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New classes of reversible cellular automata

Jan Kristian Haugland, Tron Omland

TL;DR

New families of proper liftings for arbitrary large liftings for arbitrary large k are constructed and it is discussed whether all have been identified for $k\leq 6$.

Abstract

A Boolean function $f$ on $k$~bits induces a shift-invariant vectorial Boolean function $F$ from $n$ bits to $n$ bits for every $n\geq k$. If $F$ is bijective for every $n$, we say that $f$ is a proper lifting, and it is known that proper liftings are exactly those functions that arise as local rules of reversible cellular automata. We construct new families of such liftings for arbitrary large $k$ and discuss whether all have been identified for $k\leq 6$.

New classes of reversible cellular automata

TL;DR

New families of proper liftings for arbitrary large liftings for arbitrary large k are constructed and it is discussed whether all have been identified for .

Abstract

A Boolean function on ~bits induces a shift-invariant vectorial Boolean function from bits to bits for every . If is bijective for every , we say that is a proper lifting, and it is known that proper liftings are exactly those functions that arise as local rules of reversible cellular automata. We construct new families of such liftings for arbitrary large and discuss whether all have been identified for .

Paper Structure

This paper contains 4 sections, 7 theorems, 27 equations.

Table of Contents

  1. Proper liftings and reversible cellular automata
  2. Proper liftings of diameter 6
  3. Other constructions
  4. List of new liftings with the landscape notation

Key Result

Proposition 1.2

Let $f\colon\mathbb{F}_2^k\to\mathbb{F}_2$ be a Boolean function, $s$ any integer $\geq 2$, and define the function $f_s\colon\mathbb{F}_2^{(k-1)s+1}\to\mathbb{F}_2$ by If $f$ is a $(k, n)$-lifting for all $n \geq k$, then $f_s$ is a $((k-1)s+1, n)$-lifting for all $n \geq (k-1)s+1$.

Theorems & Definitions (24)

  • Remark 1.1
  • Proposition 1.2
  • proof
  • Example 1.3
  • Proposition 1.4
  • proof
  • Example 1.5
  • Remark 1.6
  • Corollary 1.7
  • proof
  • ...and 14 more