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Turing degrees of reals of positive effective packing dimension

2008
*
Information Processing Letters
*

This leads us to examine the

doi:10.1016/j.ipl.2008.05.028
fatcat:d5sz7jw7k5hsxmn3ia5kn5zb3m
*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*? ...##
###
Constructive Dimension and Weak Truth-Table Degrees
[chapter]

2007
*
Lecture Notes in Computer Science
*

A number

doi:10.1007/978-3-540-73001-9_7
fatcat:dh4ld5ivfbfx7dt65uxl7cqgvy
*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 ...##
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Constructive Dimension and Turing Degrees
[article]

2010
*
arXiv
*
pre-print

This paper examines the constructive Hausdorff and

arXiv:cs/0701089v4
fatcat:uf33nbt7enadfbid4abicenexe
*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 ...##
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Constructive Dimension and Turing Degrees

2009
*
Theory of Computing Systems
*

This paper examines the constructive Hausdorff and

doi:10.1007/s00224-009-9170-1
fatcat:kugh7ym3jrhjxeyalpmcxjmlhe
*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 ...##
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Effective Packing Dimension and Traceability

2010
*
Notre Dame Journal of Formal Logic
*

A c.e.

doi:10.1215/00294527-2010-017
fatcat:g6tmzjqao5gylgwm7zcbecepi4
*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. ...##
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Schnorr Dimension
[chapter]

2005
*
Lecture Notes in Computer Science
*

In fact, we prove that every hyperimmune

doi:10.1007/11494645_13
fatcat:6rdw6lovw5bsrpjje4jc5cekee
*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*. ...##
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Schnorr dimension

2006
*
Mathematical Structures in Computer Science
*

In fact, we prove that every hyperimmune

doi:10.1017/s0960129506005469
fatcat:pbc3yxacuzbx3f6eritnwueo3q
*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*. ...##
###
Controlling Effective Packing Dimension of $\Delta^{0}_{2}$ Degrees

2016
*
Notre Dame Journal of Formal Logic
*

This paper presents a renement

doi:10.1215/00294527-3328401
fatcat:4ftbg4vzijgn3b3sioccycimty
*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. ...##
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Computability and Randomness

2019
*
Notices of the American Mathematical Society
*

allow us to express the

doi:10.1090/noti1905
fatcat:pg66obggazgwpj6dwy2buq6oze
*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. ...##
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Randomness extraction and asymptotic Hamming distance

2013
*
Logical Methods in Computer Science
*

Our result is that the class

doi:10.2168/lmcs-9(3:27)2013
fatcat:w23s423bzjaxnkmjbfwgonwffq
*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. ...##
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Compactness arguments with effectively closed sets for the study of relative randomness

2010
*
Journal of Logic and Computation
*

Hence they seem to play a more important role than in the context

doi:10.1093/logcom/exq036
fatcat:agg4o5dwvrhevcb2tcnt45pnhm
*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*. ...##
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Two Open Problems on Effective Dimension
[chapter]

2006
*
Lecture Notes in Computer Science
*

(Gale characterization

doi:10.1007/11780342_37
fatcat:qvyckgulfvbtjirmbv6v4o7o2i
*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 ...##
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Fractal Dimension versus Process Complexity

2016
*
Advances in Mathematical Physics
*

It turns out that there is a very strong relation between the fractal

doi:10.1155/2016/5030593
fatcat:g6ptxsbn3jfqfaaayeedmk7twm
*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. ...##
###
Diagonally non-recursive functions and effective Hausdorff dimension

2011
*
Bulletin of the London Mathematical Society
*

Hence there is a

doi:10.1112/blms/bdr003
fatcat:vwun2tti3bhdjdfhvga32kdp2m
*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 ...##
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Fractal dimension versus process complexity
[article]

2016
*
arXiv
*
pre-print

In our setting, we define fractal

arXiv:1309.1779v6
fatcat:cs5vuxm56jfvbf7wtfmq6frvre
*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. ...
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