Asynchronism in Cellular Automata

TL;DR

The work investigates how asynchronism alters cellular automata dynamics and demonstrates practical utility in clustering. It introduces Skewed Fully Asynchronous CA (SACA), in which two adjacent cells update together, and analyzes elementary CA behavior under this scheme relative to synchronous and fully asynchronous updates. It further develops a cycle-based clustering approach using reversible fully asynchronous CA, including encoding schemes and a principled selection of CA rules to obtain a target number of clusters, with results competitive to standard clustering methods. The study also derives theoretical conditions for convergence to all-0 or all-1 attractors under skewed updates and explores reversibility and convergence in α-ACA, highlighting rich dynamical phenomena and lattice-size dependencies. Overall, the work positions asynchronous CA as a versatile framework for modeling complex dynamics and solving clustering problems with potential impact in complex systems and data analysis.

Abstract

This study introduces Skewed Fully Asynchronous Cellular Automata (SACA), a novel update scheme in cellular automata that updates the states of only two consecutive and adjacent cells, such as ci and ci+1, simultaneously at each time step. The behavior and dynamics of elementary cellular automata (ECA) under this scheme are analyzed and compared with those of synchronous and fully asynchronous update methods. The comparative analysis highlights a range of phenomena, including transitions in ECAs from convergent or non-reversible dynamics to reversible, divergent behavior. The divisibility of lattice size by 2 or 4 is shown to have significant effects on the system dynamics, linked to the presence or absence of atomicity. The study also explores the convergence of ECAs to all-zero or all-one point attractors under SACA, providing theoretical insights that align with experimental findings. Additionally, the research investigates the application of fully asynchronous cellular automata in solving clustering problems. Clustering is defined as grouping objects with similar properties. The proposed method employs reversible asynchronous cellular automata to merge clusters iteratively based on their closeness, continuing until the desired number of clusters is achieved. This approach leverages a small set of rules, leading to faster convergence and efficiency in clustering tasks. The findings underscore the potential of asynchronous cellular automata as a versatile and effective framework for studying complex system dynamics and solving practical problems such as clustering.

Figures (25)

• Figure 1: neighborhood dependence in a one-dimensional cellular automaton with radious $r$.
• Figure 2: (a) Moore neighborhood dependencies for two-dimensional cellular automata; (b) John Von Numann dependencies for two-dimensional cellular automata.
• Figure 3: John Von Neumann's Self-reproducting automata banda2Neuma66
• Figure 4: Space-time diagram of a $1-$D Cellular Automaton of (Wolfram rule $30$). White cells denote state $0$, whereas black cells denote state $1$.
• Figure 5: Working of fully Asynchronous Cellular Automata under periodic boundary condition.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 4.1
• Theorem 4.1