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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 71 (23 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
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...
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 27 (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...
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...
Strong SelfReducibility Precludes Strong Immunity
, 1995
"... Do selfreducible sets inherently lack immunity from deterministic polynomial time? Though this is unlikely to be true in general, in this paper we prove that sufficiently strong selfreducibility precludes sufficiently strong immunity from deterministic polynomial time. In particular, we prove that ..."
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Cited by 6 (3 self)
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Do selfreducible sets inherently lack immunity from deterministic polynomial time? Though this is unlikely to be true in general, in this paper we prove that sufficiently strong selfreducibility precludes sufficiently strong immunity from deterministic polynomial time. In particular, we prove that NT is not P balanced immune. However, we prove that NT, a class whose sets have very strong selfreducibility properties, is P biimmune relative to a generic oracle. Thus, the previous result cannot be relativizably extended to biimmunity. We also prove that NP and \PhiP are both P balanced immune relative to a random oracle; the former provides the strongest known relativized separation of NP from P. 1 Introduction Relativization results have a long but increasingly checkered history. Today, there are conflicting views as to the extent to which relativization results give insight into the structure of feasible computation [HCRR90,All90,HCC + 92,For94], even as oracle construction has ...
The Power of Frequency Computation (Extended Abstract)
 In: Proceedings FCT'95, Lecture Notes in Computer Science
, 1995
"... ) Martin Kummer and Frank Stephan ? Universitat Karlsruhe, Institut fur Logik, Komplexitat und Deduktionssysteme, D76128 Karlsruhe, Germany. fkummer; fstephang@ira.uka.de Abstract. The notion of frequency computation concerns approximative computations of n distinct parallel queries to a set A. A ..."
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) Martin Kummer and Frank Stephan ? Universitat Karlsruhe, Institut fur Logik, Komplexitat und Deduktionssysteme, D76128 Karlsruhe, Germany. fkummer; fstephang@ira.uka.de Abstract. The notion of frequency computation concerns approximative computations of n distinct parallel queries to a set A. A is called (m; n)recursive if there is an algorithm which answers any n distinct parallel queries to A such that at least m answers are correct. This paper gives natural combinatorial characterizations of the fundamental inclusion problem, namely the question for which choices of the parameters m; n; m 0 ; n 0 , every (m;n)recursive set is (m 0 ; n 0 )recursive. We also characterize the inclusion problem restricted to recursively enumerable sets and the inclusion problem for the polynomialtime bounded version of frequency computation. Furthermore, using these characterizations we obtain many explicit inclusions and noninclusions. 1 Introduction Frequency computation is a classic...
Membership Comparable and pselective Sets
"... We show that there exists a 2membership comparable set that is not bttreducible to any pselective set. This is a rare example of an unconditional separation in computational complexity. ..."
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Cited by 1 (0 self)
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We show that there exists a 2membership comparable set that is not bttreducible to any pselective set. This is a rare example of an unconditional separation in computational complexity.
Compressibility of Infinite Binary Sequences
"... It is known that infinite binary sequences of constant Kolmogorov complexity are exactly the recursive ones. Such a kind of statement no longer holds in the presence of resource bounds. Contrary to what intuition might suggest, there are sequences of constant, polynomialtime bounded Kolmogorov comp ..."
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It is known that infinite binary sequences of constant Kolmogorov complexity are exactly the recursive ones. Such a kind of statement no longer holds in the presence of resource bounds. Contrary to what intuition might suggest, there are sequences of constant, polynomialtime bounded Kolmogorov complexity that are not polynomialtime computable. This motivates the study of several resourcebounded variants in search for a characterization, similar in spirit, of the polynomialtime computable sequences. We propose some definitions, based on Kobayashi's notion of compressibility, and compare them to both the standard resourcebounded Kolmogorov complexity of infinite strings, and the uniform complexity. Some nontrivial coincidences and disagreements are proved. The resourceunbounded case is also considered. Partially supported by the E.U. through the HCM network CHRXCT930415 (COLORET), by DGICYT, project number PB920709, and by the Accion Integrada HispanoAlemana HA119B. 1 ...