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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 66 (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...
QueryLimited Reducibilities
, 1995
"... We study classes of sets and functions computable by algorithms that make a limited number of queries to an oracle. We distinguish between queries made in parallel (each question being independent of the answers to the others, as in a truthtable reduction) and queries made in serial (each question ..."
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Cited by 41 (14 self)
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We study classes of sets and functions computable by algorithms that make a limited number of queries to an oracle. We distinguish between queries made in parallel (each question being independent of the answers to the others, as in a truthtable reduction) and queries made in serial (each question being permitted to depend on the answers to the previous questions, as in a Turing reduction). We define computability by a set of functions, and we show that it captures the informationtheoretic aspects of computability by a fixed number of queries to an oracle. Using that concept, we prove a very powerful result, the Nonspeedup Theorem, which states that 2 n parallel queries to any fixed nonrecursive oracle cannot be answered by an algorithm that makes only n queries to any oracle whatsoever. This is the tightest general result possible. A corollary is the intuitively obvious, but nontrivial result that additional parallel queries to an oracle allow us to compute additional functions; t...
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 31 (4 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...
On the Structure of Degrees of Inferability
 Journal of Computer and System Sciences
, 1993
"... Degrees of inferability have been introduced to measure the learning power of inductive inference machines which have access to an oracle. The classical concept of degrees of unsolvability measures the computing power of oracles. In this paper we determine the relationship between both notions. ..."
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Cited by 31 (18 self)
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Degrees of inferability have been introduced to measure the learning power of inductive inference machines which have access to an oracle. The classical concept of degrees of unsolvability measures the computing power of oracles. In this paper we determine the relationship between both notions. 1 Introduction We consider learning of classes of recursive functions within the framework of inductive inference [21]. A recent theme is the study of inductive inference machines with oracles ([8, 10, 11, 17, 24] and tangentially [12]; cf. [10] for a comprehensive introduction and a collection of all previous results.) The basic question is how the information content of the oracle (technically: its Turing degree) relates with its learning power (technically: its inference degreedepending on the underlying inference criterion). In this paper a definitive answer is obtained for the case of recursively enumerable oracles and the case when only finitely many queries to the oracle are allo...
On the Complexity of Finding the Chromatic Number of a Recursive Graph I: The Bounded Case
 Annals of Pure and Applied Logic
, 1989
"... We classify functions in recursive graph theory in terms of how many queries to K (or # ## or # ### ) are required to compute them. We show that (1) binary search is optimal (in terms of the number of queries to K) for finding the chromatic number of a recursive graph and that no set of Turing d ..."
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Cited by 17 (10 self)
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We classify functions in recursive graph theory in terms of how many queries to K (or # ## or # ### ) are required to compute them. We show that (1) binary search is optimal (in terms of the number of queries to K) for finding the chromatic number of a recursive graph and that no set of Turing degree less than 0 # will su#ce, (2) determining if a recursive graph has a finite chromatic number is # 2 complete, and (3) binary search is optimal (in terms of the number of queries to # ### ) for finding the recursive chromatic number of a recursive graph and that no set of Turing degree less than 0 ### will su#ce. We also explore how much help queries to a weaker set may provide. Some of our results have analogues in terms of asking p questions at a time, but some do not. In particular, (p + 1)ary search is not always optimal for finding the chromatic number of a recursive graph. Most of our results are also true for highly recursive graphs, though there are some interesting di#erenc...
Learning Recursive Functions from Approximations
, 1995
"... Investigated is algorithmic learning, in the limit, of correct programs for recursive functions f from both input/output examples of f and several interesting varieties of approximate additional (algorithmic) information about f . Specifically considered, as such approximate additional informatio ..."
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Cited by 17 (7 self)
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Investigated is algorithmic learning, in the limit, of correct programs for recursive functions f from both input/output examples of f and several interesting varieties of approximate additional (algorithmic) information about f . Specifically considered, as such approximate additional information about f , are Rose's frequency computations for f and several natural generalizations from the literature, each generalization involving programs for restricted trees of recursive functions which have f as a branch. Considered as the types of trees are those with bounded variation, bounded width, and bounded rank. For the case of learning final correct programs for recursive functions, EX learning, where the additional information involves frequency computations, an insightful and interestingly complex combinatorial characterization of learning power is presented as a function of the frequency parameters. For EX learning (as well as for BClearning, where a final sequence of cor...
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...
Bounded queries in recursion theory: A survey
 In Proc. 6th Structure in Complexity Theory
, 1991
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Learning via Queries and Oracles
 In Proc. 8th Annu. Conf. on Comput. Learning Theory
, 1996
"... Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a nonrecursive oracle. This paper combines these two types  it considers three basic models of queries to a teacher, namely QEX[Succ], QEX[!] and QEX[+], together with members ..."
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Cited by 11 (2 self)
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Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a nonrecursive oracle. This paper combines these two types  it considers three basic models of queries to a teacher, namely QEX[Succ], QEX[!] and QEX[+], together with membership queries to some oracle. The results for these three models of queryinference are very similar: If an oracle is already omniscient for queryinference, then it is already omniscient for EX. There is an oracle of trivial EXdegree, which allows nontrivial queryinference. Furthermore, queries to a teacher can not overcome differences between oracles and the queryinference degrees are a proper refinement of the EXdegrees. 1 Introduction One famous example of learning via queries to a teacher is the game Mastermind. The teacher first selects the code  a quadruple of colours  that should be learned. Then the learner tries to figure out the code. In each round, the learner makes one gue...