Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Graphs: Resolving the Periodic Reversibility of One-dimensional Finite Cellular Automata

Chen Wang, Junchi Ma, Chao Wang, Defu Lin, Weilin Chen

TL;DR

This paper performs calculations on two graphs and discovers that the reversibility of any one-dimensional FCA exhibits periodicity as the number of cells increases, and provides a method to compute the reversibility sequence that encompasses the reversibility of one-dimensional FCA with any number of cells.

Abstract

Finite cellular automata (FCA) are widely used in simulating nonlinear complex systems, and their reversibility is closely related to information loss during the evolution. However, only a relatively small portion of their reversibility problems has been solved. In this paper, we perform calculations on two graphs and discover that the reversibility of any one-dimensional FCA exhibits periodicity as the number of cells increases. We also successfully provide a method to compute the reversibility sequence that encompasses the reversibility of one-dimensional FCA with any number of cells. Additionally, the calculations in this paper are applicable to FCA with various types of boundaries. This means that we will have an efficient method to determine the reversibility of almost all one-dimensional FCA, with a complexity independent of cell number.

Two Graphs: Resolving the Periodic Reversibility of One-dimensional Finite Cellular Automata

TL;DR

This paper performs calculations on two graphs and discovers that the reversibility of any one-dimensional FCA exhibits periodicity as the number of cells increases, and provides a method to compute the reversibility sequence that encompasses the reversibility of one-dimensional FCA with any number of cells.

Abstract

Finite cellular automata (FCA) are widely used in simulating nonlinear complex systems, and their reversibility is closely related to information loss during the evolution. However, only a relatively small portion of their reversibility problems has been solved. In this paper, we perform calculations on two graphs and discover that the reversibility of any one-dimensional FCA exhibits periodicity as the number of cells increases. We also successfully provide a method to compute the reversibility sequence that encompasses the reversibility of one-dimensional FCA with any number of cells. Additionally, the calculations in this paper are applicable to FCA with various types of boundaries. This means that we will have an efficient method to determine the reversibility of almost all one-dimensional FCA, with a complexity independent of cell number.
Paper Structure (10 sections, 11 theorems, 4 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 10 sections, 11 theorems, 4 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite cellular automata
  4. Boundaries of one-dimensional FCA
  5. Reversibility problem
  6. Amoroso's algorithm
  7. Main result
  8. Reversibility graph
  9. Circuit graph and reversibility period
  10. Conclusion

Key Result

Proposition 1

For a FCA, surjective and injective are equivalent and are also referred to as bijective or reversible.

Figures (5)

  • Figure 1: The null boundary of one-dimensional FCA
  • Figure 2: The periodic boundary of one-dimensional FCA
  • Figure 3: The left side is the reversibility graph for FCA=$\{Z^1,\{0,1\},N_3,10100101,"n"\}$, while the right side is for FCA = $\{Z^1,\{0,1\},N_3,10011001,"p"\}$. Negative vertices are marked in red.
  • Figure 4: Replace the content of each vertex $v$ of $G_R$ in Fig. \ref{['RG']} with its depth $d(v)$.
  • Figure 5: The circuit graph $G_C$ of $FCA = \{Z^1,\{0,1\},N_3,10011001,"p"\}$.

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Proposition 1
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 21 more