Structural Cellular Hash Chemistry

TL;DR

This work addresses the challenge of realizing open-ended evolution with both multiscale spatial interactions and unbounded complexity in a computationally efficient framework. It introduces Structural Cellular Hash Chemistry (SCHC), where evolving higher-order entities are represented as connected components of the nearest-neighbor graph on a 2D grid, and evolve via pairwise competition using a hash-based fitness $(h(ls) mod M)/M$. Empirical results show SCHC achieves spontaneous movement, self-replication, and unbounded growth of pattern complexity, with visible spatial ecology and increased diversity, while offering substantial computational efficiency over prior Hash Chemistry variants. The study demonstrates that a simple, component-based spatial model can realize open-ended evolution across scales, with potential for GPU-accelerated scaling and exploration of alternative fitness evaluators.

Abstract

Hash Chemistry, a minimalistic artificial chemistry model of open-ended evolution, has recently been extended to non-spatial and cellular versions. The non-spatial version successfully demonstrated continuous adaptation and unbounded growth of complexity of self-replicating entities, but it did not simulate multiscale ecological interactions among the entities. On the contrary, the cellular version explicitly represented multiscale spatial ecological interactions among evolving patterns, yet it failed to show meaningful adaptive evolution or complexity growth. It remains an open question whether it is possible to create a similar minimalistic evolutionary system that can exhibit all of those desired properties at once within a computationally efficient framework. Here we propose an improved version called Structural Cellular Hash Chemistry (SCHC). In SCHC, individual identities of evolving patterns are explicitly represented and processed as the connected components of the nearest neighbor graph of active cells. The neighborhood connections are established by connecting active cells with other active cells in their Moore neighborhoods in a 2D cellular grid. Evolutionary dynamics in SCHC are simulated via pairwise competitions of two randomly selected patterns, following the approach used in the non-spatial Hash Chemistry. SCHC's computational cost was significantly less than the original and non-spatial versions. Numerical simulations showed that these model modifications achieved spontaneous movement, self-replication and unbounded growth of complexity of spatial evolving patterns, which were clearly visible in space in a highly intuitive manner. Detailed analysis of simulation results showed that there were spatial ecological interactions among self-replicating patterns and their diversity was also substantially promoted in SCHC, neither of which was present in the non-spatial version.

Figures (6)

• Figure 1: A sample simulation run of Structural Cellular Hash Chemistry. Snapshots of system configurations are arranged temporally from left to right and then top to bottom ($t = 0, 250, 500, 750, 1000, 1250, 1500, 2000$). Colors represent different element types, and blank (white) spaces represent empty cells. It is observed in these visualizations that the self-replicating patterns gradually proliferate and evolved to larger forms with more complex nontrivial structures (also see Fig. \ref{['size-growth-example']}).
• Figure 2: Example of spontaneous growth of complexity of self-replicating patterns (extracted from the simulation run shown in Fig. \ref{['sample-run']}).
• Figure 3: Box-whisker plots comparing the distributions of computational time needed to complete one simulation run for 2000 iterations among the original Hash Chemistry sayama2019 (1st), the non-spatial version sayama2024a (2nd), the prototype version of Cellular Hash Chemistry sayama2024b (3rd), and the proposed Structural Cellular Hash Chemistry (4th). The vertical axis shows the length of simulation time in minutes on a Windows 11 (64-bit) desktop workstation with an Intel i9 CPU (10 cores) at 3.70 GHz with 64 GB RAM. The four conditions are all statistically very significantly different from each other (ANOVA; $p<10^{-72}$).
• Figure 4: Fitness values (i.e., return values of the hash function) of successfully replicated patterns in SCHC simulations. Top: Maximum fitness value observed in each time step. Bottom: Average fitness value in each time step. The red curves show results of 87 independent simulation runs, while the black solid curve shows their average. The time is in log scale to show long-term trends clearly. The fitness values are visualized using $-\log_{10} | 1 - \mathrm{fitness} |$ to visualize increasingly finer improvement of the fitness that progresses over the course of simulation. The maximum fitness (top) showed a continuous increase over time, whereas the average fitness (bottom) showed an initial increase followed by a slight decrease in the long run, which was not observed in the non-spatial model sayama2024a and indicates the presence of nontrivial spatial ecological interactions among patterns. See text for more details.
• Figure 5: Maximum (top) and average (bottom) numbers of active cells in replicating patterns. The red curves show results of 87 independent simulation runs, while the black solid curve shows their average. The time is in log scale to show long-term trends clearly. Purple (dashed) and blue (solid) curves are two different growth models (purple: bounded growth, blue: unbounded growth) fitted to the average behaviors during the time period 100--2000. In both plots, the unbounded growth model (blue curve) was a significantly better fit. See Table \ref{['tab:curvefits']} for more details.