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Scaled Dimension of Individual Strings

María López-Valdés

Departamento Informática e Ingeniería de SistemasUniversity of Zaragoza

Computability in Europe 2006

Contents

1 DimensionHausdorff DimensionResource-bounded DimensionScaled Dimension

2 Discrete version of dimensionDimension of individual stringsDimension vs. Discrete Dimension

3 CharacterizationsKolmogorov complexity and discrete constructivedimensionKolmogorov complexity and constructive dimension

Dimension

Hausdorff Dimension

Contents

Dimension

Hausdorff Dimension

Why we use Hausdorff dimension?

Let X , Y complexity classes,

then X and Y are sets in the Cantor Space ({0, 1}∞)

(A ⊆ {0, 1}∗ ⇔ χA ∈ {0, 1}∞)

If dimH(X ) 6= dimH(Y ) ⇒ X 6= Y .

But most complexity classes have Hausdorff dimension 0.

Dimension

Hausdorff Dimension

(A ⊆ {0, 1}∗ ⇔ χA ∈ {0, 1}∞)

Dimension

Hausdorff Dimension

(A ⊆ {0, 1}∗ ⇔ χA ∈ {0, 1}∞)

Dimension

Hausdorff Dimension

(A ⊆ {0, 1}∗ ⇔ χA ∈ {0, 1}∞)

Dimension

Resource-bounded Dimension

Contents

Dimension

Lutz’s characterization

An s-gale is a function d : {0, 1}∗ → [0,∞) such that

d(w0) + d(w1)

2s = d(w).

dimH(X ) = inf{s ∈ [0,∞) | ∃ s − gale d s.t. X ⊆ S∞[d ]},

where X ⊆ S∞[d ] means

∀S ∈ X , lim supn

d(S[0 . . . n − 1]) = ∞.

Dimension

Lutz’s characterization

An s-gale is a function d : {0, 1}∗ → [0,∞) such that

d(w0) + d(w1)

2s = d(w).

dimH(X ) = inf{s ∈ [0,∞) | ∃ s − gale d s.t. X ⊆ S∞[d ]},

where X ⊆ S∞[d ] means

∀S ∈ X , lim supn

d(S[0 . . . n − 1]) = ∞.

Dimension

Resource-bounded dimension

Using resource bounds (∆) on d ,

dim∆(X ) = inf{s ∈ [0,∞) |∃ ∆−comp. s−gale d s.t. X ⊆ S∞[d ]}.

For example, we define constructive dimension as

cdim(X ) = inf{s ∈ [0,∞) | ∃ constructive s−gale d s.t. X ⊆ S∞[d ]}.

Dimension

Resource-bounded dimension

Using resource bounds (∆) on d ,

dim∆(X ) = inf{s ∈ [0,∞) |∃ ∆−comp. s−gale d s.t. X ⊆ S∞[d ]}.

For example, we define constructive dimension as

cdim(X ) = inf{s ∈ [0,∞) | ∃ constructive s−gale d s.t. X ⊆ S∞[d ]}.

Dimension

Scaled Dimension

Contents

Dimension

Scaled Dimension

Scaled gales

Let g : N× [0,∞) → [0,∞) an scale function,

an scaled sg-gale is a function d : {0, 1}∗ → [0,∞) such that

d(w0) + d(w1)

2g(|w |,s)−g(|w |+1,s)≤ d(w).

dimg∆(X ) = inf{s ∈ [0,∞) | ∃ ∆−comp. sg−gale d s.t. X ⊆ S∞[d ]}

Notice that if g(n, s) = ns, then dimg∆ = dim∆.

Dimension

Scaled Dimension

Scaled gales

d(w0) + d(w1)

2g(|w |,s)−g(|w |+1,s)≤ d(w).

Dimension

Scaled Dimension

Scaled gales

d(w0) + d(w1)

2g(|w |,s)−g(|w |+1,s)≤ d(w).

Dimension

Scaled Dimension

Summary

Discrete version of dimension

Dimension of individual strings

Contents

Definition of dim(w) (Lutz)

cdim(S) = cdim({S}) =

inf{s | ∃ constructive s−gale d s.t. lim supn

d(S[0 . . . n−1]) = ∞}.

To define dim(w):1 We have to replace gales by termgales.2 We have to replace “unbounded as n →∞”.3 We have to use an optimal constructive termgale to make

the definition universal.

d(S[0 . . . n−1]) = ∞}.

Scaled Dimension of individual strings

To define scaled dimension of a finite string:

1 We replace termgales by scaled termgales.2 We prove the existence of an optimal constructive scaled

termgale.3 We define scaled dimension of w using the optimal

constructive scaled termgale.

Dimension vs. Discrete Dimension

Contents

TheoremFor every S ∈ {0, 1}∞,

1 [Lutz]cdim(S) = lim inf

ndim(S[0 . . . n − 1]).

2 Let g be a scale function,

cdimg(S) = lim infn

dimg(S[0 . . . n − 1]).

1 [Lutz]cdim(S) = lim inf

ndim(S[0 . . . n − 1]).

cdimg(S) = lim infn

dimg(S[0 . . . n − 1]).

Characterizations

Kolmogorov complexity and discrete constructive dimension

Contents

Characterizations

Theorem

1 [Lutz] The Kolmogorov complexity of a string is (up anadditive constant) the product of his length and itsdimension.

|K (w)− |w |dim(w)| ≤ c

2 Let g an scale function,

|g−1(|w |, K (w))− dimg(w)| ≤ c∂g∂s (|w |, 0)

Characterizations

Theorem

1 [Lutz] The Kolmogorov complexity of a string is (up anadditive constant) the product of his length and itsdimension.

|K (w)− |w |dim(w)| ≤ c

2 Let g an scale function,

|g−1(|w |, K (w))− dimg(w)| ≤ c∂g∂s (|w |, 0)

Characterizations

Kolmogorov complexity and constructive dimension

Contents

Characterizations

1 [Mayordomo, Lutz]

cdim(S) = lim infK (S[0 . . . n − 1])

cdimg(S) = lim inf g−1(n, K (S[0, . . . n − 1])).

Characterizations

1 [Mayordomo, Lutz]

cdim(S) = lim infK (S[0 . . . n − 1])

cdimg(S) = lim inf g−1(n, K (S[0, . . . n − 1])).

Summary

Dimensions are tools that were defined to distinguishbetween complexity classes.

We can define a discrete version of constructive dimensionand constructive scaled dimension.

Constructive dimensions of (finite and infinite) sequencescan be characterized in terms of Kolmogorov complexity.

Appendix

For Further Reading

For Further Reading I

J.H. Lutz.Dimension in Complexity Classes.SIAM Journal on Computing, 32:1236–1259, 2003.

J.H. Lutz.The dimensions of individual strings and sequences.Information and Computation, 187:49–79, 2003.

E. Mayordomo.A Kolmogorov complexity characterization of constructiveHausdorff dimension.Information Processing Letters, 84(1):1–3, 2002.

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