Cellular AutomataCellular Automata

Moreno MarzollaDip. di Informatica—Scienza e Ingegneria (DISI)Università di Bologna

http://www.moreno.marzolla.name/

Complex Systems 2

Complex Systems 3

Cellular Automata

● Originally developed in the 1940s by Stanislaw Ulam and John von Neumann

● Interest in CAs expanded in 1970 after Conway's Game of Life John von Neumann

(1903—1957) Stanislaw Ulam(1909—1984)

Complex Systems 4

Cellular Automata (CA)

● A regular grid of cells, each in one of a finite number k of states (e.g., k = 2 for binary states)

● At time t = 0 an initial state for each cell is selected● A new generation is created (advancing t by 1),

according to some fixed rule that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood– The rule for updating cell states is the same for each cell

and does not change over time, though exceptions are known (e.g., stochastic CAs, asynchronous CAs...)

Complex Systems 5

Cellular Automata

● We will generally consider CAs over a linear square grid lattice (i.e., an array or matrix)

● Other topologies can be considered– Irregular– Hexagonal– …

● Example: Game of Life on an hexagonal lattice

Complex Systems 6

Elementary Cellular Automata

Complex Systems 7

1-Dimensional CA

● “Infinite” linear array of cells● Each cell can be in any of k states● The neighborhood of each cell in general consists of r

cells on the left and on the right – Usually r = 1, immediate neighbors– More complex neighborhoods can be considered

Cell

Neighbors (r = 1)

... ...

Complex Systems 8

Finite vs Infinite Grid

● Although the CA grid is logically infinite, it is usually represented using a finite array (or matrix)

● Cells on the edges can be handled in several ways– Allow the values of the cells on the border to remain

constant; or– Define neighborhoods differently for these cells (e.g., say

that they have fewer neighbors), or– Use a toroidal arrangement

1-D array → Ring

2-D array → Torus

Complex Systems 9

Elementary CA (k = 2, r = 1)

● The binary state ct+1

(i) of cell i at time t + 1 is

determined by the states of cells {i - 1, i, i + 1} at time tc

t(i - 1), c

t(i), c

t(i + 1)

i -1 i i+1

iTim

e

t

t + 1

Complex Systems 10

Example

● Rules

● Example (8 cells with wrap-around)

t = 0

t = 1

t = 2

t = 3

t = 4

ct(i-1) c

t(i) c

t(i+1)

ct+1

(i)

Complex Systems 11

Example

● Evolution of the previous CA on a larger array

Complex Systems 12

Wolfram Canonical Enumeration

● With binary state space and r=1, there are 2^8 = 256 possible elementary CAs

● We can encode the rule table with a single 8-bit number by reading the rightmost column top-to-bottom

2

4

8

16

Rule 30 CA

Complex Systems 13

Wolfram Canonical Enumeration

● With binary state space and r=1, there are 2^8 = 256 possible CAs

● We can encode the rule table with a single 8-bit number by reading the rightmost column top-to-bottom

2

4

8

32

Rule 110 CA

64

Complex Systems 14

CAs as Complex Systems:Wolfram Classification

● Class I– Always evolve to a homogeneous arrangement, with every

cell being in the same state, never to change again● Class II

– Form periodic structures that endlessly cycle through a fixed number of states

● Class III– Form aperiodic, random-like patterns similar to the static

white noise on a bad (analog) television channel● Class IV

– Form complex patterns with localized structure that move through space and time. Patterns eventually become homogeneous, like Class I, or periodic, like Class II

Elementary CA rules 0—49

Elementary CA rules 50—99

Elementary CA rules 100—149

Elementary CA rules 150—199

Elementary CA rules 200—255

Complex Systems 20

Wolfram’s Classification: Class 1

Rule 40 Rule 172 Rule 234

Slide credit: Ozalp Babaoglu

Complex Systems 21

Wolfram’s Classification: Class 2

Slide credit: Ozalp Babaoglu

Complex Systems 22

Wolfram’s Classification: Class 3

Rule 30 Rule 101 Rule 146

Slide credit: Ozalp Babaoglu

23

Wolfram’s Classification: Class 4

Slide credit: Ozalp Babaoglu

Complex Systems 24

Class II CAs

● Class II CAs are repetitive and bear resemblance to programs that execute in an infinite loop

while ( TRUE ) {for (i=0; i<10; i++)

print i;}

i = 0;while ( TRUE ) {

print i;i = i + 1;

}

Simple periodicity

Unbounded periodicity: on hypothetical machines with infinite

precision this fragment executes forever and is not periodic. In fact, this fragment does repeat when I overflows and starts back at zero; the exact value where this

happens is machine-dependent

Complex Systems 25

Langton's λ metric

● Is defined as the fraction of rules mapping into the non-quiescent state

● For elementary CAs, it is the fraction of rules mapping to one in the rule table

2

4

8

16

Rule 30 CA, λ = 4/8

Complex Systems 26

Langton's λ metric

● Is defined as the fraction of rules mapping into the non-quiescent state

● For elementary CAs, it is the fraction of rules mapping to one in the rule table

2

4

8

32

Rule 110 CA, λ = 5/8

64

Complex Systems 27

Langton's λ metric and CA classification

Complex Systems 28

Examples

● NetLogo implements elementary CAs in Sample Models → Computer Science → CA 1D Elementary

Complex Systems 29

Two-dimensional CAsGame of Life

Complex Systems 30

Two-dimensional CAs

● CAs can be defined over higher-dimensional grids● For two-dimensional CAs, different types of

neighborhoods can be defined

The eight gray cells form the Moore neighborhood for the

red cell

The four dark gray cells form the von Neumann neighborhood for the red cell; the

extended von Neumann neighborhood includes the light gray cells

Complex Systems 31

Conway's Game of Life

● Developed by the British mathematicianJohn Horton Conway in 1970

● Infinite two-dimensional orthogonal grid of square cells, each of which is in two states (alive or dead)

● Moore neighborhood● Rules:

– Any live cell with fewer than two live neighbours dies– Any live cell with two or three live neighbours lives on to the next generation– Any live cell with more than three live neighbours dies– Any dead cell with exactly three live neighbours becomes a live cell

● The initial pattern constitutes the seed of the system● Each generation is created by applying the above rules

simultaneously to every cell– The discrete moment at which this happens is sometimes called a tick

John Horton Conway(1937— )

Complex Systems 32

Some notable patterns

Still lifes

Oscillators

Spaceships

Glider Lightweight Spaceship

Source: http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

Complex Systems 33

Some notable patterns

● Conway conjectured that no pattern can grow indefinitely

● A team from the Massachusetts Institute of Technology proved he was wrong– The "Gosper glider gun" produces its first glider on the 15th

generation, and another glider every 30th generation from then on

Source: http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

Complex Systems 34

Logical operations implemented in Life

Logical Not Logical And Logical Or

Complex Systems 35

Universality● Both CA rule 110 and Life are Turing complete

– M. Cook, Universality in Elementary Cellular Automata, Complex Systems 15: 1-40.

● Below is a Turing machine implemented with a Life pattern in Golly (http://golly.sourceforge.net/), Patterns → Life → Signal Circuitry → Turing-Machine-3-state.rle

36

von Neumann Universal ConstructorPesavento, Umberto (1995) An implementation of von Neumann's self-reproducing machine. Artificial Life 2(4):337-354; See http://en.wikipedia.org/wiki/Von_Neumann_universal_constructor for details and links to the golly rule file (requires approximately 6 x 10^10 timesteps for replication)

Complex Systems 37

Another CA that replicates itself Patterns → JvN → Boustrophedon-replicator.rle

Complex Systems 38

Some practical applications of CAs

Complex Systems 39

Stochastic (Elementary) CA

● In a stochastic elementary CA, each rule is the probability that the center cell is black

● In general, in a stochastic CA the new cell states may depend on a random outcome

0.0

1.0

0.5

0.2

0.9

0.7

0.7

0.0

Complex Systems 40

Percolation model 1(site percolation)

● Three states, two-dimensional CA, von Neumann neighborhood– White: empty cell– Blue: blocked cell– Yellow: liquid

● Initial configuration– Each cell is empty with probability

p, blocked with probability 1 - p– One designated cell (e.g., at the

center of the matrix) contains liquid● One rule

– A white cell with at least one yellow neighbor turns yellow

Complex Systems 41

Percolation model 1

● Suppose we “pour” liquid on the top row. What is the probability that the liquid eventually reaches the bottom row?

Percolate Does not percolate

Complex Systems 42

Percolation model 1

● It turns out that there is a critical value p* for the parameter p (the probability that a cell is empty) s.t. the percolation probability increases sharply for p > p*– For square lattices, the site percolation threshold is p* ~ 0.59

p

Percolationprobability

1

1p*

Complex Systems 43

Percolation model 2● Percolation of a viscous liquid in a porous medium

– E.g., oil in sand● Three states, 1D stochastic CA

– 0: solid, 1: empty, 2: filled● Solid and empty squares are arranged in a

checkerboard configuration● Liquid percolates top-down through the empty squares

with probability pEmpty

Solid

Liquid

time

Percolation model 2

● NetLogo Sample Models → Earth Science → Percolation

p = 0.5 p = 0.6 p = 0.7

Complex Systems 45

Percolation model 2

● Note that for percolation model 2 we need to update the cells by row (one row at each time step, moving downwards)

● We have initially said that in a CA all cells must be updated concurrently at each timestep

● It turns out that we can implement the percolation model with global concurrent updates by using a slightly more complex cell state– Any idea?

Complex Systems 46

Percolation model 2

● Cell state is {soil, empty, newliquid, liquid}● Initial state:

– Checkerboard pattern of soil, empty– Empty cells on the first row are set in state newliquid

● Update rule

if ( this.state == newliquid ) thenthis.state ← liquid;

elseif ( this.state == empty ) thenif ( NE.state == newliquid and NW.state == newliquid ) then

this.state ← newliquid with prob. 1-(1-p)^2elseif ( NE.state == newliquid or NW.state == newliquid ) then

this.state ← newliquid with prob. pendif

endif

Complex Systems 47

Percolation model 3 (forest fire)

● A forest is modeled as a rectangular grid– Each contains a tree with probability P, is empty with

probability 1 - P ● Fire starts at the left edge

– Fire propagates along four directions: if a tree catches fire, the fire propagates to adjacent trees on the left, right, top and bottom of the current tree

– Fire does not propagate through empty patches● There is a critical density P* such that the fire can

reach the right edge of the forest● NetLogo Sample Models → Earth Science → Fire

48

Forest FireDensity P = 0.57

Density P = 0.60Density P = 0.59

Density P = 0.58

Complex Systems 49

Morphogenesis

● In 1952 Alan Turing investigated how non-uniformity (stripes, spots, spirals, etc.) may arise out of a homogeneous initial state– Reaction–diffusion theory

of morphogenesis– A. M. Turing, The Chemical

Basis of Morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Vol. 237, No. 641. (Aug. 14, 1952), pp. 37—72

Alan M. Turing (1912—1954)

Complex Systems 50

-validated-60-years-after-his-death

http://www.pnas.org/content/early/2014/03/05/1322005111

Complex Systems 51

Morphogenesis

● The original reaction-diffusion model proposed by Turing resembles a Continuous Spatial Automaton– Continuum of locations– The state of a location is any of a finite number of values– Time can be continuous, and in this case the state evolves

according to differential equations● The model can be approximated by a CA, and yields

similar patterns

Complex Systems 52

Morphogenesis

● Two states– U—undifferentiated (not colorful)– D—differentiated (colorful)

● Each D-cell secrets inhibitor (I) and activator (A)

● Each cell receives activator secreted by D-cells in its inner neighborhood, and inhibitor secreted by D-cells in its outer neighborhood– The new state is determined by the

weighted sum of inhibitor and activator received

X

Cell X receives the inhibitors secreted by D cells in this region

Cell X receives the activators secreted by D cells in this region

Complex Systems 53

Morphogenesis

● Netlogo Sample Models → Biology → Fur

Complex Systems 54

Cone Shell Patterns

● The shell grows linearly at its borders

● Pigmentation at a specific point is influenced by the pigmentation around the point of growth

● Is it a 1-D CA?– Yes, to some extent– Although pigmentation is not

really discrete, but subject to chemical diffusion

Conus textile shell

Slide Credit: prof. Franco Zambonelli Rule 30 CA

Complex Systems 55

B-Z Reaction

● The Belousov-Zhabotinsky (B-Z) reaction is an unusual chemical reaction: instead of moving towards a single equilibrium state, it oscillates between two such states

● States are integers in 0..max-state– 0 = “healthy”, max-state = “sick”, anything else = “infected”

● Rules– A cell that is sick becomes healthy.– A cell that is healthy may become infected, if enough of its

eight neighbors are infected or sick– A cell that is infected computes its new state by averaging

the states of itself and its eight neighbors

Complex Systems 56

B-Z Reaction

● Netlogo Sample Models → Chemistry & Physics → Chemical Reactions → B-Z Reaction

Complex Systems 57

An interesting book on CA

Tommaso Toffoli and Norman Margolus, Cellular Automata Machines—A new Environment for Modeling, MIT Press, 1987, ISBN 9780262200608. http://mitpress.mit.edu/books/cellular-automata-machines