Scaled Dimension of Individual Strings
Scaled Dimension of Individual Strings
María López-Valdés
Departamento Informática e Ingeniería de SistemasUniversity of Zaragoza
Computability in Europe 2006
Scaled Dimension of Individual Strings
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
Scaled Dimension of Individual Strings
Dimension
Hausdorff Dimension
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
Scaled Dimension of Individual Strings
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.
Scaled Dimension of Individual Strings
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.
Scaled Dimension of Individual Strings
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.
Scaled Dimension of Individual Strings
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.
Scaled Dimension of Individual Strings
Dimension
Resource-bounded Dimension
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
Scaled Dimension of Individual Strings
Dimension
Resource-bounded 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]) = ∞.
Scaled Dimension of Individual Strings
Dimension
Resource-bounded 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]) = ∞.
Scaled Dimension of Individual Strings
Dimension
Resource-bounded 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 ]}.
Scaled Dimension of Individual Strings
Dimension
Resource-bounded 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 ]}.
Scaled Dimension of Individual Strings
Dimension
Scaled Dimension
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
Scaled Dimension of Individual Strings
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∆.
Scaled Dimension of Individual Strings
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∆.
Scaled Dimension of Individual Strings
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∆.
Scaled Dimension of Individual Strings
Dimension
Scaled Dimension
Summary
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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.
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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.
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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.
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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.
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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.
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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.
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension of individual strings
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.
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension vs. Discrete Dimension
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
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension vs. Discrete Dimension
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]).
Scaled Dimension of Individual Strings
Discrete version of dimension
Dimension vs. Discrete Dimension
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]).
Scaled Dimension of Individual Strings
Characterizations
Kolmogorov complexity and discrete constructive dimension
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
Scaled Dimension of Individual Strings
Characterizations
Kolmogorov complexity and discrete constructive dimension
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)
Scaled Dimension of Individual Strings
Characterizations
Kolmogorov complexity and discrete constructive dimension
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)
Scaled Dimension of Individual Strings
Characterizations
Kolmogorov complexity and constructive dimension
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
Scaled Dimension of Individual Strings
Characterizations
Kolmogorov complexity and constructive dimension
TheoremFor every S ∈ {0, 1}∞,
1 [Mayordomo, Lutz]
cdim(S) = lim infK (S[0 . . . n − 1])
n.
2 Let g be a scale function,
cdimg(S) = lim inf g−1(n, K (S[0, . . . n − 1])).
Scaled Dimension of Individual Strings
Characterizations
Kolmogorov complexity and constructive dimension
TheoremFor every S ∈ {0, 1}∞,
1 [Mayordomo, Lutz]
cdim(S) = lim infK (S[0 . . . n − 1])
n.
2 Let g be a scale function,
cdimg(S) = lim inf g−1(n, K (S[0, . . . n − 1])).
Scaled Dimension of Individual Strings
Summary
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.
Scaled Dimension of Individual Strings
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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