Filters








7,663 Hits in 5.9 sec

Turing degrees of reals of positive effective packing dimension

Rod Downey, Noam Greenberg
2008 Information Processing Letters  
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]

Laurent Bienvenu, David Doty, Frank Stephan
2007 Lecture Notes in Computer Science  
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]

Laurent Bienvenu, David Doty, Frank Stephan
2010 arXiv   pre-print
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

Laurent Bienvenu, David Doty, Frank Stephan
2009 Theory of Computing Systems  
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

Rod Downey, Keng Meng Ng
2010 Notre Dame Journal of Formal Logic  
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]

Rodney Downey, Wolfgang Merkle, Jan Reimann
2005 Lecture Notes in Computer Science  
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

RODNEY DOWNEY, WOLFGANG MERKLE, JAN REIMANN
2006 Mathematical Structures in Computer Science  
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

Jonathan Stephenson
2016 Notre Dame Journal of Formal Logic  
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

Rod Downey, Denis R. Hirschfeldt
2019 Notices of the American Mathematical Society  
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

Cameron Freer, Bjoern Kjos-Hanssen, Klaus Weihrauch
2013 Logical Methods in Computer Science  
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

G. Barmpalias
2010 Journal of Logic and Computation  
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]

Elvira Mayordomo
2006 Lecture Notes in Computer Science  
(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

Joost J. Joosten, Fernando Soler-Toscano, Hector Zenil
2016 Advances in Mathematical Physics  
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

Noam Greenberg, Joseph S. Miller
2011 Bulletin of the London Mathematical Society  
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]

Joost J. Joosten, Fernando Soler-Toscano, Hector Zenil
2016 arXiv   pre-print
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
« Previous Showing results 1 — 15 out of 7,663 results