Cellular Automata

Cellular Automata Martijn van den Heuvel Models of Computation June 21st, 2011

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

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 dynamics • Biological pattern formation • Traffic models • Artifical life

Formal description • Cellular space • Lattice/grid of N identical cells • Cell i at time t has a state sit • Cell i at time t has a neighborhood nit • Transition rules • r(nit) updates cell i to next state sit+1 • Size of rule set: |R|=|s||n| • 23=8 • Simultaneous updating Rule 000  1

Elementary CA • Simplest form of CA • One-dimensional • Binary states {0,1} • 2 neighbors (n=3) • 23=8 possible configurations • 28=256 different elementary CAs

Example 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

Turing completeness

Lindenmayer systems States: { a,b,c,d,k } Rules: a  cbc b  dad c  k d  a k  k • Extensions a cbc kdadk kacbcak kcbckdadkcbck F  F[-F]F[+F][F]

L-systems vs CA • Intuitively very similar • Example: • Context sensitive L-system • <m,n>L-system <a1,..,am,b,c1…,cn>z • <1,1>L-system with binary alphabet • <a,b,c>  d • Same as CA table:

L-systems vs CA • Butler • Simulate D(m,n)L-system on a 1-dim CA • Uses registers (m per cell) containing: • Direction  or  • ‘unprocessed’ states

Butler CA: L-system to be simulated: Axiom: 1Rules: 0  00 1  101 1 101 10100101 10100101000010100101

Butler • Using registers as neighborhood • Extending CA beyond original properties? • Very simple L-system • L-system is an extended CA

Conclusion • CA and L-systems are very alike • Both: • Loop until no rule is applicable • States for data storage • Transition rules for modification • Use simultaneous rule application • Have the same computing power • Complex to interpret • Differences: • L-system usually has non-binary states • L-system rewrites multiple states in one step • CA updates only cell i • ‘Inserting cells’ • L-system == extended (elementary) 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

Cellular Automata