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Algebraic Characterization of Reversible First Degree Cellular Automata over $\mathbb{Z}_d$

Baby C. J., Kamalika Bhattacharjee

Abstract

There exists algorithms to detect reversibility of cellular automaton (CA) for both finite and infinite lattices taking quadratic time. But, can we identify a $d$-state CA rule in constant time that is always reversible for every lattice size $n\in \mathbb{N}$? To address this issue, this paper explores the reversibility properties of a subset of one-dimensional, $3$-neighborhood, $d$-state finite cellular automata (CAs), known as the first degree cellular automata (FDCAs) for any number of cells $(n\in \mathbb{N})$ under the null boundary condition. {In a first degree cellular automaton (FDCA), the local rule is defined using eight parameters. To ensure that the global transition function of $d$-state FDCA is reversible for any number of cells $(n\in \mathbb{N})$, it is necessary and sufficient to verify only three algebraic conditions among the parameter values. Based on these conditions, for any given $d$, one can synthesize all reversible FDCAs rules. Similarly, given a FDCA rule, one can check these conditions to decide its reversibility in constant time.

Algebraic Characterization of Reversible First Degree Cellular Automata over $\mathbb{Z}_d$

Abstract

There exists algorithms to detect reversibility of cellular automaton (CA) for both finite and infinite lattices taking quadratic time. But, can we identify a -state CA rule in constant time that is always reversible for every lattice size ? To address this issue, this paper explores the reversibility properties of a subset of one-dimensional, -neighborhood, -state finite cellular automata (CAs), known as the first degree cellular automata (FDCAs) for any number of cells under the null boundary condition. {In a first degree cellular automaton (FDCA), the local rule is defined using eight parameters. To ensure that the global transition function of -state FDCA is reversible for any number of cells , it is necessary and sufficient to verify only three algebraic conditions among the parameter values. Based on these conditions, for any given , one can synthesize all reversible FDCAs rules. Similarly, given a FDCA rule, one can check these conditions to decide its reversibility in constant time.
Paper Structure (8 sections, 5 theorems, 4 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 5 theorems, 4 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. First Degree Cellular Automata
  3. Reversibility of First Degree CAs
  4. Synthesis of Reversible FDCAs
  5. Reversible Rules for $d-$state FDCA where $d$ is prime
  6. Reversible Rules for $d-$state FDCA where $d$ is composite
  7. Testing Reversibility of FDCAs
  8. Conclusion

Key Result

lemma 1

If an FDCA rule is reversible for all number of cells $n\in \mathbb{N}$, then $\gcd(c_5,d)=1$ where $c_5\ne0$ and $c_7\in \mathbb{Z}_d$.

Figures (3)

  • Figure 1: Transition diagram of the 4-cell 2-state FDCA with parameters $\langle0,0,0,0,1,1,0,1\rangle$
  • Figure 2: Transition diagram of FDCA rule $\langle0,0,0,0,2,3,0,4\rangle$ for the state $d=5$
  • Figure 3: Transition diagram of FDCA rule $\langle2,0,0,0,2,3,0,4\rangle$ for the state $d=5$

Theorems & Definitions (11)

  • definition 1
  • definition 2
  • Example 1
  • lemma 1
  • Example 2
  • lemma 2
  • Example 3
  • lemma 3
  • Example 4
  • theorem 1
  • ...and 1 more