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Ternary cellular automata induced by semigroups of order 3 are solvable

Henryk Fukś

TL;DR

The paper addresses the solvability of all ternary two-input cellular automata ruled by semigroups of order 3. It develops explicit iteration formulas for each of the 18 rules by exploiting semigroup properties such as commutativity and idempotence and by employing polynomial representations of the local rules. The main contribution is a complete set of closed-form expressions for the state at site $i$ after $n$ iterations, covering cases ranging from trivial to binomial-coefficient based, product-based, and idempotent rules, including non-commutative ones. These results enable direct computation of statistical properties (e.g., expectations under Bernoulli initial distributions) and facilitate analysis of finite-size effects in periodic settings.

Abstract

The minimal number of inputs in the local function of a non-trivial cellular automaton is two. Such a function can be viewed as as a kind of binary operation. If this operation is associative, it forms, together with the set of states, a semigroup. There are 18 semigroups of order 3 up to equivalence, and they define 18 cellular automata rules with three states. We investigate these rules with respect to solvability and show that all of them are solvable, meaning that the state of a given cell after $n$ iterations can be expressed by an explicit formula. We derive the relevant formulae for all 18 rules using some additional properties possessed by particular semigroups of order 3, such as commutativity and idempotence.

Ternary cellular automata induced by semigroups of order 3 are solvable

TL;DR

The paper addresses the solvability of all ternary two-input cellular automata ruled by semigroups of order 3. It develops explicit iteration formulas for each of the 18 rules by exploiting semigroup properties such as commutativity and idempotence and by employing polynomial representations of the local rules. The main contribution is a complete set of closed-form expressions for the state at site after iterations, covering cases ranging from trivial to binomial-coefficient based, product-based, and idempotent rules, including non-commutative ones. These results enable direct computation of statistical properties (e.g., expectations under Bernoulli initial distributions) and facilitate analysis of finite-size effects in periodic settings.

Abstract

The minimal number of inputs in the local function of a non-trivial cellular automaton is two. Such a function can be viewed as as a kind of binary operation. If this operation is associative, it forms, together with the set of states, a semigroup. There are 18 semigroups of order 3 up to equivalence, and they define 18 cellular automata rules with three states. We investigate these rules with respect to solvability and show that all of them are solvable, meaning that the state of a given cell after iterations can be expressed by an explicit formula. We derive the relevant formulae for all 18 rules using some additional properties possessed by particular semigroups of order 3, such as commutativity and idempotence.
Paper Structure (13 sections, 2 theorems, 97 equations, 1 figure, 2 tables)

This paper contains 13 sections, 2 theorems, 97 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Basic results
  4. Solutions
  5. Trivial rules
  6. Rules with solutions involving binomial coefficient
  7. Rules 8 and 9
  8. Rule 10
  9. Rule 12
  10. Idempotent rules 13--16
  11. Rule 18
  12. Concluding remarks
  13. Acknowledgments.

Key Result

proposition thmcounterproposition

If the semigroup $({\cal A},\odot)$ is idempotent and $f(x_0,x_1)=x_0 \odot x_1$, then

Figures (1)

  • Figure 1: Spatiotemporal patterns of all 18 semi-group rules generated from random initial condition of 40 sites with periodic boundary conditions, iterated 25 times. Color scheme: $0=$white, $1=$gray and $2=$blue.

Theorems & Definitions (2)

  • proposition thmcounterproposition
  • proposition thmcounterproposition