Fractal dimension versus Computational Complexity

Box dimension and Turing machines

The experiment

Demo

Theoretical panorama

Fractal Dimension versus Computational
Complexity
Joost J. Joosten
Fernando Soler-Toscano
Hector Zenil
jjoosten@ub.edu, fsoler@us.es, hectorz@labores.eu

Seminari Cuc, Barcelona
January, 2014

Fractal Dimension versus Computational Complexity


The experiment

Demo


The small Turing Machine database

Small Turing machines
We consider Turing machines where the tape extends infinitely in
one direction (to the left in the diagrams)
Each tape cell contains one symbol (color)
We use just two colors: black and white
A Turing machine starts its computation with the head at the first
tape cell (beginning of the tape)
The input of the computation is written at the initial cells
The computation ends when the machine is at the beginning of
the tape and moves to the right (out of the tape)
The tape configuration upon termination of a computation is
called the output
The set of Turing machines with n states and k colors is
represented by (n, k)
We have enumerated the machines in (n, k) from 0 to (2 · n · k)n·k − 1
We present the results of an exhaustive study of (2, 2) and (3, 2)


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The small Turing Machine database

Space-time diagrams
A space-time diagram for some computation is the joint
collection of consecutive memory configurations
The top-row of each diagram
represents the input (1 to 14)
The computation starts with
the head of the TM in state 1
in the rightmost cell
Each lower row represents the
tape configuration of a next
step in the computation
These space-time diagrams define spatial objects by focussing
on the black cells. We can measure the geometrical complexity.
We wish to see if there is a relation between this geometrical
complexity and the computational complexity (space or time
usage) of the TM in question.


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Fractal dimensions

Box dimension
The notion of Box dimension is a simplification of, and an upper
bound to Hausdorff dimension
Suppose we have a mathematical object S of bounded size. The
idea is to cover S with boxes in Rn and estimate the “volume”
V(S) of S as function of the total number of boxes N(S, r) of size r
needed to cover S: V(S) = limr↓0 rd N(S, r)

Definition (Box dimension)
Let S be some spatial object that can be embedded in some Rn , let
N(S, r) denote the minimal number of boxes of size r needed to fully
cover S. The Box dimension of S is denoted by δ(S) and is defined by
δ(S) := lim
r↓0

log(N(S, r))
log( 1 )
r

in case this limit is well defined. In all other cases we shall say that
δ(S) is undefined.


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Fractal dimensions

Box dimension for Space-Time diagrams
We adapt the notion of Box dimension to space-time diagrams
Clearly, for each input on which the TM halts the corresponding
space-time diagram has dimension 2: it’s a piece of surface
It gets interesting when we consider limiting behavior of the TM

Definition (Box dimension of a Turing machine)
Let τ be a TM that converges on infinitely many input values x. In
case τ(x) ↓, let N(τ, x) denote the number of black cells in the
space-time diagram of τ on input x and let t(τ, x) denote the number
of steps needed for τ to halt on x.
We will define the Box dimension of a TM τ and denote it by d(τ). In
case t(τ, x) is constant from some x onwards, we define d(τ) := 2.
Otherwise, we define
d(τ) := lim inf
x→∞


log N(τ, x)
log t(τ, x)

.


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The Space-Time Theorem and applications

The Space-time Theorem: upper and lower bounds

Theorem (Space-time Theorem: upper bound)
Let us, for a given TM τ, denote by s(x) the number of cells visited by
τ on input x, and let t(x) denote the number of computation steps it
took τ to terminate on input x.
If lim infx→∞

log(s(x))
log(t(x))

= n then d(τ) ≤ 1 + n.

Lemma (lower bound)
For each TM τ we have that d(τ) ≥ 1 provided limx→∞


s(x)
t(x)

= 0.


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The Upper Bound Conjecture
Lemma
In case a TM τ uses polynomial space, and runs super-polynomial
time we have that d(τ) = 1.
More in general, if τ uses space sτ (x) and time tτ (x) on input x then
lim inf

log sτ (x)

x→∞

log tτ (x)

= 0 ⇐⇒ d(τ) = 1.

Conjecture (Upper Bound Conjecture)
We conjecture that for each n ∈ ω and each TM τ in (n, 2) space that
d(τ) = 1 + lim inf
x→∞


log sτ (x)
log tτ (x)


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The Space-Time Theorem and P versus NP

Using the previous Lemma, we can state a separation of P and
NP in terms of dimensions:
Let Π be some NP-complete problem
If for each PSPACE Turing machine τ that decides Π we have that
d(τ) = 1, then P NP.

Clearly, this does not constitute a real strategy since, in general, it
is undecidable whether d(τ) = 1



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Methodology

Slow convergence
Our aim is to use computer experiments to compute the Box
dimension of all TMs τ where d(τ) is not predicted by any
theoretical result.
A substantial complication is caused by the
occurrence of logarithms in d(τ)
As a consequence, increase in precision of
d(τ) requires exponentially larger inputs
For (2, 2) TM 346 we know that its Box dimension is 2, but we can
see in the picture how slow the rate of convergence is
Our way out here is to apply numerical and mathematical
analysis to the functions involved so that we can retrieve their
limit behavior.
We are interested in three diﬀerent functions:
tτ (x), number of time-steps needed for τ to halt on input x
Nτ (x), number of black cells in the space-time diagram of τ on
input x
sτ (x), number of tape cells visited by τ on input x



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Methodology

Steps followed
Each TM in (2, 2) also occurs in (3, 2) so for the ﬁnal results it
suﬃces to focus on this data-set. We isolated the TMs for which
there is no theorem that predicts the corresponding dimension.
Boxes
O(n3 )
O(n4 )
o(P)

Runtime
O(n2 )
O(n3 )
o(P)

Space
O(n)
O(n)
o(P)

Machines
3358
6
14

Per TM τ, we determined its functions sτ (x) (space), tτ (x) (time)
and Nτ (x) (black cells). We used FindSequenceFunction and
other Mathematica functions
Per TM τ, we computed its dimension d(τ) as
log(N (x))
d(τ) = lim infx→∞ log(tττ(x))
Per TM τ, we compared its dimension d(τ) to its theoretical
log(sτ (x))
upper bound 1 + lim infx→∞ log(tτ (x))


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Methodology

Alternating convergent behavior
Some machines have alternating asymptotic behavior
This is the most extreme example (TM 1,728,529):
For convenience we have changed the
orientation of the diagrams so that time
‘goes from left to right’ instead of from
‘top to bottom’.
In a sense this TM incorporates two
diﬀerent algorithms to compute this
output: one in linear time, the other, in
exponential time.

We have found alternating sequences of periodicity 2, 3 and 6
The periodicity typically reﬂects either the number of states, the
number of colors, or a divisor of their product.
Because of this alternating behavior we could not analyze the
data in a straight-forward automated fashion.


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Most salient results of the experiment

Findings in (2, 2) space
In (2, 2) there was a total of 74 diﬀerent functions. Only 5 of them
where computed by some super-linear time TMs
In total, in (2, 2) space, there are only 7 TMs that run in
super-polynomial time. Three of them run in exp-time, all
computing the tape-identity. The other four (see below) TMs
compute diﬀerent functions (that roughly double the tape input)
All these four TMs perform in quadratic time and linear space.
The dimension for these functions is 3 . This is exactly the upper
2
bound as predicted by the Space-Time Theorem.



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Findings in (3, 2) space
The (3, 2) space contains 2,985,984 many diﬀerent TMs which
compute 3,886 diﬀerent functions
Almost all TMs used at most linear space for their computations
The only exception to this was when the TM used exponential
space
Busy Beaver: we call a TM β a Busy Beaver whenever for each
TM τ, there is some value x0 so that for all x ≥ x0 we have
tβ (x) ≥ tτ (x)
Twin Machines 599,063 and 666,364 are the Busy Beavers in (3,2)
space, running in exponential space and time.
They compute the largest runtime, space and boxes sequences.
They also produce the longest output strings. Fractal dim.: 3/2



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The space-time theorem revisited
One of our most important empirical ﬁndings is that the upper bound
as given by the Space-Time Theorem is actually always attained in
(3, 2) space.

Finding 1
For all TMs τ in (3,2) space we found that
d(τ) = 1 + lim inf
x→∞

log(sτ (x))
log(tτ (x))

as conjectured in the Upper Bound Conjecture (UBC postulates this
for any number of states).



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Other findings
Finding 2
For all TMs τ in (3,2) space we found that d(τ) = 1 if and only if the
TM ran in super-polynomial time using polynomial space. We
suspect that this equivalence holds no longer true in higher spaces,
i.e., spaces (n, 2) for n > 3.

Finding 3
For all TMs τ in (3,2) space we found that d(τ) = 2 if and only if the
TM ran in at most linear time. It is unknown if this equivalence holds
true in higher spaces (the “if” part holds in general and is proven
previous lemmas)

Finding 4
s(x)

For all TMs τ in (3,2) space we found that limx→∞ t(x) = 1 so that
d(τ) ≥ 1. We conjecture this holds true also in larger space.


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Richness in the (3, 2) space
We have found two symmetric performers for even inputs
This can only occur in machines computing the tape identity and
requires strong conditions
It is surprising that such constraints can be met in (3, 2)



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Part of a larger project
H. Zenil, F. Soler-Toscano, J. J. Joosten. Empirical Encounters
with Computational Irreducibility and Unpredictability. Minds
& Machines, Volume 22, Issue 3, pages 149-165, 2012.
J. J. Joosten, F. Soler, and H. Zenil. Program-size versus Time
Complexity. Slowdown and Speed-up Phenomena in the
Micro-cosmos of Small Turing Machines. Int. Journ. of
Unconventional Computing, Vol. 7, pp. 353-387, 2011.
Joost J. Joosten, Fernando Soler-Toscano, Hector Zenil. Fractal
Dimension of Space-time Diagrams and the Runtime
Complexity of Small Turing Machines, in T. Neary and M. Cook
(Eds.): Machines, Computations and Universality (MCU 2013).
J. J. Joosten. Complexity, Universality and Intermediate Degrees.
In American Institute of Physics Conference proceedings,
Volume 1479, Pages 638-641, AIP Publishing, ISSN 0094 243X,
doi:http://dx.doi.org/10.1063/1.4756215, 2012.


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J. J. Joosten, F. Soler-Toscano, and H. Zenil. Complejidad
´
˜
descriptiva y computacional en maquinas de Turing pequenas.
Proceedings of the V Jornadas Ib´ ricas de Filosof´a de la Ciencia,
e
ı
Logica y Lenguaje, Lisbon 2010, in Logica Universal e Unidade
´
´
da Ciˆ ncia, Centro de Filosoﬁa das Ciˆ ncias da Universidade de
e
e
Lisboa, pp. 11-32, ISBN: 978-989-8247-49-0, 2011.
J. J. Joosten, F. Soler-Toscano, H. Zenil. Speedup and Slowdown
Phenomena in Turing Machines. Wolfram Demonstrations
Project, http://demonstrations.wolfram.com/
SpeedupAndSlowdownPhenomenaInTuringMachines/, 2012.
J. J. Joosten. Turing Machine Enumeration: NKS versus
Lexicographical. Wolfram Demonstrations Project,
http://demonstrations.wolfram.com/
TuringMachineEnumerationNKSVersusLexicographical/, 2010.
J. J. Joosten, F. Soler-Toscano, H. Zenil. Runtime complexity of
small Turing Machines and fractal dimension. Wolfram
Demonstrations Project, To be submitted soon, 2014.


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J. J. Joosten, H. Zenil, F. Soler-Toscano. Entropy as an indication
of the runtime of terminating discrete dynamical processes. In
Book of abstracts, European Conference on Complex Systems,
p214, S. Thurner M. Szell editors, Locker Verlag, ISBN
¨
978-3-85409-613-9, Vienna 2011.
J. J. Joosten. Complexity ﬁts the ﬁttest. In Emergence,
Complexity and Computation in Nature. Springer Verlag, I.
Zelinka, A. Sanayei, H. Zenil H., O. E. Rossler, editors, ISBN
978-3-319-00253-8, 2013.
J. J. Joosten. On the Necessity of Complexity. In Irreducibility
and Computational Equivalence: 10 Years After the Publication
of Wolfram’s A New Kind of Science, (11-24). Springer,
Heidelberg New York Dordrecht London, H. Zenil editor, ISBN
978-3-642-35481-6, 2013.


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Demo

We have prepared a demo to visualize the space-time diagrams
for several TMs in (3, 2)
It will be published in Wolfram Demonstrations Project, and is
available upon request


Demo


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Complexity measures related

Various complexity measures
Entropy, box-counyting dimension, computational complexity,
Kolmogorov complexity, Hausdorff dimension, etc.
Each such measure captures/quantifies (or aims to) the
complexity of one particular aspect of a system
On philosophical grounds we can expect relations between
different complexity measures
J. J. Joosten. Complexity fits the fittest. In Emergence,
Complexity and Computation in Nature. Springer Verlag, I.
Zelinka, A. Sanayei, H. Zenil H., O. E. Rossler, editors, ISBN
978-3-319-00253-8, 2013.
J. J. Joosten. On the Necessity of Complexity. In Irreducibility
and Computational Equivalence: 10 Years After the Publication
of Wolfram’s A New Kind of Science, (11-24). Springer,
Heidelberg New York Dordrecht London, H. Zenil editor, ISBN
978-3-642-35481-6, 2013.


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Geometrical complexity measures

Topological dimensions
Edgar divides geometrical dimensions in two main groups,
topologic versus fractal dimension.
Edgar E. G. Measure, Topology, and Fractal Geometry,
Springer-Verlag. New York, 1990.
Most basic of all topological dimensions is cover dimension also
called Lebesgue dimension.
The order of a family A of sets is ≤ n by deﬁnition when any
n + 2 of the sets have empty intersection. We say = n when ≤ n
but not ≤ n − 1.
The cover dimension of a set S is n –we write Cov(S) = n–
whenever each open covering of S has a reﬁnement of order n.
Topological measures have integer values and are invariant
under homeomorphisms.



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Fractal dimensions
A fractal dimension of some object S is an indication of how close
S is to some integer-valued dimensional space
Dimension in integer-valued dimensional space in a sense
express degrees of freedom (information theoretical focus)
Falconer: “Roughly, dimension indicates how much space a set
occupies near to each of its points.” (geometrical focus)
Falconer, K. J. Fractal Geometry, Mathematical Foundations and
Applications, Wiley, Chichester, 2003.
Most fundamental, and most common notion is that of Hausdorff
dimension
¨
F. Hausdorff Dimension und ausseres Mass. Mathematische
Annalen, 79:157–179, 1919.
¨
Building upon ideas of Carath´ odory: Carath´ odory, C. Uber das
e
e
lineare Mass von Punktmengen, eine Veralgemeinerung des
¨
Langenbegriffs. Nachrichten von der Wissenschaften zu
Gotingen, Mathematisch-Physikalische Klass, 404–426, 1914.
¨


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Hausdorff dimension
For a S some subset of some metric space we can consider
countable open coverings A of S and define
(diam A)s .

s
Hε (S) := inf
A∈A

The infimum is taken over all A that are countable open ε-covers
of S. Then
s
H s (S) := lim Hε (S).
ε→0

Main Theorem:
There is a unique s so that
t
Hε (S) = ∞ for t < s;
t
Hε (S) = 0 for t > s.

This unique s is called the Hausdorff (Mandelbrot speaks of
Hausdorff-Besicovitch) dimension of S: dimH (F).
Besicovitch, A. S. Sets of fractional dimensions. Part I:
Mathematical Annals 101, 161–193, 1929.
Part V: London Mathematical Society 12, 18–25, 1934.


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Packing dimension
Hausdorff comes with a natural dual dimension called packing
dimension.
Tricot, C. Jr. Two definitions of fractional dimension.
Mathematical Proceedings of the Cambridge Philosophical
Society, 91, 57–74, 1982.
Sullivan, D. Entropy, Hausdorff measures old and new, and limit
sets of geometrically finite Kleinian groups. Acta Mathematica,
153, 259–277, 1984.
|Bi | | {Bi }i are disjoint balls at radii ≤ δ and center in F}

Ps (F) := {sup
δ
i

Since limδ→0 Ps (F) is not a measure (consider countable dense
δ
sets) one defines
∞

Ps (F) := inf{
{Fi }i


lim Ps (Fi ) | F ⊆
δ
i

δ→0

Fi }.
i=1


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Main Theorem:
There is a unique s so that
Pt (F) = 0 for t < s;
ε
t
Hε (S) = ∞ for t > s.

This unique s is called the packing dimension of F: dimP (F).
Packing dimension is an upper bound to Hausdorﬀ dimension:
dimH (F) ≤ dimP (F)
A fundamental property: Cov(F) ≤ dimH (F)
Mandelbrot deﬁnes a fractal for any set F with Cov(F) < dimH (F)
Often considered (also by Mandelbrot) a notion of fractal that is
too broad, since it admits “true geometric chaos”
J. Taylor proposes to denote by fractals only Borel sets F for
which dimH (F) ≤ dimP (F).
Taylor, S. J. The measure theory of random fractals.
Mathematical Proceedings of the Cambridge Philosophical
Society, 100, 383–406, 1986.


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Box dimensions
The Box dimension is like Hausdorff dimension only that we
now cover by balls/boxes of fixed size.
Alternatively and equivalently, divide space into a regular mesh
with mesh-size δ and count how many cells Nδ (F) are hit by a set
F
Then define Bs (F) := Nδ (F)δs and Bs (F) := lim infδ→0 Nδ (F)δs .
δ
Again, there is a cut-off value s0 so that Bs (F) = ∞ for s < s0 and
Bs (F) = 0 for s > s0
This cut-off value is given by
lim inf
δ→0

log(Nδ (F))
.
log(1/δ)

log(Nδ (F))
and
δ→0
log(1/δ)
log(Nδ (F))
dimB := lim sup
log(1/δ)
δ→0
We define dimB := lim inf



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In case dimB (F) = dimB (F) we call this the box-counting
dimension: dimB (F)
Box dimension always provides an upper bound to Hausdorﬀ
dimension
Box dimension has desirable computational properties
but undesirable mathematical properties: a countable union of
measure zero sets can have positive box dimension
Example: dimB {0, 1 , 1 , 1 , . . .} =
2 3 4

1
2

Mathematically this can be repaired by deﬁning
∞

dimMB (F) := inf{sup dimB (Fi ) | F ⊆
{Fi }

i

Fi } and
i=1
∞

dimMB (F) := inf{sup dimB (Fi ) | F ⊆
{Fi }

i

Fi }
i=1

loosing the good computational properties of course . . .


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We have dimH (F) ≤ dimMB (F) ≤ dimMB (F) = dimP (F) ≤ dimB (F)
None of the inequalities can be replaced by equalities
Note that under Taylor’s definition of fractal, the first four
dimensions collapse and modified box dimension is an
equivalent of Hausdorff dimension
Moreover, if F has a lot of self-similarity, then modified is equal
to plane box counting dimension:
Let F ⊆ R be compact so that for any open set V we have
dimB (F) = dimB (F ∩ V), then dimB (F) = dimMB (F).
So in various situations, box counting coincides with Hausdorff
dimension (like Mandelbrot set)
There are various other situations where box-counting and
Hausdorff dimension coincide
Staiger, L. A tight upper bound on Kolmogorov complexity and
uniformly optimal prediction. Theory of Computing Systems,
31:215–229, 1998.
Staiger, L. Constructive dimension equals Kolmogorov
Complexity. Information Processing Letters, 93:149–153, 2005.


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Computational properties of fractals

Julia sets
Probably the most famous examples of fractals are Julia sets and
the corresponding “roadmap Mandelbrot set”
By FJ(f ) we denote the ﬁlled Julia set of a function f deﬁned on
the complex numbers is the set of values z in the domain of f on
which iterating f on z does not diverge. That is,
FJ(f ) := {z | lim sup |f n (z)| < ∞}
n→∞

By J(f ) –the Julia set of f – we denote the boundary of FJ(f )
Following C.T. Chong, we can consider fθ (z) = z2 + λz with
λ = e2πiθ and θ Q
Corresponding Julia sets are denoted by Jθ
Jθ being well-behaved is expressed by saying that it has a Siegel
disk at z = 0
Basically, this says that f is locally linearizable at z = 0 by a
rotation
See: Milnor, J. Dynamics in one complex variable. Introductory
lectures. Princeton University Press, 2006.


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Constructive Analysis
There are various results between the Turing degree of θ and that
of Jθ
One first has to define what the Turing degree of non-discrete
objects actually means
Braverman and Yampolsky follow an approach of Constructive
Analysis as initiated by Banach and Mazur, with influence of
Markov.
Banach, S., Mazur, S. Sur les fonctions calculables. Ann. Polon.
Math. 16, 1937.
Markov, A. A. On constructive mathematics (Russian) Tr. Mat.
Inst. Steklov. 67, 8–14; translated in Amer. Math. Soc., Trans., II
Ser. 98, 1–9, 1962.
Overview: Weihrauch, Computable Analysis, Springer, Berlin,
2000.


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Braverman, Yampolsky: b is a c.e. Turing degree if and only if it
is the degree of Jθ with θ recursive so that Jθ has a Siegel disk
Braverman, M., Yampolsky, M. Computability of Julia Sets,
Algorithms and Computation in Mathematics, Springer, 2009.
C.T. Chong Generalized this result: Let c be a Turing degree. For
every d ≥ c we have that d is c.e. in c if and only if it is the degree
of a Julia set Jθ with Siegel disk and deg(θ) = c.
C.T. Chong, unpublished; Slides Complex Dynamics and Turing
Degrees online.
Results sensitive to model of computation and change with, e.g.,
Blum-Schub-Smale model
See L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and
real computation, Springer-Verlag, New York, 1998.
This relates the Turing complexity of the fractal to the complexity
of the parameter generating it
However, no link to the corresponding dimension
This we will see in what follows


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Fractal dimension versus other complexity notions

Hausdorff dimension on strings
Let us reformulate the definition of Hausdorff dimension in the
realm of binary sequences, i.e., Cantor space
Overview can be found in Downey, R.G. and Hirschfeldt, D.R.
Algorithmic Randomness and Complexity, Chapter 13, Springer,
2010.
For σ ∈ 2<ω we denote the length of σ as |σ|
For σ ∈ 2<ω we define σ := {στ | τ ∈ 2ω } (a (sub-)basic open set)
For Σ ⊆ 2<ω we define Σ := σ∈Σ σ
Let R ⊆ 2ω . An n-cover of R is a set Σ ⊆ 2≥n so that R ⊆ Σ .
2−s|σ| | Σ an n-cover of R}

s
Hn (R) := inf{
σ∈Σ
s

H (R) := lim

n→∞

s
Hn (R)

So, as before, dimH (R) := inf{s | H s (R) = 0}
Clearly, for every r ∈ R there is R ⊆ 2ω with dimH (R) = r


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Effective Hausdorff dimension
The effective pendant is now defined via
EH s (R) := inf{
n

2−s|σ| | Σ a c.e. n-cover of R}
σ∈Σ

s

EH (R) := lim

n→∞

EHs (R)
n

So that the effective Hausdorff dimension is defined as
dimEH (R) := inf{s | EH s (R) = 0}
For every computable real r, there is a set R ⊆ 2ω with
dimEH (R) = r.
Lutz, J. H. The dimension of individual strings and sequences
Information and Computation, 187:49–79, 2003.
For important subsets F of Cantor space we have that
dimH (F) = dimEH (F):
Theorem[Hitchcock] Let F be a countable union of Π0 -definable
1
subsets of Cantor space, then dimH (F) = dimEH (F)
Hitchcock, J.M. Correspondence principles for effective
dimensions. Theory of Computing Systems, 38:559–571, 2005.
Fractal Dimension versus Computationalan equality for Σ0 classes and computable Hausdorff
Also proves Complexity


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Turing degrees and Hausdorff dimension
Note that for A ∈ 2ω we have dimH (A) = 0. We can have
dimEH (A) > 0.
In a sense, having non-zero effective Hausdorff dimension is an
indication of containing complexity
Let A ∈ 2ω . If dimEH (A) > 0, then A can compute a non-recursive
function.
In particular, A can compute a fix-point free function f (that is, a
function f so that Wf (e) We for all numbers e).
Terwijn, S.A. Complexity and Randomness Rendiconti del
Seminario Matematico di Torino, 62:1–38, 2004.
Jockush Jr., C.G., Lerman, M., Soare, R. I., and Solovay, R.M.
Recursively enumerable sets modulo iterated jumps and
Arslanov’s completeness criterion. Journal of Symbolic Logic,
54:1288–1323, 1989.


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The relation between effective dimension and computable
content is not monotone nor simple.
If dimEH (A) = α, then there exist sets B of arbitrary high Turing
degree with dimEH (B) = α
However locally, Hausdorff dimension can provide an upper
bound to Turing degrees
Let r be a left-c.e. real. There is a ∆0 -definable set R ∈ 2ω with
2
dimEH (R) = r so that moreover
A ≤T R ⇒ dimEH (A) ≤ α.
Miller, J. S. Extracting information is hard: a Turing degree of
non-integral effective Hausdorff dimension. Advances in
Mathematics.



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Kolmogorov complexity and Hausdorff dimension
For a string s ∈ 2<ω the Kolmogorov complexity K(s) is roughly
the length of the shortest program that outputs s when computed
on a particular universal Turing machine
Diﬀerent choices of a universal Turing machine only manifest
itself in an additive constant in K
K(A n)
.
n
E. Mayordomo. A Kolmogorov complexity characterization of
constructive Hausdorﬀ dimension. Information Processing
Letters, 84:1–3, 2002.
dimEH (A) = lim inf
n→∞



The experiment

Demo



Hausdorff dimension and probability
Martingales are central to probability theory and indicate
expected outcomes of betting strategies
Lutz: An s-gale is a function d : 2<ω → R≥0 such that
d(σ0)+d(σ1)
d(σ) =
2s
d(σ0)+d(σ1)

This is a generalization of ‘gales’ (Levy) where d(σ) =
2
expresses a certain fairness condition of the betting strategy.
We say that d succeeds on A whenever lim supn→∞ d(A

n) = ∞

The Success set of d is the collection of all A on which d succeeds
and is denoted by S[d]
Lutz:
dimEH (X) = inf{q ∈ Q | X ⊆ S[d] for some q-gale d}



The experiment

Demo


Our result in this landscape

Our result

Is new in that it relates geometrical complexity of object
generated by TM to the runtime complexity of the TM
All work so far dealt with Turing (or other) degrees instead of
runtime complexity
Also, we have discrete geometrical objects for which we consider
the (limiting) geometrical dimension


Fractal dimension versus Computational Complexity

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Fractal dimension versus Computational Complexity