Modelling with cellular automata

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1 Modelling with cellular automata Shan He School for Computational Science University of Birmingham Module : Computational Modelling with MATLAB

2 Outline Outline of Topics Concepts about cellular automata Elementary Cellular Automaton Langton s ant Game of life CA for Lotka-Volterra model Conclusion

3 Concepts about cellular automata What are cellular automata? cellular automaton: a discrete model consists of a regular grid of cells, each in one of a finite number of states, such as On and Off. The grid is usually in 2D, but can be in any finite number of dimensions. The cells evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells. Originated from von Neumann, populated in 70s by John Conway s Game of Life, further researched by Stephen Wolfram Can be seen as the simplest agent-based models

4 Concepts about cellular automata Why CA are important? The best tool to model and study complex system, because: They are simple; The mechanisms are completely known; They can do just about anything They might help us understand our universe: is the universe a cellular automaton? They have lots of applications, e.g., random number generator.

5 Concepts about cellular automata Some CA based biological models Pattern formation in biology, e.g., Conus textile Modelling cell interactions: brain tumor growth.

6 Elementary Cellular Automaton Elementary Cellular Automaton The simplest: One dimensional; Two possible states per cell: 0 and 1 ; A cell s neighbors defined to be the adjacent cells on either side of it. A cell and its two neighbors form a neighborhood of 3 cells: 2 3 = 8 possible patterns for a neighborhood. A rule consists of deciding, for each pattern, whether the cell will be a 1 or a 0 in the next generation: 2 8 = 256 possible rules.

7 Elementary Cellular Automaton Wolfram code Invented by Stephen Wolfram, the inventor of Mathematica and promulgator of cellular automata. Clever idea: Each possible current configuration is written on order, 111, 110,..., 001, 000, and the resulting state for each of these configurations is written in the same order and interpreted as the binary representation of an integer. Summarised as a transition rule table, for example: current pattern new state for center cell We have 2 8 = 256 possible rules, named from rule 1 to rule 255.

8 Elementary Cellular Automaton Question The previous transition table is for rule 100, because the binary number string of decimal number 110 is Using the transition rule table you derived above, show how the following initial configuration of the rule 110 cellular automaton evolves in the next three time steps: Initial configuration

9 Elementary Cellular Automaton Unique rules Many of the 256 rules are trivially equivalent to each other: simple transformation of the underlying geometry: Mirror: reflection through a vertical axis. Complement: exchange the roles of 0 and 1 in the definition. Rule 110: The mirror image is rule 124, the complement is rule 137, and the mirrored complement is rule 193. There are 88 unique rules.

10 Elementary Cellular Automaton Wolfram Classes Class 1: rapidly converge to a uniform state. Examples are rules 0, 32, 160 and 250. Class 2: rapidly converge to a repetitive or stable state. Examples are rules 4, 108, 218 and 232. Class 3: appear to remain in a random state. Examples are rules 22, 30, 126, 150, 182. Class 4: form areas of repetitive or stable states, but also form structures that interact with each other in complicated ways. Local changes to the initial pattern may spread infinitely. An example is rule 110. Rule 110 has been shown to be capable of universal computation.

11 Elementary Cellular Automaton Rule 30 and Chaos Rule 30: Discovered by Stephen Wolfram, his all-time favourite rule. Displays aperiodic, chaotic behaviour. The key to understanding how simple rules produce complex structures and behaviour in nature

12 Elementary Cellular Automaton Rule 110 and Universal Computation Also called universal Turing machine, a machine exhibits universality Universality: the property of being able to perform different tasks with the same underlying construction just by being programmed in a different way. - from Wolfram Mathworld Basically means: in principle, Rule 110 can simulated any calculation or computer program (with even a polynomial overhead!!)

13 Elementary Cellular Automaton Rule 110 and Universal Computation The proof of the universality is complicated: to proof it is capable to emulate cyclic tag system, which is known to be universal. A cyclic tag system: a binary string of finite but unbounded length evolves under the action of production rules applied in cyclic order. Requires an infinite number of localized patterns (binary string) to be embedded self-perpetuating localized patterns as background pattern (production rules) Self-perpetuating localized patterns (spaceships): Example localized patterns: , and 111.

14 Elementary Cellular Automaton Other interesting rules and an interesting article Rule 62: Appears to be class 4 in the beginning but evolves into a repetitive state (class 2). Rule 73: Class 2, a lot of repetitive states separated by walls. But some are not obvious. Rule 54: Class 4 but its capability of universal computation has not been proved. The Simplest Universal Turing Machine Is Proved

15 Langton s ant Langton s ant A very simple 2D CA: very simple rules, but complicated emergent behavior A grid of cells with two possible states per cell: 0 (white) and 1 (black); Two simple rules: At a white cell, turn 90 right, flip the color of the cell, move forward one unit At a black cell, turn 90 left, flip the color of the cell, move forward one unit Complex behaviour: Initial period of chaotic behaviour for about 10,000 steps, then builds a recurrent highway pattern of 104 steps that repeats indefinitely.

16 Langton s ant Extension of Langton s ant Instead of just black and white, more colors can be used. For each of the successive colors, a letter L or R is used to indicate whether a left or right turn should be taken. Example: Black Red Green Blue Black Black: L Red: R Green: L Blue: L Click here to see more example.

17 Game of life Game of life Also know as life. A CA invented by the John Conway in Originally devised on a board for the game of GO. A zero-player game, meaning that its evolution is determined by its initial state. Opened up a whole new field of mathematical research, the field of cellular automata. Also shown to be capable of universal computation.

18 Game of life Game of life Every cell interacts with its eight neighbours (cells are horizontally, vertically, or diagonally adjacent) following the three simple rules: Any live cell with two live neighbours lives on to the next generation. Any cell with exactly three live neighbours becomes or remains a live cell. Any other cell dies.

19 Game of life Game of life Let s try this on a very simple pattern:

20 Game of life Game of life Next step:

21 Game of life Interesting patterns in game of life Period 2 oscillator: Blinker, Toad and Beacon Period 3 oscillator: Pulsar Methuselah patterns: small seed pattern of initial live cells that take a large number of generations in order to stabilize. Spaceships: Glider Patterns grow indefinitely: Gosper glider gun Click here for common patterns in game of life

22 Game of life Other life-like CA rules Notation for rules: Wolfram Code; MCell notation: a string x/y. Digit d {0,..., 8} in the x means that a live cell with d live neighbors survives into the next generation, and the presence of d in the y string means that a dead cell with d live neighbors becomes alive in the next generation. For example: Game of Life can be denoted as 23/3 Golly notation: similar to above, but written in the form By/Sx, e.g., Game of Life can be denoted as B3/S23 Examples: Modern CA Entrance

23 Game of life Advantages of CA modelling approach They are simple and easy to be implemented. The are able to verify the relevance of physical mechanisms. They can include relationships and behaviors which are difficult to formulate as continuum equations. They reflect the intrinsic individuality of cells.

24 Game of life Disadvantages of CA modelling approach Difficult for quantitative simulations. The artificial constraints of grid (lattice discretisation) Difficult to interpret the simulation outcomes

25 CA for Lotka-Volterra model CA for Lotka-Volterra model Pick a site (only stochastic updates allowed), and a neighbour. If fox is adjacent to rabbit, rabbit gets eaten (becomes fox with probability r). Else fox dies with probability p. If rabbit is adjacent to bare ground, reproduces with probability q. If bare ground is adjacent to anything, the thing moves into bare ground.

26 Conclusion Reading and video Complex Systems TED: Computing: a theory of everything

27 Conclusion Take home messages Complex behaviour, e.g., choatic behaviour, emerges from local simple rules. By CA, we have demonstrated the four important characteristics of complex systems: Self-organisation; Non-Linearity; Chaotic behaviour; Emergent Properties. CA can be used to model complex systems, e.g., biological systems. CA are the simplest agent-based models.

28 Conclusion Assignment Write a MATLAB programme to convert patterns on Life Lexicon Home Page for our MATLAB Game of Life (Click here for Life Lexicon Home Page)