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25
Complexity Limitations on Quantum Computation
 Journal of Computer and System Sciences
, 1997
"... We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = ..."
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Cited by 98 (3 self)
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We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = BQP and the polynomialtime hierarchy is infinite.  There exists a relativized world where BQP does not have complete sets.  There exists a relativized world where P = BQP but P 6= UP " coUP and oneway functions exist. This gives a relativized answer to an open question of Simon.
On the Power of NumberTheoretic Operations with Respect to Counting
 IN PROCEEDINGS 10TH STRUCTURE IN COMPLEXITY THEORY
, 1995
"... We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain t ..."
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Cited by 32 (9 self)
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We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain the following complete characterization of these operations: #P is closed under f in every relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized case. For closure properties f of #P, we have h#Pi f = #P. The other end of the range is marked by operations f for which h#Pi f corresponds to the counting hierarchy. We call these operations counting hard and give general criteria for hardness. For many operations f we show that h#Pi f corresponds to some subclass C of the counting hierarchy. This will then imply that #P is closed under f if and only if ...
Relating Polynomial Time to Constant Depth
 THEORETICAL COMPUTER SCIENCE
, 1998
"... Going back to the seminal paper [FSS84] by Furst, Saxe, and Sipser, analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers. Oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower ..."
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Cited by 12 (2 self)
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Going back to the seminal paper [FSS84] by Furst, Saxe, and Sipser, analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers. Oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower bounds for circuit classes. In this note we show how separating oracles can be obtained uniformly from circuit lower bounds without the need of carrying out a particular diagonalization. Our technical tool is the leaf language approach to the definition of complexity classes.
Finite Automata with Generalized Acceptance Criteria
 IN THE PROCEEDINGS OF THE 26TH INTERNATIONAL COLLOQIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING, LECTURE
, 2001
"... ..."
Optimal Proof Systems Imply Complete Sets For Promise Classes
 INFORMATION AND COMPUTATION
, 2001
"... A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an hproof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned this concept in [11] and in order to compare the relative strength of dierent proof systems for ..."
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Cited by 10 (1 self)
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A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an hproof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned this concept in [11] and in order to compare the relative strength of dierent proof systems for the set TAUT of tautologies in propositional logic, they considered the notion of psimulation. Intuitively, a proof system h psimulates h if any hproof w can be translated in polynomial time into an h for h(w). Krajcek and Pudlak [18] considered the related notion of simulation between proof systems where it is only required that for any hproof w there exists an h whose size is polynomially bounded in the size of w.
On Cluster Machines and Function Classes
, 1997
"... We consider a special kind of nondeterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalence relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the ..."
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Cited by 9 (1 self)
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We consider a special kind of nondeterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalence relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the machineindependend concept of cluster sets which is the starting point to establish cluster operators. Cluster operators map complexity classes of sets into complexity classes of functions where for the domain classes only cluster sets are allowed. For the counting operator c#\Delta and the optimization operators cmax\Delta and cmin\Delta the structural relationships between images resulting from these operators on the polynomialtime hierarchy are investigated. Furthermore, we compare cluster operators with the corresponding common operators #\Delta, max\Delta and min\Delta [Tod90b, HW97].
The Boolean Hierarchy over Level 1/2 of the StraubingThérien Hierarchy
, 1998
"... For some fixed alphabet A with jAj 2, a language L ` A is in the class L 1=2 of the StraubingTherien hierarchy if and only if it can be expressed as a finite union of languages A a 1 A a 2 A \Delta \Delta \Delta A anA , where a i 2 A and n 0. The class L 1 is defined as the boo ..."
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Cited by 9 (3 self)
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For some fixed alphabet A with jAj 2, a language L ` A is in the class L 1=2 of the StraubingTherien hierarchy if and only if it can be expressed as a finite union of languages A a 1 A a 2 A \Delta \Delta \Delta A anA , where a i 2 A and n 0. The class L 1 is defined as the boolean closure of L 1=2 . It is known that the classes L 1=2 and L 1 are decidable. We give a membership criterion for the single classes of the boolean hierarchy over L 1=2 . From this criterion we can conclude that this boolean hierarchy is proper and that its classes are decidable. In finite model theory the latter implies the decidability of the classes of the boolean hierarchy over the class \Sigma 1 of the FO[!]logic. Moreover we prove a "forbiddenpattern" characterization of L 1 of the type: L 2 L 1 if and only if a certain pattern does not appear in the transition graph of a deterministic finite automaton accepting L. We discuss complexity theoretical consequences of our results. C...
On NPPartitions over Posets with an Application to Reducing the Set of Solutions of NP Problems
 In Proceedings 25th Symposium on Mathematical Foundations of Computer Science
, 2000
"... . The boolean hierarchy of kpartitions over NP for k 2 was introduced as a generalization of the wellknown boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NPpartiti ..."
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Cited by 7 (3 self)
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. The boolean hierarchy of kpartitions over NP for k 2 was introduced as a generalization of the wellknown boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NPpartitions by considering partition classes which are generated by nite labeled posets. Since we cannot prove it absolutely, we collect evidence for this extended boolean hierarchy to be strict. We give an exhaustive answer to the question of which relativizable inclusions between partition classes can occur depending on the relation between their dening posets. The study of the extended boolean hierarchy is closely related to the issue of whether one can reduce the number of solutions of NP problems. For nite cardinality types, assuming the extended boolean hierarchy of kpartitions over NP is strict, we give a complete characterization when such solution reductions are possible. 1 Introduct...
Reducing the Number of Solutions of NP Functions
, 2000
"... We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functi ..."
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Cited by 7 (4 self)
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We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses.
Succinct Inputs, Lindström Quantifiers, and a General Complexity Theoretic Operator Concept
 IN READERS OF THE NINTH EUROPEAN SUMMER SCHOOL IN LOGIC, LANGUAGE AND INFORMATION, CHAPTER CL7. CNRS AIXENPROVENCE AND THE EUROPEAN ASSOCIATION FOR LOGIC, LANGUAGE AND INFORMATION
, 1996
"... We address the question of the power of several logics with Lindstrom quantifiers over finite ordered structures. We will see that in the firstorder case this nicely fits into the framework of Barrington, Immerman, and Straubing's examination of constant depth circuit classes. In the secondorder c ..."
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Cited by 7 (2 self)
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We address the question of the power of several logics with Lindstrom quantifiers over finite ordered structures. We will see that in the firstorder case this nicely fits into the framework of Barrington, Immerman, and Straubing's examination of constant depth circuit classes. In the secondorder case we get a strong relationship to succinct encodings of languages via circuits. Some of these logics can be characterized as closures of succinct encodings under appropriate reducibilities, others by certain hierarchies of circuit classes. We will see that in a special case secondorder Lindstrom quantifiers can equivalently be expressed in firstorder logic, while in the general case this equivalence seems unlikely.