# timing and stability in cellular automata - tbi.· timing and stability in cellular automata...

Post on 18-Sep-2018

215 views

Embed Size (px)

TRANSCRIPT

Timing and stability in cellular automata

Konstantin Klemm

Bioinformatics GroupUniversity of Leipzig, Germany

Konstantin Klemm (Leipzig) Timing & Stability in CA 1 / 25

Definition

finite set of states S , often |S | = 2

mapping (rule table) f : Sz S

a lattice of n sites with coordination number z where site i hasneighbours a(i , 1), a(i , 2), . . . a(i , z)

time-discrete dynamics of lattice site i

si (t + 1) = f [sa(i ,1)(t), . . . sa(i ,z)(t)]

All lattice sites are updated in synchrony.

Konstantin Klemm (Leipzig) Timing & Stability in CA 2 / 25

Purpose

Computer Science: Models of computation,e.g. Game of Life and rule 110 are Turing-complete.

Artificial Life: Study of self-reproducing structures

Physics: Abstractions of spatio-temporal dynamics, pattern formation

Motto: Simplest rules may yield most complex patterns / computations.

Konstantin Klemm (Leipzig) Timing & Stability in CA 3 / 25

Putative CA dynamics on plant leaves

Peak et al., PNAS (2004)

Konstantin Klemm (Leipzig) Timing & Stability in CA 4 / 25

CA model for morphogenesis in Hydrasp

ace

time

Ped

Ped/CnNK2

1

rump

foot

Rohlf & Bornholdt, JSTAT (2005)

Konstantin Klemm (Leipzig) Timing & Stability in CA 5 / 25

Seashell

Konstantin Klemm (Leipzig) Timing & Stability in CA 6 / 25

Seashell and elementary CA rule 22

Konstantin Klemm (Leipzig) Timing & Stability in CA 7 / 25

Seashell and elementary CA rule 30

Konstantin Klemm (Leipzig) Timing & Stability in CA 8 / 25

Role of noise?

Are the observed patterns robust under small stochasticperturbations?

Stability analysis in continuous dynamical systems: Make a smallperturbation and check if the system returns to the fixed point / limitcycle.

Here: Discretized state space. What is a small perturbation?

Typical implementation of noise in CA:

Stochastic asynchronous update

Konstantin Klemm (Leipzig) Timing & Stability in CA 9 / 25

Rule 22, comparison sync / async update

synchronous update asynchronous update async, another realization

Konstantin Klemm (Leipzig) Timing & Stability in CA 10 / 25

Rule 150, comparison sync / async update

synchronous update asynchronous update async, another realization

Konstantin Klemm (Leipzig) Timing & Stability in CA 11 / 25

Rule 90, comparison sync / async update

synchronous update asynchronous update async, another realization

Konstantin Klemm (Leipzig) Timing & Stability in CA 12 / 25

Loss of memory

0 20 40 60 80 100 120 140 160 180 200time

0.01

0.1

1in

form

atio

n [b

its]

information of a cells state at time tmutual information between cells state at time tand initial condition

Evolution of mutual information between one cells state at time t and thefull initial configuration.

Konstantin Klemm (Leipzig) Timing & Stability in CA 13 / 25

Summary so far

Deterministic CA:Complex (interesting) spatio-temporal pattern formation

CA with noise implemented as stochastic update order:Largely irreproducible dynamics

Devastating effect of stochastic asynchronous update known for long,cf. Ingerson and Buvel, Physica D (1984)

Konstantin Klemm (Leipzig) Timing & Stability in CA 14 / 25

How to proceed

Solution 0

Use more states and more complicated interactions to implementadditional clock signalscf. Nehaniv, Int. J. Alg. Comp. (2004)

Solution 1

Stick to simple rules

Consider a less destructive type of perturbation.

Get happy.

Konstantin Klemm (Leipzig) Timing & Stability in CA 15 / 25

Framework for perturbed timing

Now time t R, no longer discrete.

Expand initial condition s(0) to unit interval:

si (t) =

{

fi(s(0)) , if t > 1 isi (0) , otherwise

where i [0, ] is drawn randomly and independently for each node i .

Let the system evolve according to

si (t + 1) =

[

(2c)1

t+c

tc

fi (s())d 1/2

]

where is the step function and 1/2 > s > 0.

Check, if system autonomously regains synchrony.

Klemm & Bornholdt, PNAS 102, 18414 (2005); Phys. Rev. E 72, 055101(R) (2005).

Konstantin Klemm (Leipzig) Timing & Stability in CA 16 / 25

CA rule 22, initial timing perturbation

Konstantin Klemm (Leipzig) Timing & Stability in CA 17 / 25

CA rule 22, spatial coarsening of time lags

Konstantin Klemm (Leipzig) Timing & Stability in CA 18 / 25

CA rule 22, timing perturbation heals

t=240

Konstantin Klemm (Leipzig) Timing & Stability in CA 19 / 25

Probability of healing

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

prob

. tha

t per

turb

atio

n he

als

out

CA rule 22

0 200 400 600 800number of cells N

CA rule 150

0 200 400 600 800

CA rule 110

Konstantin Klemm (Leipzig) Timing & Stability in CA 20 / 25

Time until healing

100 1000number of cells N

102

103

104

105

rule 22rule 150slope 2slope 1

Konstantin Klemm (Leipzig) Timing & Stability in CA 21 / 25

Unstable pattern in rule 110

Konstantin Klemm (Leipzig) Timing & Stability in CA 22 / 25

Overview of results

stable elementary rules0, 4, 22, 32, 36, 54, 72, 76, 104, 128, 132, 160, 164, 200, 204, 218,222, 236, 250,254

partially stable el. rules (strong dependence on n)90, 122, 126, 150

unstable elementary rules50, 94, 108, 110, 178

Elementary CA fail to produce sustained synchronous blinking of allcells

Game of Life: Blinkers, gliders, spaceships etc. are unstable

Konstantin Klemm (Leipzig) Timing & Stability in CA 23 / 25

Further results

Konstantin Klemm (Leipzig) Timing & Stability in CA 24 / 25

Summary

Stable and unstable CA rules can be distinguished by theirresponse to minimal timing perturbations.

Konstantin Klemm (Leipzig) Timing & Stability in CA 25 / 25

Recommended