: For any class K that contains \Sigma and ; and is closed under intersection (e.g., UP, US, and FewP), the symmetric difference hierarchy over K, the Boolean hierarchy over K, and the Boolean closure of K all are equal. On the other hand, we show that two hierarchiesthe Hausdorff hierarchy and the nested difference hierarchy which in the NP case are equal to the Boolean cl...
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 17 (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.
Manipulating the quota in weighted voting games
 AAAI
"... Weighted voting games provide a popular model of decision making in multiagent systems. Such games are described by a set of players, a list of players ’ weights, and a quota; a coalition of the players is said to be winning if the total weight of its members meets or exceeds the quota. The power of ..."
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Cited by 9 (4 self)
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Weighted voting games provide a popular model of decision making in multiagent systems. Such games are described by a set of players, a list of players ’ weights, and a quota; a coalition of the players is said to be winning if the total weight of its members meets or exceeds the quota. The power of a player in such games is traditionally identified with her Shapley– Shubik index or her Banzhaf index, two classical power measures that reflect the player’s marginal contributions under different coalition formation scenarios. In this paper, we investigate by how much the central authority can change a player’s power, as measured by these indices, by modifying the quota. We provide tight upper and lower bounds on the changes in the individual player’s power that can result from a change in quota. We also study how the choice of quota can affect the relative power of the players. From the algorithmic perspective, we provide an efficient algorithm for determining whether there is a value of the quota that makes a given player a dummy, i.e., reduces his power (as measured by both indices) to 0. On the other hand, we show that checking which of the two values of the quota makes this player more powerful is computationally hard, namely, complete for the complexity class PP, which is believed to be significantly more powerful than NP. 1