Comparing 1D and 2D Real Time on Cellular Automata

TL;DR

This paper investigates whether 2D cellular automata recognizing real-time languages can be simulated by 1D automata, showing that 2D automata can simulate multiple 1D automata linearly, with potential polynomial bounds if classes are equal.

Contribution

It establishes a connection between 1D and 2D real-time cellular automata, demonstrating linear simulation capabilities and exploring conditions for polynomial simulation.

Findings

2D CA can linearly simulate multiple 1D CA in real time

If 1D and 2D classes are equal, simulation can be polynomial

Provides bounds on simulation complexity between dimensions

Abstract

We study the influence of the dimension of cellular automata (CA) for real time language recognition of one-dimensional languages with parallel input. Specifically, we focus on the question of determining whether every language that can be recognized in real time on a 2-dimensional CA working on the Moore neighborhood can also be recognized in real time by a 1-dimensional CA working on the standard two-way neighborhood. We show that 2-dimensional CA in real time can perform a linear number of simulations of a 1-dimensional real time CA. If the two classes are equal then the number of simulated instances can be polynomial.