Non-Spatial Hash Chemistry as a Minimalistic Open-Ended Evolutionary System

TL;DR

This work addresses open-ended evolution by replacing the spatial Hash Chemistry with a non-spatial, multisets-based, well-mixed system in which replication results from pairwise matches and fitness is computed by a hash function: $f = (h \mod m)/m$, using $m = 100000$ in the original framework and $m = 100000000$ in the non-spatial variant. The proposed approach yields a substantial computational speed-up (approximately 2.25x) and leads to stronger unbounded growth of higher-order entities, demonstrating that open-endedness can arise without spatial structure. However, the non-spatial model exhibits reduced diversity and loses certain context-dependent and multiscale adaptation features inherent to the spatial version, highlighting the trade-offs between exploration efficiency and ecological richness. Overall, the study provides a minimalistic, scalable platform to study open-ended evolution and informs how spatial vs. non-spatial interactions shape the growth of complexity, with implications for efficiently exploring large possibility spaces; future work could reintroduce ecological interactions to recover diversity and context dependence while maintaining computational gains.

Abstract

There is an increasing level of interest in open-endedness in the recent literature of Artificial Life and Artificial Intelligence. We previously proposed the cardinality leap of possibility spaces as a promising mechanism to facilitate open-endedness in artificial evolutionary systems, and demonstrated its effectiveness using Hash Chemistry, an artificial chemistry model that used a hash function as a universal fitness evaluator. However, the spatial nature of Hash Chemistry came with extensive computational costs involved in its simulation, and the particle density limit imposed to prevent explosion of computational costs prevented unbounded growth in complexity of higher-order entities. To address these limitations, here we propose a simpler non-spatial variant of Hash Chemistry in which spatial proximity of particles are represented explicitly in the form of multisets. This model modification achieved a significant reduction of computational costs in simulating the model. Results of numerical simulations showed much more significant unbounded growth in both maximal and average sizes of replicating higher-order entities than the original model, demonstrating the effectiveness of this non-spatial model as a minimalistic example of open-ended evolutionary systems.

Figures (6)

• Figure 1: Schematic illustration of the outline of the proposed non-spatial Hash Chemistry model. See the main text for details.
• Figure 2: Box-whisker plots comparing the distributions of computational time needed to complete one simulation run for 2,000 iterations between the original Hash Chemistry model sayama2019 (left) and the non-spatial version proposed in this paper (right). 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 original model often exhibited extinction of particles in the very early stage of simulation, and hence those trivial extinction cases were excluded from the distribution of the original model. No such extinction occurred in the non-spatial model. Overall, the non-spatial model achieved a 69.01 / 30.73 = 2.25 times speed-up compared to the original model.
• Figure 3: Fitness values of successfully replicated multisets (higher-order entities) in simulations of non-spatial Hash Chemistry. Top: Maximum fitness value observed in each time step. Bottom: Average fitness value in each time step. In each plot, the red thin curves show results of 100 independent simulation runs, while the black solid curve shows their average. The time is in log scale to show long-term trends more 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. In both plots, the fitness increased rapidly right before $t=50$ because the population size approached the environment's carrying capacity and the selection pressure started to kick in by then. The maximum fitness and the average fitness behaved very similarly because the population was usually dominated by a single fittest multiset type most of the time. It is also noticeable that there was a large variation among the independent simulation runs (red thin curves) because the system would easily get trapped in local fitness optima for a substantially long period of time before evolution would discover a fitter multiset.
• Figure 4: Number of individual entities involved in replications of multisets in each time step. The red curves show results of 100 independent simulation runs, while the black solid curve shows their average. The time is in log scale to show long-term trends more clearly.
• Figure 5: Maximum (top) and average (bottom) numbers of individual entities in replicating multisets. The red curves show results of 100 independent simulation runs, while the black solid curve shows their average. The time is in log scale to show long-term trends more 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--2,000. In both plots, the unbounded growth model (blue curve) was a significantly better fit. See Table \ref{['tab:curvefits']} for more details.