Results 1  10
of
24
Trivial Reals
"... Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivi ..."
Abstract

Cited by 57 (31 self)
 Add to MetaCart
Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivial real. We also analyze various computabilitytheoretic properties of the Htrivial reals, showing for example that no Htrivial real can compute the halting problem. Therefore, our construction of an Htrivial computably enumerable set is an easy, injuryfree construction of an incomplete computably enumerable set. Finally, we relate the Htrivials to other classes of "highly nonrandom " reals that have been previously studied.
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 25 (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.
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.).
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.
Some fundamental issues concerning degrees of unsolvability
 In [6], 2005. Preprint
, 2007
"... Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, corecursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.
Totally ωcomputably enumerable degrees and bounding critical triples, preprint
"... Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1. ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1.
1995], Degree theoretic definitions of the low 2 recursively enumerable sets
 J. Symbolic Logic
, 1995
"... 1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) mac ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) machines, ϕe, can be used; access to
Degree structures: Local and global investigations
 Bulletin of Symbolic Logic
"... $1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead. ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.
Countable thin Π0 1 classes
 Annals of Pure and Applied Logic
, 1993
"... Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is shown that, for all ordinals α>1, no set of degree ≥ 0 ′ ′ can be a member of any thin Π0 1 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π0 1 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π0 1 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π0 1 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π0 1 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree 0 ′ ′. Introduction.
WORKING WITH STRONG REDUCIBILITIES ABOVE TOTALLY ωC.E. DEGREES
"... Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ωc.e. iff every set in it is weak truthtable reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every leftc.e. real of that degree is reducible in a computable Lipschitz way to a random leftc.e. real (an Ωnumber). 1.