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Cellular Automata the Plan What Are Cellular Automata? What Are The Plan • Automata Cellular Automata – Von Neumann to Wolfram The game of life or a new kind of • Demonstrations science? – Game of Life program – Developed by Jim Fix – Behaviors developed by high school students Richard Ladner – Sophisticated behaviors implemented by Sam Coskey 1 2 What are Cellular What are Cellular Automata? Automata? “Cellular automata have been invented many times under “When I made my first discoveries about cellular automata different names… In pure mathematics they can be recognized in the early 1980s I suspected that I had seen the beginning as a branch of topological dynamics, in electrical engineering of something important. But I had no idea just how important they are sometimes called iterative arrays, and high school it would all ultimately turn out to be. And indeed over the past kids may know them as a sort of home-computer game. twenty years I have made more discoveries than I ever thought They have been used and abused by interdisciplinary possible. And a new kind of science that I have spent so much scientists as well as interdisciplinary bumblers.” effort building has seemed an ever more central and critical direction for future intellectual development.” Toffoli and Margous Cellular Automata Machines Stephen Wolfram 1987 A New Kind of Science 3 2002 4 Automata? Automaton Example • Automata is the plural of automaton Coke machine • Simple computing device – Inputs: • Properties coins, bills, return button, choice buttons – Finite set of states – State: – Transitions from state to state money entered so far,… • Sense the environment. – Outputs: • Possibly change the environment. coke, sprite, dr. pepper, • Go to a new state, returned coins, … 5 6 1 Other Examples Cellular Automata • Automata are arranged geometrically • All automata are identical • All automata change state simultaneously cell 4 Neighbors 8 Neighbors 7 8 Communication One-Dimensional rule … … • Each cell has a left and right neighbor • All cells identical • Inputs are states of neighbors and self • Cell can be initialized to different states • Output is the state (indicated by color) 9 10 Two State Example Rule 90 Rule 254 128 64 32 16 8 4 2 Rule 90 = 64 16 8 2 11 12 2 Rule 90 Rule 30 13 14 Two-Dimensional Game of Life • Each cell is “live” or “dead” • Transition rules – N = number of live neighbors among the 8 – N 1 ç death (loneliness) – N = 2 ç no change – N = 3 ç birth – N í 4 ç death (overcrowding) • Each cell has 4 or 8 neighbors examples 15 16 Game of Life Code 494 • The Glider • The Glider gun and eater – Gosper 1970 • Alternative life games 17 18 3 Code 746 History • John von Neuman & Stanislaw Ulam (1950) – Self reproducing Machines • John Conway (1970) – The game of life – Popularized by Martin Gardner in Scientific American magazine • Stephen Wolfram (2002) – “A New Kind of Science” 19 20 Firing Squad Applications Problem • Biological systems • One-dimensional cellular automaton • Iterative arrays – parallel computer hardware • Synchronous behavior possible lieutenants • Artificial societies captain • Art and design • Computer graphics Finite number of steps • Image processing • Games All in the same state = “firing state” 21 22 Firing Squad Self-Reproducing Problem Solutions Cellular Automaton • Proposed by Myhill (1957) • Two-dimensional with 4 neighbors • Moore Solution (1962) • Initial configuration is exactly duplicated and – Called the “signal solution” spread throughout the plane – 13 states • Von Neumann Solution (1952) – 3n time – 29 states, 200,000 cell initial configuration • Mazoyer Solution (1988) • Langton Solution (1984) – Only 6 states – 8 states, 125 cell initial configuration – 2n time (minimal) • Byl Solution (1989) – 4 states impossible – 6 states, 16 cell initial configuration – 5 states unknown 23 24 4 Universality Life is Universal • There is a one-dimensional cellular • The Game of Life is universal (Gosper automaton that is a general purpose and Conway 1971) computer. – Any computation can be done by setting up program input storage the initial configuration and letting it run. 25 26 Rendell’s Universal Rule 110 is Universal Life Machine • One-dimensional Paul Rendell 1980s • Matthew Cook 1990s • Rule 110 27 28 Image Processing Follow the Scent Example Game • Gray scale to black and white • Food is the highest number • Numbers smaller farther from the food x is largest x Pick the 2x2 black and white block that Best approximates the input block 29 30 5 “A New Kind of Resources Science” • Wolfram’s thesis • Books - – Complex behaviors are often the result of simple – Martin Gardner - Wheels, Life, and Other computational rules. Mathematical Amusements – The proof: simple cellular automata and their – Toffoli and Margolus - Cellular Automata Machines variants produce such complex behavior. – Stephen Wolfram - A New Kind of Science • Corollary • Web Pages – Traditional mathematical approaches (continuous – http://nojava.cafaq.com/index.shtml mathematics) to modeling complex behavior is not – http://psoup.math.wisc.edu/ enough. – http://www.cs.washington.edu/homes/scoskey/ca/ 31 32 6.
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