Importance of Initial Conditions in Determining the Dynamical Class of Cellular Automata

In the qualitative classiication of cellular automata (CA) rules by Wolfram, there exists a class of CA rules (called Class 4) which exhibit complex pattern formation and long-lived dynamical activity (long transients). These properties of Class 4 CA's has led to the conjecture that Class 4 rules are Universal Turing machines i.e. they are bases for computational universality. We describe an embedding of a \small" universal Turing machine due to Minsky, into a cellular automaton rule-table. This produces a collection of (k = 18; r = 1) cellular automata, all of which are computationally universal. However, we observe that these rules are distributed amongst the various Wolfram classes. More precisely, we show that the identiication of the Wolfram class depends crucially on the set of initial conditions used to simulate the given CA. This work, among others, indicates that a description of complex systems and information dynamics may need a new framework for non-equilibrium statistical mechanics.