This paper investigates the structural properties of two-dimensional cellular automata (2DCAs) over rings 12 to 19, focusing on rule composition, subring hierarchy and linear evolution. We prove that rule composition is commutative across all rings and that matrix transformations preserve hierarchical relationships. The subring containment is determined by the divisors of n: prime rings have only trivial subrings, while composite rings exhibit structured hierarchies. Let I E 1knxn be a configuration matrix. For rings in 61 = {12, 14, 18}, repeated elementwise addition modulo k maps the entries of I to their immediate subrings in the containment hierarchy. For 62 = {13, 19}, triple summation modulo k similarly restricts the values to their corresponding subrings. Furthermore, we extend Moore neighborhood-based two-dimensional cellular automaton (2DCA) rules to rings from 12 to 15 and 17 to 19, proving that at time step t, rule matrices generate multiple non-overlapping replicas of the initial configuration across each ring and its subrings. Experiments in C++ across various rings and image sizes revealed two key patterns requiring further mathematical explanation. Rings with only trivial subrings-12, 13, 15 and 17-replicate the initial image at time steps t = pk, where p E {2, 3, 5, 7} and k >= 0. In contrast, rings with nontrivial subrings-14 and 18-generate multiple replicas at t = 2k, while 19 exhibits replication at t = 3k, producing both the original image and its subrings. The ring 16, with disjoint sub-rings and no clear hierarchy, shows no such structured replication.
The Collatz rule is one of the earliest examples of a simple, deterministic system that produces chaotic behavior. The rule takes any odd positive integer n to 3n+1 and any even positive integer n to n/2. Iterating this rule yields complex sequences whose dynamics are poorly understood; for example, it is unknown whether all such sequences reach 1 (the Collatz conjecture). It is reasonable to suspect that this complexity derives from the interplay of multiplication (3n) and addition (+1). However, in 2002, Monks was able to drop the +1 by constructing a 1 021 020-condition rule that simulates the Collatz rule using only multiplication. Monks’s rule greatly simplifies the Collatz dynamics but at the cost of an enormous rule. The current paper achieves the goal of removing addition with a significantly smaller 30-condition rule. We show how this rule replicates the Collatz process, and we place conditions on any cyclic trajectory purporting to be a counterexample to the Collatz conjecture.
We investigate the relationship between totalistic cellular automata (TCAs) and self-referential statements. It is well known that cellular automata (CAs) give rise to many undecidability issues related to self-referential statements. The study of self-referential statements and CAs can be traced back to the work of M. Hsiung [1], who established the connection between elementary cellular automata (ECAs) and self-referential paradoxes in terms of their evolution processes. We further deepen the connection between CAs and self-referential sentences. Specifically, we demonstrate the relationship is not only evident in one-dimensional ECAs but also extends to more complex two-dimensional TCAs. We elaborate on commonly applied TCAs, including Moore and von Neumann types. By studying their connection with self-referential sentences, we propose an algorithm for determining the fixed points of these CAs. Then, we classify them based on the (in)stability characteristics observed in their evolutionary processes. Additionally, we discuss certain specific self-referential paradoxes induced by these automata. Finally, we present a general result between TCAs and self-referential sentences.
This paper investigates phase transitions in complexity within mobile automata governed by non-local rules. Unlike traditional cellular automata, mobile automata involve a single active cell navigating and updating a one-dimensional array of binary-state cells based on a rule set. By varying the number and symmetry of dependent cells in non-local rules, we observe abrupt changes in system behavior that we identify as computational phase transitions. Using active cell growth and three complementary metrics (Shannon block entropy and estimates of Kolmogorov-Chaitin complexity based on block decomposition and lossless compression), we quantitatively analyze the complexity of automata across a range of dependent cell configurations. Our results reveal that certain increases in non-locality trigger dramatic shifts in entropy and compressibility, while other expansions produce negligible or even simplifying effects. We categorize the observed transitions into categories based on their entropy and growth patterns and demonstrate that complexity does not scale linearly with non-locality. This paper provides a formal foundation for understanding structural complexity in mobile automata and contributes to the broader theory of emergent computation in simple rule-based systems.
In an era increasingly defined by uncertainty, interconnectedness and systemic transformation, complexity science has emerged not merely as a theoretical framework, but as an essential lens for making sense of the world. From network dynamics and artificial intelligence to climate tipping points and cultural epistemologies, the field of complex systems continues to expand its relevance and application. This special issue brings together six diverse yet interwoven contributions that collectively explore how intelligent infrastructures and emergent systems are reshaping our understanding of resilience, learning and adaptation in the twenty-first century.
