Cellular Automata COMP308 Unconventional models and paradigms
Failure and Success of Complex Models New scientific paradigms bring about new insights: Research shows that complexity is a prerequisite for success of living systems (Jacobs, Langton, Wolfram, Kaufmann, Fotheringhham, Longley & Batty, Frankhauser & Sadler) Lessons learned: ◦ Complexity complication
3 Mathematical object defined as: n-dimensional homogeneous and infinite cellular space, consisting of cells of equal size (2-D CA = torus, not a plane); Cells in one of a discrete number of states; Cells change state as the result of a transition rule; Transition rule is defined in terms of the states of cells that are part of a neighbourhood; Time progresses in discrete steps. All cells change state simultaneously. What is a Cellular Automata ? Concept introduced by Von Neumann, Ulam and Burk in late 1940-ies and 1950-ies; ( Self-reproducible mechanical automata) Conway’s ‘Game of Life’ (Gardner, 1970)
What is a Cellular Automaton? A one-dimensional cellular automaton (CA) consists of two things: a row of "cells" and a set of "rules". Each of the cells can be in one of several "states". The number of possible states depends on the automaton. Think of the states as colors. In a two-state automaton, each of the cells can be either black or white. Over time, the cells can change from state to state. The cellular automaton's rules determine how the states change. It works like this: When the time comes for the cells to change state, each cell looks around and gathers information on its neighbors' states. (Exactly which cells are considered "neighbors" is also something that depends on the particular CA.) Based on its own state, its neighbors' states, and the rules of the CA, the cell decides what its new state should be. All the cells change state at the same time.
3 Black = White 2 Black = Black 1 Black = Black 3 White = White Now make your own CA
2-Dimensional Automata 2-dimensional cellular automaton consists of an infinite (or finite) grid of cells, each in one of a finite number of states. Time is discrete and the state of a cell at time t is a function of the states of its neighbors at time t-1.
Conway’s Game of Life The universe of the Game of Life is an infinite two- dimensional grid of cells, each of which is either alive or dead. Cells interact with their eight neighbors.
8 Example of a Cellular Automata: Conway’s Life (Gardner, 1970)
Conway’s Game of Life At each step in time, the following effects occur: 1. Any live cell with fewer than two neighbors dies, as if by loneliness. 2. Any live cell with more than three neighbors dies, as if by overcrowding. 3. Any live cell with two or three neighbors lives, unchanged, to the next generation. 4. Any dead cell with exactly three neighbors comes to life. The initial pattern constitutes the first generation of the system. The second generation is created by applying the above rules simultaneously to every cell in the first generation -- births and deaths happen simultaneously. The rules continue to be applied repeatedly to create further generations.
Applications of Cellular Automata Simulation of Biological Processes Simulation of Cancer cells growth Predator – Prey Models Art Simulation of Forest Fires Simulations of Social Movement …many more.. It’s a very active area of research.
Cellular Automata: Life with Simple Rules Sharks and Fish: Predator/Prey Relationships Based on the work of Bill Madden, Nancy Ricca and Jonathan Rizzo (Montclair State University) *Adapted from: Wilkinson,B and M. Allen (1999): Parallel Programming 2 nd Edition, NJ, Pearson Prentice Hall, p189 Cellular automata can be used to model complex systems using simple rules. Key features* divide problem space into cells each cell can be in one of several finite states cells are affected by neighbors according to rules all cells are affected simultaneously in a generation rules are reapplied over many generations
12 Main idea Model predator/prey relationship by CA Define set of rules 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
13 Here’s the number 2 million Fish: red; sharks: yellow; empty: black
14 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.
15 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 Initially cells contain fish, sharks or are empty Empty cells = 0 (black pixel) Fish = 1 (red pixel) Sharks = –1 (yellow pixel)
16 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)
19 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
20 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
21 Shark Random Death: Before I Sure Hope that the random number chosen is >.031
22 Shark Random Death: After YES IT IS!!! I LIVE
23 Programming logic Use 2-dimensional array to represent grid At any one (x, y) position, value is: ◦ Positive integer (fish present) ◦ Negative integer (shark present) ◦ Zero (empty cell) ◦ Absolute value of cell is age
24 Parallelism A single CPU has to do it all: ◦ Applies rules to first cell in array ◦ Repeats rules for each successive cell in array ◦ After 2 millionth cell is processed, array is updated ◦ One generation has passed ◦ Repeat this process for many generations ◦ Every 100 generations or so, convert array to red, yellow and black pixels and send results to screen
25 Parallelism How to split the work among 20 CPUs ◦ 1 CPU acts as Master (has copy of whole array) ◦ 18 CPUs act as Slaves (handle parts of the array) ◦ 1 CPU takes care of screen updates Problem: communication issue concerning cells along array boundaries among slaves
44 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
45 Generation 10020001000 40008000 Medium-sized population (1/16 of grid) 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
46 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
Elementary Cellular Automata As before think of every cell as having a left and right neighbor and so every cell and its two neighbors will be one of the following types Replace a black cell with 1 and a white cell with 0 111 110 101 100 011 010 001 000 Every yellow cell above can be filled out with a 0 or a 1 giving a total of 2 8 =256 possible update rules.
This allows any string of eight 0s and 1’s to represent a distinct update rule Example: Consider the string 0 1 1 0 1 0 1 0 it can be taken to represent the update rule 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 01101010 Or equivalently Now think of 01101010 as the binary expansion of the number 0 2 7 + 1 2 6 + 1 2 5 + 0 2 4 +1 2 3 + 0 2 2 + 1 2 1 + 0 2 0 = 64+32+8+2=106. So the update rule is rule # 106
Let’s visualize some of these rules using a Mathematica notebookMathematica notebook Rule #45=32+8+4+1 = 0 2 7 + 0 2 6 + 1 2 5 + 0 2 4 +1 2 3 + 1 2 2 + 0 2 1 + 1 2 0 =0 0 1 0 1 1 0 1 Rule #30=16+8+4+2 = 0 2 7 + 0 2 6 + 0 2 5 + 1 2 4 +1 2 3 + 1 2 2 + 1 2 1 + 0 2 0 =0 0 0 1 1 1 1 0 This naming convention of the 256 distinct update rules is due to Stephen Wolfram. He is one of the pioneers of Cellular Automata and author of the book a New Kind of Science, which argues that discoveries about cellular automata are not isolated facts but have significance for all disciplines of science.
Automata generated using Rule 30 appear in nature, on some shells. current pattern 111110101100011010001000 new state for center cell 00011110
Rule 184 Rule 184 is a one-dimensional binary cellular automaton rule, notable for solving the majority problem as well as for its ability to simultaneously describe several, seemingly quite different, particle systems: Rule 184 can be used as a simple model for traffic flow in a single lane of a highway, and forms the basis for many cellular automaton models of traffic flow with greater sophistication. In this model, particles (representing vehicles) move in a single direction, stopping and starting depending on the cars in front of them. The number of particles remains unchanged throughout the simulation. Because of this application, Rule 184 is sometimes called the "traffic rule.” Rule 184 also models a form of deposition of particles onto an irregular surface, in which each local minimum of the surface is filled with a particle in each step. At each step of the simulation, the number of particles increases. Once placed, a particle never moves.
Rule 184 In each step of its evolution, the Rule 184 automaton applies the following rule to determine the new state of each cell, in a one-dimensional array of cells: current pattern 111110101100011010001000 new state10111000 The rule set for Rule 184 may also be described intuitively, in several ways: At each step, whenever there exists in the current state a 1 immediately followed by a 0, these two symbols swap places. At each step, if a cell with value 1 has a cell with value 0 immediately to its right, the 1 moves rightwards leaving a 0 behind. A 1 with another 1 to its right remains in place, while a 0 that does not have a 1 to its left stays a 0 (traffic flow modeling) If a cell has state 0, its new state is taken from the cell to its left. Otherwise, its new state is taken from the cell to its right.
Rule 184, run for 128 steps from random configurations with each of three different starting densities: top 25%, middle 50%, bottom 75%. The view shown is a 300-pixel crop from a wider simulation.
Rule 184 dynamics and majority problem Two important properties of its dynamics may immediately be seen: First, in Rule 184, for any finite set of cells with periodic boundary conditions the number of 1s and the number of 0s in a pattern remains invariant throughout the pattern's evolution. ◦ Similarly, if the density of 1s is well-defined for an infinite array of cells, it remains invariant as the automaton carries out its steps. And second, although Rule 184 is not symmetric under left-right reversal, it does have a different symmetry: reversing left and right and at the same time swapping the roles of the 0 and 1 symbols produces a cellular automaton with the same update rule. 111110101100011010001000 10111000
Patterns in Rule 184 Patterns in Rule 184 typically quickly stabilize, either to a pattern in which the cell states move in lockstep one position leftwards at each step, or to a pattern that moves one position rightwards at each step. Specifically, if the initial density of cells with state 1 is less than 50%, the pattern stabilizes into clusters of cells in state 1, spaced two units apart, with the clusters separated by blocks of cells in state 0. Patterns of this type move rightwards. If, on the other hand, the initial density is greater than 50%, the pattern stabilizes into clusters of cells in state 0, spaced two units apart, with the clusters separated by blocks of cells in state 1, and patterns of this type move leftwards. If the density is exactly 50%, the initial pattern stabilizes (more slowly) to a pattern that can equivalently be viewed as moving either leftwards or rightwards at each step: an alternating sequence of 0s and 1s.
Majority Problem One can view Rule 184 as solving the majority problem, of constructing a cellular automaton that can determine whether an initial configuration has a majority of its cells active: if Rule 184 is run on a finite set of cells with periodic boundary conditions, and the number of active cells is less than half of all cells, then ◦ each cell will eventually see two consecutive zero states infinitely often, and two consecutive one states only finitely often, while if the number of active cells forms a majority of the cells then ◦ each cell will eventually see two consecutive ones infinitely often and two consecutive zeros only finitely often. The majority problem cannot be solved perfectly if it is required that all cells eventually stabilize to the majority state but the Rule 184 solution avoids this impossibility result by relaxing the criterion by which the automaton recognizes a majority.
Traffic flow Rule 184 interpreted as a simulation of traffic flow. Each 1 cell corresponds to a vehicle, and each vehicle moves forwards only if it has open space in front of it. Although very primitive, the Rule 184 model of traffic flow already predicts some of the familiar emergent features of real traffic: clusters of freely moving cars separated by stretches of open road when traffic is light, and waves of stop-and-go traffic when it is heavy.
Rule 184 as a model of surface deposition. In a layer of particles forming a diagonally-oriented square lattice, new particles stick in each time step to the local minima of the surface. We model a segment with slope +1 by an automaton cell with state 0, and a segment with slope -1 by an automaton cell with state 1. The local minima of the surface are the points where a segment of slope -1 lies to the left of a segment of slope +1; that is, in the automaton, a position where a cell with state 1 lies to the left of a cell with state 0. Adding a particle to that position corresponds to changing the states of these two adjacent cells from 1,0 to 0,1, which is exactly the behavior of Rule 184. 111110101100011010001000 18410111000 Surface deposition
Computer processors CA processors are a physical, not software only, implementation of CA concepts, which can process information computationally. Processing elements are arranged in a regular grid of identical cells. The grid is usually a square tiling, or tessellation, of two or three dimensions; other tilings are possible, but not yet used. Cell states are determined only by interactions with the small number of adjoining cells. Cells interact, communicate, directly only with adjoining, adjacent, neighbor cells. No means exists to communicate directly with cells farther away. Cell interaction can be via electric charge, magnetism, vibration (phonons at quantum scales), or any other physically useful means. This can be done in several ways so no wires are needed between any elements.