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Symmetric Cellular Automata

Talk at CASC 2006, Chi¸ sin˘ au, Moldova Vladimir Kornyak

Laboratory of Information Technologies Joint Institute for Nuclear Research

11 September 2006

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 1 / 17

Outline

1

Introduction Preliminaries Motivations

2

Symmetric Local Rules and Generalized Life Symmetric Rules Life Family Equivalence With Respect To Permutations Of States

3

Assembling Neighborhoods into Regular Lattices 2D Euclidean Metric Hyperbolic Plane 2D Sphere Fullerenes

4

Computer Analysis of Dynamics of Symmetric Automata

5

Summary

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 2 / 17

Preliminaries

Neighborhood contains k + 1 q-state points x1, . . . , xk, xk+1 Local rule defines one time step evolution of xk+1: xk+1 → x′

k+1

Symmetric local rule means symmetry with respect to the group Sk of all permutations of k outer points x1, . . . , xk

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 3 / 17

Motivations

Philosophical motivation

Analogy with local diffeomorphism invariance of fundamental physical theories with continuum spacetime: Sk ⇐ ⇒ Sym(M) ⊃ Diff(M)

Nontriviality

Local rule of Conway’s Life automaton is symmetric rule

Practical reasons

Numbers of general and symmetric local rules: qqk+1 vs. q(k+q−1

q−1 )q

For k = 8, q = 2 (Conway’s Life case): 10154 vs. 105

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 4 / 17

Symmetric Rules

k-valent neighborhood is a k-star graph

✉ x4

this is trivalent neighborhood

✉x1 ✉x2 ✉ x3 ✧ ✧ ❜ ❜

Local rule x′

k+1 = f (x1, . . . , xk, xk+1)

k-symmetry is symmetry over k outer points x1, . . . , xk

Number of k-symmetric rules Nq

Sk

= q(k+q−1

q−1 )q

(k +1)-symmetry is symmetry over all k +1 points x1, . . . , xk, xk+1

Number of (k + 1)-symmetric rules Nq

Sk+1

= q(k+q

q−1)

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 5 / 17

Life Family

Conway’s Life rule is defined on 3 × 3 Moore neighborhood:

Central cell    is born if it has 3 alive neighbors survives if it has 2 or 3 alive neighbors dies otherwise Symbolically: B3/S23

Another examples:

HighLife (B36/S23): replicator – self-reproducing pattern – is known Day&Night (B3678/S34678): symmetric wrt swap of states

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 6 / 17

Any k-symmetric Rule Is Generalized Life Rule And Vice Versa

Generalized k-valent Life rule is a binary rule

described by two arbitrary subsets B, S ⊆ {0, 1, . . . , k} containing conditions for the xk → x′

k transitions of the forms 0 → 1 and 1 → 1

Proposition

For any k the set of k-symmetric binary rules coincides with the set of k-valent Life rules

Proof

Number of subsets of any set A is 2|A|, hence number of pairs B/S is 2k+1 × 2k+1 = 22k+2 =

• Nq

Sk = q(k+q−1

q−1 )q

• q=2

Different pairs B/S define different rules

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 7 / 17

Equivalence With Respect To Permutations Of States

Renaming of q states of automaton leads essentially to the same rule

Burnside’s lemma counts orbits of a group G acting on a set R

|R/G| = 1 |G|

• g∈G

|Rg|

Numbers of orbits in binary case ( G = S2)

k-symmetry case: NSk/S2 = 22k+1 + 2k (k +1)-symmetry case: NSk+1/S2 = 2k+1 + 2k/2; k = 2m 2k+1 ; k = 2m + 1

For trivalent rules

NS3/S2 = 136, NS3+1/S2 = 16

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 8 / 17

Symmetric Lattices In 2D Euclidean Metric

There are only three regular lattices in E2

3-valent, {6, 3}

❜ ❜ ❜ ❜ ❜ ❜ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ✧ ❜ ❜ ❜ ❜ ❜ ❜

4-valent, {4, 4} 6-valent, {3, 6}

✧✧✧✧✧✧✧ ✧✧✧✧✧✧✧ ✧✧✧✧✧✧✧ ❜❜❜❜❜❜❜ ❜❜❜❜❜❜❜ ❜❜❜❜❜❜❜ ❜❜❜❜ ❜ ✧✧✧✧ ✧ ❜❜ ❜ ✧✧ ✧ ✧✧✧✧ ✧ ✧✧ ✧ ❜❜❜❜ ❜ ❜❜ ❜

Only two compactifications of these lattices are possible:

in torus T2 in Klein bottle K2 Schläfli symbol {p, k} denotes k-valent lattice composed of regular p-gons

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 9 / 17

Hyperbolic (Lobachevsky) Plane H2 Allows Infinitely Many Regular Lattices

Poincaré proved:

regular tilings {p, k} of H2 exist for any p, k ≥ 3 satisfying 1

p + 1 k < 1 2

Octivalent Moore neighborhood

q q q q q q q q q

• ❅

❅ ❅ ❅ ❅ ❅ ❅

is not regular in Euclidean plane

Octivalent regular lattice {3, 8} in H2

Infinitely many compactifications, i.e., for genus g > 1: V = 6(g − 1), E = 24(g − 1), F = 16(g − 1)

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 10 / 17

Regular Lattices In S2 Correspond To Platonic Solids

3-valent

• 4-valent
• 5-valent
• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 11 / 17

Fullerenes

Carbon Molecule C60 (Buckyball)

Carbon nanotubes and graphenes are other important 3-valent forms

• f large carbon molecules
• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 12 / 17

C60 Is Embodiment of Icosahedral Group A5

Felix Klein devoted a whole book (1884) to A5: Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade Presentation A5 = ˙ a, b|a5, b2, (ab)3¸ = ⇒ graph of C60 is Cayley graph of A5

Generators: a − → b − → Relations: a5 = 1 pentagons b2 = 1 ← → (ab)3 = 1 hexagons

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 13 / 17

Generalized Fullerenes

Standard fullerene

from mathematical viewpoint is 3-valent graph embeddable in M = S2 with all faces of size 5 or 6

We can slightly generalize this notion

assuming M be closed surface of other type, orientable or not

Euler–Poincaré relation leads to only possibilities for generalized fullerenes:

V = 2f6 + 20, E = 3f6 + 30, f5 = 12, sphere S2; V = 2f6 + 10, E = 3f6 + 15, f5 = 6, projective plane P2; V = 2f6, E = 3f6, f5 = 0, torus T2, Klein bottle K2. f5, f6, V, E – numbers of pentagons, hexagons, Vertices, Edges

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 14 / 17

Phase Space of Rule 86

Bit table: 01101010 Birth/Survival notation: B123/S0 F2 polynomial: x′

4 = x4 + x1 + x2 + x3 + x1x2 + x1x3 + x2x3 + x1x2x3

On Tetrahedron All trajectories

Attractor (Sink) Isolated 2-cycles

♥ ♥ ♥ ♥

7 11 13 14

✲ ✲ ✲ ✲ ♥ ♥ ♥ ♥

8 4 2 1

• ✒

❅ ❅ ❅ ❘ ✏✏ ✏ ✶ PP P q ♥

15

✲ ♥ ♥

3

♥

12

✢ ✣ ♥

5

♥

10

✢ ✣ ♥

6

♥

9

✢ ✣

Tetrahedron configurations

s s s s

→ s

s s s

→ s

s s s

→ s

s s s s s s s

→ ← s

s s s s s = 0

= 1

Equivalence classes with respect to lattice symmetries

4

✇ ✲4 ✇ ✲ ✇ ✲ ✇ ✇ ✇ ✢ ✣

× 3

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 15 / 17

Phase Space of Rule 86 on Hexahedron (Cube)

Cube is “smallest graphene”, since its graph covers torus by 4 regular hexagons: Equivalence classes of trajectories with respect to lattice symmetries Attractors

4

✈ ✟ ✟ ✯

4

✈ ✟ ✟ ✯

2

✈ ❍ ❍ ❥ ✈ ✲ ✈

Sink

4

✈ ✲4 ✈ ✲2 ✈ ✲2 ✈ ✲ ✈ ✈ ✠ ✒

× 6

Limit 2-cycles

✈ ✲ ✈ ✈

× 8

✈ ✈ q ✍ ✐ ✌

Limit 4-cycles

Isolated Cycles

✈ ✈ ✠ ✒

× 3

✈ ✈ ✠ ✒

× 12

✈ ✈

× 6

✈ ✈ q ✍ ✐ ✌ ✈ ✈ ✈ ✈ ✈ ✈

× 12

✎ ❘ ✿ ✗ ■ ✮

Total number of initial conditions 256 Number of non-equivalent trajectories 8

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 16 / 17

Summary

✞ ✝ ☎ ✆

Game of Life Family = k−symmetric binary automata

✞ ✝ ☎ ✆

Number of non-equivalent k−symmetric binary rules: 22k+1 + 2k ( = 136 in trivalent case)

✞ ✝ ☎ ✆

Highly symmetric 3-valent 2D finite lattices are:

◮ hexagonal lattices {6, 3} (graphenes) in torus T2 and Klein bottle K2 ◮ tetrahedron {3, 3}, hexahedron {4, 3}, dodecahedron {5, 3} in

sphere S2

◮ fullerenes in sphere S2 and in projective plane P2

✞ ✝ ☎ ✆

Computer program exploiting both rule and lattice symmetries for analysis of dynamics of automata is under construction

• V. V. Kornyak (LIT, JINR)

Symmetric Cellular Automata 11 September 2006 17 / 17