Cellular Automata

Martijn van den HeuvelModels of ComputationJune 21st, 2011

Overview

History Formal description Elementary CA Example of a CA Turing completeness Lindenmayer Systems

History

1940s Von Neumann - self-replicating machines

1970s Conway – Game of Life

1980s Wolfram – Investigating properties

2000s Cook – Proves Turing completeness

Use

Visualizing processes Simple components complex behavior

Fluid dynamicsBiological pattern formationTraffic modelsArtifical life

Formal description

Cellular space Lattice/grid of N identical

cells Cell i at time t has a state si

t

Cell i at time t has a neighborhood ni

t

Transition rules r(ni

t) updates cell i to next state si

t+1 Size of rule set: |R|=|s||n|

23=8 Simultaneous updating

0 0 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 0 0 0

Rule 000 1

Elementary CA

Simplest form of CA One-dimensional Binary states {0,1} 2 neighbors

23=8 possible configurations 28=256 different elementary CAs

out 0,1 0,1 0,1 0,1 0,1 0,1 0,1 0,1

s 111 110 101 100 011 010 001 000

Example

nit 111 110 101 100 011 010 001 000

sit+1 0 1 0 1 1 0 1 0

Example Java Applet

Turing completeness

Matthew Cook Rule 110 Emulation of Post Tag System

Gliders/Spaceships interacting Structures representing:

Infinite data string Infinitely repeating set of production rules Output

nit 111 110 101 100 011 010 001 000

sit+1 0 1 1 0 1 1 1 0

Turing completeness

Lindenmayer systems

Extensions

States: { a,b,c,d,k }

Rules: a cbc

b dad

c k

d a

k k

aacbccbckdadkkdadkkacbcakkacbcakkcbckdadkcbckkcbckdadkcbck

L-systems vs CA

Both: Loop until no rule is applicable States for data storage Transition rules for modification Use simultaneous rule application

Differences: L-system usually has non-binary states L-system rewrites multiple states in one step

CA updates only cell i

L-systems vs CA

Possible solutions:Abstract L-system

Allow same rules as CA

Extend CA multiple states Rule table grows fast

CA could write sequentially what L-system does in one step

L-systems vs CA

ButlerSimulate D(m,n)L-system on a 1-dim CAUses registers (m per cell) containing:

Direction or ‘unprocessed’ states

ButlerCA:

L-system to be simulated:

Axiom: 1Rules:

0 00

1 101

1

101

10100101

10100101000010100101

Butler

Using registers as neighborhoodStill 1-dimensional?

Extending CA beyond original properties? Very simple L-system

L-system is an extended CA

Conclusion

CA and L-system function in nearly the same way Using states & transition rules

Both can simulate a TM (indirectly) CA becomes complex and hard to interpret

Both apply rules as long as possible, then stop. Both apply rules simultaneously L-system could be seen as an extended CA

Reading

Wolfram's publications Simulation of TM on a CA

Universality in Elementary Cellular Automata Matthew Cook; Complex Systems 15 (2004) p.1-40