Results 21  30
of
346
Array Nonrecursive Degrees and Genericity
 London Mathematical Society Lecture Notes Series 224
, 1996
"... A class of r.e. degrees, called the array nonrecursive degrees, previously studied by the authors in connection with multiple permitting arguments relative to r.e. sets, is extended to the degrees in general. This class contains all degrees which satisfy a (i.e. a 2 GL 2 ) but in addition ..."
Abstract

Cited by 24 (7 self)
 Add to MetaCart
A class of r.e. degrees, called the array nonrecursive degrees, previously studied by the authors in connection with multiple permitting arguments relative to r.e. sets, is extended to the degrees in general. This class contains all degrees which satisfy a (i.e. a 2 GL 2 ) but in addition there exist low r.e. degrees which are array nonrecursive (a.n.r.).
Some ComputabilityTheoretical Aspects of Reals and Randomness
 the Lect. Notes Log. 18, Assoc. for Symbol. Logic
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. ..."
Abstract

Cited by 24 (7 self)
 Add to MetaCart
We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.
Enumeration Reducibility, Nondeterministic Computations and Relative . . .
 RECURSION THEORY WEEK, OBERWOLFACH 1989, VOLUME 1432 OF LECTURE NOTES IN MATHEMATICS
, 1990
"... ..."
Kolmogorov complexity and instance complexity of recursively enumerable sets
 SIAM Journal on Computing
, 1996
"... We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the wellknown 2log n upper bound on the Kolmogorov complexity of initial segments of r.e. sets is optimal and characterize the Tdegrees of r.e. sets which attain this bound. The main pa ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the wellknown 2log n upper bound on the Kolmogorov complexity of initial segments of r.e. sets is optimal and characterize the Tdegrees of r.e. sets which attain this bound. The main part of the paper is concerned with instance complexity of r.e. sets. We construct a nonrecursive r.e. set with instance complexity logarithmic in the Kolmogorov complexity. This refutes a conjecture of Ko, Orponen, Schöning, and Watanabe. In the other extreme, we show that all wttcomplete set and all Qcomplete sets have infinitely many hard instances.
A lower cone in the wtt degrees of nonintegral effective dimension
 In Proceedings of IMS workshop on Computational Prospects of Infinity
, 2006
"... ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the e ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the existence of a wttlower cone of effective dimension r. 1.
Embeddings into the Medvedev and Muchnik lattices of Π0 1 classes
 Archive for Mathematical Logic
, 2004
"... Archive for Mathematical Logic Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of nonempty Π 0 1 subsets of 2 ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are latticeembeddable below any nonzero ele ..."
Abstract

Cited by 22 (17 self)
 Add to MetaCart
Archive for Mathematical Logic Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of nonempty Π 0 1 subsets of 2 ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are latticeembeddable below any nonzero element of Pw. We show that many countable distributive lattices are latticeembeddable below any nonzero element of PM. 1
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 ..."
Abstract

Cited by 22 (16 self)
 Add to MetaCart
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.
Specification Structures and PropositionsasTypes for Concurrency
 Logics for Concurrency: Structure vs. AutomataProceedings of the VIIIth Banff Higher Order Workshop, volume 1043 of Lecture Notes in Computer Science
, 1995
"... Many different notions of "property of interest" and methods of verifying such properties arise naturally in programming. A general framework of "Specification Structures" is presented for combining different notions and methods in a coherent fashion. This is then applied to concurrency in the se ..."
Abstract

Cited by 21 (5 self)
 Add to MetaCart
Many different notions of "property of interest" and methods of verifying such properties arise naturally in programming. A general framework of "Specification Structures" is presented for combining different notions and methods in a coherent fashion. This is then applied to concurrency in the setting of Interaction Categories.
Randomness Space
 Automata, Languages and Programming, Proceedings of the 25th International Colloquium, ICALP’98
, 1998
"... MartinL#of de#ned in#nite random sequences over a #nite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After stud ..."
Abstract

Cited by 21 (4 self)
 Add to MetaCart
MartinL#of de#ned in#nite random sequences over a #nite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After studying basic results and constructions for such randomness spaces a general invariance result is proved which gives conditions under which a function between randomness spaces preserves randomness. This corrects and extends a result bySchnorr. Calude and J#urgensen proved that the randomness notion for real numbers obtained by considering their bary representations is independent from the base b. We use our invariance result to show that this notion is identical with the notion which one obtains by viewing the real number space directly as a randomness space. Furthermore, arithmetic properties of random real numbers are derived, for example that every computable analytic function pres...
Lowness properties and approximations of the jump
 Proceedings of the Twelfth Workshop of Logic, Language, Information and Computation (WoLLIC 2005). Electronic Lecture Notes in Theoretical Computer Science 143
, 2006
"... ..."