Symmetry Analysis of Discrete Dynamical SystemsTalk at CADE 2007, February 20-23, Turku, Finland

Vladimir Kornyak

Laboratory of Information TechnologiesJoint Institute for Nuclear Research

February 23, 2007

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 1 / 26

Outline

1 Introduction

2 Discrete Relations

3 Lattices and Their Symmetries

4 Influence of Lattice Symmetry On Dynamics

5 Symmetric Local Rules

6 Computer Program And Its Functionality

7 Reversibility

8 Lattice Models On Symmetric Lattices

9 Summary

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 2 / 26

Motivations

• Symmetry analysis of finite discrete systems can be completein contrast to continuous systems where only negligible small part ofall thinkable symmetries is considered: point and contact Lie, Bäcklundand Lie–Bäcklund, sporadic instances of non-local symmetries etc.

• Many hints from quantum mechanics and quantum gravity thatdiscreteness is more suitable for physics at small distancesthan continuity which arises only as a logical limit in considering largecollections of discrete structures

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 3 / 26

Preliminaries

Discrete Relations on Abstract Simplicial ComplexesOur recent observation. Any collection of discrete pointsvalued by states from finite set posseses some kind of locality:such collection has naturally a structure of abstract simplicial complex.This construction is generalization of both

system of polynomial equationscellular automaton

The talk is aboutdiscrete dynamical systems on symmetric latticesMore specifically, we consider cellular automata with symmetric localrules – a special case of discrete relations on abstract simplicialcomplex

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 4 / 26

Discrete Relations

Relations (basic definitions)Collection of N “points”: symbolically δ = {x1, . . . , xN}qi states of i th point: Qi =

{s1

i , . . . , sqii

}or in standard notation

Qi = {0, . . . , qi − 1}Relation on δ is arbitrary subset Rδ ⊆ Qδ

Qδ is symbolical notation for Cartesian product Q1 × · · · ×QN

Extension of relation Rδ to larger set of points τ ⊇ δ:Rτ = Rδ ×Qτ\δ

Consequence of relation Rδ is arbitrary subset Sδ ⊆ Rδ

Proper consequence of Rδ is relation Pα on proper subset α ⊂ δsuch that Pα ×Qδ\α ⊆ Rδ (extention of projection is usualconsequence)

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 5 / 26

Decomposition of RelationCanonical DecompositionRδ can be represented as combination of its proper consequencesPδ1 , . . . , Pδm and principal factor PRδ

Rδ = PRδ⋂(

m⋂i=1

Pδi ×Qδ\δi

)

Principal Factor

PRδ = Rδ⋃(

Qδ \m⋂

i=1

Pδi ×Qδ\δi

)

Canonical decomposition imposes structure of AbstractSimplicial Complex on Rδ

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 6 / 26

Compatibility of System of Relations

Compatibility Condition (= Base Relation)of any set of relations Rδ1 , . . . , Rδn is intersection of their extentions to

common domain δ =n⋃

i=1δi

Bδ =n⋂

i=1

Rδi ×Qδ\δi

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 7 / 26

Decomposition And Compatibility Analysis Algorithmsare very simple and efficient

main loops include only bitwise operations

NOT OR XOR ANDoperators in C ∼ | ∧ &

ExampleCompatibility analysis of relations defining rules for Conway’sGame of Life takes < 1 sec (in fact, “zero” within timer uncertainty)Maple Gröbner basis package applied to these relationsexpressed by polynomials over F2 produces essentially the sameresult for 0.5 – 1.5 hours depending on ordering

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 8 / 26

When Polynomials Work

Any system of discrete relations Rδ1 , . . . , Rδn can be represented as aset of polynomial equations if

1 The sets of states at all points in union δ = {x1, . . . , xN} ofdomains δ1, . . . , δn are the same:

Q1 = · · · = QN = Q2 Number of elements in Q is a power of a prime:

q = pk

Then relations correspond to sets of zeros for some polynomials fromring Fq[x1, . . . , xN ] and Gröbner basis technique can be applied tocompatibility analysis of the system of relations

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 9 / 26

When System Of Relations Is Cellular Automaton

1 collection of points x1, . . . , xN can be represented as union ofconnected congruent simplices with the same relation (local rule)on each simplex

2 the local rule is functional relation, i.e., value of one of the pointsin simplex is determined unambiguously by values of other points

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 10 / 26

Lattices and Their Symmetries

Space of cellular automaton islattice L ∼= k -valent (k -regular) graph GL

Symmetry of lattice L is graph automorphism group Aut (GL)

In applications it is assumed often that lattice is embedded in somecontinuous space — in this case notion of ‘dimension’ of lattice makessense

Reasons to study trivalent two-dimensional latticescarbon molecules of special interest known as fullerenes,nanotubes and graphenes are 3-valent 2-dimensional structuresthis is the simplest case except bivalent Wolfram’s elementaryautomata whose ‘geometry’ is rather primitive

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 11 / 26

Examples of Symmetric Trivalent Lattices

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 12 / 26

Remark. Cube Is “Smallest Graphene”Distinction between lattice and its graph

The same graph forms

4-gonal (6 tetragons) lattice in sphere S2 and6-gonal (4 hexagons) lattice in torus T2

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 13 / 26

Functions On Lattices And Their Orbits

Q = {0, . . . , q − 1} is set of states of lattice verticesQL is space of Q-valued functions on LAut (GL) acts non-transitively on QL splitting this spaceinto orbits of different sizesBurnside’s lemma counts total number of orbits

Norbits =1

|Aut(GL)|∑

g∈Aut(GL)

qNgcycles ,

Ngcycles is number of cycles in group element g

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 14 / 26

Some Numerical InformationOn Symmetric 3-valent Lattices, Their Groups And Orbits

Lattice V (GL) |Aut(GL)| Nmultistates = qV (GL) Norbits

Tetrahedron 4 24 16 5Cube 8 48 256 22Dodecahedron 20 120 1048576 9436

Buckyball 60 1201152921504606846976

≈ 10189607679885269312

≈ 1016

Graphene 3×4Torus 24 48 16777216 355353

Graphene 3×4Klein bottle 24 16 16777216 1054756

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 15 / 26

Influence of Lattice Symmetry On Dynamics“Second Law of Thermodynamics”

For Any Causal Dynamical System

trajectories can be directed only from larger orbits to smaller orbitsor to orbits of the same size

periodic trajectories must lie within the orbits of equal size

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 16 / 26

Symmetric Local Rules

k -valent neighborhood is a k -star graph

u x4this is trivalent neighborhoodux1

ux2

u x3

"" bb

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 NqSk

= q(k+q−1q−1 )q

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 17 / 26

k -symmetric Rules And Generalized “Game Of Life”

Generalized k -valent Life rule is a binary ruledescribed by two arbitrary subsets B, S ⊆ {0, 1, . . . , k} containingconditions for the xk → x ′

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

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

Hence there are 3 ways to express binary local rulex ′

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

by bit string (Example: Rule 86 = 01101010)by Birth/Survival lists (B123/S0)by polynomial over F2(x ′

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

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 18 / 26

Equivalence With Respect To Permutations Of StatesRenaming of q states of automaton leads essentially to the same rule

Burnside’s lemma counts number of non-equivalent rules

in k -valent binary case this number is Nk = 22k+1 + 2k

For trivalent rules

N3 = 136

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 19 / 26

Computer Program And Its FunctionalityProgram is written in C language

InputGraph of lattice GL = {N1, . . . , Nn}Set of local rules R = {r1, . . . , rm}Control parameters

Program constructsautomorphism group Aut (GL)

phase portraits of automata modulo Aut(GL) for all rules from R

Manipulating control parameters we canselect automata with specified properties, e.g., reversibility,Hamiltonian conservation etc.search automata producing specified structures, e.g., limit cycles,isolated cycles, Gardens of Eden etc.

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 20 / 26

Small Lattices: Phase Portraits For Rule 86Tetrahedron

Isolated 2-cyclesAttractor (Sink)4w -

4w -1w -

1w × 6w w�

�

Hexahedron (Cube)Attractors

4v��*4v��*2vHHj1v-1v

Sink

24v-24v-12v-12v-6v 6v

�Limit 2-cycles

v- v v× 8vvq �

i

Limit 4-cycles

Isolated cycles

v v

�× 6 v v

�×24 v v×12

vvq �

i v v v vvv

× 24�

R:

�

I

)

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 21 / 26

Large Lattices: Concise Outputs Instead Of Phase PortraitsDodecahedron Graphene 3×4

Torus Klein BottlePeriod Nisolated Nlimit Nisolated Nlimit

1 1 12 116 78 154 1114 267 552 655 14236 66 232 162 5418 1 4 1 5

10 12 50 24 8412 17 84 42 26214 2 1 1 1016 5 18 11 2718 4 620 14 3022 2 2 424 126 3 428 1 9 4 1030 4 532 1 334 2 236 1 1 138 140 142 1 244 246 148 160 162 164 3

128 1144 1416 1

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 22 / 26

Quantum Gravity Difficulties:Irreversibility of Gravity: information loss (=dissipation) at horizon of black hole

Reversibility and Unitarity of standard Quantum Mechanics

G. ’t Hooft’s approach to reconcile Gravity with Quantum Mechanics1 Discrete degrees of freedom at Planck distance scales2 States of these degrees of freedom form primordial basis (’t

Hooft’s term) of (nonunitary) Hilbert space3 Equivalence classes of states: two states are equivalent if they

evolve into the same state after some lapse of time4 Equivalence classes form basis of unitary Hilbert space and now

evolution is described by time-reversible Schrödinger equationIn our terminology this corresponds to transition to limit cycles:after short time of evolution limit cycles become physicallyindistinguishable from reversible isolated cycles

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 23 / 26

Search For ReversibilityReversibility is rare property tending to disappear with growth of complexity of latticeTwo rules trivially reversible on all lattices

85 or x ′4 = x4 + 1170 or x ′4 = x4

Tetrahedron, 6 additional reversible rules43 or x ′4 =x4(x2x3+x1x3+x1x2+x3+x2+x1)+x1x2x3+x2x3+x1x3+x1x2+x3+x2+x1+1

51 or x ′4 =x3 + x2 + x1 + 1

77 or x ′4 =x4(x2x3 + x1x3 + x1x2 + x3 + x2 + x1 + 1) + x1x2x3 + x2x3 + x1x3 + x1x2 + 1

178 or x ′4 =x4(x2x3 + x1x3 + x1x2 + x3 + x2 + x1 + 1) + x1x2x3 + x2x3 + x1x3 + x1x2

204 or x ′4 =x3 + x2 + x1

212 or x ′4 =x4(x2x3+x1x3+x1x2+x3+x2+x1)+x1x2x3+x2x3+x1x3+x1x2+x3+x2+x1

Cube, 2 additional reversible rules51 or x ′4 =x3 + x2 + x1 + 1204 or x ′4 =x3 + x2 + x1

Dodecahedron and Graphene 3×4, no additional reversible rules

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 24 / 26

Lattice Models On Symmetric LatticesDistribution function via summation of group orbit sizes

in statistical physics e−|H|kT

Ising model: si ∈ Q = {−1, 1} H = −∑〈i,k〉

sisk

Dodecahedron Graphene 3×4 o Torus + Klein bottle

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 25 / 26

Main Results

C program for symmetry analysis of finite discrete dynamicalsystems has been created.Trajectories of dynamical systems with non-trivial symmetry goalways in the direction of non-increasing sizes of group orbits.Cyclic traectories lie within orbits of the same size.Reversibility is rare property.Taking into account lattice symmetries facilitates work withphysical lattice models.

V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, 2007 26 / 26