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50
Splittings, Robustness and Structure of Complete Sets
, 1993
"... We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. To address this question for a given hard set A, we construct a sparse set S, ..."
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Cited by 16 (4 self)
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We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. To address this question for a given hard set A, we construct a sparse set S, and ask whether A \Gamma S is still hard. It turns out that for most of the reductions considered and for an arbitrary given sparseness condition, there is a single subexponential time computable set S that meets this condition, such that A \Gamma S is not hard for any A. Not only is this set S subexponential time computable, but a slight modification of the construction can make the complexity of S meet any reasonable superpolynomial function. On the other hand we show that for any polynomialtime computable sparse set S, the set A \Gamma S remains hard. There are other properties than time complexity that make a set `almost' polynomialtime computable. For sparse pselective sets...
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 13 (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...
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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Cited by 9 (2 self)
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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
A Guided Tour of Minimal Indices and Shortest Descriptions
 Archives for Mathematical Logic
, 1997
"... The set of minimal indices of a G#del numbering ' is deøned as MIN' = fe : (8i ! e)[' i 6= 'e ]g. It has been known since 1972 that MIN' jT ; 00 , but beyond this MIN' has remained mostly uninvestigated. This thesis collects the scarce results on MIN' from the literature and adds some new observa ..."
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Cited by 9 (3 self)
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The set of minimal indices of a G#del numbering ' is deøned as MIN' = fe : (8i ! e)[' i 6= 'e ]g. It has been known since 1972 that MIN' jT ; 00 , but beyond this MIN' has remained mostly uninvestigated. This thesis collects the scarce results on MIN' from the literature and adds some new observations including that MIN' is autoreducible, but neither regressive nor (1; 2) computable. We also study several variants of MIN' that have been deøned in the literature like sizeminimal indices, shortest descriptions, and minimal indices of decision tables. Some challenging open problems are left for the adventurous reader. 1 Introduction How long is the shortest program that solves your problem? There are at least two ways to interpret this question depending on the type of problem involved. If the program's task is to output one speciøc object, we are looking for a shortest description of that object. This interpretation is closely related to Kolmogorov complexity. Although we have sev...
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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Cited by 8 (0 self)
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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 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 declares to be most ..."
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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 Σ^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.
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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Cited by 6 (4 self)
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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...
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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Cited by 6 (3 self)
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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...
NPhard sets are superterse unless NP is small
 Information Processing Letters
, 1997
"... Introduction One of the important questions in computational complexity theory is whether every NP problem is solvable by polynomial time circuits, i.e., NP `?P=poly. Furthermore, it has been asked what the deterministic time complexity of NP is if NP ` P=poly. That is, if NP is easy in the nonunif ..."
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Cited by 6 (0 self)
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Introduction One of the important questions in computational complexity theory is whether every NP problem is solvable by polynomial time circuits, i.e., NP `?P=poly. Furthermore, it has been asked what the deterministic time complexity of NP is if NP ` P=poly. That is, if NP is easy in the nonuniform complexity measure, how easy is NP in the uniform complexity measure? Let P T (SPARSE) be the class of languages that are polynomial time Turing reducible to some sparse sets. Then it is well known that P T (SPARSE) = P=poly. Hence the above question is equivalent to the following question. NP `?PT (SPARSE): It has been shown by Wilson [18] that thi
Effective Hausdorff dimension
 In Logic Colloquium ’01
, 2005
"... ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective sdimensional Hausdorff measures, similar to the effectivization ..."
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Cited by 5 (2 self)
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ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective sdimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTINLÖF. It turns out that effective Hausdorff dimension allows to classify sequences according to their ‘degree ’ of algorithmic randomness, i.e., their algorithmic density of information. Earlier the works of STAIGER and RYABKO showed a deep connection between Kolmogorov complexity and Hausdorff dimension. We further develop this relationship and use it to give effective versions of some important properties of (classical) Hausdorff dimension. Finally, we determine the effective dimension of some objects arising in the context of computability theory, such as degrees and spans. 1.