Automates cellulaires :
structures
Nicolas Ollinger
LIP, ENS Lyon, France
19 décembre 2002 / soutenance
◦I×
Cellular Automata:
Structures
Nicolas Ollinger
LIP, ENS Lyon, France
19 december 2002 / Ph.D. defence
◦I×
Approach
• A main concern of Complex Systems :
a relatively simple microscopic rule
completely de ned local rule (given)
may produce
a very complex macroscopic behavior
far more complex global rule (induced)
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Approach
• A main concern of Complex Systems :
a relatively simple microscopic rule
completely de ned local rule (given)
may produce
a very complex macroscopic behavior
far more complex global rule (induced)
• Cellular Automata provide a simple not simplistic
and uniform model for studying this problem.
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Table of Content
1.
2.
3.
4.
5.
De nitions
Classi cations
Geometrical Transformations
Abstract Bulking
Exploration
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Cellular Automata
De nition. A d•CA A is a 4•uple Z , S, N, δ where:
d
• S is the nite state set of A;
• N ⊂ Zd , nite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
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Cellular Automata
De nition. A d•CA A is a 4•uple Z , S, N, δ where:
d
• S is the nite state set of A;
• N ⊂ Zd , nite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
A con guration C is a mapping from Zd to S.
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Cellular Automata
De nition. A d•CA A is a 4•uple Z , S, N, δ where:
d
• S is the nite state set of A;
• N ⊂ Zd , nite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
A con guration C is a mapping from Zd to S.
The global rule applies δ uniformly according to N:
∀p ∈ Zd , G(C)p = δ (Cp+N1 , . . . , Cp+Nν )
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Examples (1)
σ = (Z, {
, } , {−1} , q 7→ q)
Σ2 = (Z, {
, } , J−1, 0K, (q, q 0 ) 7→ q ⊕ q 0 ),
where ({
, } , ⊕) is isomorphic to (Z2 , +)
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Examples (2)
(Z, {
, } , J−1, 1K, maj),
where maj is majority between 3
(Z, {
, , , , , } , J−1, 1K, δ6 )
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Topological Charact.
• Endow S with the trivial topology.
• Endow S
Zd
with the induced product topology.
• The shift σv : S
Zd
→S
Zd
is de ned as
σv (C)p+v = Cp .
8 CC J ◦ B BB ×
Topological Charact.
• Endow S with the trivial topology.
• Endow S
Zd
with the induced product topology.
• The shift σv : S
Zd
→S
Zd
is de ned as
σv (C)p+v = Cp .
Zd
Zd
Theorem[Hedlund 69]. A map G : S → S is the
global rule of a d•CA if and only if it is continue and
commutes with shifts.
8 CC C ◦ B BB ×
Topological Charact.
• Endow S with the trivial topology.
• Endow S
Zd
with the induced product topology.
• The shift σv : S
Zd
→S
Zd
is de ned as
σv (C)p+v = Cp .
Zd
Zd
Theorem[Hedlund 69]. A map G : S → S is the
global rule of a d•CA if and only if it is continue and
commutes with shifts.
Consequences. We can freely compose CA and
invert b¼ective CA to obtain new CA.
8 CC C ◦ I BB ×
Subautomaton
∼ B) if there exists
• A CA A is isomorphic to a CA B (A =
a b¼ective map ϕ : SA → SB such that
ϕ ◦ GA = GB ◦ ϕ
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Subautomaton
∼ B) if there exists
• A CA A is isomorphic to a CA B (A =
a b¼ective map ϕ : SA → SB such that
ϕ ◦ GA = GB ◦ ϕ
De nition. A ⊆ B if there exists an injective map
ϕ : SA → SB such that this diagram commutes:
C


GA y
ϕ
−−−−→
ϕ(C)

G
y B
GA (C) −−−−→ ϕ(GA (C))
ϕ
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Closure (1)
• An autarkic CA ψ is a CA with neighborhood {0} and
local rule ψ : S → S. (notice that ψ is ultimately periodic)
• An elementary shift is a shift σv such that kvk1 = 1.
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Closure (1)
• An autarkic CA ψ is a CA with neighborhood {0} and
local rule ψ : S → S. (notice that ψ is ultimately periodic)
• An elementary shift is a shift σv such that kvk1 = 1.
• The composition A ◦ B of two CA A and B satis es
GA◦B = GA ◦ GB .
• The Cartesian product A × B of two CA satis es
GA×B = GA × GB .
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Closure (2)
• A new characterization of CA
Theorem. The set of CA is the closure of the set of
autarkic CA and elementary shifts by the operations of
composition, Cartesian product and subautomaton.
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Closure (2)
• A new characterization of CA
Theorem. The set of CA is the closure of the set of
autarkic CA and elementary shifts by the operations of
composition, Cartesian product and subautomaton.
Theorem. The set of reversible (b¼ective) CA is the
closure of the set of b¼ective autarkic CA and elemen•
tary shifts by the operations of composition, Cartesian
product and subautomaton.
11 CC C ◦ I BB ×
Table of Content
1.
2.
3.
4.
5.
De nitions
Classi cations
Geometrical Transformations
Abstract Bulking
Exploration
12 CC J ◦ I BB ×
Experimental Work
Wolfram (1984) First classi cation.
[. . . ] In class 1, the behavior is very
simple, and almost all initial condi•
tions lead to exactly the same uniform
nal state.
In class 2, there are many di er•
ent possible nal states, but all of
them consist just of a certain set of
simple structures that either remain
the same forever or repeat every few
steps.
In class 3, the behavior is more com•
plicated, and seems in many respects
random, although triangles and other
small•scale structures are essentially
always at some level seen.
And nally [. . . ] class 4 involves
a mixture of order and random•
ness: localized structures are pro•
duced which on their own are fairly
simple, but these structures move
around and interact with each other
in very complicated ways. [. . . ]
S. Wolfram [ANKOS, chapter 6, pp. 231 235]
13 CC J ◦ B BB ×
Experimental Work
Wolfram (1984) First unformal classi cation.
[. . . ] In class 1, the behavior is very
simple, and almost all initial condi•
tions lead to exactly the same uniform
nal state.
In class 2, there are many di er•
ent possible nal states, but all of
them consist just of a certain set of
simple structures that either remain
the same forever or repeat every few
steps.
In class 3, the behavior is more com•
plicated, and seems in many respects
random, although triangles and other
small•scale structures are essentially
always at some level seen.
And nally [. . . ] class 4 involves
a mixture of order and random•
ness: localized structures are pro•
duced which on their own are fairly
simple, but these structures move
around and interact with each other
in very complicated ways. [. . . ]
S. Wolfram [ANKOS, chapter 6, pp. 231 235]
13 CC C ◦ B BB ×
Experimental Work
Wolfram (1984) First unformal classi cation.
[. . . ] In class 1, the behavior is very
simple, and almost all initial condi•
tions lead to exactly the same uniform
nal state.
Nilpotency
In class 2, there are many di er•
ent possible nal states, but all of
them consist just of a certain set of
simple structures that either remain
the same forever or repeat every few
steps.
In class 3, the behavior is more com•
plicated, and seems in many respects
random, although triangles and other
small•scale structures are essentially
always at some level seen.
And nally [. . . ] class 4 involves
a mixture of order and random•
ness: localized structures are pro•
duced which on their own are fairly
simple, but these structures move
around and interact with each other
in very complicated ways. [. . . ]
S. Wolfram [ANKOS, chapter 6, pp. 231 235]
13 CC C ◦ B BB ×
Experimental Work
Wolfram (1984) First unformal classi cation.
[. . . ] In class 1, the behavior is very
simple, and almost all initial condi•
tions lead to exactly the same uniform
nal state.
Nilpotency
In class 2, there are many di er•
ent possible nal states, but all of
them consist just of a certain set of
simple structures that either remain
the same forever or repeat every few
steps.
Ult. Periodic
(up to a shift)
In class 3, the behavior is more com•
plicated, and seems in many respects
random, although triangles and other
small•scale structures are essentially
always at some level seen.
And nally [. . . ] class 4 involves
a mixture of order and random•
ness: localized structures are pro•
duced which on their own are fairly
simple, but these structures move
around and interact with each other
in very complicated ways. [. . . ]
S. Wolfram [ANKOS, chapter 6, pp. 231 235]
13 CC C ◦ B BB ×
Experimental Work
Wolfram (1984) First unformal classi cation.
[. . . ] In class 1, the behavior is very
simple, and almost all initial condi•
tions lead to exactly the same uniform
nal state.
Nilpotency
In class 2, there are many di er•
ent possible nal states, but all of
them consist just of a certain set of
simple structures that either remain
the same forever or repeat every few
steps.
Ult. Periodic
(up to a shift)
In class 3, the behavior is more com•
plicated, and seems in many respects
random, although triangles and other
small•scale structures are essentially
always at some level seen.
Chaoticity
And nally [. . . ] class 4 involves
a mixture of order and random•
ness: localized structures are pro•
duced which on their own are fairly
simple, but these structures move
around and interact with each other
in very complicated ways. [. . . ]
S. Wolfram [ANKOS, chapter 6, pp. 231 235]
13 CC C ◦ B BB ×
Experimental Work
Wolfram (1984) First unformal classi cation.
[. . . ] In class 1, the behavior is very
simple, and almost all initial condi•
tions lead to exactly the same uniform
nal state.
Nilpotency
In class 2, there are many di er•
ent possible nal states, but all of
them consist just of a certain set of
simple structures that either remain
the same forever or repeat every few
steps.
In class 3, the behavior is more com•
plicated, and seems in many respects
random, although triangles and other
small•scale structures are essentially
always at some level seen.
Chaoticity
Ult. Periodic Complexity
(up to a shift)
And nally [. . . ] class 4 involves
a mixture of order and random•
ness: localized structures are pro•
duced which on their own are fairly
simple, but these structures move
around and interact with each other
in very complicated ways. [. . . ]
S. Wolfram [ANKOS, chapter 6, pp. 231 235]
13 CC C ◦ B BB ×
Experimental Work
Wolfram (1984) First unformal classi cation.
[. . . ] In class 1, the behavior is very
simple, and almost all initial condi•
tions lead to exactly the same uniform
nal state.
Nilpotency
In class 2, there are many di er•
ent possible nal states, but all of
them consist just of a certain set of
simple structures that either remain
the same forever or repeat every few
steps.
In class 3, the behavior is more com•
plicated, and seems in many respects
random, although triangles and other
small•scale structures are essentially
always at some level seen.
Chaoticity
Ult. Periodic Complexity
(up to a shift)
And nally [. . . ] class 4 involves
a mixture of order and random•
ness: localized structures are pro•
duced which on their own are fairly
simple, but these structures move
around and interact with each other
in very complicated ways. [. . . ]
S. Wolfram [ANKOS, chapter 6, pp. 231 235]
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Understanding class 4
• Only one proposition of classi cation
(to our knowledge)
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Understanding class 4
• Only one proposition of classi cation
(to our knowledge)
J. Mazoyer and I. Rapaport. Inducing an order on cellular
automata by a grouping operation. Discrete Applied Mathe•
matics 91(1•3):177 196. 1999
• Grouping relies on an algebraic approach
14 CC C ◦ B BB ×
Understanding class 4
• Only one proposition of classi cation
(to our knowledge)
J. Mazoyer and I. Rapaport. Inducing an order on cellular
automata by a grouping operation. Discrete Applied Mathe•
matics 91(1•3):177 196. 1999
• Grouping relies on an algebraic approach
Idea. De ne a quasi•order on CA using the subau•
tomaton relation, up to some geometrical transforma•
tion of these CA.
14 CC C ◦ I BB ×
Example • Particles (1)
How to eliminate the periodic background pattern ?
You can zoom out and use shades of grey...
X
0
Cp = 1/9
C3p+v
v∈J0,2K2
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Example • Particles (2)
How to eliminate the periodic background pattern ?
...but also make blocks of bottom cells of the squares
0
Cp = C3p+(0,0) , C3p+(1,0) , C3p+(2,0)
17 CC J ◦ I BB ×
Grouping
We consider 1D CA with neighborhood J−1, 1K.
• De ne the kth power Ak of a CA A.
De nition. A CA B simulates a CA A, A 6 B, if
there exists m and n such that Am ⊆ Bn .
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Grouping
We consider 1D CA with neighborhood J−1, 1K.
• De ne the kth power Ak of a CA A.
De nition. A CA B simulates a CA A, A 6 B, if
there exists m and n such that Am ⊆ Bn .
Theorem. The relation 6 is a quasi•order.
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Grouping
We consider 1D CA with neighborhood J−1, 1K.
• De ne the kth power Ak of a CA A.
De nition. A CA B simulates a CA A, A 6 B, if
there exists m and n such that Am ⊆ Bn .
Theorem. The relation 6 is a quasi•order.
It admits a global minimum, some equivalence classes
at the bottom of the order correspond to simple known
CA families. It admits no global maximum.
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Table of Content
1.
2.
3.
4.
5.
De nitions
Classi cations
Geometrical Transformations
Abstract Bulking
Exploration
19 CC J ◦ I BB ×
Extension
Claim. The grouping operation doesn't take into ac•
count some classical geometrical transformations of
the literature, natural in the context of:
• Transformation from CA to OCA,
• Nilpotency,
• Intrinsic Universality.
20 CC J ◦ I BB ×
Classical transform.
• Classical transformations are usually of the type:
k
qn,1+k
...
...
q1,m
qn,m+k qn,m+1+k
...
qn,2m+k
n
q1,1
q1,m+1
...
q1,2m
m
hm,n,ki
GA
n
= o−1
◦
σ
◦
G
k
m
A ◦ om
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Formalization (1)
• A geometrical transformation on space•time dia•
grams transforms a cellular automaton into a
new one by combining cells of a space•time dia•
gram of the rst one to construct a space•time diagram
of the second one.
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Formalization (1)
• A geometrical transformation on space•time dia•
grams transforms a cellular automaton into a
new one by combining cells of a space•time dia•
gram of the rst one to construct a space•time diagram
of the second one.
• Formally, it is a pair (k, Λ) where
Λ:
d
N×Z
−→
7−→
d k
N×Z
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Formalization (2)
• To apply a transformation (k, Λ) to a space•time di•
d
d
N×Z
agram ∆ over S, we de ne ΛS : SN×Z → Sk
by
ΛS (∆)(t, p) = (∆(Λ(t, p)1 ), . . . , ∆(Λ(t, p)k ))
.
23 CC J ◦ B BB ×
Formalization (2)
• To apply a transformation (k, Λ) to a space•time di•
d
d
N×Z
agram ∆ over S, we de ne ΛS : SN×Z → Sk
by
ΛS (∆)(t, p) = (∆(Λ(t, p)1 ), . . . , ∆(Λ(t, p)k ))
.
• We de ne an operation rather similar to composition:
(k 0 , Λ 0 ) ◦ (k, Λ) = (kk 0 , Λ 0 ◦ Λ)
where
0
0
0
Λ ◦ Λ (t, p) = Λ Λ (t, p)1 1 . . . , Λ Λ (t, p)k 0 k
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Formalization (3)
e as
• We also introduce Λ
e :
Λ
N×Zd
2
X
−→
7−→
N×Zd
2 S
{Λ(t, p)1 , . . . , Λ(t, p)k }
(t,p)∈X
24 CC J ◦ B BB ×
Formalization (3)
e as
• We also introduce Λ
e :
Λ
N×Zd
2
X
−→
7−→
N×Zd
2 S
{Λ(t, p)1 , . . . , Λ(t, p)k }
(t,p)∈X
• A good geometrical transformation satis es
1. ∀A, ∃B,
ΛSA (∆) ∆∈Diag(A) = Diag(B)
;
24 CC C ◦ B BB ×
Formalization (3)
e as
• We also introduce Λ
e :
Λ
N×Zd
2
X
−→
7−→
N×Zd
2 S
{Λ(t, p)1 , . . . , Λ(t, p)k }
(t,p)∈X
• A good geometrical transformation satis es
1. ∀A, ∃B,
ΛSA (∆) ∆∈Diag(A) = Diag(B)
e {t + 1} × Z
2. ∀t ∈ N, Λ
d
e {t} × Z
*Λ
d
;
.
24 CC C ◦ I BB ×
Packing
v1
v0
PF,v (t, p) = t ~ (F ⊕ (p v))
Transformed CA global rule:
o−1
F,v ◦ G ◦ oF,v
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Cut ting
7−→
CT (t, p) = (tT, p)
Transformed CA global rule:
GT
26 CC J ◦ I BB ×
Shifting
7−→
Ss (t, p) = (t, p ⊕ ts)
Transformed CA global rule:
σs ◦ G
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Composition
We de ne PCS transformations as
PCSF,v,T,s = PF,v ◦ Ss ◦ CT
PCSF,v,T,s (t, p) = tT ~ (F ⊕ (p v ⊕ ts))
Transformed CA global rule:
T
o−1
◦
σ
◦
G
◦ oF,v
s
F,v
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Composition
We de ne PCS transformations as
PCSF,v,T,s = PF,v ◦ Ss ◦ CT
PCSF,v,T,s (t, p) = tT ~ (F ⊕ (p v ⊕ ts))
Transformed CA global rule:
T
o−1
◦
σ
◦
G
◦ oF,v
s
F,v
PCS transformations are closed under composition.
28 CC C ◦ I BB ×
Characterization
Theorem. A geometrical transformation is a good
geometrical transformation if and only if it can be ex•
pressed as a PCS transformation.
The proof highly relies on the uniformity of cellular automata
and the construction of counter•examples.
29 CC J ◦ I BB ×
Table of Content
1.
2.
3.
4.
5.
De nitions
Classi cations
Geometrical Transformations
Abstract Bulking
Exploration
30 CC J ◦ I BB ×
Abstract Bulking
• We don't want to reproof that we have a quasi•order
for each kind of grouping we introduce.
• Some properties are generic and do not rely on
painful computation at the level of geometrical trans•
formations but come from more abstract properties.
• We introduce a logical theory to uniformize the work
with grouping.
31 CC J ◦ I BB ×
De nition
De nition. An abstract bulking A is a logical theory
on the signature
Obj, Trans; apply : Obj × Trans → Obj,
divide ⊆ Obj × Obj,
combine : Trans × Trans → Trans .
Notation. An object y simulates an object x if they
satisfy the formula
α
β
x 4 y ≡ ∃α∃β x | y
32 CC J ◦ I BB ×
A xioms (1)
Combination. (Trans, ·) is a monoid.
A ` ∃1∀α (α · 1 = α ∧ 1 · α = α)
∧ ∀α∀β∀γ ((α · β) · γ = α · (β · γ))
Compatibility. (Trans, ·) acts on Obj through apply.
x
x = x1
A`
1
∀x x = x
∧
xα
xα·β
α β
∀x∀α∀β (x ) = x
α·β
33 CC J ◦ I BB ×
A xioms (2)
Divisibility. divide is a quasi•order on Obj.
A`
∀x (x | x)
∧
∀x∀y∀z ((x | y ∧ y | z) → x | z)
Transitivity. apply is compatible with divide.
yα
xα
x
A`
y
∀x∀y∀α (x | y → x | y )
α
α
34 CC J ◦ I BB ×
A xioms (3)
Surjectivity. apply preserve the richness of objects.
yα
y
A`
x
∀α∀x∃y (x | yα )
35 CC J ◦ I BB ×
A xioms (4)
Proximity. apply keeps objects nearby. There exists
two functions ζ and ξ such that
ζ(x,β)
(xα )
xα
x
x
A`
β
x
β ξ(x,α,β)
ζ(x,β)
∀x∀α∀β (xα )
| x
β ξ(x,α,β)
36 CC J ◦ I BB ×
Properties
Theorem. 4 is a quasi•order is a bulking property.
A`
∀x (x 4 x) ∧
∀x∀y∀z ((x 4 y ∧ y 4 z) → x 4 z)
37 CC J ◦ B BB ×
Properties
Theorem. 4 is a quasi•order is a bulking property.
A`
∀x (x 4 x) ∧
∀x∀y∀z ((x 4 y ∧ y 4 z) → x 4 z)
• u is universal if ∀x (x 4 u).
• u is strongly universal if ∀x∃α (x | uα ).
37 CC C ◦ B BB ×
Properties
Theorem. 4 is a quasi•order is a bulking property.
A`
∀x (x 4 x) ∧
∀x∀y∀z ((x 4 y ∧ y 4 z) → x 4 z)
• u is universal if ∀x (x 4 u).
• u is strongly universal if ∀x∃α (x | uα ).
Theorem. If there exists a strongly universal objet
then each universal object is strongly universal is a
bulking property.
37 CC C ◦ I BB ×
Table of Content
1.
2.
3.
4.
5.
De nitions
Classi cations
Geometrical Transformations
Abstract Bulking
Exploration
38 CC J ◦ I BB ×
First try
Idea. Use abstract bulking theory with:
Obj
Trans
apply
divide
combine
the set of d•CA,
the set of PCS transformations,
the transformation operator,
the subautomaton relation,
the composition of transformations.
39 CC J ◦ B BB ×
First try
Idea. Use abstract bulking theory with:
Obj
Trans
apply
divide
combine
the set of d•CA,
the set of PCS transformations,
the transformation operator,
the subautomaton relation,
the composition of transformations.
Argh! The Proximity axiom is not satis ed.
39 CC C ◦ I BB ×
Regular Packing
P̃: restriction on P transformations.
P̃(m1 ,...,md ),τ = PQd
(στ(1) ,...,στ(d) )⊗m
i=1 J0,mi −1K,
v0
v1
40 CC J ◦ I BB ×
Second try
Idea. Use abstract bulking theory with:
Obj
Trans
apply
divide
combine
the set of d•CA,
the set of P̃CS’ transformations,
the transformation operator,
the subautomaton relation,
the composition of transformations.
41 CC J ◦ B BB ×
Second try
Idea. Use abstract bulking theory with:
Obj
Trans
apply
divide
combine
the set of d•CA,
the set of P̃CS’ transformations,
the transformation operator,
the subautomaton relation,
the composition of transformations.
(P̃CS’ and P̃CS de ne the same relation of simulation)
41 CC C ◦ B BB ×
Second try
Idea. Use abstract bulking theory with:
Obj
Trans
apply
divide
combine
the set of d•CA,
the set of P̃CS’ transformations,
the transformation operator,
the subautomaton relation,
the composition of transformations.
(P̃CS’ and P̃CS de ne the same relation of simulation)
It works: All the axioms are satis ed.
41 CC C ◦ I BB ×
Summary
Applying a P̃CS transformation hmτ , n, ki to a CA A:
hmτ ,n,ki
GA
n
= o−1
◦
σ
◦
G
k
mτ
A ◦ omτ
42 CC J ◦ B BB ×
Summary
Applying a P̃CS transformation hmτ , n, ki to a CA A:
hmτ ,n,ki
GA
n
= o−1
◦
σ
◦
G
k
mτ
A ◦ omτ
De nition. A CA A is simulated by a CA B, A 6 B,
if there exists two P̃CS transformations hmτ , n, ki and
hmτ0 , n 0 , k 0 i such that:
A
hmτ ,n,ki
mτ0 ,n 0 ,k 0 i
h
⊆B
42 CC C ◦ B BB ×
Summary
Applying a P̃CS transformation hmτ , n, ki to a CA A:
hmτ ,n,ki
GA
n
= o−1
◦
σ
◦
G
k
mτ
A ◦ omτ
De nition. A CA A is simulated by a CA B, A 6 B,
if there exists two P̃CS transformations hmτ , n, ki and
hmτ0 , n 0 , k 0 i such that:
A
hmτ ,n,ki
mτ0 ,n 0 ,k 0 i
h
⊆B
Theorem. The relation 6 is induced by an abstract
bulking model.
42 CC C ◦ I BB ×
Basic properties
Corollary. The relation 6 is a quasi•order.
43 CC J ◦ B BB ×
Basic properties
Corollary. The relation 6 is a quasi•order.
• In dimension 1, the relation 6 re nes 6.
43 CC C ◦ B BB ×
Basic properties
Corollary. The relation 6 is a quasi•order.
• In dimension 1, the relation 6 re nes 6.
• × corresponds to a local maximum.
43 CC C ◦ B BB ×
Basic properties
Corollary. The relation 6 is a quasi•order.
• In dimension 1, the relation 6 re nes 6.
• × corresponds to a local maximum.
• We still have in nite chains.
n
6≷
>
ξ m,n
<
6≷
m
43 CC C ◦ I BB ×
Bottom of the order
Nil
(Zp , ⊕)
σ-Per
···
⊥
level 1
level 0
44 CC J ◦ I BB ×
Top of the order
U
∞
There is no quasi•universal CA.
45 CC J ◦ I BB ×
Universality
Theorem. Given a CA, deciding whether it is intrinsi•
cally universal is undecidable.
46 CC J ◦ B BB ×
Universality
Theorem. Given a CA, deciding whether it is intrinsi•
cally universal is undecidable.
Theorem. There exists no real•time intrinsically uni•
versal CA (∀A, ∃n, A ⊆ U hn,n,0i ).
46 CC C ◦ B BB ×
Universality
Theorem. Given a CA, deciding whether it is intrinsi•
cally universal is undecidable.
Theorem. There exists no real•time intrinsically uni•
versal CA (∀A, ∃n, A ⊆ U hn,n,0i ).
• We can construct very small intrinsically universal CA
(ex. 1D, von Neumann neighborhood, 6 states)
46 CC C ◦ I BB ×
Structure
Qk
• The structure of products of shifts, i=1 σvi , and CA
they simulate can be completely described.
47 CC J ◦ B BB ×
Structure
Qk
• The structure of products of shifts, i=1 σvi , and CA
they simulate can be completely described.
<
(2)
<
<
(2, 4)
<
(2, 3)
(3)
The relation 6 induces no semi•lattice structure.
47 CC C ◦ B BB ×
Structure
Qk
• The structure of products of shifts, i=1 σvi , and CA
they simulate can be completely described.
<
(2)
<
<
(2, 4)
<
(2, 3)
(3)
The relation 6 induces no semi•lattice structure.
Idea. Modify bulking so that × de nes a supremum.
47 CC C ◦ I BB ×
A new bulking (1)
• New transformations: (k, l, Λ) where
Λ:
d
N×Z
−→
7−→
d k
J1, lK × N × Z
48 CC J ◦ B BB ×
A new bulking (1)
• New transformations: (k, l, Λ) where
Λ:
d
N×Z
−→
7−→
d k
J1, lK × N × Z
• PCST(Fi ,vi ,Ti ,si )i∈J1,lK transforms A into
T1
Tl
−1
−1
oF1 ,v1 ◦σs1 ◦GA ◦oF1 ,v1 ×· · ·× oFl ,vl ◦σsl ◦GA ◦oFl ,vl
48 CC C ◦ I BB ×
A new bulking (2)
Idea. Use abstract bulking theory with:
Obj
Trans
apply
divide
combine
the set of d•CA,
the set of P̃CST’ transformations,
the transformation operator,
the subautomaton relation,
the composition of transformations.
49 CC J ◦ B BB ×
A new bulking (2)
Idea. Use abstract bulking theory with:
Obj
Trans
apply
divide
combine
the set of d•CA,
the set of P̃CST’ transformations,
the transformation operator,
the subautomaton relation,
the composition of transformations.
• P̃CST transformations are de ned like P̃CS ones.
• All the axioms are satis ed.
• The relation of simulation induces a sup•semi•lattice
with × as a supremum operator.
49 CC C ◦ I BB ×
Ideals
• An ideal is a set of equivalence classes stable by ×
and lower element by 6.
surj.
UR
inj.
ult. per.
⊥
50 CC J ◦ I BB ×
Perspectives
• CA at the bottom and the top of the order seem to
correspond to CA which are easy to describe. What
about CA in the middle ?
• Links between stuctural properties of bulking and de•
cidability questions have been presented. What about
topological properties ?
• Study abstract bulking in the case of a di erent kind
of dynamical system, re ne the choice of axioms, gen•
erals properties.
51 CC J ◦ I BB ×