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
7/8/2019
Ali Pirnar

2. Purpose In Theory: In Practice:
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!”
May 5, 1998
Ali Pirnar

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 t+1 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
May 5, 1998
Ali Pirnar

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 t+1 is a function of the values of neighboring cells at time t
May 5, 1998
Ali Pirnar

5. Von Neumann’s 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
May 5, 1998
Ali Pirnar

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
May 5, 1998
Ali Pirnar

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 it’s 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)
May 5, 1998
Ali Pirnar

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.
May 5, 1998
Ali Pirnar

9. Universality and Complexity (Wolfram)
Four classes
Limit points
Limit cycles
Chaotic attractors
Undecidable infinite time behavior
(probably universal computation capable)
May 5, 1998
Ali Pirnar

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?
What higher level descriptions of information processing in CA can be given?
May 5, 1998
Ali Pirnar

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
May 5, 1998
Ali Pirnar

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
May 5, 1998
Ali Pirnar

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)
May 5, 1998
Ali Pirnar

14. Web Sites Used for Demos
May 5, 1998
Ali Pirnar