Cellular Automataautomata

CellularCellular AutomataAutomata Moreno Marzolla Dip. di Informatica—Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ Complex Systems 2 Cellular Automata ● Originally developed in the 1940s by Stanislaw Ulam and John von Neumann ● Interest in CAs expanded in 1970 after Conway's Game of Life Stanislaw Ulam John von Neumann (1909—1984) (1903—1957) Complex Systems 3 Cellular Automata (CA) ● A regular grid of cells, each in one of a finite number k of states (e.g., k = 2 for binary states) ● At time t = 0 an initial state for each cell is selected ● A new generation is created (advancing t by 1), according to some fixed rule that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood – The rule for updating cell states is the same for each cell and does not change over time, though exceptions are known (e.g., stochastic CAs, asynchronous CAs...) Complex Systems 4 Cellular Automata ● We will generally consider CAs over a linear square grid lattice (i.e., an array or matrix) ● Other topologies can be considered – Irregular – Hexagonal – … ● Example: Game of Life on an hexagonal lattice Complex Systems 5 Elementary Cellular Automata Complex Systems 6 1-Dimensional CA ● “Infinite” linear array of cells ● Each cell can be in any of k states ● The neighborhood of each cell in general consists of r cells on the left and on the right – Usually r = 1, immediate neighbors – More complex neighborhoods can be considered Cell ... ... Neighbors (r = 1) Complex Systems 7 Finite vs Infinite Grid ● Although the CA grid is logically infinite, it is usually represented using a finite array (or matrix) ● Cells on the edges can be handled in several ways – Allow the values of the cells on the border to remain constant; or – Define neighborhoods differently for these cells (e.g., say that they have fewer neighbors), or – Use a toroidal arrangement 1-D array → Ring 2-D array → Torus Complex Systems 8 Elementary CA (k = 2, r = 1) ● The binary state c (i) of cell i at time t + 1 is t+1 determined by the states of cells {i - 1, i, i + 1} at time t c (i - 1), c (i), c (i + 1) t t t t i -1 i i+1 e m i t + 1 i T Complex Systems 9 Example c (i-1) c (i) c (i+1) ● Rules t t t c (i) ● Example (8 cells with wrap-around) t+1 t = 0 t = 1 t = 2 t = 3 t = 4 Complex Systems 10 Example ● Evolution of the previous CA on a larger array Complex Systems 11 Wolfram Canonical Enumeration ● With binary state space and r=1, there are 2^8 = 2 256 possible elementary 4 CAs ● We can encode the rule 8 table with a single 8-bit number by reading the 16 rightmost column top-to- bottom Complex Systems Rule 30 CA 12 Wolfram Canonical Enumeration ● With binary state space and r=1, there are 2^8 = 2 256 possible CAs 4 ● We can encode the rule table with a single 8-bit 8 number by reading the rightmost column top-to- bottom 32 64 Complex Systems Rule 110 CA 13 CAs as Complex Systems: Wolfram Classification ● Class I – Always evolve to a homogeneous arrangement, with every cell being in the same state, never to change again ● Class II – Form periodic structures that endlessly cycle through a fixed number of states ● Class III – Form aperiodic, random-like patterns similar to the static white noise on a bad (analog) television channel ● Class IV – Form complex patterns with localized structure that move through space and time. Patterns eventually become homogeneous, like ClassComplex I, Systemsor periodic, like Class II 14 Elementary CA rules 0—49 Elementary CA rules 50—99 Elementary CA rules 100—149 Elementary CA rules 150—199 Elementary CA rules 200—255 Wolfram’s Classification: Class 1 Rule 40 Rule 172 Rule 234 Slide credit: Ozalp Babaoglu Complex Systems 20 Wolfram’s Classification: Class 2 Slide credit: Ozalp Babaoglu Complex Systems 21 Wolfram’s Classification: Class 3 Rule 30 Rule 101 Rule 146 Slide credit: Ozalp Babaoglu Complex Systems 22 Wolfram’s Classification: Class 4 23 Slide credit: Ozalp Babaoglu Class II CAs ● Class II CAs are repetitive and bear resemblance to programs that execute in an infinite loop while ( TRUE ) { i = 0; for (i=0; i<10; i++) while ( TRUE ) { print i; print i; } i = i + 1; } Unbounded periodicity: on hypothetical machines with infinite Simple periodicity precision this fragment executes forever and is not periodic. In fact, this fragment does repeat when I overflows and starts back at zero; the exact value where this happens is machine-dependent Complex Systems 24 Langton's λ metric ● Is defined as the fraction of rules mapping into the 2 non-quiescent state 4 ● For elementary CAs, it is the fraction of rules 8 mapping to one in the rule table 16 Complex Systems Rule 30 CA, λ = 4/8 25 Langton's λ metric ● Is defined as the fraction of rules mapping into the 2 non-quiescent state 4 ● For elementary CAs, it is the fraction of rules 8 mapping to one in the rule table 32 64 Complex Systems Rule 110 CA, λ = 5/8 26 Langton's λ metric and CA classification Complex Systems 27 Examples ● NetLogo implements elementary CAs in Sample Models → Computer Science → CA 1D Elementary Complex Systems 28 Two-dimensional CAs Game of Life Complex Systems 29 Two-dimensional CAs ● CAs can be defined over higher-dimensional grids ● For two-dimensional CAs, different types of neighborhoods can be defined The eight gray cells form the The four dark gray cells form the von Moore neighborhood for the Neumann neighborhood for the red cell; the red cell extended von Neumann neighborhood includes the light gray cells Complex Systems 30 Conway's Game of Life ● Developed by the British mathematician John Horton Conway in 1970 ● Infinite two-dimensional orthogonal grid of square cells, each of which is in two states (alive or dead) ● Moore neighborhood John Horton Conway ● Rules: (1937— ) – Any live cell with fewer than two live neighbours dies – Any live cell with two or three live neighbours lives on to the next generation – Any live cell with more than three live neighbours dies – Any dead cell with exactly three live neighbours becomes a live cell ● The initial pattern constitutes the seed of the system ● Each generation is created by applying the above rules simultaneously to every cell – The discrete moment at which this happens is sometimes called a tick Complex Systems 31 Some notable patterns Still lifes Oscillators Spaceships Glider Lightweight Spaceship Source: http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life Complex Systems 32 Some notable patterns ● Conway conjectured that no pattern can grow indefinitely ● A team from the Massachusetts Institute of Technology proved he was wrong – The "Gosper glider gun" produces its first glider on the 15th generation, and another glider every 30th generation from then on Source: http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life Complex Systems 33 Logical operations implemented in Life Logical Not Logical And Logical Or Complex Systems 34 Universality ● Both CA rule 110 and Life are Turing complete – M. Cook, Universality in Elementary Cellular Automata, Complex Systems 15: 1-40. ● Below is a Turing machine implemented with a Life pattern in Golly (http://golly.sourceforge.net/), Patterns → Life → Signal Circuitry → Turing-Machine-3-state.rle Complex Systems 35 von Neumann Universal Constructor Pesavento, Umberto (1995) An implementation of von Neumann's self-reproducing machine. Artificial Life 2(4):337-354; See http://en.wikipedia.org/wiki/Von_Neumann_universal_constructor for details and links to the golly rule file (requires approximately 6 x 10^10 timesteps for replication) 36 Another CA that replicates itself Patterns → JvN → Boustrophedon-replicator.rle Complex Systems 37 Some practical applications of CAs Complex Systems 38 Stochastic (Elementary) CA 0.0 ● In a stochastic elementary CA, each rule is the probability that the 1.0 center cell is black ● In general, in a stochastic CA the 0.5 new cell states may depend on a random outcome 0.2 0.9 0.7 0.7 0.0 Complex Systems 39 Percolation model 1 (site percolation) ● Three states, two-dimensional CA, von Neumann neighborhood – White: empty cell – Blue: blocked cell – Yellow: liquid ● Initial configuration – Each cell is empty with probability p, blocked with probability 1 - p – One designated cell (e.g., at the center of the matrix) contains liquid ● One rule – A white cell with at least one yellow neighbor turns yellow Complex Systems 40 Percolation model 1 ● Suppose we “pour” liquid on the top row. What is the probability that the liquid eventually reaches the bottom row? Percolate Does not percolate Complex Systems 41 Percolation model 1 ● It turns out that there is a critical value p* for the parameter p (the probability that a cell is empty) s.t. the percolation probability increases sharply for p > p* – For square lattices, the site percolation threshold is p* ~ 0.59 1 Percolation probability p* 1 p Complex Systems 42 Percolation model 2 ● Percolation of a viscous liquid in a porous medium – E.g., oil in sand ● Three states, 1D stochastic CA – 0: solid, 1: empty, 2: filled ● Solid and empty squares are arranged in a checkerboard configuration ● Liquid percolates top-down through the empty squares with probability p Empty time Solid Complex Systems 43 Liquid Percolation model 2 ● NetLogo Sample Models → Earth Science → Percolation p = 0.5 p = 0.6 p = 0.7 Percolation model 2 ● Note that for percolation model 2 we need to update the cells by row (one row at each time step, moving downwards) ● We have initially said that in a CA all cells must be updated concurrently at each timestep ● It turns out that we can implement the percolation model with global concurrent updates by using a slightly more complex cell state – Any idea? Complex Systems 45 Percolation model 2 ● Cell state is {soil, empty, newliquid, liquid} ● Initial state: – Checkerboard pattern of soil, empty – Empty cells on the first row are set in state newliquid ● Update rule if ( this.state == newliquid ) then this.state ← liquid; elseif ( this.state == empty ) then if ( NE.state == newliquid and NW.state == newliquid ) then this.state ← newliquid with prob.

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