Exploring Periodic Boundary Cellular Automaton Rules for One and Two Fixed Points

Baisakhi Das
Institute of Engineering and Management
University of Engineering and Management
Kolkata, West Bengal, India
Mamata Dalui
Mousumi Saha
National Institute of Technology
Durgapur, West Bengal, India
Kasturi Ghosh
University Institute of Technology
Burdwan, West Bengal, India
Nilanjana Das
Biplab K. Sikdar
Indian Institute of Engineering Science and Technology
Shibpur, West Bengal, India

Abstract

This paper characterizes cellular automaton (CA) rules with a periodic boundary. We find the rules that have the potential to form single length cycle attractors (fixed points) in the CA state space. The characterization enables synthesizing periodic boundary cellular automata (PBCAs), for arbitrary length, that only have fixed points. A tool referred to as the NSRT diagram (NSRTD) has been developed for PBCAs to provide the basis for such characterization and synthesis. The NSRTD identifies rules $R_{S} / R_{T}$ that form a uniform single length cycle single-attractor CA (SACA) (with one fixed point) or a single length cycle two-attractor CA (TACA) (with two fixed points) for any length. The NSRTD is also capable of determining the presence of multi- length attractor basins in a CA state space. The framework is further employed to develop the schemes for exploring rules $R_{h}$ that enable splitting and merging fixed-point attractor basins (required for devising different CA-based solutions) in a uniform SACA or TACA when hybridized by $R_{h}$ .

Keywords: periodic boundary cellular automata; attractor; fixed point; NSRT diagram; SACA; TACA

Cite this publication as:
B. Das, M. Dalui, M. Saha, K. Ghosh, N. Das and B. K. Sikdar, “Exploring Periodic Boundary Cellular Automaton Rules for One and Two Fixed Points,” Complex Systems, 33(1), 2024 pp. 31–60.
https://doi.org/10.25088/ComplexSystems.33.1.31