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 Introduction 1 Preliminaries Motivations Symmetric Local Rules and Generalized Life 2 Symmetric Rules Life Family Equivalence With Respect To Permutations Of States Assembling Neighborhoods into Regular Lattices 3 2D Euclidean Metric Hyperbolic Plane 2D Sphere Fullerenes Computer Analysis of Dynamics of Symmetric Automata 4 Summary 5 V. V. Kornyak (LIT, JINR) Symmetric Cellular Automata 11 September 2006 2 / 17
Preliminaries Neighborhood contains k + 1 q -state points x 1 , . . . , x k , x k + 1 Local rule defines one time step evolution of x k + 1 : x k + 1 → x ′ k + 1 Symmetric local rule means symmetry with respect to the group S k of all permutations of k outer points x 1 , . . . , x k 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: S k ⇐ ⇒ Sym ( M ) ⊃ Diff ( M ) Nontriviality Local rule of Conway’s Life automaton is symmetric rule Practical reasons Numbers of general and symmetric local rules: q ( k + q − 1 q − 1 ) q q q k + 1 vs. 10 154 vs. 10 5 For k = 8 , q = 2 ( Conway’s Life case): V. V. Kornyak (LIT, JINR) Symmetric Cellular Automata 11 September 2006 4 / 17
Symmetric Rules k -valent neighborhood is a k -star graph ✉ x 3 this is trivalent neighborhood ✉ x 4 ✧ ❜ ✧ ❜ ✉ x 1 ✉ x 2 x ′ Local rule k + 1 = f ( x 1 , . . . , x k , x k + 1 ) k -symmetry is symmetry over k outer points x 1 , . . . , x k q ( k + q − 1 q − 1 ) q N q Number of k -symmetric rules = S k ( k + 1 ) -symmetry is symmetry over all k + 1 points x 1 , . . . , x k , x k + 1 q ( k + q q − 1 ) N q Number of ( k + 1 ) -symmetric rules = S k + 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: is born if it has 3 alive neighbors Central cell 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 x k → 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 q − 1 ) q �� � S k = q ( k + q − 1 B / S is 2 k + 1 × 2 k + 1 = 2 2 k + 2 = N 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 � | R g | | G | g ∈ G Numbers of orbits in binary case ( G = S 2 ) N S k / S 2 = 2 2 k + 1 + 2 k k -symmetry case: � 2 k + 1 + 2 k / 2 ; k = 2 m ( k + 1 ) -symmetry case: N S k + 1 / S 2 = 2 k + 1 ; k = 2 m + 1 For trivalent rules N S 3 / S 2 = 136 , N S 3 + 1 / S 2 = 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 E 2 3-valent , { 6 , 3 } 4-valent , { 4 , 4 } 6-valent , { 3 , 6 } ✧✧✧✧✧✧✧ ❜❜❜❜❜❜❜ ✧ ❜❜❜❜ ✧ ❜❜ ✧✧ ✧✧✧✧ ❜ ❜ ❜ ✧ ✧ ❜ ❜ ✧ ✧ ❜ ❜ ✧ ✧ ❜❜❜❜❜❜❜ ✧✧✧✧✧✧✧ ❜ ✧ ✧ ❜ ❜ ✧ ✧ ❜ ❜ ✧ ✧ ❜ ❜ ✧✧✧✧✧✧✧ ❜❜❜❜❜❜❜ ❜❜❜❜ ✧ ❜ ❜ ✧ ✧ ❜ ❜ ✧ ✧ ❜ ❜ ✧ ✧ ✧✧✧✧ ❜❜ ✧ ✧✧ ✧ ✧ ❜ ❜ ✧ ✧ ❜ ❜ ✧ ✧ ❜ ❜ ❜ ❜ Only two compactifications of these lattices are possible: in torus T 2 in Klein bottle K 2 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 H 2 Allows Infinitely Many Regular Lattices Poincaré proved: regular tilings { p , k } of H 2 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 H 2 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 S 2 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 C 60 (Buckyball) Carbon nanotubes and graphenes are other important 3-valent forms of large carbon molecules V. V. Kornyak (LIT, JINR) Symmetric Cellular Automata 11 September 2006 12 / 17
C 60 Is Embodiment of Icosahedral Group A 5 Felix Klein devoted a whole book (1884) to A 5 : Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade a , b | a 5 , b 2 , ( ab ) 3 ¸ ˙ Presentation A 5 = = ⇒ graph of C 60 is Cayley graph of A 5 Generators: a − → − → b Relations: a 5 = 1 pentagons b 2 = 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 = S 2 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: sphere S 2 ; V = 2 f 6 + 20 , E = 3 f 6 + 30 , f 5 = 12 , projective plane P 2 ; V = 2 f 6 + 10 , E = 3 f 6 + 15 , f 5 = 6 , torus T 2 , Klein bottle K 2 . V = 2 f 6 , E = 3 f 6 , f 5 = 0 , f 5 , f 6 , 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 x ′ F 2 polynomial: 4 = x 4 + x 1 + x 2 + x 3 + x 1 x 2 + x 1 x 3 + x 2 x 3 + x 1 x 2 x 3 On Tetrahedron All trajectories ✲ ♥ ♥ ✢ ✢ 14 1 ❅ ♥ ♥ ♥ ♥ 5 3 12 10 Attractor (Sink) ✲ ❅ ♥ ♥ PP 13 2 ❘ ❅ ✣ ✣ P q ♥ ✲ ♥ ✶ ✏ ✏✏ 15 0 Isolated 2-cycles ✲ ♥ ♥ ✢ � ✒ 11 4 � ♥ ♥ 6 9 ✲ � ♥ ♥ 7 8 ✣ Tetrahedron configurations s s s s s s s s = 0 s s s s s s → → s → s → s s s ← s = 1 s s s s s s 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 2 ✈ ✐ ✈ 4 ✲ 4 ✲ 2 ✲ 2 ❍ ✠ ✌ ❍ ❥ ✈ ✲ ✈ ✲ ✈ ✲ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ 4 × 6 × 8 ✯ ✟ ✟ ✈ ✒ ✍ q 4 ✟ ✯ Sink Limit 2-cycles ✈ ✟ ✈ Limit 4-cycles Isolated Cycles ✮ ✈ ✈ ✈ ✐ ■ ✌ ✎ ✠ ✠ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ × 3 × 12 × 6 × 12 ✒ ✒ ✍ ✗ q ❘ ✈ ✈ ✈ ✿ 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: ✝ ✆ 2 2 k + 1 + 2 k ( = 136 in trivalent case ) ✞ ☎ Highly symmetric 3-valent 2D finite lattices are: ✝ ✆ ◮ hexagonal lattices { 6 , 3 } (graphenes) in torus T 2 and Klein bottle K 2 ◮ tetrahedron { 3 , 3 } , hexahedron { 4 , 3 } , dodecahedron { 5 , 3 } in sphere S 2 ◮ fullerenes in sphere S 2 and in projective plane P 2 ✞ ☎ 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
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