Universality in Freezing Cellular Automata

Abstract

Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this "freezing-order". We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.Context-sensitive simulation can get us over this hurdle as we show below; it is akin to the notion of conjugacy in symbolic dynamics [14].

Cellular Automata and Bootstrap Percolation

“…• for some p ∈ P and µ-almost every x, we have f n (x) z = p for all z ∈ Z 2 and all large enough n ∈ N (depending on z).…”

“…If there exists N ∈ F (f ), then by Lemma 4.3 there is a closed rational half-plane H with 0 ∈ H and N ∩ H = ∅. Translating H so that 0 lies on its border, we have 2 . Let V N be the set of directions of the faces of the polygon A N , and r N = diam(A N ).…”

“…The dynamics of freezing cellular automata have been studied explicitly in e.g. [7,6,2]. We note that in the literature it is common to require freezing CA to be decreasing rather than increasing, but here we choose to follow the opposite convention of percolation theory.…”

Freezing, Bounded-Change and Convergent Cellular Automata

“…They were in particular considered as theoretical models of bootstrap percolation and a lot of work was dedicated to the experimental and rigorous analysis of the phase transitions they exhibit (see for instance [21]). The fact that these examples are monotone (with respect to the order extended to configurations) is to be taken into account when studying their computational complexity [15,3].…”

“…One of our inspiration is the profusion of examples in the literature, from the seminal works of S. Ulam [42] to the recent and fast development of self-assembly models [44], which all share the convergence property: a larger and larger zone of the configuration gets frozen by the dynamics while changes continue outside the zone. On the other hand, some previous works explicitly studied the class of freezing CA [14,3] or bounded-change CA [43] and established that universal computation is possible in any dimension but "slowed down" by the bounded-change constraint in dimension 1. This was done by studying the short-term prediction problem on one hand, and by giving an explicit encoding of Minsky machines into such CA on the other hand.…”