A Symmetric Identity-Rule-Variation-Based Method For Enumerating And Building Radius-1 Two-State Ncca Rules

The class of cellular automata that preserve quantities, referred to as Number-Conserving Cellular Automata (NCCA), serves as a crucial tool for modeling various complex systems that exhibit the preservation of specific physical properties. In this paper, we first present some well-known necessary and/or sufficient conditions that must satisfy any NCCA rule. These conditions can be used to find NCCA rules using a brute-force method. However, the process of examining the set of all rules becomes impractical for complex cases with larger neighborhoods, dimensions, or number of CA states. To address this challenge, we propose a new approach to constructing and writing radius-1 two-state NCCA rules. The main idea of our contribution is the use of symmetric variations injected into the CA identity rule, which allows us to efficiently find and write NCCA rules. The proposed method has successfully reproduced the well-known 1D- and 2D-NCCA with the von Neumann neighborhood. Moreover, it has also been able to give the codes of the seventeen 2D-NCCA conservative rules with the Moore neighborhood. We believe that our approach could be generalized for higher dimensions and larger neighborhood radius.