Linear cellular automata with boundary conditions

Additive Cellular Automata

“…This is called a cylindrical cellular automata [77] because evolution of the rule can be represented as taking place on a cylinder. If the lattice is isomorphic to f0; : : : ; n 1g, null, or Dirchlet boundary conditions are set [78,79,80]. That is, the symbol assigned to all sites in L outside of this set is the null symbol.…”

Jointly Periodic Points in Cellular Automata: Computer Explorations and Conjectures

“…Σ N is a group under coordinatewise addition (mod N ), and some c.a. are group endomorphisms of this group; these are the linear cellular automata whose jointly periodic points were studied in [24] and later in a number of papers (see [9,10,15,21,26,31] and their references). The algebraic structure allowed a number theoretic description of the way that f -periods of jointly periodic points of S N period n vary (irregularly) with n. We show now that when f is a linear c.a., it is easy to see that ν(f, S N ) = log N .…”

“…In the case f is linear (f (x) + f (y) = f (x + y)), Martin, Odlyzko and Wolfram [24] (see also the further work in [9,10,15,21,26,31] and their references) gave an algebraic analysis of f -periods and preperiods for points of a given shift period, and also provided some numerical data. One key feature for linear f is an easy observation: among the jointly periodic points of shift period k, there will be a point (generally many points) whose least f -period will be an integer multiple of all the least f -periods of the jointly periodic points of shift period k. In contrast, a very special case of a powerful theorem of Ashley [1] has the following statement: for any K, N and any shift-commuting map g from ∪ 1≤k≤K P k (S N ) to itself, there will exist surjective c.a.…”

“…The map F is the flip involution F = 1 + x 0 . The data on F • j are copied in from Table 4 for contrast with D • j. j 10-dense at j 10-dense at j 10-dense at j 10-dense at 18-24 9 1. For such a map j, let p j (x 0 , x 1 , x 2 , x 3 ) be the polynomial such that (jx) 0 = p j (x 0 , x 1 , x 2 , x 3 ).…”

“…There is by this time a lot of work addressing periodic and jointly periodic points for linear one-dimensional c.a. ; we refer to [9,10,15,21,24,26,31] and their references. Also see [25] regarding the structure of periodic points for these and more general algebraic maps in the setting of [18,30].…”