Nonsynchronous updating in the multiverse of cellular automata

In this paper we study updating effects on cellular automata rule space. We consider a subset of 6144 order-3 automata from the space of 262144 bidimensional outer-totalistic rules. We compare synchronous to asynchronous and sequential updatings. Focusing on two automata, we discuss how update changes destroy typical structures of these rules. Besides, we show that the first-order phase transition in the multiverse of synchronous cellular automata, revealed with the use of a recently introduced control parameter, seems to be robust not only to changes in update schema but also to different initial densities.

Figure 1

Snapshot at steady state of Life with synchronous 1, asynchronous 1, and sequential 1 updates in a 2D lattice with $L_m\times L_n=100\times 50$. The system evolved to steady state from a randomly initial condition with $\rho \left(0\right)=0.50$.

Figure 2

Relation between ${\rho }_{2\text{D}}/{\rho }^{*}$ and $\sigma$. Results were obtained for $L=100$ and averaged over 30 runs. We chose $\rho \left(0\right)={\rho }^{*}$ to see the effect of $\sigma$ on competition of alive and dead cells in 2D with synchronous update in panel 2, with asynchronous update in panel 2, and sequential update in panel 2. The Life $\sigma =1.006$ and its position is marked with a large gray dot in each panel (pointed by the gray arrow).

Figure 3

Relation between ${\rho }_{2\text{D}}/{\rho }^{*}$ and $\sigma$. Results were obtained for $L=100$ and averaged over 30 runs. We chose $\rho \left(0\right)=0+\Delta \rho$, where $\Delta \rho =0.05$, to see the effect of $\sigma$ on competition of alive and dead cells in 2D with synchronous update in panel 3, with asynchronous update in panel 3, and sequential update in panel 3. The Life $\sigma =1.006$ and its position is marked with a large gray dot in each panel (pointed by the gray arrow).

Figure 4

Snapshot at steady state of rule B4678S2345 with synchronous 4, asynchronous 4, and sequential 4 updates in a 2D lattice with $L_m\times L_n=100\times 50$. The system evolved to steady state from a randomly initial condition with $\rho \left(0\right)={\rho }^{*}$. Figures 4–4 show steady states for the three schemas when $\rho \left(0\right)=0+\Delta \rho$, where $\Delta \rho =0.05$. Figures 4–4 show steady states for updates synchronous, asynchronous, and sequential, respectively, when $\rho \left(0\right)=0.20$.