Cellular Automata (CA) Overview

1
Cellular Automata (CA)Overview

• Introduction and Purpose
• von Neumann and generalized CA results
• Equivalences with Turing Machines and Shape
  Grammars
• More general theory
• Example applications
• Simulations

2
Purpose

• In Theory
• Computation of all computable functions
• Construction of (also non-homogenous) automata by
  other automata, the offspring being at least as
  powerful (in some well-defined sense) as the
  parent
• In Practice
• Exploring how complex systems with emergent
  patterns seem to evolve from purely local
  interactions of agents. I.e. Without a master
  plan!

3
Original von Neumann CA

• Infinite 2-D Cartesian grid of cells
• Synchronous time in the universe of cells
• Each cell has same simple finite automaton (state
  machine)
• Each cell sees immediate neighborhood of 8 cells
• State of each cell at time t1 is a function of
  the values of neighboring cells at time t
• Each cell can have 29 states
• There is a quiescent state Vo, where F(Vo)Vo
• Most of the universe is quiescent

4
Generalized CA

• Infinite or connected d-dimensional space of
  cells
• e.g. Torus, but usually 1 plus time, or 2 plus
  time
• Synchronous time in the universe of cells
• Each cell sees m-neighborhood of cells
• Each cell can have n-states
• Each cell has a finite deterministic (FDA) or
  non-deterministic automaton (state machine)
• State of each cell at time t1 is a function of
  the values of neighboring cells at time t

5
Von Neumanns results

• A Turing machine can be embedded in the space
• It is possible to embed an automaton A in the
  space. which can then build any other properly
  specified independent automaton B
• A can equal B (self reproduction)
• Further Results (Codd and others)
• What other n-state, m-neighbor spaces are
  computation-universal in the above sense?
• Minimum was found to be 8-state, 5-neighbor
  space.
• Related to more general Holland iterative circuit
  computers where cell neighborhoods vary over
  space and time. (Explaining some quantum
  interactions at a distance might require this
  or a model with multiple CA spaces in coexistence
  etc.)
• Turing Machines can simulate CA

6
Correspondence of Turing Machines (TM) and CA

• Consider TM that can handle 2-D tapes (or a long
  1-D tape with infinite segments)
• The blank symbol is the quiescent state
• The FDA is the state machine of the head
• TM is more general than CA
• and Shape Grammars
• CA can be expressed as a shape grammar (just draw
  the neighborhood as a shape)
• What about reverse? Apply correspondence of TM
  and Shape Grammars (Stiny) for the rest

7
Some definitions

• Configuration The state of all the cells in the
  space of interest
• (The catch is What are allowable states?
  Reachability problem rears its head)
• Computation
• Set up a correspondence with TM that preserves
  the tape/head distinction. Not all CA are TM and
  therefore not all CA are universal computers.
• Construction
• Stable or dying out configurations are not
  computationally interesting
• Self-Reproduction A special case of construction
• Symmetries of Cellular Spaces
• Symmetries of neighborhood functions, and those
  of transition functions.
• Propagation
• Does the CA go to infinity? Is it unbounded,
  bounded, asymptotic?
• Universality Back to TM
• Paths and Signals The states may be complex
  vectors with semantics (see wire world example)

8
Statistical Mechanics of CA (Wolfram)

• Using an elementary CA on a tape with 0,1
• Using nearest neighbor deterministic rules
• Simple initial configurations CA tend either to
  homogenous states, or generate self-similar
  patterns with fractal dimensions (1.59 1.69)
• Random initial configurations tend to two
  universality classes, independent of the
  properties of the initial state or the rules.
• Algebraic properties (Wolfram et. al.) include
  (usually) irreversibility, evolution through
  transients to attractors consisting of cycles
  sometimes containing a large number of
  configurations.

9
Universality and Complexity (Wolfram)

• Four classes
• Limit points
• Limit cycles
• Chaotic attractors
• Undecidable infinite time behavior
• (probably universal computation capable)

10
20 Problems (Wolfram)

• What overall classification of CA can be given?
• What are the exact relations between entropies
  and Lyapunov exponents for CA?
• What is the analogue of geometry for the
  configuration space of a CA?
• What statistical quantities characterize CA
  behavior?
• What invariants are there in CA evolution?
• How does thermodynamics apply to CA? (broken time
  symmetry problem)
• How is different behavior distributed in the
  space of CA rules?
• What are the scaling properties of CA?
• What is the correspondence between CA and
  continuous systems?
• What is the correspondence between CA and
  stochastic systems?
• How are CA affected by noise and other
  perturbations?
• Is regular language complexity generically
  non-decreasing with time in 1-D CA?
• What limit sets can CA produce?
• What are the connections between the
  computational and statistical characteristics of
  CA?
• How random are the sequences generated by CA?
• How common are computational universality and
  undecidability in CA?
• What is the nature of the infinite size limit of
  CA?
• How common is computational irreducability in CA?
• How common are computationally intractable
  problems about CA?

11
Example Application Budworm Infestation

• Spruce budworm is a pest that defoliates and
  kills balsam and spruce trees
• After the trees die, they are replaced by beech
  trees which do not support budworms
• After the budworms are gone, the spruce and
  balsam eventually displace the beeches through
  competition for sunlight and soil, and the cycle
  can repeat
• The CA is (2-dimensional, 3-state, 4-neighborhood
  DFA)
• A defoliated site becomes green the next season
• An infested site becomes defoliated the next
  season
• An infested site will next season infest those of
  its four nearest neighbors that are green
• Eradicating an infestation is equivalent
  mathematically to finding the smallest
  self-reproducing patterns in an established
  pattern

12
Other applications

• Lattice models for solidification and aggregation
  (crystal formation etc.)
• Chemical reactions
• Chemical and physical Turbulence
• Soliton-like behavior
• Lattice gas Navier-Stokes equation solution
• Thermodynamics, hydrodynamics
• Vertebrate skin patterns
• Forestry, urban planning, system dynamics

13
The Game of Life (Conway)

• A live cell with 2 or 3 live neighbors continues
  to live
• A live cell with 0,1,4,5,6,7,8 neighbors dies
• A vacant cell becomes live if it has 3 live
  neighbors
• Questions Which forms of excitation or initial
  state persist as stable configurations, which
  recur periodically, which die out? (equivalence
  to the halting problem)

14
Web Sites Used for Demos

• http//ourworld.compuserve.com/homepages/cdosborn/
  
• http//www.student.nada.kth.se/d95-aeh/lifeeng.ht
  ml
• http//lcs.www.media.mit.edu/groups/el/projects/em
  ergence/index.html
• http//alife.santafe.edu/alife/topics/ca/caweb/
• http//www.aridolan.com/