Reversibility Problem of Multidimensional Finite Cellular Automata

TL;DR

This paper introduces a new criterion and algorithm for testing the reversibility of multidimensional linear cellular automata under null boundary conditions, linking the problem to block Toeplitz matrices and improving computational efficiency.

Contribution

It provides the first practical criterion and algorithm for reversibility of multidimensional linear cellular automata with null boundary conditions, extending previous theoretical results.

Findings

The criterion reduces computational cost for large systems.

The method applies to both null and periodic boundary conditions.

Reversibility can be tested efficiently using block Toeplitz matrix properties.

Abstract

While the reversibility of multidimensional cellular automata is undecidable and there exists a criterion for determining if a multidimensional linear cellular automaton is reversible, there are only a few results about the reversibility problem of multidimensional linear cellular automata under boundary conditions. This work proposes a criterion for testing the reversibility of a multidimensional linear cellular automaton under null boundary condition and an algorithm for the computation of its reverse, if it exists. The investigation of the dynamical behavior of a multidimensional linear cellular automaton under null boundary condition is equivalent to elucidating the properties of block Toeplitz matrix. The proposed criterion significantly reduce the computational cost whenever the number of cells or the dimension is large; the discussion can also apply to cellular automata under periodic boundary condition with a minor modification.