Geometrical
Transformations
of Space-Time Diagrams
Nicolas Ollinger
LIP, ENS Lyon, France
13 september 2002 / AUTOMATA’2002 / Prague, Czechia
◦I×
Cellular Automata (1)
Definition. A d-CA A is a 4-uple Z , S, N, δ where:
d
• S is the finite state set of A;
• N ⊂ Zd , finite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
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Cellular Automata (1)
Definition. A d-CA A is a 4-uple Z , S, N, δ where:
d
• S is the finite state set of A;
• N ⊂ Zd , finite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
A configuration C is a mapping from Zd to S.
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Cellular Automata (1)
Definition. A d-CA A is a 4-uple Z , S, N, δ where:
d
• S is the finite state set of A;
• N ⊂ Zd , finite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
A configuration 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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Cellular Automata (2)
• The space-time diagram ∆ of A starting from C is the
mapping from N × Zd to S defined for any time t and any
position p by ∆(t, p) = Gt (C)p where G0 (C) = C.
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Cellular Automata (2)
• The space-time diagram ∆ of A starting from C is the
mapping from N × Zd to S defined for any time t and any
position p by ∆(t, p) = Gt (C)p where G0 (C) = C.
Zd
Zd
Theorem[Hedlund]. A map G : S → S is the global
rule of a cellular automaton if and only if G is continuous
for the product topology of the trivial topology on S and G
commutes with shifts (for all v, we have σv ◦ G = G ◦ σv ).
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Cellular Automata (2)
• The space-time diagram ∆ of A starting from C is the
mapping from N × Zd to S defined for any time t and any
position p by ∆(t, p) = Gt (C)p where G0 (C) = C.
Zd
Zd
Theorem[Hedlund]. A map G : S → S is the global
rule of a cellular automaton if and only if G is continuous
for the product topology of the trivial topology on S and G
commutes with shifts (for all v, we have σv ◦ G = G ◦ σv ).
Consequences. If we compose cellular automata, or
inverse bijective ones, we obtain cellular automata.
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Example - Fisher (1)
P. C. Fisher, Generation of Primes by a One-Dimensional Real-Time Iterative Array,
Journal of the ACM, 1965
Idea:
,
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Example - Fisher (2)
Pb: it recognizes 3p such that p is prime... not p!
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Example - Fisher (3)
Fisher scales the CA by making 3 × 3 blocks of cells...
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Example - Fisher (4)
...but we can also keep just the bottom cells.
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Example - Particles (1)
How to eliminate the periodic background pattern ?
You can zoom out and use shades of grey...
X
C3p+v
Cp0 = 1/9
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)
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Example - from CA to OCA (1)
δ(ql ,qc ,qr )
δ(ql ,qc ,qr )
ql
qc
qr
ql
(ql , qc )
qc
(qc , qr )
qr
C. Choffrut and K. Culik II, On real-time cellular automata and trellis automata,
Acta Informatica, 1984
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Example - from CA to OCA (2)
δ(qa ,qb ,qc ) δ(qb ,qc ,qd )
qa
qb
qc
(δ(qa ,qb ,qc ),δ(qb ,qc ,qd ))
qd
(qa , qb )
(qc , qd )
O. H. Ibarra, Sequential machine characterizations of trellis and cellular automata and
applications, SIAM Journal on Computing, 1985
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Formalization (1)
• A geometrical transformation on space-time diagrams
transforms a cellular automaton into a new one by
combining cells of a space-time diagram of the first one
to construct a space-time diagram of the second one.
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Formalization (1)
• A geometrical transformation on space-time diagrams
transforms a cellular automaton into a new one by
combining cells of a space-time diagram of the first 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 diagram
d
d
N×Z
by
∆ over S, we define ΛS : SN×Z → Sk
ΛS (∆)(t, p) = (∆(Λ(t, p)1 ), . . . , ∆(Λ(t, p)k ))
.
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Formalization (2)
• To apply a transformation (k, Λ) to a space-time diagram
d
d
N×Z
by
∆ over S, we define ΛS : SN×Z → Sk
ΛS (∆)(t, p) = (∆(Λ(t, p)1 ), . . . , ∆(Λ(t, p)k ))
.
• We define 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 )k 0
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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
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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
• A good geometrical transformation satisfies
1. ∀A, ∃B,
ΛSA (∆) ∆∈Diag(A) = Diag(B)
;
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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
• A good geometrical transformation satisfies
1. ∀A, ∃B,
ΛSA (∆) ∆∈Diag(A) = Diag(B)
e {t + 1} × Zd * Λ
e {t} × Zd
2. ∀t ∈ N, Λ
;
.
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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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Cutting
7−→
CT (t, p) = (tT, p)
Transformed CA global rule:
GT
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Shifting
7−→
Ss (t, p) = (t, p ⊕ ts)
Transformed CA global rule:
σs ◦ G
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Composition
We define 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 define 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.
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Characterization
Theorem. A geometrical transformation is a good geometrical transformation if and only if it can be expressed
as a PCS transformation.
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