Baby C. J. | Searcharxiv

arXiv2026

Algebraic Characterization of Reversible First Degree Cellular Automata over $\mathbb{Z}_d$

There exists algorithms to detect reversibility of cellular automaton (CA) for both finite and infinite lattices taking quadratic time. But, can we identify a $d$-state CA rule in constant time that is always reversible for every lattice size $n\in \mathbb{N}$? To address this issue, this paper explores the reversibility properties of a subset of one-dimensional, $3$-neighborhood, $d$-state finite cellular automata (CAs), known as the first degree cellular automata (FDCAs) for any number of cells $(n\in \mathbb{N})$ under the null boundary condition. {In a first degree cellular automaton (FDCA), the local rule is defined using eight parameters. To ensure that the global transition function of $d$-state FDCA is reversible for any number of cells $(n\in \mathbb{N})$, it is necessary and sufficient to verify only three algebraic conditions among the parameter values. Based on these conditions, for any given $d$, one can synthesize all reversible FDCAs rules. Similarly, given a FDCA rule, one can check these conditions to decide its reversibility in constant time.

cs.FL↗

arXiv2024

Hierarchical Clustering using Reversible Binary Cellular Automata for High-Dimensional Data

This work proposes a hierarchical clustering algorithm for high-dimensional datasets using the cyclic space of reversible finite cellular automata. In cellular automaton (CA) based clustering, if two objects belong to the same cycle, they are closely related and considered as part of the same cluster. However, if a high-dimensional dataset is clustered using the cycles of one CA, closely related objects may belong to different cycles. This paper identifies the relationship between objects in two different cycles based on the median of all elements in each cycle so that they can be grouped in the next stage. Further, to minimize the number of intermediate clusters which in turn reduces the computational cost, a rule selection strategy is taken to find the best rules based on information propagation and cycle structure. After encoding the dataset using frequency-based encoding such that the consecutive data elements maintain a minimum hamming distance in encoded form, our proposed clustering algorithm iterates over three stages to finally cluster the data elements into the desired number of clusters given by user. This algorithm can be applied to various fields, including healthcare, sports, chemical research, agriculture, etc. When verified over standard benchmark datasets with various performance metrics, our algorithm is at par with the existing algorithms with quadratic time complexity.

cs.FL↗