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| {{One source|text=The article relies on sources from one research group.|date=November 2013}}
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| In [[mathematics]], '''effective dimension''' is a modification of [[Hausdorff dimension]] and other [[fractal dimension]]s which places it in a [[computability theory]] setting. There are several variations (various notions of effective dimension) of which the most common is '''effective Hausdorff dimension'''. [[Dimension]], in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size). Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, [[List of fractals by Hausdorff dimension|fractal]] subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smaller" than filled circles or rectangles. Effective dimension modifies Hausdorff dimension by requiring that objects with small effective dimension be not only small but also locatable (or partially locatable) in a computable sense. As such, objects with large Hausdorff dimension also have large effective dimension, and objects with small effective dimension have small Hausdorff dimension, but an object can have small Hausdorff but large effective dimension. An example is an [[algorithmically random sequence|algorithmically random]] point on a line, which has Hausdorff dimension 0 (since it's a point) but effective dimension 1 (because, roughly speaking, it can't be effectively localized any better than a small interval, which has Hausdorff dimension 1).
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| == Rigorous definitions ==
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| This article will define effective dimension for subsets of [[Cantor space]] '''2'''<sup>ω</sup>; closely related definitions exist for subsets of [[Euclidean space]] '''R'''<sup>''n''</sup>. We will move freely between considering a set ''X'' of natural numbers, the infinite sequence <math>\chi_X</math> given by the characteristic function of ''X'', and the real number with binary expansion 0.''X''.
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| === Martingales and other gales ===
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| <!-- This isn't properly part of the article on effective dimension, but is necessary background information. It could, potentially be moved to a different article (perhaps on algorithmic randomness? That whole area of Wikipedia seems somewhat disorganized and I didn't want to redo it.) -->
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| A ''[[martingale (probability theory)|martingale]]'' on Cantor space '''2'''<sup>ω</sup> is a function ''d'': '''2'''<sup>ω</sup> → '''R'''<sup>≥ 0</sup> from Cantor space to nonnegative reals which satisfies the fairness condition:
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| : <math>d(\sigma)=\frac12 (d(\sigma 0)+d(\sigma 1))</math>
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| A martingale is thought of as a betting strategy, and the function <math>d(\sigma)</math> gives the capital of the better after seeing a sequence σ of 0s and 1s. The fairness condition then says that the capital after a sequence σ is the average of the capital after seeing σ0 and σ1; in other words the martingale gives a betting scheme for a bookie with 2:1 odds offered on either of two "equally likely" options, hence the name fair.
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| (Note that this is subtly different from the probability theory notion of [[martingale (probability theory)|martingale]].<ref name=wccrsm>{{cite journal | author = John M. Hitchcock and Jack H. Lutz | title = Why computational complexity requires stricter martingales | journal = Theory of Computing Systems | year = 2006}}</ref> That definition of martingale has a similar fairness condition, which also states that the expected value after some observation is the same as the value before the observation, given the prior history of observations. The difference is that in probability theory, the prior history of observations just refers to the capital history, whereas here the history refers to the exact sequence of 0s and 1s in the string.)
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| A ''supermartingale'' on Cantor space is a function ''d'' as above which satisfies a modified fairness condition:
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| : <math>d(\sigma) \geq \frac12 (d(\sigma 0)+d(\sigma 1))</math>
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| A supermartingale is a betting strategy where the expected capital after a bet is no more than the capital before a bet, in contrast to a martingale where the two are always equal. This allows more flexibility, and is very similar in the non-effective case, since whenever a supermartingale ''d'' is given, there is a modified function ''d''' which wins at least as much money as ''d'' and which is actually a martingale. However it is useful to allow the additional flexibility once one starts talking about actually giving algorithms to determine the betting strategy, as some algorithms lend themselves more naturally to producing supermartingales than martingales.
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| An ''s''-''gale'' is a function ''d'' as above of the form
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| : <math>d(\sigma) = \frac{e(\sigma)}{2^{(1-s)|\sigma|}}</math>
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| for ''e'' some martingale. | |
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| An ''s''-''supergale'' is a function ''d'' as above of the form
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| : <math>d(\sigma) = \frac{e(\sigma)}{2^{(1-s)|\sigma|}}</math>
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| for ''e'' some supermartingale.
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| An ''s''-(super)gale is a betting strategy where some amount of capital is lost to inflation at each step. Note that ''s''-gales and ''s''-supergales are examples of supermartingales, and the 1-gales and 1-supergales are precisely the martingales and supermartingales.
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| Collectively, these objects are known as "gales".
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| A gale ''d'' ''succeeds'' on a subset ''X'' of the natural numbers if <math>\limsup_n d(X|n)=\infty</math> where <math>X|n</math> denotes the ''n''-digit string consisting of the first ''n'' digits of ''X''.
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| A gale ''d'' ''succeeds strongly'' on ''X'' if <math>\liminf_n d(X|n)=\infty</math>.
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| All of these notions of various gales have no effective content, but one must necessarily restrict ones self to a small class of gales, since some gale can be found which succeeds on any given set. After all, if one knows a sequence of coin flips in advance, it is easy to make money by simply betting on the known outcomes of each flip. A standard way of doing this is to require the gales to be either computable or close to computable:
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| A gale ''d'' is called ''constructive'', ''c.e.'', or ''lower semi-computable'' if the numbers <math>d(\sigma)</math> are uniformly left-c.e. reals (i.e. can uniformly be written as the limit of an increasing computable sequence of rationals).
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| The '''effective Hausdorff dimension''' of a set of natural numbers ''X'' is <math>\inf \{s : \mathrm{some\ c.e.}\ s\mathrm{-gale\ succeeds\ on\ } X \}</math>.<ref name=dicc>{{cite journal | author = Jack H. Lutz | title = Dimension in complexity classes | journal = SIAM Journal on Computing | year = 2003}}</ref>
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| The '''effective packing dimension''' of ''X'' is <math>\inf \{s : \mathrm{some\ c.e.}\ s\mathrm{-gale\ succeeds\ strongly\ on\ } X\}</math>.<ref name=efsdiaiacc>{{cite journal | author = Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz and Elvira Mayordomo | title = Effective strong dimension in algorithmic information and computational complexity | journal = SIAM Journal on Computing | year = 2007}}</ref>
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| === Kolmogorov complexity definition ===
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| [[Kolmogorov complexity]] can be thought of as a lower bound on the algorithmic compressibility of a finite sequence (of characters or binary digits). It assigns to each such sequence ''w'' a natural number ''K(w)'' that, intuitively, measures the minimum length of a computer program (written in some fixed programming language) that takes no input and will output ''w'' when run.
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| The '''effective Hausdorff dimension''' of a set of natural numbers ''X'' is <math>\liminf_n \frac{K(X|n)}n</math>.<ref name=akccochd>{{cite journal | author = Elvira Mayordomo | title = A Kolmogorov complexity characterization of constructive Hausdorff dimension.|doi=10.1016/S0020-0190(02)00343-5 | journal = Information Processing Letters | year = 2002}}</ref>
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| The '''effective packing dimension''' of a set ''X'' is <math>\limsup_n \frac{K(X|n)}n</math>.<ref name="efsdiaiacc" />
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| From this one can see that both the effective Hausdorff dimension and the effective packing dimension of a set are between 0 and 1, with the effective packing dimension always at least as large as the effective Hausdorff dimension. Every [[algorithmically random sequence| random sequence]] will have effective Hausdorff and packing dimensions equal to 1, although there are also nonrandom sequences with effective Hausdorff and packing dimensions of 1.
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| == Comparison to classical dimension ==
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| If ''Z'' is a subset of '''2'''<sup>ω</sup>, its Hausdorff dimension is <math>\inf \{s : \mathrm{some}\ s\mathrm{-gale\ succeeds\ on\ all\ elements\ of\ } Z \}</math>.<ref name="dicc" />
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| The packing dimension of ''Z'' is <math>\inf \{s : \mathrm{some}\ s\mathrm{-gale\ succeeds\ strongly\ on\ all\ elements\ of\ } Z \}</math>.<ref name="efsdiaiacc" />
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| Thus the effective Hausdorff and packing dimensions of a set <math>X</math> are simply the classical Hausdorff and packing dimensions of <math>\{X\}</math> (respectively) when we restrict our attention to c.e. gales.
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| Define the following:
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| :<math>H_{\beta} := \{X \in 2^\omega : X\ \mathrm{has\ effective\ Hausdorff\ dimension\ } \beta \}</math>
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| :<math>H_{\leq \beta} := \{X \in 2^\omega : X\ \mathrm{has\ effective\ Hausdorff\ dimension\ } \leq \beta \}</math>
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| :<math>H_{< \beta} := \{X \in 2^\omega : X\ \mathrm{has\ effective\ Hausdorff\ dimension\ } < \beta \}</math>
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| :<math>P_{\beta} := \{X \in 2^\omega : X\ \mathrm{has\ effective\ packing\ dimension\ } \beta \}</math>
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| :<math>P_{\leq \beta} := \{X \in 2^\omega : X\ \mathrm{has\ effective\ packing\ dimension\ } \leq \beta \}</math>
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| :<math>P_{< \beta} := \{X \in 2^\omega : X\ \mathrm{has\ effective\ packing\ dimension\ } < \beta \}</math>
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| A consequence of the above is that these all have Hausdorff dimension <math>\beta</math>.
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| <math>H_{\beta}, H_{\leq \beta}</math> and <math>H_{< \beta}</math> all have packing dimension 1.
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| <math>P_{\beta}, P_{\leq \beta}</math> and <math>P_{< \beta}</math> all have packing dimension <math>\beta</math>.
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| == References ==
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| {{reflist}}
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| *{{cite journal
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| | author = J. H. Lutz
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| | title = Effective fractal dimensions
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| | journal = Mathematical Logic Quarterly
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| | volume = 51
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| | issue = 1
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| | pages = 62–72
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| | year = 2005
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| | doi = 10.1002/malq.200310127}} [http://www.cs.iastate.edu/~lutz/papers.html]
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| *{{Cite document
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| | author = J. Reimann
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| | title = Computability and fractal dimension, PhD thesis
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| | publisher = Ruprecht-Karls Universität Heidelberg
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| | year = 2004
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| | postscript = <!--None-->}} [http://www.math.uni-heidelberg.de/logic/reimann/publications.html]
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| [[Category:Fractals]]
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| [[Category:Measure theory]]
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| [[Category:Metric geometry]]
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| [[Category:Dimension theory]]
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| [[Category:Computable analysis]]
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Nice to meet you, my title is Ling and I completely dig that title. Her family members life in Delaware but she needs to transfer because of her family. The occupation he's been occupying for many years is a messenger. To perform badminton is something he truly enjoys doing.
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