In early 2000, classical Hausdorff dimension Haus19 was shown to admit a new characterization in terms of betting strategies called martingales DCCcon . This characterization enabled the development of various effective, i.e., algorithmic, versions of Hausdorff dimension obtained by imposing computability and complexity constraints on these martingales. These algorithmic versions included resource-bounded dimensions, which impose dimension structure on various complexity classes DCC , the (constructive) dimensions of infinite binary sequences, which interact usefully with algorithmic information theory DISS , and the finite-state dimensions of infinite binary sequences, which interact usefully with data compression and Borel normality FSD . Soon thereafter, classical packing dimension Tric82 ; Sull84 was shown to admit a new characterization in terms of martingales that is exactly dual to the martingale characterization of Hausdorff dimension ESDAICC . This led immediately to the development of strong resource-bounded dimensions, strong (constructive) dimension, and strong finite-state dimension ESDAICC , which are all algorithmic versions of packing dimension. In the years since these developments, hundreds of research papers by many authors have deepened our understanding of these algorithmic dimensions.
Most work to date on effective dimensions has been carried out in the Cantor space, which consists of all infinite binary sequences. This is natural, because effective dimensions speak to many issues that were already being investigated in the Cantor space. However, the classical fractal dimensions from which these effective dimensions arose–Hausdorff dimension and packing dimension–are powerful quantitative tools of geometric measure theory that have been most useful in Euclidean spaces and other metric spaces that have far richer structures than the totally disconnected Cantor space.
This chapter surveys research results to date on algorithmic fractal dimensions in geometric measure theory, especially fractal geometry in Euclidean spaces. This is a small fraction of the existing body of work on algorithmic fractal dimensions, but it is substantial, and it includes some exciting new results.
It is natural to identify a real number with its binary expansion and to use this identification to define algorithmic dimensions in Euclidean spaces in terms of their counterparts in Cantor space. This approach works for some purposes, but it becomes a dead end when algorithmic dimensions are used in geometric measure theory and computable analysis. The difficulty, first noted by Turing in his famous correction Tur38 , is that many obviously computable functions on the reals (e.g., addition) are not computable if reals are represented by their binary expansions Wei00 . We thus take a principled approach from the beginning, developing algorithmic dimensions in Euclidean spaces in terms of the quantity in the following paragraph, so that the theory can seamlessly advance to sophisticated applications.
Algorithmic dimension and strong algorithmic dimension are the most extensively investigated effective dimensions. One major reason for this is that these algorithmic dimensions were shown by the second author and others May02 ; ESDAICC ; LM08 to have characterizations in terms of Kolmogorov complexity, the central notion of algorithmic information theory. In Section 2 below we give a brief introduction to the Kolmogorov complexity of a point in Euclidean space at a given precision .
In Section 3 we use the above Kolmorogov complexity notion to develop the algorithmic dimension and the strong algorithmic dimension of each point in Euclidean space. This development supports the useful intuition that these dimensions are asymptotic measures of the density of algorithmic information in the point . We discuss how these dimensions relate to the local dimensions that arise in the so-called thermodynamic formalism of fractal geometry; we discuss the history and terminology of algorithmic dimensions; we review the prima facie case that algorithmic dimensions are geometrically meaningful; and we discuss what is known about the circumstances in which algorithmic dimensions agree with their classical counterparts. We then discuss the authors’ use of algorithmic dimensions to analyze self-similar fractals LM08 . This analysis gives us a new, information-theoretic proof of the classical formula of Moran Mora46 for the Hausdorff dimensions of self-similar fractals in terms of the contraction ratios of the iterated function systems that generate them. This new proof gives a clear account of “where the dimension comes from” in the construction of such fractals. Section 3 concludes with a survey of the dimensions of points on lines in Euclidean spaces, a topic that has been surprisingly challenging until a very recent breakthrough by N. Lutz and Stull LutStu17 .
We survey interactive aspects of algorithmic fractal dimensions in Euclidean spaces in Section 4, starting with the mutual algorithmic dimensions developed by Case and the first author CasLut15 . These dimensions, and , are analogous to the mutual information measures of Shannon information theory and algorithmic information theory. Intuitively, and are asymptotic measures of the density of the algorithmic information shared by points and in Euclidean spaces. We survey the fundamental properties of these mutual dimensions, which are analogous to those of their information-theoretic analogs. The most important of these properties are those that govern how mutual dimensions are affected by functions on Euclidean spaces that are computable in the sense of computable analysis Wei00 . Specifically, we review the information processing inequalities of CasLut15 , which state that and hold for all computable Lipschitz functions , i.e., that applying such a function to a point cannot increase the density of algorithmic information that it contains about a point . We also survey the conditional dimensions and recently developed by the first author and N. Lutz LutLut17 . Roughly speaking, these conditional dimensions quantify the density of algorithmic information in beyond what is already present in .
It is rare for the theory of computing to be used to answer open questions in mathematical analysis whose statements do not involve computation or related aspects of logic. In Section5 we survey exciting new developments that do exactly this. We first describe new characterizations by the first author and N. Lutz LutLut17 of the classical Hausdorff and packing dimensions of arbitrary sets in Euclidean spaces in terms of the relativized dimensions of the individual points that belong to them. These characterizations are called point-to-set principles because they enable one to use a bound on the relativized dimension of a single, judiciously chosen point in a set in Euclidean space to prove a bound on the classical Hausdorff or packing dimension of the set . We illustrate the power of the point-to-set principle by giving an overview of its use in the new, information-theoretic proof LutLut17 of Davies’s 1971 theorem stating that the Kakeya conjecture holds in the Euclidean plane Davi71 . We then discuss two very recent uses of the point-to-set principle to solve open problems in classical fractal geometry. These are N. Lutz and D. Stull’s strengthened lower bounds on the Hausdorff dimensions of generalized Furstenberg sets LutStu17 and N. Lutz’s extension of the fractal intersection formulas for Hausdorff and packing dimensions in Euclidean spaces from Borel sets to arbitrary sets. These are, to the best of our knowledge, the first uses of algorithmic information theory to solve open problems in classical mathematical analysis.
We briefly survey promising directions for future research in Section 6. These include extending the algorithmic analysis of self-similar fractals LM08 to other classes of fractals, extending algorithmic dimensions to metric spaces other than Euclidean spaces, investigating algorithmic fractal dimensions that are more effective than constructive dimensions (e.g., polynomial-time or finite-state fractal dimensions) in fractal geometry, and extending algorithmic methods to rectifiability and other aspects of geometric measure theory that do not necessarily concern fractal geometry. In each of these we begin by describing an existing result that sheds light on the promise of further inquiry.
Overviews of algorithmic dimensions in Cantor space appear in DowHir10 ; EFDAIT , though these are already out of date. Even prior to the development of algorithmic fractal dimensions, a rich network of relationships among gambling strategies, Hausdorff dimension, and Kolmogorov complexity was uncovered by reserach of Ryabko Ryab84 ; Ryab86 ; Ryab93 ; Ryab94 , Staiger Stai93 ; Stai98 ; Stai00 , and Cai and Hartmanis CaiHar94 . A brief account of this “prehistory” of algorithmic fractal dimensions appears in section 6 of DISS .
2 Algorithmic Information in Euclidean Spaces
Algorithmic information theory has most often been used in the set of all finite binary strings. The conditional Kolmogorov complexity (or conditional algorithmic information content) of a string given a string is
is a fixed universal Turing machine, andis the length of a binary “ program” . Hence is the minimum number of bits required to specify to , when is provided as side information. We refer the reader to any of the standard texts LiVit09 ; DowHir10 ; Nies12 ; ShUsVe17 for the history and intuition behind this notion, including its essential invariance with respect to the choice of the universal Turing machine . The Kolmogorov complexity (or algorithmic information content) of a string is then
where is the empty string.
Routine binary encoding enables one to extend the definitions of and to situations where and range over other countable sets such as , , , etc.
The key to “lifting” algorithmic information theory notions to Euclidean spaces is to define the Kolmogorov complexity of a set to be
(Shen and Vereshchagin SheVer02 used a very similar notion for a very different purpose.) Note that is the amount of information required to specify not the set itself, but rather some rational point in . In particular, this implies that
Note also that, if contains no rational point, then .
The Kolmogorov complexity of a point at precision is
where is the open ball of radius about , i.e., the number of bits required to specify some rational point satisfying , where is the Euclidean distance of from the origin.
3 Algorithmic Dimensions
3.1 Dimensions of Points
We now define the (constructive) dimension of a point to be
and the strong (constructive) dimension of to be
We note that and were originally defined in terms of algorithmic betting strategies called gales DISS ; ESDAICC . The identities (3.1) and (3.2) were subsequent theorems proven in LM08 , refining very similar results in May02 ; ESDAICC . These identities have been so convenient for work in Euclidean space that it is now natural to regard them as definitions.
Since is the amount of information required to specify a rational point that approximates to within (i.e., with bits of precision), and are intuitively the lower and upper asymptotic densities of information in the point . This intuition is a good starting point, but the fact that and are geometrically meaningful will only become evident in light of the mathematical consequences of (3.1) and (3.2) surveyed in this chapter.
It is an easy exercise to show that, for all ,
If is a computable point in , then , so . On the other hand, if is a random point in (i.e., a point that is algorithmically random in the sense of Martin-Löf MarL66 ), then , so . Hence the dimensions of points range between 0 and the dimension of the Euclidean space that they inhabit. In fact, for every real number , the dimension level set
and the strong dimension level set
Classical fractal geometry has local, or pointwise, dimensions that are useful, especially in connection with dynamical systems. Specifically, if is an outer measure on , i.e., a function satisfying , monotonicity (), and countable subadditivity (), and if is locally finite (i.e., every has a neighborhood with ), then the lower and upper local dimensions of at a point are
respectively, where Falc14 .
Until very recently, no relationship was known between the dimensions and and the local dimensions (3.6) and (3.7). However, N. Lutz recently observed that a very non-classical choice of the outer measure remedies this. For each , let
where is defined as in (2.1). Then is easily seen to be an outer measure on that is finite (i.e., ), hence certainly locally finite, whence the local dimensions and are well defined. In fact we have the following.
(N. LutzNLut16 ) For all ,
The Hausdorff dimension of a set was introduced by Hausdorff Haus19 before 1920 and is arguably the most important notion of fractal dimension. Its classical definition, which may be found in standard texts such as SteSha05 ; Falc14 ; BisPer17 , involves covering the set by families of sets with diameters vanishing in the limit. In all cases, .
At the beginning of the present century, in order to formulate versions of Hausdorff dimensions that would work in complexity classes and other algorithmic settings, the first author DCC gave a new characterization of Hausdorff dimension in terms of betting strategies, called gales, on which it is easy to impose computability and complexity conditions. Of particular interest here, he then defined the constructive dimension of a set exactly like the gale characterization of , except that the gales were now required to be lower semicomputable DISS . He then defined the dimension of a point to be the constructive dimension of its singleton, i.e., . The existence of a universal Turing machine made it immediately evident that constructive dimension has the absolute stability property that
Two things should be noted about the preceding paragraph. First, these early papers were written entirely in terms of binary sequences, rather than points in Euclidean space. However, the most straightforward binary encoding of points bridges this gap. (In this survey we freely use those results from Cantor space that do extend easily to Euclidean space.) Second, although the gale characterization is essential for polynomial time and many other stringent levels of effectivization, constructive dimension can be defined equivalently by effectivizing Hausdorff’s original formulation Reim04 .
3.2 The Correspondence Principle
In 2001, the first author conjectured that there should be a correspondence principle (a term that Bohr had used analogously in quantum mechanics) assuring us that for sufficiently simple sets , the constructive and classical dimensions agree, i.e.,
Hitchcock Hitchcock:CPED confirmed this conjecture, proving that (3.10) holds for any set that is a union of sets that are computably closed, i.e., that are in Kleene’s arithmetical hierarchy. (This means that (3.10) holds for all sets, and also for sets that are nonuniform unions of sets.) Hitchcock also noted that this result is the best possible in the arithmetical hierarchy, because there are sets (e.g., , where is a Martin-Löf random point that is ) for which (3.10) fails.
which is a very nonclassical, pointwise characterization of the classical Hausdorff dimensions of sets that are unions of sets. Since most textbook examples of fractal sets are , (3.11) is a strong preliminary indication that the dimensions of points are geometrically meaningful.
The packing dimension of a set was introduced in the early 1980s by Tricot Tric82 and Sullivan Sull84 . Its original definition is a bit more involved that that of Hausdorff dimension Falc14 ; BisPer17 and implies that for all .
After the development of constructive versions of Hausdorff dimension outlined above, Athreya, Hitchcock, and the authors ESDAICC undertook an analogous development for packing dimension. The gale characterization of turns out to be exactly dual to that of , with just one limit superior replaced by a limit inferior. The strong constructive dimension of a set is defined by requiring the gales to be lower semicomputable, and the strong dimension of a point is . The absolute stability of strong constructive dimension,
holds for all , as does the Kolmogorov complexity characterization (3.2). All this was shown in ESDAICC , but a correspondence principle for strong constructive dimension was left open. In fact, Conidis Coni08 subsequently used a clever priority argument to construct a set for which . It is still not known whether some simple, logical definability criterion for implies that . Staiger’s proof that regular -languages satisfy this identity is an encouraging step in this direction Stai07 .
3.3 Self-Similar Fractals
The first application of algorithmic dimensions to fractal geometry was the authors’ investigation of the dimensions of points in self-similar fractals LM08 . We give a brief exposition of this work here, referring the reader to LM08 for the many missing details.
Self-similar fractals are the most widely known and best understood classes of fractals Falc14 . Cantor’s middle-third set, the von Koch curve, the Sierpinski triangle, and the Menger sponge are especially well known examples of self-similar fractals.
Briefly, a self-similar fractal in a Euclidean space is generated from an initial nonempty closed set by an iterated function system (IFS), which is a finite list of contracting similarities . Each of these similarities is coded by the symbol in the alphabet , and each has a contraction ratio . The IFS is required to satisfy Moran’s open set condition Mora46 , which says that there is a nonempty open set whose images , for , are disjoint subsets of .
For example, the Sierpinski triangle is generated from the set consisting of the triangle with vertices , , and , together with this triangle’s interior, by the IFS , where each is defined by
for . Note that and in this example. Note also that the open set condition is satisfied here by letting be the topological interior of . Each infinite sequence codes a point that is obtained by applying the similarities coded by the successive symbols in in a canonical way. (See Figure 1.) The Sierpinski is the attractor (or invariant set) of and , which consists of all points for .
The main objective of LM08 was to relate the dimension and strong dimension of each point in a self-similar fractal to the corresponding dimensions of the coding sequence . As it turned out, the algorithmic dimensions in had to be extended in order to achieve this.
The similarity dimension of an IFS with contraction ratios is the unique solution of the equation
The similarity probability measure of
is the probability measure onthat is implicit in (3.13), i.e., the function defined by
for each . If the contraction ratios of are all the same, then is the uniform probability measure on , but this is not generally the case. We extend to the domain by setting
for each . We define the Shannon -self-information of each string to be the quantity
Finally, we define the dimension of a sequence with respect to the IFS to be
Similarly, the strong dimension of with respect to is
A set is a computably self-similar fractal if it is the attractor of some and as above such that the contracting similarities are all computable in the sense of computable analysis.
The following theorem gives a complete analysis of the dimensions of points in computably self-similar fractals.
(J. Lutz and Mayordomo LM08 ) If is a computably self-similar fractal and is an IFS testifying to this fact, then, for all points and all coding sequences for ,
The proof of Theorem 3.2 is nontrivial. It combines some very strong coding properties of iterated function systems with some geometric Kolmogorov complexity arguments.
The following characterization of continuous functions on the reals is one of the oldest and most beautiful theorems of computable analysis.
Using Lacombe’s theorem it is easy to derive the classical analysis of self-similar fractals (which need not be computably self-similar) from Theorem 3.2.
Let and be as given. By Lacombe’s theorem there is an oracle relative to which is computable. It follows by a theorem by Kamo and Kawamura KK99 that the set is relative to , whence the relativization of (3.11) tells us that
We then have
so (3.21) holds.
3.4 Dimension Level Sets
Hitchcock, Terwijn, and the first author HiLuTe07 investigated the complexities of these dimension level sets from the viewpoint of descriptive set theory. Following standard usage Mosc80 , we write and for the classes at the th level () of the Borel hierarchy of subsets of . That is, is the class of all open subsets of , each is the class of all complements of sets in , and each is the class of all countable unions of sets in . We also write and for the classes of the th level of Kleene’s arithmetical hierarchy of subsets of . That is, is the class of all computably open subsets of , each is the class of all complements of sets in , and each is the class of all effective (computable) unions of sets in .
Recall that a real number is -computable if there is a computable function such that .
The following facts were proven in HiLuTe07 .
is but not .
For all , is (and if is -computable) but not .
is and but not .
For all , is (and if is -computable) but not .
Weihrauch and the first author LutWei08 investigated the connectivity properties of sets of the form
where is an interval. After making the easy observation that each of the sets and is totally disconnected, they proved that each of the sets and is path-connected. These results are especially intriguing in the Euclidean plane, where they say that extending either of the sets and to include the level set transforms it from a totally disconnected set to a path-connected set. This suggests that is somehow a very special subset of .
Turetsky Ture11 investigated this matter further and proved that is a connected set in . He also proved that is not a path-connected subset of .
3.5 Dimensions of Points on Lines
Since effective dimension is a pointwise property, it is natural to study the dimension spectrum of a set , i.e., the set . This study is far from obvious even for sets as apparently simple as straight lines. We review in this section the results obtained so far, mainly for the case of straight lines in .
As noted in section 3.4, the set of points in of dimension exactly one is connected, while the set of points in with dimension less than 1 is totally disconnected. Therefore every line in contains a point of dimension 1. Despite the surprising fact that there are lines in every direction that contain no random points LL15 , the first author and N. Lutz have shown that almost every point on any line with random slope has dimension 2 LutLut17 . Still all these results leave open fundamental questions about the structure of the dimension spectra of lines, since they don’t even rule out the possibility of a line having the singleton set as its dimension spectrum.
Very recently this latest open question has been answered in the negative. N. Lutz and Stull LutStu17 have proven the following general lower bound on the dimension of points on lines in .
(N. Lutz and Stull LutStu17 ) For all ,
In particular, for almost every , .
Taking and a Martin-Löf random real relative to , Theorem 3.4 gives us two points in the line, and , whose dimensions differ by at least one, so the dimension spectrum cannot be a singleton.
We briefly sketch here the main intuitions behind the (deep) proof of Theorem 3.4, fully based on algorithmic information theory. Theorem 3.4’s aim is to connect with (i.e., a dimension in with a dimension in ). Notice that in the case the theorem’s conclusion is close to saying .
The proof is based on the property that says that under the following two conditions
whenever , either is large or is close to
it holds that is close to .
There is an extra ingredient to finish this intuition.While condition (ii) can be shown to hold in general, condition (i) can only be proven in a particular relativized world whereas the conclusion of the theorem still holds for every oracle.
N. Lutz and Stull LS17 have also shown that the dimension spectrum of a line is always infinite, proving the following two results. The first theorem proves that if then the corresponding line contains a length one interval.
(N. Lutz and Stull LS17 ) Let satisfy that . Then for every there is a point such that .
The second result proves that all spectra of lines are infinite.
(N. Lutz and Stull LS17 ) Let be any line in . Then the dimension spectrum is infinite.
4 Mutual and Conditional Dimensions
Just as the dimension of a point in Euclidean space is the asymptotic density of the algorithmic information in , the mutual dimension between two points and in Euclidean spaces is the asymptotic density of the algorithmic information shared by and
. In this section, we survey this notion and the data processing inequalities, which estimate the effect of computable functions on mutual dimension. We also survey the related notion of conditional dimension.
4.1 Mutual Dimensions
The mutual (algorithmic) information between two rational points and is
This notion, essentially due to Kolmogorov Kolm65 , is an analog of mutual entropy in Shannon information theory Shan48 ; CovTho06 ; LiVit09 . Intuitively, is the amount of information in not contained in , so is the amount of information in that is contained in . It is well known LiVit09 that, for all and ,
in the sense that the magnitude of the difference between the two sides of (4.2) is . This fact is called symmetry of information, because it immediately implies that .
This is the amount of information that rational points and must share in order to be in and , respectively. Note that, for all and ,
The mutual information between two points and at precision is
This is the amount of information that rational approximations of and must share, merely due to their proximities (distance less than ) to and .
The following theorem shows that the mutual dimensions mdim and Mdim have many of the properties that one should expect them to have. The proof is involved and includes a modest generalization of Levin’s coding theorem Levi73a ; Levi74 .
(Case and J. Lutz CasLut15 ) For all and , the following hold.
If and are independently random, then .
4.2 Data Processing Inequalities
The data processing inequality of Shannon information theory CovTho06 says that, for any two probability spaces and , any set , and any function ,
i.e., the induced probability space obtained by “processing the information in through ” has no greater mutual entropy with than has with . More succintly, tells us no more about than tells us about . The data processing inequality of algorithmic information theory LiVit09 says that, for any computable partial function , there is a constant (essentially the number of bits in a program that computes ) such that, for all strings and ,
That is, modulo the constant , contains no more information about than contains about .
The data processing inequality for the mutual dimension should say that every nice function has the property that, for all and ,
But what should “nice” mean? A nice function certainly should be computable in the sense of computable analysis BC06 ; Ko91 ; Wei00 . But this is not enough. For example, there is a function that is computable and space-filling in the sense that Saga94 ; CDM12 . For such a function, choose such that , and let . Then
so (4.6) fails.
Intuitively, the above failure of (4.6) occurs because the function is extremely sensitive to its input, a property that “ nice” functions do not have. A function is Lipschitz if there is a real number such that, for all ,
The following data processing inequalities show that computable Lipschitz functions are “nice”.
(Case and J. Lutz CasLut15 ) If is computable and Lipschitz, then, for all and ,
Several more theorems of this type and applications of these appear in CasLut15 .
4.3 Conditional Dimensions
A comprehensive theory of the fractal dimensions of points in Euclidean spaces requires not only the dimensions and and the mutual dimensions and , but also the conditional dimensions and formulated by the first author and N. Lutz LutLut17 . We briefly describe these formulations here.
The conditional Kolmogorov complexity , defined for rational points and , is lifted to the conditional dimensions in the following four steps.
For , , and , the conditional Kolmogorov complexity of at precision given is
For , , and , the conditional Kolmogorov complexity of at precision given at precision is
For , , and , the conditional Kolmogorov complexity of given at precision is
For and , the lower and upper conditional dimensions of given are
Steps 1, 2, and 4 of the above lifting are very much in the spirit of what has been done in section 2, 3.1, and 4.1 above. Step 3 appears to be problematic, because using the same precision bound for both and makes the definition seem arbitrary and “brittle”. However, the following result shows that this is not the case.
(LutLut17 ) Let . If , then, for all and ,
The following result is useful for many purposes.
(chain rule for
(chain rule for) For all and ,
An oracle for a point is a function such that, for all , . The Kolmogorov complexity of a rational point relative to a point is
where is the Kolmogorov complexity of when the universal machine has access to the oracle . The purpose of the maximum here is to prevent from using oracles that code more than into their behaviors. For and , the dimension relative to is defined from exactly as was defined from in Sections 2 and 3.1 above. The relativized strong dimension is defined analogously.
The following result captures the intuition that conditioning on a point is a restricted form of oracle access to .
(LutLut17 ) For all and , and .
The remaining results in this section confirm that conditional dimensions have the correct information-theoretic relationships to dimensions and mutual dimensions.
(LutLut17 ) For all and ,
(chain rule for dimension LutLut17 ) For all and ,