13 November 2018

Séminaire CaNa : Exploring attractor invariance in elementary cellular automata : complexité algorithmique, Stéphanie MacLean

We are interested in the study of the set of attractors of all 256 Elementary Cellular Automata (ECA) , i.e., one-dimensional, binary 3-neighbourhood cellular automata, dened on an n-cell lattice (n is the length of the automaton). Recent works have studied the invariance of their attractors against dierent update schemes, and how these aect their dynamics. Specically, in [1] was introduced the notion of k-invariance: an ECA rule r is k-invariant if its set of attractors, denoted by Fr(s), remains invariant for all update schemes s having blocks of length at most k (notice that k=1 stands for sequential update schemes, while k = n is the parallel, or synchronous one). In this context, the 1-invariance was studied in [2], where 104 ECA rules showed to have this kind of invariance. In [1] and [3] the authors studied the k-invariance, for 2 < k ≤ n, for all 104 ECA rules above mentioned. In this work, we explore another notion of attractor invariance "in between ECA rules", we say that two ECA rules u and v are attractor equivalent over a set of update schemes S, if given any s in S, Fu(s) = Fv(s). We begin our study with the 1-invariant rules, searching for equivalences between their sets of attractors, for all 4 ≤ n ≤ 14, so as to have a set of rules that are "candidates" to be attractor equivalent, since we know that if u and v are both 1-invariant ECA rules, then their sets of attractors remain invariant, but not necessarily the same, for all sequential update schemes s. Attractor equivalence gives us new insights of the dynamical behaviour of a rule (and its equivalent rules) under dierent update schemes.
13 November 2018, 14h0015h00
Salle de réunion du bâtiment modulaire, Luminy

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