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70
Propositional Circumscription and Extended Closed World Reasoning are $\Pi^P_2$complete
 Theoretical Computer Science
, 1993
"... Circumscription and the closed world assumption with its variants are wellknown nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction prob ..."
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Cited by 108 (20 self)
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Circumscription and the closed world assumption with its variants are wellknown nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is $\Pi^P_2$complete, i.e., complete for a class of the second level of the polynomial hierarchy. We answer this question by proving these problems $\Pi^P_2$complete, and we show how this result applies to other variants of closed world reasoning.
Structure in Approximation Classes
, 1996
"... this paper we obtain new results on the structure of several computationallydefined approximation classes. In particular, after defining a new approximation preserving reducibility to be used for as many approximation classes as possible, we give the first examples of natural NPOcomplete problems ..."
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Cited by 77 (13 self)
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this paper we obtain new results on the structure of several computationallydefined approximation classes. In particular, after defining a new approximation preserving reducibility to be used for as many approximation classes as possible, we give the first examples of natural NPOcomplete problems and the first examples of natural APXintermediate problems. Moreover, we state new connections between the approximability properties and the query complexity of NPO problems.
Notes on Polynomially Bounded Arithmetic
"... We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The p ..."
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Cited by 60 (1 self)
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We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2 The axioms of secondorder bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.3 Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.5 Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.6 Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.7 Relations with Buss' bounded arithmetic. : : : :...
On the Computational Complexity of Qualitative Coalitional Games
 Artificial Intelligence
, 2004
"... We study coalitional games in which agents are each assumed to have a goal to be achieved, and where the characteristic property of a coalition is a set of choices, with each choice denoting a set of goals that would be achieved if the choice was made. Such qualitative coalitional games (QCGs) are a ..."
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Cited by 52 (15 self)
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We study coalitional games in which agents are each assumed to have a goal to be achieved, and where the characteristic property of a coalition is a set of choices, with each choice denoting a set of goals that would be achieved if the choice was made. Such qualitative coalitional games (QCGs) are a natural tool for modelling goaloriented multiagent systems. After introducing and formally defining QCGs, we systematically formulate fourteen natural decision problems associated with them, and determine the computational complexity of these problems. For example, we formulate a notion of coalitional stability inspired by that of the core from conventional coalitional games, and prove that the problem of showing that the core of a QCG is nonempty is D 1 complete. (As an aside, we present what we believe is the first "natural" problem that is proven to be complete for D 2 .) We conclude by discussing the relationship of our work to other research on coalitional reasoning in multiagent systems, and present some avenues for future research.
The Boolean Hierarchy and the Polynomial Hierarchy: a Closer Connection
 SIAM Journal on Computing
, 1993
"... We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH 3 (k), where BH 3 (k) is the k th level of the Boolean hierarchy over \Sigma P 2 . This is an improvement over the known results [3], which show that the polynomial hierarchy would collapse t ..."
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Cited by 51 (13 self)
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We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH 3 (k), where BH 3 (k) is the k th level of the Boolean hierarchy over \Sigma P 2 . This is an improvement over the known results [3], which show that the polynomial hierarchy would collapse to P NP NP [O(log n)] . This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. Also, this result points to some previously unexplored connections between the Boolean and query hierarchies of \Delta P 2 and \Delta P 3 . Namely, BH(k) = coBH(k) =) BH 3 (k) = coBH 3 (k) P NPk[k] = P NPk[k+1] =) P NP NP k[k+1] = P NP NP k[k+2] : Key words: polynomial time hierarchy, Boolean hierarchy, polynomial time Turing reductions, oracle access, nonuniform algorithms, sparse sets AMS (MOS) subject classifications: 68Q15, 03D15, 03D20 1 Introduction The Boolean hierarchy (BH) w...
A Relationship between Difference Hierarchies and Relativized Polynomial Hierarchies
, 1993
"... Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal the kth level of the difference hierarchy over \Sigma p 2 . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP c ..."
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Cited by 40 (8 self)
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Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal the kth level of the difference hierarchy over \Sigma p 2 . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level k, then PH collapses to i P NP (k\Gamma1)tt j NP , the class of sets recognized in polynomial time with k \Gamma 1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class C that has p m complete sets and is closed under p conj  and NP m reductions (alternatively, closed under p disj  and coNP m reductions), if the difference hierarchy over C collapses to level k, then PH C = i P NP (k\Gamma1)tt j C . Then we show that the exact counting class C=P is closed under p disj  and coNP m  reductions. Consequently, if the difference hiera...
Two queries
 In CCC
, 1999
"... We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits. ..."
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Cited by 31 (7 self)
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We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits.
NPhard Sets are PSuperterse Unless R = NP
, 1992
"... A set A is pterse (psuperterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of ..."
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Cited by 28 (5 self)
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A set A is pterse (psuperterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of functions reducible to A via a polynomialtime Turing reduction that makes at most q queries. A set A is pterse if PF A qtt 6` PF A (q\Gamma1)T for all constants q. A is psuperterse if PF A qtt 6` PF X qT for all constants q and sets X . We show that all NPhard sets (under p tt reductions) are psuperterse, unless it is possible to distinguish uniquely satisfiable formulas from satisfiable formulas in polynomial time. Consequently, all NPcomplete sets are psuperterse unless P = UP (oneway functions fail to exist), R = NP (there exist randomized polynomialtime algorithms for all problems in NP), and the polynomialtime hierarchy collapses. This mostly solves the main open...
The random oracle hypothesis is false
, 1990
"... The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which holdforalmost all relativized worlds must also hold in the unrelativized case. Although this paper is not the rst to provideacounterexample to the Random Oracle Hy ..."
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Cited by 28 (2 self)
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The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which holdforalmost all relativized worlds must also hold in the unrelativized case. Although this paper is not the rst to provideacounterexample to the Random Oracle Hypothesis, it does provide a most compelling counterexample by showing that for almost all oracles A, IP A 6=PSPACE A. If the Random Oracle Hypothesis were true, it would contradict Shamir's result that IP = PSPACE. In fact, it is shown that for almost all oracles A, coNP A 6 IP A. These results extend to the multiprover proof systems of BenOr, Goldwasser, Kilian and Wigderson. In addition, this paper shows that the Random Oracle Hypothesis is sensitive to small changes in the de nition. A class IPP, similar to IP, is de ned. Surprisingly, the IPP = PSPACE result holds for all oracle worlds. Warning: Essentially this paper has been published in Information and Computation and is hence subject to copyright restrictions. It is for personal use only. 1
Guarantees for the success frequency of an algorithm for finding Dodgsonelection winners
 In Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science
, 2006
"... Dodgson’s election system elegantly satisfies the Condorcet criterion. However, determining the winner of a Dodgson election is known to be Θ p 2complete ([HHR97], see also [BTT89]), which implies that unless P = NP no polynomialtime solution to this problem exists, and unless the polynomial hiera ..."
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Cited by 23 (7 self)
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Dodgson’s election system elegantly satisfies the Condorcet criterion. However, determining the winner of a Dodgson election is known to be Θ p 2complete ([HHR97], see also [BTT89]), which implies that unless P = NP no polynomialtime solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates (although the number of voters may still be polynomial in the number of candidates), a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner. 1