Cellular Automata This is week 7 of Biologically Inspired Computing Various credits for these slides, which have in part been adapted from slides by: Ajit Narayanan, Rod Hunt, Marek Kopicki.

Cellular Automata A CA is a spatial lattice of N cells, each of which is one of k states at time t. Each cell follows the same simple rule for updating its state. The cell's state s at time t+1 depends on its own state and the states of some number of neighbouring cells at t. For one-dimensional CAs, the neighbourhood of a cell consists of the cell itself and r neighbours on either side. Hence, k and r are the parameters of the CA. CAs are often described as discrete dynamical systems with the capability to model various kinds of natural discrete or continuous dynamical systems

SIMPLE EXAMPLE Suppose we are interested in understanding how a forest fire spreads. We can do this with a CA as follows. Start by defining a 2D grid of `cells’, e.g.: This will be a spatial representation of our forest.

SIMPLE EXAMPLE continued Now we define a suitable set of states. In this case, it makes sense for a cell to be either empty, ok_tree, or fire_tree – meaning: empty: no tree here ok_tree: there is a tree here, and it’s healthy fire_tree: there is a tree here, and it’s on fire. When we visualise the CA, we will use colours to represent the states. In these cases; white, green and red seem the right Choices.

SIMPLE EXAMPLE continued Next we define the neighbourhood structure – when we run our CA, cells will change their state under the influence of their neighbours, so we have to define what counts as a “neighbour”. You’ll see example neighbourhoods in a later slide, but usually you just use a cell’s 9 immediately surrounding neighbours. Let’s do that in this case. Next we decide what the neighbourhood will be like at the boundaries of the grid.

Example of 1-D cellular automaton For a binary input N long, are there more 1s than 0s? Set k=2 and r=1 with the following rule: Cell + 2 neighbours: Result: That is, the value of a cell at time t+1 will depend on its value and the values of its two immediate neighbours at time t. This is a form of ‘majority voting’ between all three cells.

Density classification In the above example, we have assumed wrap-around, and r=1. In this case, the CA has reached a ‘limit point’ from which no escape is possible. CAs have been used for simulating fluid dynamics, chemical oscillations, crystal growth, galaxy formation, stellar accretion disks, fractal patterns on mollusc shells, parallel formal language recognition, plant growth, traffic flow, urban segregation, image processing tasks, etc …

See HIV CA demo – My Documents\students\hivca java Main Rule 1 - If an H cell has at least one I1 neighbour, or if has at least 2 I2 neighbours, then it becomes I1. Otherwise, it stays healthy. Rule 2 – An I1 cell becomes I2 after 4 time steps (simulated weeks). (to operate this the CA maintains a counter associated with each I1 cell). Rule 3 - An I2 cell becomes D. Rule 4 – A D cell becomes H, with probability ; I1, with probability ; otherwise, it remains D 4 states: Healthy, Infected1, Infected2, Dead

Types of neighbourhood Many more neighbourhood techniques exist - see and follow the link to ‘neighbourhood survey’

Classes of cellular automata (Wolfram) Class 1: after a finite number of time steps, the CA tends to achieve a unique state from nearly all possible starting conditions (limit points) Class 2: the CA creates patterns that repeat periodically or are stable (limit cycles) – probably equivalent to a regular grammar/finite state automaton Class 3: from nearly all starting conditions, the CA leads to aperiodic-chaotic patterns, where the statistical properties of these patterns are almost identical (after a sufficient period of time) to the starting patterns (self-similar fractal curves) – computes ‘irregular problems’ Class 4: after a finite number of steps, the CA usually dies, but there are a few stable (periodic) patterns possible (e.g. Game of Life) - Class 4 CA are believed to be capable of universal computation

John Conway’s Game of Life 2D cellular automata system. Each cell has 8 neighbors - 4 adjacent orthogonally, 4 adjacent diagonally. This is called the Moore Neighborhood.

Simple rules, executed at each time step: –A live cell with 2 or 3 live neighbors survives to the next round. –A live cell with 4 or more neighbors dies of overpopulation. –A live cell with 1 or 0 neighbors dies of isolation. –An empty cell with exactly 3 neighbors becomes a live cell in the next round.

Is it alive? Compare it to the definitions…

Glider

Sequences

More Sequence leading to Blinkers Clock Barber’s pole

A Glider Gun

Assumptions –Computation universality not required Characteristics –8 states, 2D Cellular automata –Needed CA grid of 100 cells –Self Reproduction into identical copy –Input tape with data and instructions –Concept of Death Significance – Could be modeled through computer programs Loops

Langton’s Loop 0 – Background cell state3, 5, 6 – Phases of reproduction 1 – Core cell state4 – Turning arm left by 90 degrees 2 – Sheath cell state state 7 – Arm extending forward cell state

Loop Reproduction

Loop Death

Langton’s Loops Chris Langton formulated a much simpler form of self-rep structure - Langton's loops - with only a few different states, and only small starting structures.

There remains debate and interest about the `essentials of life’ issue with CAs, but their main BIC value is as modelling techniques. Modelling Sharks and Fish: Predator/Prey Relationships Bill Madden, Nancy Ricca and Jonathan Rizzo Graduate Students, Computer Science Department Research Project using Department’s 20-CPU Cluster We’ve seen HIV – here are some more examples.

This project modeled a predator/prey relationship Begins with a randomly distributed population of fish, sharks, and empty cells in a 1000x2000 cell grid (2 million cells) Initially, –50% of the cells are occupied by fish –25% are occupied by sharks –25% are empty

Here’s the number 2 million Fish: red; sharks: yellow; empty: black

Rules A dozen or so rules describe life in each cell: birth, longevity and death of a fish or shark breeding of fish and sharks over- and under-population fish/shark interaction Important: what happens in each cell is determined only by rules that apply locally, yet which often yield long-term large-scale patterns.

Do a LOT of computation! Apply a dozen rules to each cell Do this for 2 million cells in the grid Do this for 20,000 generations Well over a trillion calculations per run! Do this as quickly as you can

Rules in detail: Initial Conditions Initially cells contain fish, sharks or are empty Empty cells = 0 (black pixel) Fish = 1 (red pixel) Sharks = –1 (yellow pixel)

Rules in detail: Breeding Rule Breeding rule: if the current cell is empty If there are >= 4 neighbors of one species, and >= 3 of them are of breeding age, »Fish breeding age >= 2, »Shark breeding age >=3, and there are <4 of the other species: then create a species of that type »+1= baby fish (age = 1 at birth) »-1 = baby shark (age = |-1| at birth)

Breeding Rule: Before EMPTY

Breeding Rule: After

Rules in Detail: Fish Rules If the current cell contains a fish: Fish live for 10 generations If >=5 neighbors are sharks, fish dies (shark food) If all 8 neighbors are fish, fish dies (overpopulation) If a fish does not die, increment age

Rules in Detail: Shark Rules If the current cell contains a shark: Sharks live for 20 generations If >=6 neighbors are sharks and fish neighbors =0, the shark dies (starvation) A shark has a 1/32 (.031) chance of dying due to random causes If a shark does not die, increment age

Shark Random Death: Before I Sure Hope that the random number chosen is >.031

Shark Random Death: After YES IT IS!!! I LIVE

Spring 2005JR36 Sample Code (C++): Breeding

Results Next several screens show behavior over a span of 10,000+ generations

Spring 2005BM38 Generation: 0

Spring 2005BM39 Generation: 100

Generation: 500

Generation: 1,000

Generation: 2,000

Generation: 4,000

Generation: 8,000

Generation: 10,500

Long-term trends Borders tended to ‘harden’ along vertical, horizontal and diagonal lines Borders of empty cells form between like species Clumps of fish tend to coalesce and form convex shapes or ‘communities’

Variations of Initial Conditions Still using randomly distributed populations: –Medium-sized population. Fish/sharks occupy: 1/16 th of total grid Fish: 62,703; Sharks: 31,301 –Very small population. Fish/sharks occupy: 1/800 th of total grid Initial population: Fish: 1,298; Sharks: 609

Generation Medium-sized population (1/16 of grid)

Very Small Populations Random placement of very small populations can favor one species over another Fish favored: sharks die out Sharks favored: sharks predominate, but fish survive in stable small numbers

Gen ,00012,00014,000 Ultimate welfare of sharks depends on initial random placement of fish and sharks Very Small Populations

Very small populations Fish can live in stable isolated communities as small as A community of less than 200 sharks tends not to be viable

Forest Fire Model (FFM) During each time step the system is updated according to the rules: Forest Fire Model is a stochastic 3-state cellular automaton defined on a d-dimensional lattice with L d sites. Each site is occupied by a tree, a burning tree, or is empty. 1.empty site  tree with the growth rate probability p 2.tree  burning tree with the lightning rate probability f, if no nearest neighbour is burning 3.tree  burning tree with the probability 1-g, if at least one nearest neighbour is burning, where g defines immunity. 4.burning tree  empty site

The application

Simulation forest density 45% fire is not visible The average cluster size is small in comparison to lattice size L.

Simulation forest density 60% first signs of fire Forest density reaches the critical value 59% - the percolation threshold for square lattice. The average cluster size goes to infinity for infinite lattice size.

Simulation Fire spreads quickly burning down all connected tree clusters. A variety of global structures emerges. The whole process repeats and after some time forest reaches the steady state in which the mean number of growing trees equals the mean number of burning trees.

Next time A bit more CAs, and L systems