Rule 110: universality and catenations

Cellular automata are a simple model of parallel computation. Many people wonder about the computing power of such a model. Following an idea of S. Wolfram [16], M. Cook [3] has proved than even one of the simplest cellular automata can embed any Turing computation. In this paper, we give a new high-level version of this proof using particles and collisions as introduced in [10]. Introduced in the forties by J. Von Neumann as a parallel model of computation [13], cellular automata consist of many simple entities (cells) disposed on a regular grid. All cells evolve synchronously by changing their state according to the ones of their neighbours. Despite being completely known at the local level, global behavior of a cellular automata is often impossible to predict (see J. Kari [6]). This comes from the fact that even “simple” cellular automata can exhibit a wide range of complex behaviors. Among those behavior, one often refers as emergence the fact that “complexity” of the whole system seems far greater than complexity of elements. Elementary cellular automata are an example of subclass of “simple” cellular automata. They are obtained by considering only a one dimensional grid (i.e., a line), two possible states and nearest neighbours (i.e., left and right one in addition to the cell itself). Although very restrictive, some elements of this class do exhibit very complex behaviors including emergence. One way to assert such a claim is to prove that some of those cellular automata can embed any Turing computation. Among elementary cellular automata, more likely candidate to this property were though to be the ones that exhibit meta-structures with predictable behavior. Those meta-structures have been studied with regards to their combinatorial aspect (see N. Boccara et al. [1] or J. P. Crutchfield et al. [5]) and widely used as support for constructions. In fact, M. Cook [3] manages to embed any Turing computation in an elementary cellular automaton (namely rule 110) using these structures. However, lack of formalism on those meta-structures forced the author to develop long and complex combinatorial arguments to prove that intuition on behavior is correct. In this paper, we shall use a new formalism on these meta-structures developed in [10] to provide a complete and high-level proof of Turing universality of rule 110 without the need of complex combinatorial concerns. 2000 ACM Subject Classification: 68Q80,68Q05,37F99.