Introducing Cellular Automata

1. Artificial Life IntroducingCellular Automata

2. Introduction Why living organisms with inanimate components differ from non-livingorganisms with the same components? Lehninger

3. An Introduction • Studying Artificial Life • Self-reproduction, a special ability on living molecules • A definition of Automaton: • Informal: • Object that performs movements from one position to another • Mathematical: • Abstract object characterized by its state at any instant and changing state at different point of time. • Cellular automaton is a large collection of automaton.

4. Cellular Automata 1 3 2 The Foundations The Game of Life What is it about? Studying Artificial Life with Cellular Automata

5. 1 The Foundations Turing Work

6. Alan Turing • Alan Turing (1912-1954) • British mathematician • Father of modern Computer Science • Introduced the concepts of Algorithm and Computation. • Inspired John von Neumann • Defined AL in terms of Automata Theory • Formulated Church-Turing thesis • Studied abstract machines called Turing machines even before computers existed

7. Church-Turing Thesis • Church-Turing Thesis • Alonzo Church and Alan Turing • Alan Turing Definition: • Computability by Turing machine. • The theory states that whenever there is an effective method for obtaining the values of a mathematical function, the function can be computed by a Turing machine.

8. Automata Theory • Automata Theory 1936 • A theoretical branch of Computer Science • Study of abstract computing devices, or “machines” • Automaton= an abstract model of computing devices that performs computations on an input by moving through a series of states. • Four families of automatons: • Finite-state machine • Pushdown automata • Linear bounded automata • Turing Machine

9. The Turing Machine • Turing Machine • Designed in 1931 • Mathematical tool to prove limits of computability • Performs any mathematical computation represented as an algorithm. • Universal Turing Machines

10. 2 Cellular Automata What is it about? Definition, Rules, and Behaviors

11. Cellular Automata (CA) • Discrete model in computability theory. • An infinite n-dimensional space divided up into an array of unit squares, called cells. • Each cell contains an identical copy of the same finite automaton • Each cell take on k-states. • A value is assigned depending on the value of the sites around it. • Three components: • The array dimension • The neighborhood structure • The rules Definition

12. John Von Neumann Theory of Self-Reproducing Automata

13. John von Neumann • John von Neumann • Hungarian mathematician • Pioneer of applications of operator theory • Inventor of cellular automata, universal constructor and the digital computer. • Analysis of self-structure precedes the discovery of structure of DNA

14. von Neumann Neighborhood • 2D cellular automaton • Each cell could take on 29 states. • Dependable on update rule f Definition The state of each cell C( t + 1) at time-step t + 1 is determined by this cell’s neighboring cells at the previous time t C(t + 1) = f[ N(t), E(t), S(t), W(t), C(t)]

15. The Transition Rules • Define the operation of the Cellular Automata • Radius • Neighbors are crucial • The radius r of the automaton = number of neighbors Transition Rules The function Ø is called the CA. In 1D, each site has two nearest neighbors. • 1D (one-dimensional) CA with radius r = 1 • 1D CA with radius r = 2 ai(t + 1) = Ø( ai-1(t), ai(t), ai+1(t) )

16. The States • States can become quite large • There are kn different possible states • k = number of states • n = neighbors and the center cell itself • There are kkn total number of different transitions functions Transition Rules • An Example: Elementary Cellular Automata • 1D • Radius 1 = two neighbors and the center = 3 cells • States 0 or 1 = 2 states • kn= 23 = 8 • kkn= 28 = 256

17. Working out rule 30 in 1D • 111 110 101 100 011 010 001 000 128 64 32 16 8 4 2 1 Rule 30 00 0 1 1 1 1 0 00011110 = 0*128 + 0*64 + 0*32 + 1*16 + 1*8 + 1*4 + 1*2 + 0*1 = 30

18. Rule 30 Rule 30 after 22 Generations

19. Rule 30 after MANY Generations Rule 30 Cone snail species conus cipher

20. Stephen Wolfram Specific Rules and Classification

21. Stephen Wolfram • Stephen Wolfram • British Scientist • Known for his work in theoretical physics • Published his first scientific paper at the age of 15 • PhD at the age of 20 • Investigated transition rules operating in CA • Classified CA Behaviors

22. Type of transition rules. • Symmetric • Legalistic (32 rules) • Must allow an initial state of all 0’s to remain at all 0’s • Partial isotropy: rules must be symmetric (For instance 1,0,0 and 0,0,1 must map to the same state) • Totalistic (8 rules) • The next state of a cell depends on the sum of the values on the neighboring cells. • Peripheral • Depend only on the state of ai-1, and ai+1 (not on ai) • Asymmetric Types of Rules

23. Symmetric Rules

24. Asymmetric Rule 110

25. Cellular Automaton Behavior • Class 1 • Displays limit point behavior, evolves to a homogeneous state Types of Rules

26. Cellular Automaton Behavior • Class 2 • Cellular automata evolve to simple separated periodic structures, or limit cycles. Types of Rules

27. Cellular Automaton Behavior • Class 3 • Leads to a chaotic state Types of Rules

28. Cellular Automaton Behavior • Class 4 • Yields complex patterns, capable of universal computation Types of Rules

29. 3 The Game of Life John Horton Conway

30. The Game of Life: Rules • Any live cell with fewer than two live neighbors dies, as if caused by under-population. • Any live cell with two or three live neighbors lives on to the next generation. • Any live cell with more than three live neighbors dies, as if by overcrowding. • Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction. An infinite universe is divided into cells. Cells have two states, live and dead. Rules based on the number of neighbors cells

31. Important Patterns

32. Simulator • Let’s check out the • Javascript implementation of Conway’s Game of Life • Javascript Conway's GoL

33. Cellular Automata Questions?