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29
Using random sets as oracles
"... Let R be a notion of algorithmic randomness for individual subsets of N. We say B is a base for R randomness if there is a Z �T B such that Z is R random relative to B. We show that the bases for 1randomness are exactly the Ktrivial sets and discuss several consequences of this result. We also sho ..."
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Cited by 34 (15 self)
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Let R be a notion of algorithmic randomness for individual subsets of N. We say B is a base for R randomness if there is a Z �T B such that Z is R random relative to B. We show that the bases for 1randomness are exactly the Ktrivial sets and discuss several consequences of this result. We also show that the bases for computable randomness include every ∆ 0 2 set that is not diagonally noncomputable, but no set of PAdegree. As a consequence, we conclude that an nc.e. set is a base for computable randomness iff it is Turing incomplete. 1
Enumeration Reducibility, Nondeterministic Computations and Relative . . .
 RECURSION THEORY WEEK, OBERWOLFACH 1989, VOLUME 1432 OF LECTURE NOTES IN MATHEMATICS
, 1990
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Infinitary Self Reference in Learning Theory
, 1994
"... Kleene's Second Recursion Theorem provides a means for transforming any program p into a program e(p) which first creates a quiescent self copy and then runs p on that self copy together with any externally given input. e(p), in effect, has complete (low level) self knowledge, and p represents how ..."
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Cited by 18 (6 self)
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Kleene's Second Recursion Theorem provides a means for transforming any program p into a program e(p) which first creates a quiescent self copy and then runs p on that self copy together with any externally given input. e(p), in effect, has complete (low level) self knowledge, and p represents how e(p) uses its self knowledge (and its knowledge of the external world). Infinite regress is not required since e(p) creates its self copy outside of itself. One mechanism to achieve this creation is a self replication trick isomorphic to that employed by singlecelled organisms. Another is for e(p) to look in a mirror to see which program it is. In 1974 the author published an infinitary generalization of Kleene's theorem which he called the Operator Recursion Theorem. It provides a means for obtaining an (algorithmically) growing collection of programs which, in effect, share a common (also growing) mirror from which they can obtain complete low level models of themselves and the other prog...
The Baire category theorem in weak subsystems of secondorder arithmetic
 THE JOURNAL OF SYMBOLIC LOGIC
, 1993
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Defining the Turing Jump
 MATHEMATICAL RESEARCH LETTERS
, 1999
"... The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle" for a ..."
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Cited by 10 (6 self)
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The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle" for a subset A of the natural numbers #, i.e. an external procedure that answers questions of the form "is n in A", they define the basic notion of relative computability or Turing reducibility (from Turing (1939)). We say that A is computable from (or recursive in) B if there is a Turing machine which, when equipped with an oracle for B, computes (the characteristic function of) A, i.e. for some e, # B e = A. We denote this relation by A # T<F10
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
δuniform BSS machines
 J. Complexity
, 1998
"... A δuniform BSS machine is a standard BSS machine which does not rely on exact equality tests. We prove that, for any real closed archimedean field R, a set is δuniformly semidecidable iff it is open and semidecidable by a BSS machine which is locally time bounded; we also prove that the local ti ..."
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Cited by 6 (4 self)
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A δuniform BSS machine is a standard BSS machine which does not rely on exact equality tests. We prove that, for any real closed archimedean field R, a set is δuniformly semidecidable iff it is open and semidecidable by a BSS machine which is locally time bounded; we also prove that the local time bound condition is nontrivial. This entails a number of results about BSS machines, in particular the existence of decidable sets whose interior (closure) is not even semidecidable without adding constants. Finally, we show that the sets semidecidable by Turing machines are the sets semidecidable by δuniform machines with coefficients in Q or T, the field of Turing computable numbers. 1
A basis theorem for Π0 1 classes of positive measure and jump inversion for random reals
 Proceedings of the American Mathematical Society
, 2006
"... We extend the Shoenfield jump inversion theorem to the members of any Π0 1 class P⊆2ω with nonzero measure; i.e., for every Σ0 2 set S ≥T ∅ ′, there is a ∆0 2 real A ∈Psuch that A ′ ≡T S. In particular, we get jump inversion for ∆0 2 1random reals. This paper is part of an ongoing program to stud ..."
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Cited by 6 (1 self)
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We extend the Shoenfield jump inversion theorem to the members of any Π0 1 class P⊆2ω with nonzero measure; i.e., for every Σ0 2 set S ≥T ∅ ′, there is a ∆0 2 real A ∈Psuch that A ′ ≡T S. In particular, we get jump inversion for ∆0 2 1random reals. This paper is part of an ongoing program to study the relationship between two fundamental notions of complexity for real numbers. The first is the computational complexity of a real as captured, for example, by its Turing degree. The second is the intrinsic randomness of a real. In particular, we are interested in the 1random reals, which were introduced by MartinLöf [13] and represent the most widely studied randomness class. For the purposes of this introduction, we will assume that the reader is somewhat familiar with basic algorithmic randomness, as per LiVitányi [12], and with computability theory [18]. A review of notation and terminology will be given in Section 1. Intuitively, a 1random real is very complex. This complexity can be captured formally in terms of unpredictability or incompressibility, but is it reflected in the
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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Upper SemiLattice of Binary Strings With the Relation "x is Simple Conditional to y"
"... In this paper we construct a structure R that is a "finite version" of the semilattice of Turing degrees. Its elements are strings (technically, sequences of strings) and x <=y means that K(xy) = (conditional Kolmogorov complexity of x relative to y) is small. We construct two elements... ..."
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Cited by 6 (1 self)
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In this paper we construct a structure R that is a "finite version" of the semilattice of Turing degrees. Its elements are strings (technically, sequences of strings) and x <=y means that K(xy) = (conditional Kolmogorov complexity of x relative to y) is small. We construct two elements...