Abstract

 
 
 We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.
 
 
Highlights
We investigate one-dimensional cellular automata (CA), a class of temporally and spatially discrete dynamical systems, in which the update rule is selected uniformly at random, and thereafter applied deterministically
In the literature [3], such a configuration is called doubly or jointly periodic. Since these are the only objects we study, we refer to such a configuration as a periodic solution (PS)
If we have ξτ (x) = ξ0(x), for all x ∈ Z and that σ and τ are both minimal, we have found a periodic solution of temporal period τ and spatial period σ
Summary
We investigate one-dimensional cellular automata (CA), a class of temporally and spatially discrete dynamical systems, in which the update rule is selected uniformly at random, and thereafter applied deterministically. A method of finding temporally periodic trajectories is discussed in [21], which reiterates the utility of the relation between periodic configurations and cycles on graphs induced by the CA rules, introduced in [14]. This approach is useful in the present paper as well. Let Pτ,σ,n be the random set of PS with temporal period τ and spatial period σ of such a uniformly chosen CA rule. We will determine lim P (Pτ,σ,n = ∅), the limiting probability that a random CA rule has a PS with given temporal and spatial periods.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
