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Cellular Automata Various credits for these slides: Ajit Narayanan, Rod Hunt, several more.

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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

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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: 000000001001010010011011100100101101110110111111 00010111 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.

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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 …

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Types of neighbourhood Many more neighbourhood techniques exist - see http://cell-auto.com and follow the link to ‘neighbourhood survey’ http://cell-auto.com

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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

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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.

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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.

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Is it alive? http://www.bitstorm.org/gameoflife/ Compare it to the definitions…

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Glider

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Sequences

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More Sequence leading to Blinkers Clock Barber’s pole

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A Glider Gun

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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

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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

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Loop Reproduction

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Loop Death

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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.

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Example: 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

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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

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

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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.

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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

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Do a LOT of computation! We used a 20-CPU cluster in the Computer Science Department (Galaxy) ‘Gal’ is the smaller of two clusters run by the Department (larger one has 64 CPUs) 15x faster than a single PC Longest runs still took about 45 minutes GO PARALLEL !!!

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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)

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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)

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Breeding Rule: Before EMPTY

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Breeding Rule: After

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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

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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

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

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

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Spring 2005JR33 Sample Code (C++): Breeding

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

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Spring 2005BM35 Generation: 0

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Spring 2005BM36 Generation: 100

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Generation: 500

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Generation: 1,000

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Generation: 2,000

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Generation: 4,000

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Generation: 8,000

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Generation: 10,500

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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’

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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

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Generation 10020001000 40008000 Medium-sized population (1/16 of grid)

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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

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Gen. 100 40006000 8000 1500 10,00012,00014,000 Ultimate welfare of sharks depends on initial random placement of fish and sharks Very Small Populations

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Very small populations Fish can live in stable isolated communities as small as 20-30 A community of less than 200 sharks tends not to be viable

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Community image This is what a community of virtual plants looks like Contrasting tones show patches of resource depletion

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CSR type, frame 1 This is a single propagule of a virtual plant It is about to grow in a resource-rich above- and below-ground environment

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ditto f. 2

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ditto f. 3

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ditto f. 4

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ditto f. 5

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ditto f. 6

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ditto f. 7

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ditto f. 8

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ditto f. 9

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ditto f. 10

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ditto f. 11

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ditto f. 12

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ditto f. 13

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ditto f. 14

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ditto f. 15

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ditto f. 16

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ditto f. 17

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ditto f. 18

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ditto f. 19

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ditto f. 20 The plant has produced abundant growth above- and below-ground and zones of resource depletion have appeared

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See Rod Hunt at http://www.ex.ac.uk/~rh203/ for lots more about the plant life CA and its uses http://www.ex.ac.uk/~rh203/

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Chapter 5 Relations and Functions

Chapter 5 Relations and Functions

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