Cellular Automata PDF

Cellular Automata -----------------------------------  Cellular Automata A CA is a spatial lattice of N cells, each of which is at one of the 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...

Cellular Automata -----------------------------------  Cellular Automata A CA is a spatial lattice of N cells, each of which is at one of the 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 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. To visualise the CA, colours are used to represent the states. In this case; white, green and red seem the right Choices. A fairly dense forest with a couple of trees on fire -- maybe from lightning strikes Types of neighbourhood Many more neighbourhood techniques exist - see http://cell-auto.com and follow the link to ‘neighbourhood survey’  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”. Usually we just use a cell’s 8 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.  CA Rules Now, the main thing: how do we update the states at the next time step? We use sensible rules. E.g.  If a tree is not on fire, and has n neighbours on fire, it catches fire in next step with probabilty n/8.  If a tree has been on fire for 3 steps, it dies Step 0 Step 1 Step 2 Step 3 Step 4 … and so on  CA Rules:  A small number of sensible rules, for any given suitable application, usually leads to convincing behaviour.  Every CA rule says: A cell in state X changes to a cell of state Y if certain neighbourhood conditions are satisfied  CAs are increasingly used to simulate a wide number of complex systems, to see “what would happen if…”, and generally investigate the effects of various strategies  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 live neighbors becomes a live cell in the next round.  Is it alive? http://www.bitstorm.org/gameoflife/ Compare it to the definitions… Glider  Langton’s Loops  CA are a main part of the research area “Artificial Life”.  A common definition of “life” involves that the living organism(s) must be capable of self-reproduction. Langton’s “Loops” achieve that.  Characteristics  8 states, 2D Cellular automata  Needed CA grid of 100 cells  Self Reproduction into identical copy  A simple set of rules produces self-reproducing “organism” – a deep connection between Life and Computation.  Langton’s Loop 0 – Background cell state 3, 5, 6 – Phases of reproduction 1 – Core cell state 4 – Turning arm left by 90 degrees 2 – Sheath cell state state 7 – Arm extending forward cell state Langton’s Loops  Boundary conditions Rules for a cell’s state transitions are usually defined in terms of the cell’s neighbourhood. E.g. this is the Moore neighbourhood: But what about cells on the edge? The common approach in 2D is to treat the CA surface as a Toroid This just means wraparound in the way indicated by the blue and green neighbourhoods illustrated There remains debate and interest about the `essentials of life’ issue with CAs, but their main BIC value is as modelling techniques. Some more examples. 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  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 =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   Results  Next several screens show behavior over a span of 10,000+ generations  Generation: 0 Spring 2005 BM 40  Generation: 100 Spring 2005 BM 41 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’  What can be discovered by simulating 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  Forest Fire Model (FFM) Forest Fire Model is a stochastic 3-state cellular automaton defined on a d-dimensional lattice with Ld sites. Each site is occupied by a tree, a burning tree, or is empty. During each time step the system is updated according to the rules: 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 Eventually After some time forest reaches the steady state in which the mean number of growing trees equals the mean number of burning trees.  Modelling brain tumour growth Kansal et al, 2000, Journal of Theoretical Biology Incidence of primary malignant brain tumours is 8/100,000 p.a. 3D CA, modelling brain tumour growth Shows that Macroscopic tumour behaviour can be predicted via microscopic parameters Uses only 4 parameters Makes predictions that match the biological reality MRI scan showing a tumour; the white area Represents blood leakage around the tumour Kansal et al use the Delaunay Tesselation as their lattice – on the right we see blackened cells representing the tumour, in a simplified 2D version  States and Rules Not easy to glean from the paper, but: cells are either healthy (empty lattice site) or tumour. Tumour cells are either proliferative (they divide into additional tumour cells) or not. When a proliferative tumour cell wants to divide, it fills a healthy space with a new tumour cell if it can find one within delta_p of its position. If it can’t find one, it becomes non- proliferative. 1.5M lattice sites initial tumour is 1000 proliferative cells at centre of lattice Result seems realistic Very good fit to real data; The lines are the CA model predictions of tumour radius and volume against time The plotted points are measurements from real cases of untreated tumours  Read about various applications for yourself. Google for the: Influenza CA paper Tumour CA paper A Traffic Simulation CA paper Historic urban growth in the San Francisco bay area CA Not examinable reading, but recommended