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80
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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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...
A guided tour of minimal indices and shortest descriptions
 Archive for Mathematical Logic
, 1998
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DegreeTheoretic Aspects of Computably Enumerable Reals
 in Models and Computability
, 1998
"... A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequ ..."
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Cited by 12 (0 self)
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A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to #: For example, every representation A of # is Turing reducible to L###: Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. reals are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below deg T L### necessarily contains a representation of #: 1 Introduction Computability theory essentially studies the relative computability of sets of natural numbers. Since G#odel introduced a method for coding s...
Asymptotic density and computably enumerable sets
"... We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in ..."
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Cited by 11 (6 self)
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We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: (i) The degrees of such sets A are precisely the nonlow c.e. degrees. (ii) There is a c.e. set A of density 1 with no computable subset of nonzero density. (iii) There is a c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A \ B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of “computable at density r ” where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness.
Turing degrees of certain isomorphic images of computable relations
 Ann. Pure Appl. Logic
, 1998
"... This paper is dedicated to Chris Ash, who invented αsystems. Abstract. A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A. That is, R is not namedinthelanguag ..."
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This paper is dedicated to Chris Ash, who invented αsystems. Abstract. A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A. That is, R is not namedinthelanguageofA. Wedefine DgA(R) to be the set of Turing degrees of the images f(R) under all isomorphisms f from A to computable models. We investigate conditions on A and R which are sufficient and necessary for DgA(R) to contain every Turing degree. These conditions imply that if every Turing degree ≤ 0 00 can be realized in DgA(R) via an isomorphism of the same Turing degree as its image of R, thenDgA(R) contains every Turing degree. We also discuss an example of A and R whose DgA(R) coincides with the Turing degrees which are ≤ 0 0. 1. Introduction and
On Sets Bounded TruthTable Reducible to Pselective Sets
, 1996
"... We show that if every NP set is polynomialtime bounded truthtable reducible to some Pselective set, then NP is contained in DTIME(2 n O(1= p log n) ). In the proof, we implement a recursive procedure that reduces the number of nondeterministic steps of a given nondeterministic computation. ..."
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We show that if every NP set is polynomialtime bounded truthtable reducible to some Pselective set, then NP is contained in DTIME(2 n O(1= p log n) ). In the proof, we implement a recursive procedure that reduces the number of nondeterministic steps of a given nondeterministic computation.
The Communication Complexity of Enumeration, Elimination, and Selection
"... Let k, n ∈ N and f: {0, 1} n × {0, 1} n → {0, 1}. Assume Alice has x1,..., xk ∈ {0, 1} n, Bob has y1,..., yk ∈ {0, 1} n, and they want to compute f k (x1x2 · · · xk, y1y2 · · · yk) = (f(x1, y1), · · · , f(xk, yk)) (henceforth f(x1, y1) · · · f(xk, yk)) communicating as few bits as possibl ..."
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Let k, n ∈ N and f: {0, 1} n × {0, 1} n → {0, 1}. Assume Alice has x1,..., xk ∈ {0, 1} n, Bob has y1,..., yk ∈ {0, 1} n, and they want to compute f k (x1x2 · · · xk, y1y2 · · · yk) = (f(x1, y1), · · · , f(xk, yk)) (henceforth f(x1, y1) · · · f(xk, yk)) communicating as few bits as possible. The Direct Sum Conjecture (henceforth DSC) of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x1, y1), then f(x2, y2), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits since a variant of it implies NC 1 � = NC 2. We consider two related problems. Enumeration: Alice and Bob output e ≤ 2 k − 1 elements of {0, 1} k, one of which is f(x1, y1) · · · f(xk, yk). Elimination: Alice and Bob output � b such that � b � = f(x1, y1) · · · f(xk, yk). Selection: (k = 2) Alice and Bob output i ∈ {1, 2} such that if f(x1, y1) = 1 ∨ f(x2, y2) = 1 then f(xi, yi) = 1. a) We devise the Enumeration Conjecture (henceforth ENC) and the Elimination
The Complexity of Finding TopTodaEquivalenceClass Members
, 2003
"... We identify two properties that for Pselective sets are effectively computable. Namely we show that, for any Pselective set, finding a string that is in a given length's top Toda equivalence class (very informally put, a string from Σ^n that the set's Pselector function declar ..."
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Cited by 7 (4 self)
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We identify two properties that for Pselective sets are effectively computable. Namely we show that, for any Pselective set, finding a string that is in a given length's top Toda equivalence class (very informally put, a string from &Sigma;^n that the set's Pselector function declares to be most likely to belong to the set) is FP computable, and we show that each Pselective set contains a weaklyP rankable subset.
Nondeterministic Bounded Query Reducibilities
 ANNALS OF PURE AND APPLIED LOGIC
, 1989
"... A querybounded Turing machine is an oracle machine which computes its output function from a bounded number of queries to its oracle. In this paper we investigate the behavior of nondeterministic querybounded Turing machines. In particular we study how easily such machines can compute the function ..."
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A querybounded Turing machine is an oracle machine which computes its output function from a bounded number of queries to its oracle. In this paper we investigate the behavior of nondeterministic querybounded Turing machines. In particular we study how easily such machines can compute the function F A n (x 1 , . . . , x n ) from A, where A # N and F A n (x 1 , . . . , x n ) = ##A (x 1 ), . . . , #A (x n )#. We show that each truthtable degree contains a set A such that, F A n can be nondeterministically computed from A by asking at most one question per nondeterministic branch; and that every set of the form A # also has this property. On the other hand, we show that if A is a 1generic set then F A n cannot be nondeterministically computed from A in less that n queries to A; and that each nonzero r.e. Turing degree contains an r.e. set A with the same property. If the machines involved can only make queries that are part of their input then all sets such that F A n ca...
Frequency Computation and Bounded Queries
 Theoretical Computer Science
, 1995
"... There have been several papers over the last ten years that consider the number of queries needed to compute a function as a measure of its complexity. The following function has been studied extensively in that light: F A a (x 1 ; : : : ; x a ) = A(x 1 ) 1 1 1 A(x a ): We are interested in t ..."
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There have been several papers over the last ten years that consider the number of queries needed to compute a function as a measure of its complexity. The following function has been studied extensively in that light: F A a (x 1 ; : : : ; x a ) = A(x 1 ) 1 1 1 A(x a ): We are interested in the complexity (in terms of the number of queries) of approximating F A a . Let b a and let f be any function such that F A a (x 1 ; : : : ; x a ) and f(x 1 ; : : : ; x a ) agree on at least b bits. For a general set A we have matching upper and lower bounds that depend on coding theory. These are applied to get exact bounds for the case where A is semirecursive, A is superterse, and (assuming P 6= NP) A = SAT. We obtain exact bounds when A is the halting problem using different methods. 1 Introduction The complexity of a function can be measured by the number of queries (to some oracle) needed to compute it. This notion has been studied in both a 3 Dept. of Computer Sc...