Representing Families of Cellular Automata Rules

A Fully Operational Framework for Handling Cellular Automata Templates

“…So, in its origin, there was a still immature but promising idea that was conveyed at the time to two distinct audiences: the early technical details of the implementation to Mathematica users [8] and the actual idea and its prospects as a short communication to the cellular automata community [12]. From then on, the work developed, until reaching the required degree of maturation that it deserved, as embodied in the present report.…”

“…The template concept was first described in [8,12] and has gone through significant improvements since then. For instance, the template's core, essentially what was described as the template itself in the original papers, is now a part of a tuple composed by important attributes, as shown in the definition below.…”

“…Currently, six template builders for static properties are implemented, associated to rules of the following kinds: captive; totalistic; outer-totalistic; number conserving; colorblind; and maximum internal symmetry values (see [1]) for reflection, conjugacy, and their composition. The latter two are not discussed herein partly because we have already discussed them previously (see [8]), and, at least in the context number conservation, the corresponding template builder simply did not change since its conception. However, the builder for rule table symmetries has undergone a generalisation that now allows for any value of symmetry to be accounted for, not only maximum.…”

“…A particularly powerful representational tool in this context is the template. First introduced in [8], templates are data structures which represent potentially extremely large sets of CAs, further allowing to combine and manipulate them with operations such as the intersection, before effectively enumerating the whole set.…”

Fundamental collapse in cellular automaton process

“…Consider an automaton that is a one-dimensional grid of cells labeled i, where each can be turned on or off. In our examined model, the evolution of the system is determined by the following rules [6]:…”