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

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 s i t  Cell i at time t has a neighborhood n i t Transition rules  r(n i t ) updates cell i to next state s i t+1  Size of rule set: |R|=|s| |n| 2 3 =8 Simultaneous updating Rule000  1

Elementary CA Simplest form of CA  One-dimensional  Binary states {0,1}  2 neighbors 2 3 =8 possible configurations 2 8 =256 different elementary CAs out0,1 s

Example nitnit s i t 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 nitnit s i t

Turing completeness

Lindenmayer systems Extensions States: { a,b,c,d,k } Rules: a  cbc b  dad c  k d  a k  k acbckdadkkacbcakkcbckdadkcbck

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 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: 1 Rules: 0  00 1 

Butler Using registers as neighborhood  Still 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