On undecidability of equicontinuity classification for cellular automata

Cellular automata are simple model for the study of complex phenomena produced by simple local interactions. They consist of a regular lattice of cells. Each cell contain finite automaton which has a state chosen from a finite set of states. Updates are made according to a local rule which takes into account the current state of the cell and those of a fixed finite set of its neighboring cells. All cells are updated synchronously. A snapshot of the state of all cells at the same time is called a configuration. Despite of their definition simplicity, cellular automata exhibit a wide range of dynamical behaviors. The classification of such behaviors is known as “the classification problem”. It has captivated researchers for years and a complete solution does not appear to be on coming. The first empirical classification was proposed about twenty years ago by Wolfram after an extensive experimental work. Successive studies of researchers in the field, tried to give this classification a formal justification. The first attempts ofČulik et al. [CY88, CPY89], made immediately clear were the problem is: CA behavior is so complex that almost any question about their long-term behavior is undecidable. Unfortunately we do not know how to formalize (and then prove) this idea. Two attempts in this direction are [Kar94] (a Rice theorem with variable states) and [CD00] (a partial Rice theorem with fixed states). In the middle of nineties, P. Kůrka proposed to classify CA according to their degree of equicontinuity [Kur97]. Denote E the set of equicontinuity points of a CA and X the set of all configurations. Kůrka devised the following four classes