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40
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 33 (6 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.
Proving SAT does not have Small Circuits with an Application to the Two Queries Problem
, 2002
"... We show that if SAT does not have small circuits, then there must exist a small number of formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P , then the polynomialtime hierarchy collapses to S ..."
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Cited by 19 (2 self)
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We show that if SAT does not have small circuits, then there must exist a small number of formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P , then the polynomialtime hierarchy collapses to S 2 2 . Even showing that the hierarchy collapsed to 2 remained open prior to this paper.
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 18 (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...
The Power of the Middle Bit of a #P Function
, 1995
"... This paper studies the class MP of languages which can be solved in polynomial time with the additional information of one bit from a #P function f . The middle bit of f(x) is shown to be as powerful as any other bit, whereas the O(log n) bits at either end are apparently weaker. The polynomial hie ..."
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Cited by 17 (3 self)
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This paper studies the class MP of languages which can be solved in polynomial time with the additional information of one bit from a #P function f . The middle bit of f(x) is shown to be as powerful as any other bit, whereas the O(log n) bits at either end are apparently weaker. The polynomial hierarchy and the classes Mod k P, k 2, are shown to be low for MP. They are also low for a class we call AmpMP which is defined by abstracting the "amplification" methods of Toda (SIAM J. Comput. 20 (1991), 865877). Consequences of these results for circuit complexity are obtained using the concept of a MidBit gate, which is defined to take binary inputs x 1 ; : : : ; xw and output the blog 2 (w)=2c th bit in the binary representation of the number P w i=1 x i . Every language in ACC can be computed by a family of depth2 deterministic circuits of size 2 (log n) O(1) with a MidBit gate at the root and ANDgates of fanin (log n) O(1) at the leaves. This result improves the known ...
What’s up with downward collapse: Using the easyhard technique to link boolean and polynomial hierarchy collapses
 DEPARTMENT OF COMPUTER SCIENCE, UNIVERSITY OF ROCHESTER
, 1998
"... During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or boundedquery hierarchies collapse. The final four papers of this ninepaper progression actually achieve downward collapse—that is, they show that highlevel collapses induce col ..."
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Cited by 13 (7 self)
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During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or boundedquery hierarchies collapse. The final four papers of this ninepaper progression actually achieve downward collapse—that is, they show that highlevel collapses induce collapses at (what beforehand were thought to be) lower complexity levels. For example, for each k ≥ 2 it is now known that if PΣp k [1] = PΣp k [2] then PH = Σ p k. This article surveys the history, the results, and the technique—the socalled easyhard method—of these nine papers.
Bounded queries in recursion theory: A survey
 In Proc. of the 6th Annu. Conference on Structure in Complexity Theory
, 1991
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Raising NP Lower Bounds to Parallel NP Lower Bounds
 SIGACT News
, 1997
"... A decade ago, a beautiful paper by Wagner [Wag87] developed a "toolkit" that in certain cases allows one to prove problems hard for parallel access to NP. However, the problems his toolkit applies to most directly are not overly natural. During the past year, problems that previously were ..."
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Cited by 9 (9 self)
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A decade ago, a beautiful paper by Wagner [Wag87] developed a "toolkit" that in certain cases allows one to prove problems hard for parallel access to NP. However, the problems his toolkit applies to most directly are not overly natural. During the past year, problems that previously were known only to be NPhard or coNPhard have been shown to be hard even for the class of sets solvable via parallel access to NP. Many of these problems are longstanding and extremely natural, such as the Minimum Equivalent Expression problem [GJ79] (which was the original motivation for creating the polynomial hierarchy), the problem of determining the winner in the election system introduced by Lewis Carroll in 1876 [Dod76], and the problem of determining on which inputs heuristic algorithms perform well. In the present article, we survey this recent progress in raising lower bounds. 1 Introduction Suppose you are given some nice, challenging problem and you are able to prove an NPhardness lower boun...
Bounded queries, approximations and the Boolean hierarchy
 Electronic Colloquium on Computational Complexity
, 1997
"... This paper investigates nondeterministic bounded query classes in relation to the complexity of NPhard approximation problems and the Boolean Hierarchy. Nondeterministic bounded query classes turn out be rather suitable for describing the complexity of NPhard approximation problems. The results in ..."
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Cited by 7 (3 self)
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This paper investigates nondeterministic bounded query classes in relation to the complexity of NPhard approximation problems and the Boolean Hierarchy. Nondeterministic bounded query classes turn out be rather suitable for describing the complexity of NPhard approximation problems. The results in this paper take advantage of this machinebased model to prove that in many cases, NPapproximation problems have the upward collapse property. That is, a reduction between NPapproximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level. For example, if MaxClique reduces to (log n)approximating MaxClique using manyone reductions, then the Traveling Salesman Problem (TSP) is equivalent to MaxClique under manyone reductions. Several upward collapse theorems are presented in this paper. The proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes. In fact, these results depend on a surprising connection between the Boolean Hierarchy and nondeterministic bounded query classes.
ON THE STRUCTURE OF NP COMPUTATIONS UNDER BOOLEAN OPERATORS
, 1991
"... This thesis is mainly concerned with the structural complexity of the Boolean Hierarchy. The Boolean Hierarchy is composed of complexity classes constructed using Boolean operators on NP computations. The thesis begins with a description of the role of the Boolean Hierarchy in the classification of ..."
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Cited by 7 (0 self)
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This thesis is mainly concerned with the structural complexity of the Boolean Hierarchy. The Boolean Hierarchy is composed of complexity classes constructed using Boolean operators on NP computations. The thesis begins with a description of the role of the Boolean Hierarchy in the classification of the complexity of NP optimization problems. From there, the thesis goes on to motivate the basic definitions and properties of the Boolean Hierarchy. Then, these properties are shown to depend only on the closure of NP under the Boolean operators, AND2 and OR2. A central theme of this thesis is the development of the hard/easy argument which shows intricate connections between the Boolean Hierarchy and the Polynomial Hierarchy. The hard/easy argument shows that the Boolean Hierarchy cannot collapse unless the Polynomial Hierarchy also collapses. The results shown in this regard are improvements over those previously shown by Kadin. Furthermore, it is shown that the hard/easy argument can be adapted for Boolean hierarchies over incomplete NP languages. That is, under the assumption that certain incom