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47
Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
"... \Lambda \Lambda ..."
Lowness for the Class of Random Sets
, 1998
"... A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the noti ..."
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Cited by 31 (3 self)
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A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the notion of randomness as originally defined by P. MartinLof in [8]. A set is MartinLofrandom, or 1random for short, if it cannot be approximated in measure by recursive means. These sets have played a central role in the study of algorithmic randomness. One can relativize the definition of randomness to an arbitrary oracle. Relativized randomness has been studied by several authors. The intuitive meaning of "A is 1random relative to B" is that A is independent of B. A justification for this interpretation is given by M. van Lambalgen [7]. In this introduction we review some of the basic properties of sets which are 1random and we state the main problem. We work in the Cantor space 1 The fi...
An extension of the recursively enumerable Turing degrees
 Journal of the London Mathematical Society
, 2006
"... Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overco ..."
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Cited by 22 (16 self)
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Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with nonempty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally nonrecursive functions, and rn is the weak degree of the nrandom reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is onetoone, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1.
Randomness, computability, and density
 SIAM Journal of Computation
, 2002
"... 1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ..."
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Cited by 13 (6 self)
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1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ff for any n 2! and ff 2 (0; 1], and we do this below without further comment.
On the Autoreducibility of Random Sequences
, 2001
"... A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition ..."
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Cited by 12 (1 self)
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A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truthtableautoreducible.
Degree Spectra of Relations on Computable Structures
 J. Symbolic Logic
, 1999
"... Abstract We give some new examples of possible degree spectra of invariant relations on \Delta 02categorical computable structures that demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the kinds of sets of degrees that can ..."
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Cited by 11 (5 self)
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Abstract We give some new examples of possible degree spectra of invariant relations on \Delta 02categorical computable structures that demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the kinds of sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a \Delta 02categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [22] to establish the same result for computable relations on computable linear orderings.
On Schnorr and computable randomness, martingales, and machines
 Mathematical Logic Quarterly
, 2004
"... examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines. ..."
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Cited by 10 (6 self)
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examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines.
Every Set Has a Least Jump Enumeration
 Journal of the London Mathematical Society
, 1998
"... Given a computably enumerable set B; there is a Turing degree which is the least jump of any set in which B is computably enumerable, namely 0 : Remarkably, this is not a phenomenon of computably enumerable sets. We show that for every subset A of N; there is a Turing degree, c (A); which ..."
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Cited by 10 (0 self)
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Given a computably enumerable set B; there is a Turing degree which is the least jump of any set in which B is computably enumerable, namely 0 : Remarkably, this is not a phenomenon of computably enumerable sets. We show that for every subset A of N; there is a Turing degree, c (A); which is the least degree of the jumps of all sets X for which A is \Sigma 1 (X): 1
Degree spectra of relations on Boolean algebras
, 2002
"... Abstract We show that every computable relation on a computable Boolean algebra B is either definable by a quantifierfree formula with constants from B (in which case it is obviously intrinsically computable) or has infinite degree spectrum. Computable mathematics has been the focus of a large amou ..."
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Cited by 8 (3 self)
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Abstract We show that every computable relation on a computable Boolean algebra B is either definable by a quantifierfree formula with constants from B (in which case it is obviously intrinsically computable) or has infinite degree spectrum. Computable mathematics has been the focus of a large amount of research in the past few decades. Computable model theory in particular has seen vigorous and varied activity, leading to the discovery and intensive investigation of a number of central recurring themes. Among these is the study of the computabilitytheoretic properties of the images of a relation on a structure in different computable copies of the structure. In this paper, we investigate computable relations on Boolean algebras from this point of view. Boolean algebras are very interesting to computable model theorists because, like linear orderings, they are a natural, nontrivial, and wellstudied class of structures that exhibits much more structure than is present in the general case. Thus, studying computable Boolean algebras can give us insight into the nature of computation under constraints. We will define the relevant concepts from computable model theory below. A valuable recent reference covering a wide range of topics in computable mathematics is The first and second authors ' research was partially supported by the Marsden Fund of New Zealand.