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Turing degrees of reals of positive effective packing dimension
2008
Information Processing Letters
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This leads us to examine the Turing degrees of reals with positive effective packing dimension. ...
We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. ...
Theorem 1.1 raises a more general question: what kind of Turing degrees contain reals with positive effective packing dimension? ...
doi:10.1016/j.ipl.2008.05.028
fatcat:d5sz7jw7k5hsxmn3ia5kn5zb3m
Constructive Dimension and Weak Truth-Table Degrees
[chapter]
2007
Lecture Notes in Computer Science
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A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). ...
This paper examines the constructive Hausdorff and packing dimensions of weak truth-table degrees. ...
We also thank the American Institute of Mathematics which generously invited us to the Workshop on Effective Randomness; this paper is a result of a workgroup discussing open questions during this workshop ...
doi:10.1007/978-3-540-73001-9_7
fatcat:dh4ld5ivfbfx7dt65uxl7cqgvy
Constructive Dimension and Turing Degrees
[article]
2010
arXiv
pre-print
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This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. ...
A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. ...
We also thank the American Institute of Mathematics which generously invited us to the Workshop on Effective Randomness; this paper is a result of a workgroup discussing open questions during this workshop ...
arXiv:cs/0701089v4
fatcat:uf33nbt7enadfbid4abicenexe
Constructive Dimension and Turing Degrees
2009
Theory of Computing Systems
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This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. ...
A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. ...
We also thank the American Institute of Mathematics which generously invited us to the Workshop on Effective Randomness; this paper is a result of a workgroup discussing open questions during this workshop ...
doi:10.1007/s00224-009-9170-1
fatcat:kugh7ym3jrhjxeyalpmcxjmlhe
Effective Packing Dimension and Traceability
2010
Notre Dame Journal of Formal Logic
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A c.e. degree contains a real with positive effective packing dimension iff it is array non-computable. ...
It is still open if every degree of effective packing dimension one contains a real of effective packing dimension one, and this seems to be a difficult problem. ...
doi:10.1215/00294527-2010-017
fatcat:g6tmzjqao5gylgwm7zcbecepi4
Schnorr Dimension
[chapter]
2005
Lecture Notes in Computer Science
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In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1. ...
Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. ...
In fact, every hyperimmune Turing degree contains a set of computable packing dimension 1 and this set can be chosen to be c.e. in the special case of a c.e. Turing degree. ...
doi:10.1007/11494645_13
fatcat:6rdw6lovw5bsrpjje4jc5cekee
Schnorr dimension
2006
Mathematical Structures in Computer Science
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In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1. ...
Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. ...
In fact, every hyperimmune Turing degree contains a set of computable packing dimension 1 and this set can be chosen to be c.e. in the special case of a c.e. Turing degree. ...
doi:10.1017/s0960129506005469
fatcat:pbc3yxacuzbx3f6eritnwueo3q
Controlling Effective Packing Dimension of $\Delta^{0}_{2}$ Degrees
2016
Notre Dame Journal of Formal Logic
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This paper presents a renement of a result by Conidis [3], who proved that there is a real X of eective packing dimension 0 < α < 1 which cannot compute any real of eective packing dimension 1. ...
Controlling Eective Packing Dimension of ∆ 0 2 Degrees 3 1961 Sacks [12] noted that the construction could also be carried out below ...
Turing degree which contains a real of nonzero eective packing dimension contains reals of eective packing dimension arbitrarily close to 1. ...
doi:10.1215/00294527-3328401
fatcat:4ftbg4vzijgn3b3sioccycimty
Computability and Randomness
2019
Notices of the American Mathematical Society
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allow us to express the dimensions of sets of reals in terms of the effective dimensions of their elements. ...
Hausdorff used work of Carathéodory on -dimensional measures to generalize the notion of dimension to possibly nonintegral values, leading to concepts such as Hausdorff dimension and packing dimension. ...
Publications of Hindustan Book Agency are distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for commercial channels. ...
doi:10.1090/noti1905
fatcat:pg66obggazgwpj6dwy2buq6oze
Randomness extraction and asymptotic Hamming distance
2013
Logical Methods in Computer Science
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Our result is that the class of stochastically bi-immune sets is not Medvedev reducible to the class of sets having complex packing dimension 1. ...
We obtain a non-implication result in the Medvedev degrees by studying sequences that are close to Martin-L\"of random in asymptotic Hamming distance. ...
each Turing reduction Φ, of a set Y of complex packing dimension 1 for which Φ Y is not stochastically bi-immune.
1.2. ...
doi:10.2168/lmcs-9(3:27)2013
fatcat:w23s423bzjaxnkmjbfwgonwffq
Compactness arguments with effectively closed sets for the study of relative randomness
2010
Journal of Logic and Computation
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Hence they seem to play a more important role than in the context of the Turing degrees, where they were originally applied by Jockusch and Soare in their study of Π 0 1 classes and degrees of theories ...
We present a variety of compactness arguments with Π 0 1 classes which yield results about relative randomness, and in particular properties of the LR degrees. ...
Turing degrees of positive effective dimension. However this no longer holds for effective packing dimension. ...
doi:10.1093/logcom/exq036
fatcat:agg4o5dwvrhevcb2tcnt45pnhm
Two Open Problems on Effective Dimension
[chapter]
2006
Lecture Notes in Computer Science
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(Gale characterization of fractal dimension) Let X be a set of sequences. ...
The dimension of a class X inside a base class C is a real number in [0,1] corresponding to the relative size of X ∩ C inside C. ...
A third aspect of effective dimension is as a formal tool in Computational Complexity, allowing us to consider new working hypothesis such as "NP has positive dimension in exponential time", that can imply ...
doi:10.1007/11780342_37
fatcat:qvyckgulfvbtjirmbv6v4o7o2i
Fractal Dimension versus Process Complexity
2016
Advances in Mathematical Physics
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It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. ...
In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. ...
So by this identification, it makes sense to speak of the Turing degree of a real number. ...
doi:10.1155/2016/5030593
fatcat:g6ptxsbn3jfqfaaayeedmk7twm
Diagonally non-recursive functions and effective Hausdorff dimension
2011
Bulletin of the London Mathematical Society
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Hence there is a real of effective Hausdorff dimension 1 that does not compute a Martin-Löf random real. This answers a question of Reimann and Terwijn. ...
We prove that every sufficiently slow growing DNR function computes a real with effective Hausdorff dimension one. ...
We remark that a weaker version of the answer we give was proved by Downey and Greenberg [4] , where they show that there is a set of minimal Turing degree whose effective packing dimension is 1; such ...
doi:10.1112/blms/bdr003
fatcat:vwun2tti3bhdjdfhvga32kdp2m
Fractal dimension versus process complexity
[article]
2016
arXiv
pre-print
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In our setting, we define fractal dimension of a Turing machine as the limiting fractal dimension of the corresponding space-time diagram. ...
It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. ...
So by this identification it makes sense to speak of the Turing degree of a real number. ...
arXiv:1309.1779v6
fatcat:cs5vuxm56jfvbf7wtfmq6frvre
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