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

2. 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

3. Review of Cellular Automata

4. 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) )

5. 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

6. 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 { , }

7. 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

8. 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

9. Typical 2-D neighborhood shapes
von Neumann neighborhood
Moore neighborhood

10. 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

11. Phase Space of Cellular Automata

12. 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

13. 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

14. 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

15. Example
|S|=2, N=16
© Andy Wuenshe

16. 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)

17. Wolfram’s classification

18. 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

19. 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

20. Converting PDE Models into CA with Real-Valued States

21. 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)

22. 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)

23. 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)

24. 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

25. 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

26. Other Extensions of Cellular Automata

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

28. 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 …

29. 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

30. 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

31. 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

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

33. 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

34. 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

35. 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)

36. 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