Hiroki Sayama sayama@binghamton.edu NECSI Summer School 2008 Week 3: Methods for the Study of Complex Systems Cellular Automata Hiroki Sayama sayama@binghamton.edu

Four approaches to complexity Nonlinear Dynamics Complexity = No closed-form solution, Chaos Information Complexity = Length of description, Entropy Computation Complexity = Computational time/space, Algorithmic complexity Collective Behavior Complexity = Multi-scale patterns, Emergence

Review of Cellular Automata

Cellular automata (CA) A regular grid model made of many “automata” whose states are finite and discrete ( nonlinearity) Their states are simulta-neously updated by a uniform state-transition function that refers to states of their neighbors st+1(x) = F ( st(x+x0), st(x+x1), ... , st(x+xn-1) )

Some terminologies Configuration Local situation Quiescent state ( ) A mapping from spatial coordinates to states (st(x)); global arrangement of states at time t Local situation A specific arrangement of states within a local neighborhood, to be given to the state-transition function as an input Quiescent state ( ) A state that represents “empty” space; never changes if surrounded by other quiescent states Some CA have no quiescent state

State-transition function How CA works Configuration Configuration Neighborhood T C R B L Local situation State set State-transition function C T R B L C T R B L C T R B L C T R B L { , }

Boundary conditions Periodic boundary condition Cells at the edge of the space is connected to the cells at the other edge Cut-off boundary condition Cells at the edge of the space do not have neighbors outside the space Fixed boundary condition Cells at the edge of the space are fixed to specific states

Exercise: 1-D majority rule Each cell looks at its own and two nearest neighbors’ states, and then change to the local majority state Periodic boundary conditions assumed Complete time evolution of this CA t=0 t=1 t=2 t=3

Typical 2-D neighborhood shapes von Neumann neighborhood Moore neighborhood

Exercise: 2-D parity rule Each cell looks at states of cells in its von Neumann neighborhood, and then change to the state whose number was odd Quiescent states assumed outside the space Complete time evolution of this CA t=0 t=1 t=2

Phase Space of Cellular Automata

Phase space of cellular automata For a fixed number of cells, the total number of possible configurations is also fixed and finite: |S|N S: State set N: Total number of cells For deterministic CA, each configuration is always mapped to just one configuration You can draw a state-transition diagram

Exercise Draw a phase space of the binary-state 1-D majority CA with 5 cells and periodic boundary conditions Use symmetries to reduce the number of configurations

Features in phase space Self-loop → fixed point Cycle → periodic attractor Configuration with no predecessor → “Garden of Eden” states Just one big basin of attraction → Initial configurations don’t matter Many separate basins of attraction → Sensitive to initial configurations

Example |S|=2, N=16 © Andy Wuenshe http://www.ddlab.com/

Wolfram’s classification Wolfram (1984) classified binary-state 1-D CA based on their typical attractors’ properties I: Fixed point of homogeneous states II: Fixed point or small cycle involving heterogeneous states III: Chaotic attractor (very long cycle) IV: Complex, long-lived localized structures (attractor preceded by very long tree-like basins of attraction)

Wolfram’s classification

Reversible CA Some CA rules are reversible Every configuration has one and only one predecessor configuration May or may not be locally reversible Reversible CA has a particular type of phase space There are only cycles or fixed points; no trees, branches All the fixed points are isolated

Exercise Draw a phase space of the binary-state 1-D parity CA with 5 cells and periodic boundary conditions Use symmetries to reduce the number of configurations

Converting PDE Models into CA with Real-Valued States

CA with real-valued states CA may be used as a discrete analog of PDE-based models States are extended to real numbers (or even to vectors if PDEs involve multiple state variables)

Discretizing spatial derivatives f/x = limh0 { f(x+h,t) – f(x,t) } / h If we make a discrete analog of this by letting h = 1: f/x (in PDEs) ~ ft(x+1) – ft(x) (in CA) If both sides of neighbors are used: f/x (in PDEs) ~ { (ft(x+1) – ft(x)) + (ft(x) – ft(x-1)) } / 2 = (ft(x+1) – ft(x-1)) / 2 (in CA)

Discretizing Laplacians (1) 2f/x2 = limh0 { f’(x+h,t) – f’(x-h,t) } / 2h = limh0 { limk0 (f(x+h+k,t) – f(x+h-k,t)) / 2k – limk0 (f(x-h+k,t) – f(x-h-k,t)) / 2k } / 2h = limh0 { f(x+2h,t) + f(x-2h,t) – 2f(x,t) } / (2h)2 If we make a discrete analog of this by letting h = 1/2: 2f/x2 (in PDEs) ~ ft(x+1) + ft(x-1) – 2ft(x) (in CA)

Discretizing Laplacians (2) Similarly, for 2-D space: 2f = 2f/x2 + 2f/y2 (in PDEs) ~ ft(x+1,y) + ft(x-1,y) + ft(x,y+1) + ft(x,y-1) – 4ft(x,y) (in CA) Note: These are qualitatively similar, but not quantitatively

Exercise 2f = 2f/x2 + 2f/y2 (in PDEs) ~ ft(x+1,y) + ft(x-1,y) + ft(x,y+1) + ft(x,y-1) – 4ft(x,y) (in CA) Create a simple cell aggregation model on real-valued CA using the above analog

Other Extensions of Cellular Automata

Stochastic CA CA with stochastic state-transition functions Stochastic growth models Ecological/epidemiological models (1-p)k 1-(1-p)k

Multi-field CA CA whose cells have composite states Multiple “fields” in a state Similar to vector states, but each field can have non-numerical values as well Represents multiple CA spaces interacting with each other State for field 1 State for field 2 State for field 3 …

Multi-field CA Examples Models of spatially distributed agents with complex internal states Models of interaction between agents and environment CA that depend on more than one step past configurations

Locally reversible CA Reversible CA whose reversed dynamics can also be written as CA One needs local information only in order to determine the previous state of a cell All the information must be kept locally, typically attained by “block-to-block” mapping

Example: CA with Margolus neighborhoods Partition of space alternates Information can propagate over space with this alteration Can be used for irreversible CA as well t = even t = odd

Exercise Think about CA with Margolus neighborhoods where each block is rotated clockwise by 90 degrees What kind of behavior will happen?

Example: Lattice gas automata CA that simulates fluid flows using many particles with conservation laws Hardy-Pomeau-Pazzis (HPP) model Square lattice, reversible, anisotropic Frisch-Hasslacher-Pomeau (FHP) model Hexagonal lattice, irreversible (stochastic), isotropic, capable of simulating Navier-Stokes equation at macroscopic limit

Asynchronous CA CA where cells are updated asynchronously Random updating, sequential updating, etc. Sometimes works better as a model of physical systems Can simulate any synchronous CA

FYI: Emergent evolution on CA Evoloops (Sayama 1998) Self-replicating worms (Sayama 2000) Ecology of self-replicating worms (Suzuki et al. 2003) Genetic evolution of self-replicating loops (Salzberg et al. 2003, 2004) http://bingweb.binghamton.edu/~sayama/

Summary Cellular automata are extensively used for complex systems modeling Phase space structure of CA determines their dynamical properties CA may be used as a substitute for PDE-based models Several non-traditional model extensions are possible