Cellular Automata. History. 1-Dimensional CA. 1-Dimensional CA. Ozalp Babaoglu

Transcription

1 History Cellular Automata Ozalp Babaoglu Developed by John von Neumann as a formal tool to study mechanical self replication Studied extensively by Stephen Wolfram ALMA MATER STUDIORUM UNIVERSITA DI BOLOGNA 2 1-Dimensional CA 1-Dimensional CA t An (infinite) array of cells Each cell has a value from a k-ary state (assume binary) Each cell has has a position in the array and has r left and r right neighbors (assume r =1)

2 State Transitions (Look-up Table) Wolfram Canonical Enumeration Xt Xt+1 With a binary state and radius r =1, there are 2 23 =256 possible CAs Read off the final state column of the look-up table as a binary number Each possible CA identified through an integer Wolfram Canonical Enumeration Wolfram Canonical Enumeration 2 4 #8 2 4 #8 #16 #32 #64 Rule = 30 Rule =

3 Wolfram s Classification Wolfram s Classification: Class 1 Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state (fixed point) Class 2: Nearly all initial patterns evolve quickly into stable or oscillating structures (periodic) Class 3: Nearly all initial patterns evolve in a pseudo-random or chaotic manner (chaotic) Class 4: Nearly all initial patterns evolve into structures that interact in complex and interesting ways. This class is capable of universal computation Rule 40 Rule 172 Rule 234 Source: Wolfram s Classification: Class 2 Wolfram s Classification: Class 3 Rule 30 Rule 101 Rule

4 Wolfram s Classification: Class 4 NetLogo Rule 110 CA 1D Elementary Wolfram s Classification Langdon s λ Metric Seek a compact characterization of the CA behavior class Count the number of ones in the look-up table final state column Fixed Periodic Complex Chaotic 15 16

5 Langdon s λ Metric Conway s Game of Life λ ALL#Rules Class#III Class#IV Dimensional Cellular Automata Developed by British mathematician John Conway Similar to Schelling s model Each cell has eight neighbors Each cell can be alive or dead Instead of moving or staying, cells come alive, die or survive Conway s Game of Life Conway s Game of Life X Dead (off) Alive (on) Each cell has eight neighbors Each cell can be alive or dead Rules: a live cell with fewer than 2 live neighbors dies (loneliness) a live cell with 2 or 3 live neighbors survives (stasis) a live cell with more than 3 live neighbors dies (over crowding) a dead cell with exactly 3 live neighbors come alive (reproduction) 19 20

6 Fixed point

7

8 Glider Periodic Fish NetLogo Game of Life t=0 t=1 t=2 t=3 t=

9 Universal Computation Building blocks: Both Conway s game of life and CA rule 110 are capable of universal computation Prove by showing that the game of life is equivalent to a Turing Machine Babaoglu Complex Systems 33 Logical Operators from Game of Life NOT Babaoglu AND Complex Systems Logical Operators from Game of Life OR 35 Babaoglu Complex Systems 34