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Maximal Length Cellular Automata : A Survey

Sumit Adak, Sukanta Das

TL;DR

The main goal of this survey is to provide a tutorial on maximal length CA theory to researchers with classical and new results on maximality, and to suggest some open problems.

Abstract

This article surveys some theoretical aspects of Cellular Automata (CAs) research. In particular, we discuss on maximal length CA. An n-cell CA is a maximal length CA, if all the configurations except one form a single cycle. There is a bonding between maximal length CA and primitive polynomial. So, primitive polynomials occupy a good amount of space in this survey. The main goal of this survey is to provide a tutorial on maximal length CA theory to researchers with classical and new results on maximality. We also give a compact collection of known results with references to their proofs, and to suggest some open problems. Additionally, some new theorems and corollaries are added to bridge the gaps among several known results.

Maximal Length Cellular Automata : A Survey

TL;DR

The main goal of this survey is to provide a tutorial on maximal length CA theory to researchers with classical and new results on maximality, and to suggest some open problems.

Abstract

This article surveys some theoretical aspects of Cellular Automata (CAs) research. In particular, we discuss on maximal length CA. An n-cell CA is a maximal length CA, if all the configurations except one form a single cycle. There is a bonding between maximal length CA and primitive polynomial. So, primitive polynomials occupy a good amount of space in this survey. The main goal of this survey is to provide a tutorial on maximal length CA theory to researchers with classical and new results on maximality. We also give a compact collection of known results with references to their proofs, and to suggest some open problems. Additionally, some new theorems and corollaries are added to bridge the gaps among several known results.
Paper Structure (28 sections, 28 theorems, 51 equations, 10 figures, 7 tables, 3 algorithms)

This paper contains 28 sections, 28 theorems, 51 equations, 10 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. One-dimensional Hybrid Cellular Automata
  3. Basics
  4. Maximal Length Cellular Automata
  5. Matrix Algebra : Characterization Tool
  6. Characteristic Polynomial
  7. Analysis of Maximality
  8. Boundary Condition Dependence
  9. Palindromic and Uniform CA
  10. Minimal Cost Maximal Length CA
  11. Phase Shift Operation
  12. Searching for Pattern
  13. Synthesis of Maximal Length CA from Primitive Polynomial
  14. Solutions to the Quadratic Congruence
  15. The Formula for the Solutions
  16. ...and 13 more sections

Key Result

Lemma 1

Let $T_{k}$ denote the $k\times k$ submatrix of $T$ and ${\Delta}_{k-1}=\det(xI+T_k)$. Then, ${\Delta}_{k-1}$ satisfies the following recurrence: ${\Delta}_{-1} = 1$, ${\Delta}_0 = (x+d_0)$${\Delta}_{k-1} = (x+d_{k-1}){\Delta}_{k-2} + b_{k-2} a_{k-1} {\Delta}_{k-3}$

Figures (10)

  • Figure 1: Transition diagram of CA $(150, 90, 90, 90)$
  • Figure 2: Transition diagram of CA $(150, 150, 90, 150)$
  • Figure 3: Block diagram of null, periodic and intermediate boundary CAs
  • Figure 4: Sub-rule vector related by the concatenation relation
  • Figure 5: Space-time diagram of $CA(90')$ and $CA(150')$. Figures (a) and (b) for $CA(90')$ where the evolutions have started from configuration $p^0(0^{n-1}1)$ and figures (c) and (d) for $CA(150')$ where evolutions have started from $p^{n-1}(10^{n-1})$. Here, white is for state $0$ and black is for state $1$.
  • ...and 5 more figures

Theorems & Definitions (57)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • Example 1
  • Definition 8
  • ...and 47 more