Convergence of Asynchronous Cellular Automata (Under Null Boundary Condition) and Their Application in Pattern Classification

Abstract

This paper studies the convergence of one-dimensional 3-neighborhood two-state asynchronous cellular automata (ACA), and utilizes the convergence property of ACA in designing pattern classifier. Here, we consider the cellular automata as fully asynchronous and the boundary condition of ACA as null. As a result of our current study, a directed graph, named fixed-point graph (FPG) is proposed to identify the fixed-point attractors in an automaton. A theorem is reported to characterize ACA having fixed-points in their state space. 184 (out of 256) ACA are identified with fixed-point attractors utilizing the theorem. All the ACA with fixed-points, however, may not converge always to fixed-point attractors. We report another theorem to understand the convergence of ACA. Following this theorem, 137 (out of 184) ACA are identified, which always converge to some fixed-point attractors. Then, we exploit the convergence property of ACA in designing pattern classifier. However, ACA with multiple fixed-point attractors can only be the candidate in the said design. Exploring the FPG, we further identify 74 ACA (out of 137) having at least two fixed-points under null boundary condition, which are utilized in design of an efficient 2-class pattern classifier. Finally, the designed classifier is tested with real-life data sets, and the performance of the classifier is compared with other well-known classifiers. It is observed that the proposed classifier performs better than many other well-known classifiers, and also better than synchronous Cellular automata (CA) based classifier.