Universality in Elementary Cellular Automata

The purpose of this paper is to prove that one of the simplest one dimensional cellular automata is computationally universal, implying that many questions concerning its behavior, such as whether a particular sequence of bits will occur, or whether the behavior will become periodic, are formally undecidable. The cellular automaton we will prove this for is known as “Rule 110” according to Wolfram’s numbering scheme [2]. Being a one dimensional cellular automaton, it consists of an infinitely long row of cells "Ci # i $ !%. Each cell is in one of the two states "0, 1%, and at each discrete time step every cell synchronously updates itself according to the value of itself and its nearest neighbors: &i, Ci ( F(Ci)1, Ci, Ci*1), where F is the following function: