Local maps inducing surjective global maps of one-dimensional tessellation automata

Abstract

Abstract. In this paper, we investigate some combinatorial aspects of C-surjective local maps, i.e., local maps inducing surjective global maps, CF-surjective local maps, i.e., local maps inducing surjective restrictions of global maps on the set CF of finite configurations, and C-injective local maps, i.e., local maps inducing injective local maps, of one-dimensional tessellation automata.We introduce a pair of right and left bundle-graphs and a pair of right and left X-bundle-graphs for every C-surjective local map. We give characterizations for CF-surjectivity, C-injectivity and some other properties of C-surjective local maps in relation to these bundle-graphs. We also establish some properties of the inverse of a C-injective local map.

Unconventional Invertible Behaviors in Reversible One-Dimensional Cellular Automata

“…Welch set of w with regard of ϕ; analogously let R ϕ (w) = {s ∈ S | ∃ v ∈ S m such that vs ∈ Λ(w)} be the right Welch set of w. Reversible automata are fully characterized by Hedlund [1969] and Nasu [1978]; in particular demonstrating the following three properties:…”

“…A complete local characterization of reversible cellular automata is developed for the onedimensional case by Hedlund [1969] and Nasu [1978] using a combinatorial, topological and graph-theoretical approach. In these systems, reversibility is given in the temporal sense; Boykett [2003], Hillman [2004] and Boykett [2006] have studied reversible automata from an algebraic perspective, some of their results suggest the existence of other types of reversible behaviors different from the classical one.…”

Procedures for calculating reversible one-dimensional cellular automata

“…This problem was resolved independently by Nasu [12] and Kari [6] with a length at most of k − 1 if a Welch index is equal to 1. But the question stills open when both indices are different from 1.…”

“…Reversible one-dimensional cellular automata were carefully studied by Gustav A. Hedlund as automorphisms of the shift dynamical system [4]. Another important papers in the theory of reversible automata are those made by Masakazu Nasu [12] using a graph-theoretical approach and by Jarkko Kari using block permutations [7].…”

Weak gardens of Eden for 1‐dimensional tessellation automata

“…We note also that information about parallel maps which are h-to-one can be found in [73, [8], and [9] and that if h -> 2, then WG(T) PROOF. Suppose there are natural numbers m and n such that aoma and alna each contain a third copy of a.…”