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39
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 70 (22 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
Complexity Classes Defined By Counting Quantifiers
, 1991
"... We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other com ..."
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Cited by 59 (0 self)
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We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded complexity classes.
On PolynomialTime Bounded TruthTable Reducibility of NP Sets to Sparse Sets
, 1991
"... We prove that if P ≠ NP, then there exists a set in NP that is not polynomial time bounded truthtable reducible (in short, p btt reducible) to any sparse set. In other words, we prove that no sparse p btt hard set exists for NP unless P = NP. By using the technique proving this result, w ..."
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Cited by 46 (4 self)
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We prove that if P &ne; NP, then there exists a set in NP that is not polynomial time bounded truthtable reducible (in short, p btt reducible) to any sparse set. In other words, we prove that no sparse p btt hard set exists for NP unless P = NP. By using the technique proving this result, we investigate intractability of several number theoretic decision problems, i.e., decision problems defined naturally from number theoretic problems. We show that for these number theoretic decision problems, if it is not in P, then it is not p btt reducible to any sparse set.
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...
Lower Bounds for the Low Hierarchy
"... this paper. The low hierarchy, as defined in [Sc83], can only be used to classify the complexity of sets in NP. In order to talk about related sets that are not in NP, the extended low hierarchy was introduced in [BBS86]. (The levels of this hierarchy are labeled EL 2 ,EL 2 , A preliminary ver ..."
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Cited by 34 (4 self)
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this paper. The low hierarchy, as defined in [Sc83], can only be used to classify the complexity of sets in NP. In order to talk about related sets that are not in NP, the extended low hierarchy was introduced in [BBS86]. (The levels of this hierarchy are labeled EL 2 ,EL 2 , A preliminary version of this paper was presented at the 16th International Colloquium on Automata, Languages, and Programming [AH89a]
PolynomialTime Membership Comparable Sets
, 1994
"... This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m j ..."
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Cited by 32 (5 self)
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This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomialtime membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomialtime membership comparable sets have polynomialsize circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)tt reducible to a Pselective set, then the set is polynomialtime (1 + c) log f(n)membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` Pmc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...
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...
On The Computational Complexity of Inferring Evolutionary Trees
, 1993
"... The process of reconstructing evolutionary trees can be viewed formally as an optimization problem. Recently, decision problems associated with the most commonly used approaches to reconstructing such trees have been shown to be NPcomplete [Day87, DJS86, DS86, DS87, GF82, Kri88, KM86]. In this t ..."
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Cited by 20 (4 self)
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The process of reconstructing evolutionary trees can be viewed formally as an optimization problem. Recently, decision problems associated with the most commonly used approaches to reconstructing such trees have been shown to be NPcomplete [Day87, DJS86, DS86, DS87, GF82, Kri88, KM86]. In this thesis, a framework is established that incorporates all such problems studied to date. Within this framework, the NPcompleteness results for decision problems are extended by applying theorems from [CT91, Gas86, GKR92, JVV86, KST89, Kre88, Sel91] to derive bounds on the computational complexity of several functions associated with each of these problems, namely ffl evaluation functions, which return the cost of the optimal tree(s), ffl solution functions, which return an optimal tree, ffl spanning functions, which return the number of optimal trees, ffl enumeration functions, which systematically enumerate all optimal trees, and ffl randomselection functions, which return a random...
Circuit size relative to pseudorandom oracles
 THEORETICAL COMPUTER SCIENCE A 107
, 1993
"... Circuitsize complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspacerandom oracle A, and relative toalmost every oracle A 2 ESPACE. (i) NP A is not contained in SIZE ..."
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Cited by 17 (4 self)
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Circuitsize complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspacerandom oracle A, and relative toalmost every oracle A 2 ESPACE. (i) NP A is not contained in SIZE A (2 n) for any real < 1 3. (ii) E A is not contained in SIZE A ( 2n n). Thus, neither NP A nor E A is contained in P A /Poly. In fact, these separations are shown to hold for almost every n. Since a randomly selected oracle is pspacerandom with probability one, (i) and (ii) immediately imply the corresponding random oracle separations, thus improving a result of Bennett and Gill [9] and answering open questions of Wilson [47].
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 ...