Dynamics of Non-convergent Skewed Elementary Cellular Automata

Abstract

This study explores the notion of skewed asynchronous cellular automata (ACA) which violates the atomicity property of fully ACA. In an early work, Roy et al. [18] have identified the convergent elementary cellular automata (ECA) rules under skewed update. In this direction, this study explores the remaining 28 non-convergent skewed ECA rules. To understand the dynamics of these rules, this study considers space-time pictures, communication class structure and empirical tests. First, we classify the non-convergent skewed systems following space-time dynamics. Hereafter, we record the number of recurrent configurations and transient configurations for these rules following finite size experiments which identify the reversible skewed systems. Moreover, the communication class dynamics of these rules also show connection with different non-trivial OEIS sequences (like, A001644, A001608, A001639 etc.). According to the results, ECA 90 acts as a good randomness enhancer following the Dieharder tests. Therefore, we theoretically analyse the communication class of ECA 90 which shows recurrent or reversible phenomenon under skewed update. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.