Mitchell Chapter 10. Living systems are open systems that exchange energy, materials & information

Transcription

1 Living systems compute Mitchell Chapter 10 Living systems are open systems that exchange energy, materials & information E.g. Erwin Shrodinger (1944) & Lynn Margulis (2000) books: What is Life? discuss how life creates order from disorder Computation is what a complex system does to adapt to its environment Life is computation? Cellular Automata: simplified demonstrations of how life might compute

2 Introduction to Cellular Automata (CA) Invented by John von Neumann (circa~1950). A cellular automata consists of: A regular arrangement (lattice) of cells. Cells can be in one of a finite number of states at each time step. All cells have the same synchronous update rule. Cells have a local interaction neighborhood. Example: The game of life. Many cellular automata applications: Hydrodynamics, fluid dynamics Forest fire simulation Models of ecosystems and epidemics Ferro-magnetic modeling (ISING models)

3 The Game of Life The number of states, k=2. The alphabet, Σ = {0,1} Call 0 dead and 1 alive. The Moore neighborhood: NW W N NE E SW S SE Transition rules: Loneliness: If a live cell has less than 2 live neighbors, then it dies. Overcrowding: If a live cell has more than 3 live neighbors, then it dies. Birth: If an empty (dead) cell has 3 live neighbors, then it becomes alive. Otherwise: If a cell has 2 or 3 live neighbors, then it remains unchanged.

4 Moore vs. Von Neumann Neighborhoods NW N NE N W E W E SW S SE S Moore Von Neumann

5 Example Transition Center square changes to one (birth). Just 3 for birth, 2 or 3 for survival.

6 Possible Life Histories (Dynamical behaviors that enable universal computation) Static structures: Beehive, Loaf, Pond, etc. (Figure 15.11) Periodic structures: Blinkers (Figure 15.12) Moving structures: Gliders (next slide) Glider guns Logical gates (and, or, not) Self-reproducing structures.

7 Static Objects

8 Periodic Objects

9 Gliders (Moving Objects)

10 GOLLY

11 2D CAs Cyclic Cellular Automaton Conus textile

12 Rule Table f : Neighborhood: Output bit: 1 Dimensional CAs Lattice: Periodic boundary conditions Neighborhood h t = t =

13 One-Dimensional CA Illustration r t i - r... i - 1 i i+1... i+r h i f t+1 i

14 0 0 0 à à à à 1 Rule 110 Read the bit string BOTTOM to TOP 111 is leftmost (most significant) bit 000 is the least significant bit à à à à = = Rule 110

15 One-Dimensional Cellular Automaton (CA) Consists of a linear array of identical cells (called a lattice), each of which can be in a finite number of k states. The (local) state of cell i at time t is denoted: s i t Σ = { 0,1,..., k 1} The (global) state s t at time t is the configuration of the entire array, s t = ( s 0 t, s 1 t,..., s N 1 t ) Σ where N is the (possibly infinite) size of the array. N

16 One-Dimensional CA cont. At each time step, all cells in the array update their state simultaneously, according to a local update rule φ : η s. This update rule takes as input the local neighborhood configuration η of a cell. A local neighborhood configuration η consists of s i and its 2r nearest neighbors (r cells on either side): η = ( s i i r,..., s,..., s i+ r ) r is called the radius of the CA. The local update rule φ is the same for every cell in the array Represented as a lookup table, with all neighborhood configurations The state of each cell at time t+1 is determined by applying to each cell at time t: s i t+ 1 = φ ( η i t ) φ

17 0 0 0 à à à à à à 1 Rule 110 The number of rules(rows) in the rule table is the number of neighborhood configurations ( η = k 2 r+1 Each rule can map to one of k states, so The number of possible rules is k 2r+1 k ) à à = = Rule 110 Read the bit string BOTTOM to TOP

18 Example One-Dimensional CA Rule 110 The number of states, k=2. The alphabet Σ = {0,1} Σ The size of the array, N=11. The configuration space Σ N = {(0,0,0,0,0,0,0,0,0,0,0),(0,0,0,0,0,0,0,0,0,0,11),...} The radius r = 1. The rule table : i i s k 2r+1 t+ 1 = φ ( η t) neighborhood configurations in φ In Wolfram notation this is rule 110 (base 10) because the output states are: (base 2). Rule 110 supports universal computation. η = k

19 Rule 110 Space-Time Plot

20 Rule 30 current pattern new state for center cell

21 Comments on Rule 30 Generates apparent randomness, despite being finite Wolfram uses the central column as a pseudo-random number generator in Mathematica Passes many tests for randomness, but many inputs produce regular patterns: All zeroes repeated infinitely (try separating by 6 1s)

22 Wolfram s CA Classification Class I: Eventually every cell in the array settles into one state, never to change again. Analogous to computer programs that halt after a few steps dynamical systems that have fixed-point attractors Class II: Eventually the array settles into a periodic cycle of states computer programs that execute infinite loops dynamical systems that fall into limit cycles. Class III: The array forms aperiodic random-like patterns. computer programs that are pseudo-random number generators (pass most tests for randomness, highly sensitive to seed, or initial condition). Analogous to chaotic dynamical systems. Determinisitic but almost never repeat themselves, sensitive to initial conditions, embedded unstable limit cycles.

23 Wolfram s Classification cont. Class IV: The array forms complex patterns with localized structure that move through space and time: Difficult to describe. Not regular, not periodic, not random. Speculate that it is interesting computation. Hypothesis: The most interesting and complex behavior occurs in Class IV CA--- the edge of chaos. Example: Rule 110

24 Wolfram Class I

25 Wolfram Class II Wolfram s Class II

26 Wolfram Class III

27 Wolfram Class IV

28 4 classes Cellular Automata as models of complexity Wolfram Nature 1984 I: Fixed point II: Limit cycle III: Random or chaotic (unpredictable) IV: Complex Class IV supports universal computation (?) Long-term behavior undecidable and intractable Simulation is necessary ping-future-stephen-wolfram-ray.html