Density Of Ones Research Articles

Exponential Convergence to Equilibrium in Cellular Automata Asymptotically Emulating Identity

We consider the problem of finding the density of ones in a configuration obtained by n iterations of a given cellular automaton (CA) rule, starting from a disordered initial condition. While this problem is intractable in full generality for a general CA rule, we argue that for some sufficiently simple classes of rules it is possible to express the density in terms of elementary functions. Elementary CA rules (k ! 2, r ! 1) asymptotically emulating identity are one example of such a class, and density formulas have been previously obtained for several of them. We show how to obtain formulas for density for two further rules in this class, 160 and 168, and postulate likely expressions for density for eight other rules. Our results are valid for arbitrary initial density. Finally, we conjecture that the density of ones for CA rules asymptotically emulating identity always approaches the equilibrium point exponentially fast.

Orbits of the Bernoulli measure in single-transition asynchronous cellular automata

We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.

Response Curves for Cellular Automata in One and Two Dimensions

In this paper, the authors consider the problem of computing a response curve for binary cellular automata, that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. The authors demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns; therefore, it is possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration n times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. The authors also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.

Probabilistic initial value problem for cellular automaton rule 172

We present a method of solving of the probabilistic initial value problem for cellular automata (CA) using CA rule 172 as an example. For a disordered initial condition on an infinite lattice, we derive exact expressions for the density of ones at arbitrary time step. In order to do this, we analyze topological structure of preimage trees of finite strings of length 3. Level sets of these trees can be enumerated directly using classical combinatorial methods, yielding expressions for the number of $n$-step preimages of all strings of length 3, and, subsequently, probabilities of occurrence of these strings in a configuration obtained from the initial one after $n$ iterations of rule 172. The density of ones can be expressed in terms of Fibonacci numbers, while expressions for probabilities of other strings involve Lucas numbers. Applicability of this method to other CA rules is briefly discussed.

The density of ones in Pascal's rhombus

Pascal's rhombus is a variation of Pascal's triangle in which values are computed as the sum of four terms, rather than two. It is shown that the limiting ratio of the number of ones to the number of zeros in Pascal's rhombus, taken modulo 2, approaches zero. An asymptotic formula for the number of ones in the rhombus is also shown.