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22
A Taxonomy of Complexity Classes of Functions
 Journal of Computer and System Sciences
, 1992
"... This paper comprises a systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP. There are three components to this work. ffl A taxonomy is presented that demonstrates all known inclusion relations of these cla ..."
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Cited by 88 (12 self)
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This paper comprises a systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP. There are three components to this work. ffl A taxonomy is presented that demonstrates all known inclusion relations of these classes. For (nearly) each inclusion that is not shown to hold, evidence is presented to indicate that the inclusion is false. As an example, consider FewPF, the class of multivalued functions that are nondeterministically computable in polynomial time such that for each x, there is a polynomial bound on the number of distinct output values of f(x). We show that FewPF ` PF NP tt . However, we show PF NP tt ` FewPF if and only if NP = coNP, and thus PF NP tt ` FewPF is likely to be false. ffl Whereas it is known that P NP (O(log n)) = P NP tt ` P NP [Hem87, Wagb, BH88], we show that PF NP (O(log n)) = PF NP tt implies P = FewP and R = NP. Also, we show that PF NP tt = PF ...
Bounded Queries to SAT and the Boolean Hierarchy
 Theoretical Computer Science
, 1991
"... We study the complexity of decision problems that can be solved by a polynomialtime Turing machine that makes a bounded number of queries to an NP oracle. Depending on whether we allow some queries to depend on the results of other queries, we obtain two (probably) different hierarchies. We present ..."
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Cited by 64 (12 self)
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We study the complexity of decision problems that can be solved by a polynomialtime Turing machine that makes a bounded number of queries to an NP oracle. Depending on whether we allow some queries to depend on the results of other queries, we obtain two (probably) different hierarchies. We present several results relating the bounded NP query hierarchies to each other and to the Boolean hierarchy. We also consider the similarlydefined hierarchies of functions that can be computed by a polynomialtime Turing machine that makes a bounded number of queries to an NP oracle. We present relations among these two hierarchies and the Boolean hierarchy. In particular we show for all k that there are functions computable with 2 k parallel queries to an NP set that are not computable in polynomial time with k serial queries to any oracle, unless P = NP. As a corollary k + 1 parallel queries to an NP set allow us to compute more functions than are computable with only k parallel queries to a...
PSelective Sets, and Reducing Search to Decision vs. SelfReducibility
, 1993
"... We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to de ..."
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Cited by 39 (9 self)
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We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not selfreducible. Funding for this research was provided by the National Science Foundation under grant CCR9002292. y Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 z Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 x Research performed while visiting the Department of Computer Science, State University of New York at Buffalo, Jan. 1992Dec. 1992. Current address: Department of Computer Science, University of ElectroCommunications, Chofushi, Tokyo 182, Japan.  Department of Computer Science, State University of New York at Buffalo, 226...
On Using Oracles That Compute Values
 In Proc. 10th Annual Symp. on Theoret. Aspects of Computer Science, Lecture Notes in Computer Science
, 1993
"... This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown th ..."
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Cited by 17 (6 self)
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This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown that 1. A multivalued partial function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable via 2 k \Gamma 1 nonadaptive queries to NPMV. 2. A characteristic function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable with k adaptive queries to NP. 3. Unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV is different than k + 1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the...
Quantifying the Amount of Verboseness
, 1995
"... We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explic ..."
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Cited by 16 (6 self)
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We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explicit description we consider several interesting combinatorial problems. 1 Introduction In the theory of bounded queries, we measure the complexity of a function by the number of queries to an oracle which are needed to compute it. The field has developed in various directions, both in complexity theory and in recursion theory; see Gasarch [21] for a recent survey. One of the original concerns is the classification of sets A of natural numbers by their "query complexity," i.e., according to the number of oracle queries that are needed to compute the nfold characteristic function F A n = x 1 ; : : : ; x n : (ØA (x 1 ); : : : ; ØA (x n )). In [3, 8] a set A is called verbose iff F A n is com...
Bounded Query Classes and the Difference Hierarchy
 Archive for Mathematical Logic
, 1995
"... Let A be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible to A based on bounding the number of queries to A that an oracle machine can make. When A is the halting problem K our hierarchy of sets interleaves with the difference hierarch ..."
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Cited by 15 (12 self)
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Let A be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible to A based on bounding the number of queries to A that an oracle machine can make. When A is the halting problem K our hierarchy of sets interleaves with the difference hierarchy Current address: Department of Computer Science, Yale University, 51 Prospect Street, P.O. Box 2158 Yale Station, New Haven, CT 06520. Supported in part by NSF grant CCR8808949. Part of this work was completed while this author was a student at Stanford University supported by fellowships from the National Science Foundation and from the Fannie and John Hertz Foundation. y Supported in part by NSF grant CCR8803641. z Part of this work was completed while this author was on sabbatical leave at the University of California, Berkeley. on the r.e. sets in a logarithmic way; this follows from a tradeoff between the number of parallel queries and the number of serial queries needed to...
On Membership Comparable Sets
 Journal of Computer and System Sciences
, 1999
"... A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(log n) membership comparable, then Unique ..."
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Cited by 15 (1 self)
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A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(log n) membership comparable, then UniqueSAT 2 P. This extends the work of Ogihara; Beigel, Kummer, and Stephan; and Agrawal and Arvind [Ogi94, BKS94, AA94], and answers in the affirmative an open question suggested by Buhrman, Fortnow, and Torenvliet [BFT97]. Our proof also shows that if SAT is o(n) membership comparable, then UniqueSAT can be solved in deterministic time 2 o(n) . Our main technical tool is an algorithm of Ar et al. [ALRS92] to reconstruct polynomials from noisy data through the use of bivariate polynomial factorization.
On the Query Complexity of Clique Size and Maximum Satisfiability
 In Proceedings of the 9th Structure in Complexity Theory Conference
, 1996
"... This paper explores the bounded query complexity of approximating the size of the maximum clique in a graph (Clique Size) and the number of simultaneously satisfiable clauses in a 3CNF formula (MaxSat). The results in the paper show that for certain approximation factors, approximating Clique Size a ..."
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Cited by 12 (6 self)
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This paper explores the bounded query complexity of approximating the size of the maximum clique in a graph (Clique Size) and the number of simultaneously satisfiable clauses in a 3CNF formula (MaxSat). The results in the paper show that for certain approximation factors, approximating Clique Size and MaxSat are complete for corresponding bounded query classes under metric reductions. The completeness result is important because it shows that queries and approximation are interchangeable: NP queries can be used to solve NPapproximation problems and solutions to NPapproximation problems answer queries to NP oracles. Completeness also shows the existence of approximation preserving reductions from many NPapproximation problems to approximating Clique Size and MaxSat (e.g., from approximating Chromatic Number to approximating Clique Size). Since query complexity is a quantitative complexity measure, these results also provide a framework for comparing the complexities of approximating C...
Six Hypotheses in Search of a Theorem
, 1997
"... We consider the following six hypotheses: ffl P = NP. ffl SAT is truthtable reducible to a Pselective set. ffl SAT is truthtable reducible to a k approximable set for some k. ffl FP NP jj = FP NP[log] ffl SAT is O(log n)approximable. ffl Solving SAT is in P on formulae with at most one ..."
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Cited by 12 (3 self)
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We consider the following six hypotheses: ffl P = NP. ffl SAT is truthtable reducible to a Pselective set. ffl SAT is truthtable reducible to a k approximable set for some k. ffl FP NP jj = FP NP[log] ffl SAT is O(log n)approximable. ffl Solving SAT is in P on formulae with at most one assignment. We discuss their importance and relationships among them. URL: http://www.cwi.nl/cwi/people/Harry.Buhrman.html. Email: buhrman@cwi.nl. Partially supported by the Dutch foundation for scientific research (NWO) by SION project 61234002, and by the European Union through NeuroCOLT ESPRIT Working Group Nr. 8556, and HC&M grant nr. ERB4050PL930516. CWI, Kruislaan 413, 1098SJ Amsterdam, The Netherlands. y URL: http://www.cs.uchicago.edu/~fortnow. Email: fortnow @cs.uchicago.edu. Supported in part by NSF grant CCR 9253582, the Dutch Foundation for Scientific Research (NWO) and a Fulbright Scholar award. CWI and University of Chicago, Department of Computer Science 1100 E. 58t...
Oracles That Compute Values
, 1997
"... . This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown ..."
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Cited by 11 (4 self)
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. This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown that 1. A multivalued partial function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable via 2 k \Gamma 1 nonadaptive queries to NPMV. 2. A characteristic function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable with k adaptive queries to NP. 3. Unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV is different than k + 1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses. Key words. computational complexity, complexity classes, relativized computation, bounded query classes, Boolean hierarchy, multivalued functions, NPMV AMS subject classifications. 68Q05, 68Q10, 68Q15, 03D10, 03D15 1.