fractal dimension versus computational complexity

Box dimension and Turing machines The experiment Demo Theoretical panorama

Fractal Dimension versus ComputationalComplexity

Joost J. JoostenFernando Soler-Toscano

Hector [email protected], [email protected], [email protected]

Seminari Cuc, BarcelonaJanuary, 2014

Fractal Dimension versus Computational Complexity


The small TuringMachine database

Small Turing machines

We consider Turing machines where the tape extends infinitely inone direction (to the left in the diagrams)Each tape cell contains one symbol (color)We use just two colors: black and whiteA Turing machine starts its computation with the head at the firsttape cell (beginning of the tape)The input of the computation is written at the initial cellsThe computation ends when the machine is at the beginning ofthe tape and moves to the right (out of the tape)The tape configuration upon termination of a computation iscalled the outputThe set of Turing machines with n states and k colors isrepresented by (n, k)We have enumerated the machines in (n, k) from 0 to (2 ·n · k)n·k

−1We present the results of an exhaustive study of (2, 2) and (3, 2)



The small TuringMachine database

Space-time diagrams

A space-time diagram for some computation is the jointcollection of consecutive memory configurations

The top-row of each diagramrepresents the input (1 to 14)The computation starts withthe head of the TM in state 1in the rightmost cellEach lower row represents thetape configuration of a nextstep in the computation

These space-time diagrams define spatial objects by focussingon the black cells. We can measure the geometrical complexity.We wish to see if there is a relation between this geometricalcomplexity and the computational complexity (space or timeusage) of the TM in question.



Fractal dimensions

Box dimension

The notion of Box dimension is a simplification of, and an upperbound to Hausdorff dimensionSuppose we have a mathematical object S of bounded size. Theidea 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 rneeded to cover S: V(S) = limr↓0 rdN(S, r)

Definition (Box dimension)Let S be some spatial object that can be embedded in some Rn, letN(S, r) denote the minimal number of boxes of size r needed to fullycover S. The Box dimension of S is denoted by δ(S) and is defined by

δ(S) := limr↓0

log(N(S, r))

log( 1r )

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



Fractal dimensions

Box dimension for Space-Time diagrams

We adapt the notion of Box dimension to space-time diagramsClearly, for each input on which the TM halts the correspondingspace-time diagram has dimension 2: it’s a piece of surfaceIt 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. Incase τ(x) ↓, let N(τ, x) denote the number of black cells in thespace-time diagram of τ on input x and let t(τ, x) denote the numberof steps needed for τ to halt on x.We will define the Box dimension of a TM τ and denote it by d(τ). Incase t(τ, x) is constant from some x onwards, we define d(τ) := 2.Otherwise, we define

d(τ) := lim infx→∞

log(N(τ, x)

)log

(t(τ, x)

) .Fractal Dimension versus Computational Complexity


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 byτ on input x, and let t(x) denote the number of computation steps ittook 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.




The Upper Bound Conjecture

LemmaIn case a TM τ uses polynomial space, and runs super-polynomialtime we have that d(τ) = 1.More in general, if τ uses space sτ(x) and time tτ(x) on input x then

lim infx→∞

log(sτ(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 infx→∞

log(sτ(x)

)log

(tτ(x)

)Fractal Dimension versus Computational Complexity



The Space-Time Theorem and P versus NP

Using the previous Lemma, we can state a separation of P andNP in terms of dimensions:

Let Π be some NP-complete problemIf for each PSPACE Turing machine τ that decides Π we have thatd(τ) = 1, then P , NP.

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



Methodology

Slow convergence

Our aim is to use computer experiments to compute the Boxdimension of all TMs τ where d(τ) is not predicted by anytheoretical result.A substantial complication is caused by theoccurrence of logarithms in d(τ)As a consequence, increase in precision ofd(τ) requires exponentially larger inputs

For (2, 2) TM 346 we know that its Box dimension is 2, but we cansee in the picture how slow the rate of convergence isOur way out here is to apply numerical and mathematicalanalysis to the functions involved so that we can retrieve theirlimit behavior.We are interested in three different functions:

tτ(x), number of time-steps needed for τ to halt on input xNτ(x), number of black cells in the space-time diagram of τ oninput xsτ(x), number of tape cells visited by τ on input x



Methodology

Steps followed

Each TM in (2, 2) also occurs in (3, 2) so for the final results itsuffices to focus on this data-set. We isolated the TMs for whichthere is no theorem that predicts the corresponding dimension.

Boxes Runtime Space MachinesO(n3) O(n2) O(n) 3358O(n4) O(n3) O(n) 6o(P) o(P) o(P) 14

Per TM τ, we determined its functions sτ(x) (space), tτ(x) (time)and Nτ(x) (black cells). We used FindSequenceFunction andother Mathematica functionsPer TM τ, we computed its dimension d(τ) asd(τ) = lim infx→∞

log(Nτ(x))log(tτ(x))

Per TM τ, we compared its dimension d(τ) to its theoreticalupper bound 1 + lim infx→∞

log(sτ(x))log(tτ(x))



Methodology

Alternating convergent behavior

Some machines have alternating asymptotic behaviorThis is the most extreme example (TM 1,728,529):

For convenience we have changed theorientation of the diagrams so that time‘goes from left to right’ instead of from‘top to bottom’.

In a sense this TM incorporates twodifferent algorithms to compute thisoutput: one in linear time, the other, inexponential time.

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



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 themwhere computed by some super-linear time TMsIn total, in (2, 2) space, there are only 7 TMs that run insuper-polynomial time. Three of them run in exp-time, allcomputing the tape-identity. The other four (see below) TMscompute 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

2 . This is exactly the upperbound as predicted by the Space-Time Theorem.




Findings in (3, 2) space

The (3, 2) space contains 2,985,984 many different TMs whichcompute 3,886 different functionsAlmost all TMs used at most linear space for their computationsThe only exception to this was when the TM used exponentialspaceBusy Beaver: we call a TM β a Busy Beaver whenever for eachTM τ, there is some value x0 so that for all x ≥ x0 we havetβ(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




The space-time theorem revisited

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

Finding 1For all TMs τ in (3,2) space we found that

d(τ) = 1 + lim infx→∞

log(sτ(x))log(tτ(x))

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




Other findings

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

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

Finding 4For all TMs τ in (3,2) space we found that limx→∞

s(x)t(x) = 1 so that

d(τ) ≥ 1. We conjecture this holds true also in larger space.Fractal Dimension versus Computational Complexity



Richness in the (3, 2) space

We have found two symmetric performers for even inputsThis can only occur in machines computing the tape identity andrequires strong conditionsIt is surprising that such constraints can be met in (3, 2)




Part of a larger project

H. Zenil, F. Soler-Toscano, J. J. Joosten. Empirical Encounterswith 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 TimeComplexity. Slowdown and Speed-up Phenomena in theMicro-cosmos of Small Turing Machines. Int. Journ. ofUnconventional Computing, Vol. 7, pp. 353-387, 2011.Joost J. Joosten, Fernando Soler-Toscano, Hector Zenil. FractalDimension of Space-time Diagrams and the RuntimeComplexity 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.





J. J. Joosten, F. Soler-Toscano, and H. Zenil. Complejidaddescriptiva y computacional en maquinas de Turing pequenas.Proceedings of the V Jornadas Ibericas de Filosofıa de la Ciencia,Logica y Lenguaje, Lisbon 2010, in Logica Universal e Unidadeda Ciencia, Centro de Filosofia das Ciencias da Universidade deLisboa, pp. 11-32, ISBN: 978-989-8247-49-0, 2011.J. J. Joosten, F. Soler-Toscano, H. Zenil. Speedup and SlowdownPhenomena in Turing Machines. Wolfram DemonstrationsProject, http://demonstrations.wolfram.com/SpeedupAndSlowdownPhenomenaInTuringMachines/, 2012.J. J. Joosten. Turing Machine Enumeration: NKS versusLexicographical. Wolfram Demonstrations Project,http://demonstrations.wolfram.com/TuringMachineEnumerationNKSVersusLexicographical/, 2010.J. J. Joosten, F. Soler-Toscano, H. Zenil. Runtime complexity ofsmall Turing Machines and fractal dimension. WolframDemonstrations Project, To be submitted soon, 2014.





J. J. Joosten, H. Zenil, F. Soler-Toscano. Entropy as an indicationof the runtime of terminating discrete dynamical processes. InBook of abstracts, European Conference on Complex Systems,p214, S. Thurner M. Szell editors, Locker Verlag, ISBN978-3-85409-613-9, 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, ISBN978-3-319-00253-8, 2013.J. J. Joosten. On the Necessity of Complexity. In Irreducibilityand Computational Equivalence: 10 Years After the Publicationof Wolfram’s A New Kind of Science, (11-24). Springer,Heidelberg New York Dordrecht London, H. Zenil editor, ISBN978-3-642-35481-6, 2013.



Demo

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



Demo



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) thecomplexity of one particular aspect of a systemOn philosophical grounds we can expect relations betweendifferent complexity measuresJ. 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, ISBN978-3-319-00253-8, 2013.J. J. Joosten. On the Necessity of Complexity. In Irreducibilityand Computational Equivalence: 10 Years After the Publicationof Wolfram’s A New Kind of Science, (11-24). Springer,Heidelberg New York Dordrecht London, H. Zenil editor, ISBN978-3-642-35481-6, 2013.



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 alsocalled Lebesgue dimension.The order of a familyA of sets is ≤ n by definition when anyn + 2 of the sets have empty intersection. We say = n when ≤ nbut 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 invariantunder homeomorphisms.




Fractal dimensions

A fractal dimension of some object S is an indication of how closeS is to some integer-valued dimensional spaceDimension in integer-valued dimensional space in a senseexpress degrees of freedom (information theoretical focus)Falconer: “Roughly, dimension indicates how much space a setoccupies near to each of its points.” (geometrical focus)Falconer, K. J. Fractal Geometry, Mathematical Foundations andApplications, Wiley, Chichester, 2003.Most fundamental, and most common notion is that of HausdorffdimensionF. Hausdorff Dimension und ausseres Mass. MathematischeAnnalen, 79:157–179, 1919.Building upon ideas of Caratheodory: Caratheodory, C. Uber daslineare Mass von Punktmengen, eine Veralgemeinerung desLangenbegriffs. Nachrichten von der Wissenschaften zuGotingen, Mathematisch-Physikalische Klass, 404–426, 1914.




Hausdorff dimension

For a S some subset of some metric space we can considercountable open coveringsA of S and define

Hsε(S) := inf

∑A∈A

(diam A)s.

The infimum is taken over allA that are countable open ε-coversof S. Then

Hs(S) := lim

ε→0H

sε(S).

Main Theorem:There is a unique s so that

Htε(S) = ∞ for t < s;H

tε(S) = 0 for t > s.

This unique s is called the Hausdorff (Mandelbrot speaks ofHausdorff-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.




Packing dimension

Hausdorff comes with a natural dual dimension called packingdimension.Tricot, C. Jr. Two definitions of fractional dimension.Mathematical Proceedings of the Cambridge PhilosophicalSociety, 91, 57–74, 1982.Sullivan, D. Entropy, Hausdorff measures old and new, and limitsets of geometrically finite Kleinian groups. Acta Mathematica,153, 259–277, 1984.

Psδ(F) := {sup

∑i

|Bi| | {Bi}i are disjoint balls at radii ≤ δ and center in F}

Since limδ→0Psδ(F) is not a measure (consider countable dense

sets) one defines

Ps(F) := inf

{Fi}i

{

∑i

limδ→0P

sδ(Fi) | F ⊆

∞⋃i=1

Fi}.




Main Theorem:There is a unique s so that

Ptε(F) = 0 for t < s;H

tε(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 istoo broad, since it admits “true geometric chaos”J. Taylor proposes to denote by fractals only Borel sets F forwhich dimH(F) ≤ dimP(F).Taylor, S. J. The measure theory of random fractals.Mathematical Proceedings of the Cambridge PhilosophicalSociety, 100, 383–406, 1986.




Box dimensions

The Box dimension is like Hausdorff dimension only that wenow cover by balls/boxes of fixed size.Alternatively and equivalently, divide space into a regular meshwith mesh-size δ and count how many cells Nδ(F) are hit by a setFThen define Bs

δ(F) := Nδ(F)δs and Bs(F) := lim infδ→0 Nδ(F)δs.Again, there is a cut-off value s0 so that Bs(F) = ∞ for s < s0 andB

s(F) = 0 for s > s0

This cut-off value is given by

lim infδ→0

log(Nδ(F))log(1/δ)

.

We define dimB := lim infδ→0


and

dimB := lim supδ→0





In case dimB(F) = dimB(F) we call this the box-countingdimension: dimB(F)Box dimension always provides an upper bound to HausdorffdimensionBox dimension has desirable computational propertiesbut undesirable mathematical properties: a countable union ofmeasure zero sets can have positive box dimensionExample: dimB{0, 1

2 ,13 ,

14 , . . .} =

12

Mathematically this can be repaired by defining

dimMB(F) := inf{Fi}{sup

idimB(Fi) | F ⊆

∞⋃i=1

Fi} and

dimMB(F) := inf{Fi}{sup

idimB(Fi) | F ⊆

∞⋃i=1

Fi}

loosing the good computational properties of course . . .




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



Computational properties of fractals

Julia sets

Probably the most famous examples of fractals are Julia sets andthe corresponding “roadmap Mandelbrot set”By FJ(f ) we denote the filled Julia set of a function f defined onthe complex numbers is the set of values z in the domain of f onwhich iterating f on z does not diverge. That is,

FJ(f ) := {z | lim supn→∞

|f n(z)| < ∞}

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 withλ = e2πiθ and θ < QCorresponding Julia sets are denoted by JθJθ being well-behaved is expressed by saying that it has a Siegeldisk at z = 0Basically, this says that f is locally linearizable at z = 0 by arotationSee: Milnor, J. Dynamics in one complex variable. Introductorylectures. Princeton University Press, 2006.




Constructive Analysis

There are various results between the Turing degree of θ and thatof JθOne first has to define what the Turing degree of non-discreteobjects actually meansBraverman and Yampolsky follow an approach of ConstructiveAnalysis as initiated by Banach and Mazur, with influence ofMarkov.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., IISer. 98, 1–9, 1962.Overview: Weihrauch, Computable Analysis, Springer, Berlin,2000.





Braverman, Yampolsky: b is a c.e. Turing degree if and only if itis the degree of Jθ with θ recursive so that Jθ has a Siegel diskBraverman, 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. Forevery d ≥ c we have that d is c.e. in c if and only if it is the degreeof a Julia set Jθ with Siegel disk and deg(θ) = c.C.T. Chong, unpublished; Slides Complex Dynamics and TuringDegrees online.Results sensitive to model of computation and change with, e.g.,Blum-Schub-Smale modelSee L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity andreal computation, Springer-Verlag, New York, 1998.This relates the Turing complexity of the fractal to the complexityof the parameter generating itHowever, no link to the corresponding dimensionThis we will see in what follows



Fractal dimension versus other complexity notions

Hausdorff dimension on strings

Let us reformulate the definition of Hausdorff dimension in therealm of binary sequences, i.e., Cantor spaceOverview 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�.

Hsn(R) := inf{

∑σ∈Σ

2−s|σ|| Σ an n-cover of R}

Hs(R) := lim

n→∞H

sn(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




Effective Hausdorff dimension

The effective pendant is now defined via

EHsn(R) := inf{

∑σ∈Σ

2−s|σ|| Σ a c.e. n-cover of R}

EHs(R) := lim

n→∞EHs

n(R)

So that the effective Hausdorff dimension is defined asdimEH(R) := inf{s | EH s(R) = 0}For every computable real r, there is a set R ⊆ 2ω withdimEH(R) = r.Lutz, J. H. The dimension of individual strings and sequencesInformation and Computation, 187:49–79, 2003.For important subsets F of Cantor space we have thatdimH(F) = dimEH(F):Theorem[Hitchcock] Let F be a countable union of Π0

1-definablesubsets of Cantor space, then dimH(F) = dimEH(F)Hitchcock, J.M. Correspondence principles for effectivedimensions. Theory of Computing Systems, 38:559–571, 2005.Also proves an equality for Σ0

2 classes and computable Hausdorffdimension (covers are required to be computable rather than c.e.)




Turing degrees and Hausdorff dimension

Note that for A ∈ 2ω we have dimH(A) = 0. We can havedimEH(A) > 0.In a sense, having non-zero effective Hausdorff dimension is anindication of containing complexityLet A ∈ 2ω. If dimEH(A) > 0, then A can compute a non-recursivefunction.In particular, A can compute a fix-point free function f (that is, afunction f so that Wf (e) ,We for all numbers e).Terwijn, S.A. Complexity and Randomness Rendiconti delSeminario 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 andArslanov’s completeness criterion. Journal of Symbolic Logic,54:1288–1323, 1989.




The relation between effective dimension and computablecontent is not monotone nor simple.If dimEH(A) = α, then there exist sets B of arbitrary high Turingdegree with dimEH(B) = α

However locally, Hausdorff dimension can provide an upperbound to Turing degreesLet r be a left-c.e. real. There is a ∆0

2-definable set R ∈ 2ω withdimEH(R) = r so that moreover

A ≤T R ⇒ dimEH(A) ≤ α.

Miller, J. S. Extracting information is hard: a Turing degree ofnon-integral effective Hausdorff dimension. Advances inMathematics.




Kolmogorov complexity and Hausdorff dimension

For a string s ∈ 2<ω the Kolmogorov complexity K(s) is roughlythe length of the shortest program that outputs s when computedon a particular universal Turing machineDifferent choices of a universal Turing machine only manifestitself in an additive constant in K

dimEH(A) = lim infn→∞

K(A�n)n

.

E. Mayordomo. A Kolmogorov complexity characterization ofconstructive Hausdorff dimension. Information ProcessingLetters, 84:1–3, 2002.




Hausdorff dimension and probability

Martingales are central to probability theory and indicateexpected outcomes of betting strategiesLutz: An s-gale is a function d : 2<ω → R≥0 such thatd(σ) =

d(σ0)+d(σ1)2s

This is a generalization of ‘gales’ (Levy) where d(σ) =d(σ0)+d(σ1)

2expresses 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 succeedsand is denoted by S[d]Lutz:

dimEH(X) = inf{q ∈ Q | X ⊆ S[d] for some q-gale d}



Our result in this landscape

Our result

Is new in that it relates geometrical complexity of objectgenerated by TM to the runtime complexity of the TMAll work so far dealt with Turing (or other) degrees instead ofruntime complexityAlso, we have discrete geometrical objects for which we considerthe (limiting) geometrical dimension


fractal dimension versus computational complexity

Education

box dimension of s

notion of box dimension

rdefinition box dimension

tmdefinition box dimension

limxfractal dimension

hausdor dimension

input x

space s x