On the Complexity of the Stability Problem of Binary Freezing Totalistic Cellular Automata

TL;DR

This paper classifies two-state totalistic freezing cellular automata on triangular and square grids, analyzing the computational complexity of their stability problem across different rule classes, revealing polynomial-time and NC complexities.

Contribution

It provides a complete classification of TFCA rules into five classes and determines the complexity of the stability problem for each class.

Findings

Algebraic and Topological rules have the stability problem in NC.

Turing Universal rules have a P-Complete stability problem.

Classification aids in understanding computational complexity of cellular automata.

Abstract

In this paper we study the family of two-state Totalistic Freezing Cellular Automata (TFCA) defined over the triangular and square grids with von Neumann neighborhoods. We say that a Cellular Automaton is Freezing and Totalistic if the active cells remain unchanged, and the new value of an inactive cell depends only on the sum of its active neighbors. We classify all the Cellular Automata in the class of TFCA, grouping them in five different classes: the Trivial rules, Turing Universal rules,Algebraic rules, Topological rules and Fractal Growing rules. At the same time, we study in this family the Stability problem, consisting in deciding whether an inactive cell becomes active, given an initial configuration.We exploit the properties of the automata in each group to show that: - For Algebraic and Topological Rules the Stability problem is in $\text{NC}$. - For Turing Universal rules the Stability problem is $\text{P}$-Complete.