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Generalization of Compositions of Cellular Automata on Groups

Y. Mizoguchi∗, M. Fujio∗∗, S. Inokuchi∗∗∗

∗ Institute of Mathematics for Industry,

Kyushu University, JAPAN∗∗ Dept. of Systems Design and Informatics,

Kyushu Institute of Technology, JAPAN∗∗∗ Faculty of Mathematics,

Kyushu University, JAPAN

WACSJTU 2011, Shanghai, China (2011/09/15-18)

1-D Cellular Automaton (CA) (1)

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horizontal ↔ one-dimensional space (c ∈ 2Z)

vertical ↔ time line (ct+1 = F (ct))

f

Let 2 = {0, 1} and f : 2 × 2 × 2 → 2 be a local function of 1-D CA.The global function F : 2Z → 2

Z induced by f is defined by

F (c)(i) = f(c(i − 1), c(i), c(i + 1)).

1-D Cellular Automaton (CA) (2)

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CA(3,105)

The local function f105 : 2 × 2 × 2 → 2 denoted by

f105 =

(

111 110 101 100 011 010 001 0000 1 1 0 1 0 0 1

)

means f105(1, 1, 1) = 0, f105(1, 1, 0) = 1, · · · , f105(0, 0, 0) =1. The number 105 (Wolfram Number) is corresponding to thebinary number interpretation of the lower line.

105 = 27×0+26×1+25×1+24×0+23×1+22×0+21×0+20×1

Each function f : 2 × 2 × 2 → 2 has a unique number R ∈{0, 1, 2, · · · , 255} which is called the Wolfram Number and f

is denoted by f = fR.The cellular automaton defined by fR is denoted by CA(3, R).

1-D Cellular Automaton (CA) (3)

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CA(3,204) CA(3,51) CA(3,240) CA(3,15)

CA(3,90) CA(3,102) CA(3,105) CA(3,166)

2-D Cellular Automaton (1)

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Let 2 = {0, 1}, 3 = {0, 1, 2} and f : 23×3 → 2 be a local function of 2-DCA. The global function F : 2Z×Z → 2

Z×Z induced by f is defined by

F (c)(i, j) = f

c(i − 1, j − 1) c(i, j − 1) c(i + 1, j − 1)c(i − 1, j) c(i, j) c(i + 1, j)

c(i − 1, j + 1) c(i, j + 1) c(i + 1, j + 1)

The local function used for ’Conway’s Life’ is

f

a(0, 0) a(0, 1) a(0, 2)a(1, 0) a(1, 1) a(1, 2)a(2, 0) a(2, 1) a(2, 2)

=

0 (∑

(i,j)∈3×3

a(i, j) ≤ 2)

1 (∑

(i,j)∈3×3

a(i, j) = 3)

a(1, 1) (∑

(i,j)∈3×3

a(i, j) = 4)

0 (∑

(i,j)∈3×3

a(i, j) ≥ 5)

.

2-D Cellular Automaton (2)

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(Movie (Glider.m4v))

■ A 2-D cellular automaton is considered as a (parallel) computer.■ An initial configuration is considered as a program.■ Using patterns, we can implement Boolean circuits (logical gates) and

flows of information.■ The movie is an example of creating the flows of information which is

called a glider-gun.■ We can also create the AND-gate, OR-gate and NOT-gate by using some

pattern arrangements. (Computational Universal)

Historical Background

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■ Alan Turing (1936)Formalize the concept of computability. Introduce the universal Turingmachine.

■ John von Neumann (1948), Stanislaw Ulam(1950)Introduce self-reproducing automata. A reductionist model for biologicalevolution.

■ John Conway (1970)Introduce the Life Game. A simple model proven to be computationaluniversal.

■ Doyne Farmer, Tommaso Toffoli, Stephen Wolfram (1984)The first conference devoted solely to research in CA.

■ IFIP (International Federation for Information Processing)Technical Committee 1 (Foundations of Computer Science)Working Group 1.5 (Cellular Automata and Discrete Complex Systems)

Requirements of a CA models

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A CA models usually possesses following five characteristics:

■ Discrete lattice of cells:Space consists of a one-, two- or three-dimensional lattice of cells.

■ Homogeneity:All cells are equivalent. (uniform)

■ Discrete states:Each cell takes one of a finite number of possible states.

■ Local interactions:Each cell interacts only with cells that are in its local neighborhood.

■ Discrete dynamics:At each discrete unit time, each cell updates its state.

Characteristic of CA

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Characteristic

■ Generate a rich spectrum of very complex patterns of behavior out of setsof relatively simple underlying rules

■ Capture many essential features of complex self-organizing cooperativebehavior observed in real systems.

■ Used as discrete versions of partial differential equations.The real power lies in the fact that they represent a large class of exactlycomputable models.

Specific examples

■ Fluid and chemical turbulence■ Plant growth, Dendritic growth of crystals, DNA evolution■ Ecological theory, Social dynamics■ Propagation of infectious diseases, Forest fires■ Cryptography, Natural Computing

Belousov-Zhabotinski Reaction

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The Belousov-Zhabotinski (BZ) reaction is a chemical reaction consisting ofsimple organic molecules that is characterized by spectacular oscillatingtemporal and spatial patterns. The spatial and temporal patterns that emergefrom the initially random mixture of states are a good general example of howCA can be used to model self-organization.

M.Gerhardt, H.Schuster, A cellular automaton describing the formation of spatiallyordered structures in chemical systems, Phys. D 36 (1989), 209―221.A. Ilachinski, Cellular Automata: A Discrete Universe, World Scientific Pub Co Inc(2001), 840pages.

Our Objective

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■ Introduce a generalized theory for CA’s which local functions may havedifferent domains. (2-neighborhood, 3-neighborhood, ...)

■ Introduce an operations between CA’s which local functions may havedifferent domains. (composition, union, division, ...)

■ Consider generalization of cell spaces:Lattice (1-D,2-D,..) → Caley-Graph → Group

■ Decompose a complex CA to some simple CA’s using defined operations.■ Investigate relationship between defined operations and dynamics of CA’s.

CA on group

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Definition 1 Let G be a group. A cellular automaton is a tripleC = (G, V, L) of a group G, subsets V ⊂ G and L ⊂ 2V . For L, we definefunctions lL : 2V → {φ, {e}} by

lL(X) =

{

φ (X 6∈ L)

{e} (X ∈ L),

and FC : 2G → 2G by FC(c) =⋃

g∈G

glL(g−1c ∩ V ). We call the map lL a

local transition function and FC a global transition function.

■ 2 = {0, 1}: States, G: Cell space, V : Neighborhood.■ L, lL: Local function, FC : Global function.■ Note(1) glL(g−1

c ∩ V ) = φ or {g}.■ Note(2) X = g−1

c ∩ V ⇔ gX = c ∩ gV

CA(k, n)

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Definition 2 For k ≥ 1 and n ∈ {0, 1, · · · , 22k

− 1},we define cellular automata CA(k, n) on Z byCA(k, n) = (Z, V , Ln) where V = {0, 1, · · · , k − 1},

and Ln is the subset of 2V which satisfies n =∑

X∈Ln2

P

i∈X2i

.

Example 3 Recall f90(x, y, z) = (x + z) mod 2. Since 90 = 2 + 23 + 24 + 26

= 220

+ 220+21

+ 222

+ 221+22

, we have

CA(3, 90) = (Z, {0, 1, 2}, {{0}, {2}, {0, 1}, {1, 2}}).

The elements X in Ln represents the state of neighborhood which induce thenext states ’1’. For a rule number 90, we have the following table:

Neighborhood 111 110 101 100 011 010 001 000X ∈ Ln {0, 1, 2} {1, 2} {0, 2} {2} {0, 1} {1} {0} φ

lL(X) φ {e} φ {e} {e} φ {e} φ

Conway’s Life

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Example 4 LIFE = (Z2, VLIFE, LLIFE) is a cellular automata on group Z2,

where

VLIFE = {

(

−1−1

)

,

(

0−1

)

,

(

+1−1

)

,

(

−10

)

,

(

00

)

,

(

+10

)

,

(

−1+1

)

,

(

0+1

)

,

(

+1+1

)

}

LLIFE = {v ∈ 2VLIFE | (#v = 3) ∨ (#v = 4 ∧

(

00

)

∈ v)}.

We note that #v is the number of elements in a set v.

Union and Division

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Definition 5 (Union) Let C1 = (G, V1, L1) and C2 = (G, V2, L2) be cellularautomata on G. The union C1 ∪ C2 of C1 and C2 is defined byC1 ∪ C2 = (G, V1 ∪ V2, L1 ∪ L2).

Definition 6 (Division) Let C = (G, V, L) be a cellular automaton on G. Ifthere exist C1 = (G, V1, L1) and C2 = (G, V2, L2) be cellular automata on G

such that V = V1 ∪ V2 and L = L1 ∪ L2, then we call C1 and C2 are divisionof C and C is dividable by C1 and C2.

CA(2, n) ∪ CA(2,m) (n,m = 0, .., 15)

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n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 152 2 3 2 3 6 7 6 7 10 11 10 11 14 15 14 153 3 3 3 3 7 7 7 7 11 11 11 11 15 15 15 154 4 5 6 7 4 5 6 7 12 13 14 15 12 13 14 155 5 5 7 7 5 5 7 7 13 13 15 15 13 13 15 156 6 7 6 7 6 7 6 7 14 15 14 15 14 15 14 157 7 7 7 7 7 7 7 7 15 15 15 15 15 15 15 158 8 9 10 11 12 13 14 15 8 9 10 11 12 13 14 159 9 9 11 11 13 13 15 15 9 9 11 11 13 13 15 1510 10 11 10 11 14 15 14 15 10 11 10 11 14 15 14 1511 11 11 11 11 15 15 15 15 11 11 11 11 15 15 15 1512 12 13 14 15 12 13 14 15 12 13 14 15 12 13 14 1513 13 13 15 15 13 13 15 15 13 13 15 15 13 13 15 1514 14 15 14 15 14 15 14 15 14 15 14 15 14 15 14 1515 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

Table of unions : CA(2, n) ∪ CA(2, m)

Composition

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Definition 7 (Composition) Let C1 = (G, V1, L1) and C2 = (G, V2, L2) becellular automata on G. The composition C1♦C2 of C1 and C2 is defined byC1♦C2 = (G, V1 · V2, L1♦L2) where

V1 · V2 = {v1v2 ∈ G | v1 ∈ V1, v2 ∈ V2} and

L1♦L2 = {X ∈ 2V1·V2 | {v ∈ V1 | v−1X ∩ V2 ∈ L2} ∈ L1}.

Example 8 Let CA(2, 6) = (Z, {0, 1}, {{0}, {1}}), V = {0, 1} andL = {{0}, {1}}. Since 1−1{0, 1} ∩ {0, 1} = {0− 1, 1− 1} ∩ {0, 1} = {0} ∈ L

and 0−1{1, 2} ∩ {0, 1} = {1 − 0, 2 − 0} ∩ {0, 1} = {1} ∈ L, we haveL♦L = {{0}, {2}, {0, 1}, {1, 2}}.So we have CA(2, 6)♦CA(2, 6) = CA(3, 90).

V · V = {v1 + v2 ∈ Z | v1 ∈ {0, 1}, v2 ∈ {0, 1}} = {0, 1, 2}

L♦L = {X ∈ 2{0,1,2} | {v ∈ {0, 1} | (v−1X ∩ {0, 1, 2}) ∈ L} ∈ L}

CA(2, n)♦CA(2,m) (n,m = 0, .., 15)

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n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 255 236 209 192 139 136 129 128 55 36 17 0 3 0 1 02 0 16 34 48 68 68 66 64 8 24 34 48 12 12 2 03 255 252 243 240 207 204 195 192 63 60 51 48 15 12 3 04 0 2 12 12 48 34 24 8 64 66 68 68 48 34 16 05 255 238 221 204 187 170 153 136 119 102 85 68 51 34 17 06 0 18 46 60 116 102 90 72 72 90 102 116 60 46 18 07 255 254 255 252 255 238 219 200 127 126 119 116 63 46 19 08 0 1 0 3 0 17 36 55 128 129 136 139 192 209 236 2559 255 237 209 195 139 153 165 183 183 165 153 139 195 209 237 25510 0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 25511 255 253 243 243 207 221 231 247 191 189 187 187 207 221 239 25512 0 3 12 15 48 51 60 63 192 195 204 207 240 243 252 25513 255 239 221 207 187 187 189 191 247 231 221 207 243 243 253 25514 0 19 46 63 116 119 126 127 200 219 238 255 252 255 254 25515 255 255 255 255 255 255 255 255 255 255 255 255 255 255 255 255

Example 9 The rule numbers of the 3-neighborhood cellular automata generated bycomposing 2-neighborhood cellular automata is {0, 1, 2, 3, 8, 12, 15, 16, 17, 18, 19, 24,34, 36, 46, 48, 51, 55, 60, 63, 64, 66, 68, 72, 85, 90, 102, 116, 119, 126, 127, 128, 129,136, 139, 153, 165, 170, 183, 187, 189, 191, 192, 195, 200, 204, 207, 209,219, 221, 231, 236, 237, 238, 239, 240, 243, 247, 252, 253, 254, 255}. There are 62kinds of 3-neighborhood cellular automata.

Main Theorem

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The composition C1♦C2 of cellular automata corresponds to the cellularautomaton which global transition function is the composition FC1

◦ FC2of

global transition functions of original cellular automata C1 and C2.

Theorem 10FC1

◦ FC2= FC1♦C2

Proof. Since FC2(c) ∩ V1 = {v ∈ V1 | v

−1c ∩ V2 ∈ V ′

2}, we have

e ∈ FC1(FC2

(c)) ⇔ FC2(c) ∩ V1 ∈ V ′

1

⇔ FC2(c ∩ V1 · V2) ∩ V1 ∈ V ′

1

⇔ {v1 ∈ V1 | v−1(c ∩ V1 · V2) ∩ V2 ∈ V ′

2} ∈ V ′1

⇔ c ∩ V1 · V2 ∈ V ′1♦V ′

2

⇔ e ∈ FC1♦C2(c)

�

Distributive Law

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Theorem 11 Let C1 = (G, V, L1), C2 = (G, V, L2) and C3 = (G, V3, L3) becellular automata on a group G. Then,

(C1 ∪ C2)♦C3 = (C1♦C3) ∪ (C2♦C3)

We note that C1♦(C2 ∪ C3) = (C1♦C2) ∪ (C1♦C3) does not always holds forcellular automata C1, C2 and C3. For exampleCA(2, 6)♦(CA(2, 2) ∪ CA(2, 4)) = CA(2, 6)♦CA(2, 6) = CA(3, 90), and(CA(2, 6)♦CA(2, 2)) ∪ (CA(2, 6)♦CA(2, 4)) = CA(3, 46) ∪ CA(3, 116)= CA(3, 126).

CA(3, 3)♦CA(3, 102) = CA(3, 18)

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Example 12 We note CA(3, 3) = (Z, {0, 1, 2}, {φ, {0}}) and CA(3, 102) = (Z,{0, 1, 2}, {{0}, {1}, {0, 2}, {1, 2}}). The composition

CA(3, 3)♦CA(3, 102) = (Z, {0, 1, 2, 3, 4},

{{1}, {0, 1}, {1, 4}, {0, 1, 4},

{3}, {0, 3}, {3, 4}, {0, 3, 4}})

= (Z, {0, 1, 2, 3, 4},⋃

{{{0} ∪ s, s ∪ {4}, {0} ∪ s ∪ {4}} | s ∈ {{1}, {3}}})

= CA(3, 18)51.

Figure 1: CA(3, 3) Figure 2: CA(3, 102) Figure 3: CA(3, 18)

CA(2, 1)♦CA(2, 1) = CA(3, 17)♦CA(3, 17)

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Example 13 A 2-neighborhood cellular automaton is considered as3-neighborhood cellular automaton and 3-neighborhood cellular automaton isalso considered as 5-neighborhood cellular automaton. The followings is anobservation of the embeddings and compositions.

CA(2, 1) = (Z, {0, 1}, {φ})

CA(2, 1)30 = (Z, {0, 1, 2}, {φ, {2}})

= CA(3, 17)

CA(2, 1)♦CA(2, 1) = (Z, {0, 1, 2}, {{0, 1}, {0, 2}, {1, 2}, {1}})

= CA(3, 236)

CA(3, 17)♦CA(3, 17) = (Z, {0, 1, 2, 3, 4, 5}, L)

= CA(5, 3974950124) = CA(3, 236)50

L =⋃

{{s, s ∪ {3}, s ∪ {4}, s ∪ {3, 4}}|s ∈ CA(3, 236)}

Generalization

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A subset V of G is considered as a characteristic function V : G → 2 where2 = {0, 1}. That is V is a function which values are

V (g) =

{

0 (g 6∈ V )

1 (g ∈ V ).

Sometimes V is represented as an injection iV : V → G where iV (g) = g.Extending our 2-states cellular automata on groups to many-states cellularautomata on groups, we replace the set 2 = {0, 1} to a finite set S.

Definition 14 Let G be a group, S a finite set. A generalized cellularautomaton on G is a four-tuple C = (G, S, iV , L) of the group G, aninjection iv : V → G, and a function L : SV → S where SV is the set of allfunctions from V to S. A configuration c : G → S is a function. The globaltransition function FC : SG → SG is defined by FC(c)(g) = L(c ◦ g ◦ iV ).

Conclusion (Abstract)

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■ We introduced the notion of ’Composition’, ’Union’ and ’Division’ of CAon groups (Caley Graph).

■ We observed all ♦ compositions generated by one-dimensional2-neighborhood CA over Z2 including non-linear CA and found that 62kinds of 3-neighborhood CA are generated by composing 2-neighborhoodCA’s.

■ Next we proved that the composition is right-distributive over union, butis not left-distributive.

■ Finally, we concluded by showing the generalization of our definition ofcellular automata on group which admit more than three states.

■ We also showed our formulation contains the representation using formalpower series for linear cellular automata.

■ This is an on going work and is just introducing the new framework forinvestigating the class of CAs which domain of local function may not beequal.