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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...
Algorithmic randomness of closed sets
 J. LOGIC AND COMPUTATION
, 2007
"... We investigate notions of randomness in the space C[2 N] of nonempty closed subsets of {0, 1} N. A probability measure is given and a version of the MartinLöf test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that any random 4 clos ..."
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Cited by 11 (8 self)
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We investigate notions of randomness in the space C[2 N] of nonempty closed subsets of {0, 1} N. A probability measure is given and a version of the MartinLöf test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that any random 4 closed set is perfect, has measure 0, and has box dimension log2. A 3 random closed set has no nc.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If Tn = T ∩ {0, 1} n, then for any random closed set [T] where T has no dead ends, K(Tn) ≥ n − O(1) but for any k, K(Tn) ≤ 2 n−k + O(1), where K(σ) is the prefixfree complexity of σ ∈ {0, 1} ∗.
On the Turing degrees of weakly computable real numbers
 Journal of Logic and Computation
, 1986
"... The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others ..."
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Cited by 6 (3 self)
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The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1
ON THE NUMBER OF INFINITE SEQUENCES WITH TRIVIAL INITIAL SEGMENT COMPLEXITY
"... Abstract. The sequences which have trivial prefixfree initial segment complexity are known as Ktrivial sets, and form a cumulative hierarchy of length ω. We show that the problem of finding the number of Ktrivial sets in the various levels of the hierarchy is ∆0 3. This answers a question of Down ..."
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Cited by 3 (3 self)
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Abstract. The sequences which have trivial prefixfree initial segment complexity are known as Ktrivial sets, and form a cumulative hierarchy of length ω. We show that the problem of finding the number of Ktrivial sets in the various levels of the hierarchy is ∆0 3. This answers a question of Downey/Miller/Yu (see [DH10, Section 10.1.4]) which also appears in [Nie09, Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give shorter initial segment complexity compared to the computable oracles. In both cases the classification ∆0 3 is sharp. 1.
On Genericity and Ershov's Hierarchy
 Mathematical Logic Quarterly
"... It is natural to wish to study miniaturisations of Cohen forcing suitable to sets of low arithmetic complexity. We consider extensions of the work of Schaeer[9] and Jockusch and Posner[6] by looking at genericity notions within the 2 sets. ..."
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It is natural to wish to study miniaturisations of Cohen forcing suitable to sets of low arithmetic complexity. We consider extensions of the work of Schaeer[9] and Jockusch and Posner[6] by looking at genericity notions within the 2 sets.
The nr.e. degrees: undecidability and Σ1 substructures
, 2012
"... We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1 ..."
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We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1