Cellular Automata Using Infinite Computations

This paper presents an application of the Infinite Unit Axiom, introduced by Yaroslav Sergeyev, (see [11] - [14]) to the development of one-dimensional cellular automata. This application allows the establishment of a new and more precise metric on the space of definition for one-dimensional cellular automata, whereby accuracy of computations is increased. Using this new metric, open disks are defined and the number of points in each disk is computed. The forward dynamics of a cellular automaton map are also studied via defined equivalence classes. Using the Infinite Unit Axiom, the number of configurations that stay close to a given configuration under the shift automaton map can now be computed.

💡 Research Summary

The paper introduces a novel quantitative framework for one‑dimensional cellular automata (CA) by exploiting Yaroslav Sergeyev’s Infinite Unit Axiom (IUA), commonly known as the “grossone” (ℵ) methodology. The grossone treats the first infinite natural number as a concrete numeral, allowing arithmetic with infinite quantities in a rigorous, non‑standard way. By embedding ℵ into the definition of a distance function on the configuration space Σ^ℤ, the authors construct a metric that simultaneously accounts for discrepancies at every lattice site while weighting them so that contributions from far‑away cells decay exponentially as ℵ^{‑|i|}. The resulting metric

 d(x,y)=∑_{i∈ℤ} ℵ^{‑|i|}·δ(x_i,y_i)

is symmetric, satisfies the triangle inequality, and is complete. Unlike the traditional Hamming‑type metrics that only handle finitely many mismatches, this metric can meaningfully compare configurations that differ on infinitely many sites, because the infinite sum converges to a finite hyperreal value expressed in terms of ℵ.

With this metric in hand, the authors define open balls B_r(x)={y∈Σ^ℤ | d(x,y)<r}. By choosing the radius r as a finite real number, a ball restricts freedom to a finite window around the origin; the number of configurations inside is exactly |Σ|^{2k+1} where k depends on r. When r is taken as a positive multiple of ℵ, the ball admits infinitely many independent degrees of freedom, and its cardinality becomes |Σ|^{ℵ}=ℵ^{ℵ}. Thus the grossone calculus turns the vague statement “infinitely many configurations” into a precise cardinality.

The dynamical analysis proceeds by examining the forward orbit of a CA map F. The authors introduce equivalence classes