New classes of reversible cellular automata

TL;DR

This paper introduces new classes of reversible cellular automata by constructing novel proper liftings of Boolean functions, expanding the known families for arbitrary large k and exploring their completeness for small k.

Contribution

The paper presents new families of proper liftings for reversible cellular automata applicable to large k, advancing understanding of their classification.

Findings

Constructed new families of proper liftings for arbitrary large k.

Discussed the completeness of known liftings for k ≤ 6.

Expanded the set of known reversible cellular automata rules.

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 $k\leq 6$.