Aperiodicity In The Elementary Cellular Automaton

There are many complex systems in nature, the sturctures of their components may be quite simple, however, they can display very rich and complex global behaviors since there are local interactions among the components.Cellular automata are ideal mathematical models in studying of complex systems. Historically,the first cellular automaton was proposed by Von Neumann to simulate the self-reproductive phenomena in living systems. Since then they have been used widely to simulate various natural and life phenomena. The aperiodicity of all temporal sequences generated by rule 90 are already clear,and as has been noted rules 126 and 122 can simulate rule 90 in that their behaviors coincide when resticted to certain spatial subsequences. This paper proved that:(1)Every temporal sequence genarated by rule 126 with arbitrary finite inital conditions on an infinite lattice is aperiodic.(2)With the exception of the case that the temporal sequences genarated by the finite initial conditions ofare periodic,every temporal sequence genarated by rule 122 with arbitrary finite inital conditionson an infinite lattice is aperiodic.