CELLULAR AUTOMATON Presented by Rajini Singh.

CELLULAR AUTOMATON: Discrete Model Infinite Regular Grid of cells. Finite number of States. State of a cell is a function of the States of its neighborhood. Every cell has the same rule for updating. New generation is created every time rules are applied to the whole grid.

CELLULAR AUTOMATON Simulated on a Finite Grid. In Two Dimensions, the universe would be a rectangle. The edge cells are handled with a toroidal arrangement.

EXAMPLE Infinite sheet of graph paper. Every cell (square) has 2 states. Neighborhood are the 8 squares. 29=512 patterns.

CELLULAR AUTOMATON Simplest non trivial CA is one-dimensional, with two States per cell. Every cell’s neighborhood are the cells on adjacent sides of it. A cell and its 2 neighbors form a neighborhood of 3 cells, so there are 23 = 8 possible patterns for a neighborhood and 28 = 256 possible rules. These 256 CA’s are referred to using a standard naming convention invented by Wolfram.

CELLULAR AUTOMATON The name of the CA is the decimal number, which, in binary, gives the rule table, with the eight possible neighborhoods listed in reverse counting order. Examples are: Rule 30 CA (binary - 11110) Rule 110 CA (binary – 1101110)

111 110 101 100 011 010 001 000 1 EXAMPLES OF CELLULAR AUTOMATON RULE 30 CELLULAR AUTOMATION CURRENT PATTERN 111 110 101 100 011 010 001 000 NEW STATE FOR CENTER CELL 1

RULE 30 CELLULAR AUTOMATON

RULE 110 CELLULAR AUTOMATON CURRENT PATTERN 111 110 101 100 011 010 001 000 NEW STATE FOR CENTER CELL 1

RULE 110 CELLULAR AUTOMATION

CELLULAR AUTOMATION Table completely defines a CA rule. For example, Rule 30 table says that if 3 adjacent cells in the CA currently have the pattern 100, then the middle cell will become 1 on the next time step Rule 110 table says the opposite of it for that particular case.

CATEGORIES OF CELLULAR AUTOMATON CELLULAR AUTOMATONS CATEGORIES OF CELLULAR AUTOMATON Reversible Totalistic REVERSIBLE

REVERSIBLE A CA is said to be Reversible if for every configuration of the CA there is exactly one past configuration (preimage) For one dimensional CA, preimages can be found, and any 1D rule can be proved either reversible or irreversible. For CA of two or more dimensions, reversibility is undecidable for arbitrary rules.

TOTALISTIC The State of each cell in a Totalistic CA is represented by a number, which is a value, and this value of the cell at time ‘t’ depends on the sum of the values of the cells in its neighborhood (including itself) at time ‘t-1’. If the state of the cell at time ‘t’ does depend on its own state at time ‘t-1’ then the CA is called outer totalistic. An example of the above is Conway’s Game of Life with cell values 0 and 1.

CONWAYS GAME OF LIFE

CONWAYS GAME OF LIFE Devised by a British Mathematician- John Horton Conway. The evolution of the game is determined by its initial state. Its universe is a 2-D square grid. Every cell has a state - live or dead, and interacts with its 8 neighbors.

3. In all other cases, a cell dies or remains dead. At each step in time, A dead cell with exactly 3 live neighbors comes to life. 2. A live cell with two or three live neighbors stay alive. 3. In all other cases, a cell dies or remains dead.

Initial pattern constitutes first Generation of the system. The above rules are applied to every cell in the first generation, and the discrete moment at which this happens is called a ‘tick’. Births and deaths happen simultaneously in this phase. The rules continue to be applied repeatedly to create further generations.

The kinds of Objects that emerge in Life: Still Life Objects. Block – 2 x 2 square Beehive Boat Ship Loaf Oscillators Objects that change but eventually repeat themselves. Gliders Moving patterns consisting of 5 cells. Guns Generates an endless stream of new patterns.

Guns and Gliders – Turing Complete.

CONCLUSION Behavior of cells or animals can be better understood using simple rules. Computer viruses are also examples of Cellular Automaton. Finding the cure could be hidden in the patterns of this game. Human diseases could be cured if we better understand why cells live and die. Cryptography.

REFERENCES http://www.stephenwolfram.com/publications/articles/ca/ http://www.santafe.edu/~shlizi/notebooks/cellular-automata.html http://www.stephenwolfram.com/publications/articles/ca/85-cryptography/1/text.html

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