Cellular Automata: Reversibility, Semi-reversibility and Randomness

TL;DR

This dissertation investigates the properties of 1D cellular automata, introducing new tools for reversibility analysis, classifying CAs into reversible, semi-reversible, and irreversible, and developing CA-based pseudo-random number generators with competitive performance.

Contribution

It introduces the reachability tree for efficient reversibility analysis, redefines reversibility with semi-reversible CAs, and proposes CA-based PRNGs with improved performance over existing generators.

Findings

Reachability tree effectively characterizes reversible CAs.

Semi-reversible CAs bridge finite and infinite CA reversibility.

CA-based PRNGs outperform SFMT19937-64 in tests.

Abstract

In this dissertation, we study two of the global properties of 1-dimensional cellular automata (CAs) under periodic boundary condition, namely, reversibility and randomness. To address reversibility of finite CAs, we develop a mathematical tool, named reachability tree, which can efficiently characterize those CAs. A decision algorithm is proposed using minimized reachability tree which takes a CA rule and size n as input and verifies whether the CA is reversible for that n. To decide reversibility of a finite CA, we need to know both the rule and the CA size. However, for infinite CAs, reversibility is decided based on the local rule only. Therefore, apparently, these two cases seem to be divergent. This dissertation targets to construct a bridge between these two cases. To do so, reversibility of CAs is redefined and the notion of semi-reversible CAs is introduced. Hence, we propose a new classification of finite CAs -(1) reversible CAs, (2) semi-reversible CAs and (3) strictly irreversible CAs. Finally, relation between reversibility of finite and infinite CAs is established. This dissertation also explores CAs as source of randomness and build pseudo-random number generators (PRNGs) based on CAs. We identify a list of properties for a CA to be a good source of randomness. Two heuristic algorithms are proposed to synthesize candidate (decimal) CAs which have great potentiality as PRNGs. Two schemes tare developed o use these CAs as window-based PRNGs - (1) as decimal number generators and as (2) binary number generators. We empirically observe that in comparison to the best PRNG SFMT19937-64, average performance of our proposed PRNGs are slightly better. Hence, our decimal CAs based PRNGs are one of the best PRNGs today.