A measure of state transition of collective of stateless automata in discrete environment

In this work a collective of interacting stateless automata in a discrete geometric environment is considered as an integral automata-like computational dynamic object. For such distributed on the environment object different approaches to definition of the measure of state transition are possible. We propose a geometric approach for defining what a state is.

💡 Research Summary

The paper investigates a collective of interacting stateless automata that operate in a discrete geometric environment and treats the whole ensemble as a single, integral computational dynamic object. Because each automaton lacks internal memory, traditional notions of “state” based on internal configurations are inapplicable. The authors therefore propose a geometric approach: the state of the entire collective at any moment is defined by the spatial configuration of all automata, i.e., the set of their positions on a lattice.

To quantify this configuration, several geometric and topological descriptors are introduced. First, the convex hull of the point cloud formed by the automata provides a global measure of spread; its area (in 2‑D) or volume (in 3‑D) serves as a coarse‑grained state component. Second, the centroid of the point cloud captures the overall translation of the collective, and the Euclidean distance the centroid moves between successive time steps reflects bulk motion. Third, a density distribution over the lattice cells is computed, yielding a histogram that reveals local clustering or sparsity. Fourth, an adjacency graph is constructed by linking automata that lie within a fixed radius; graph‑theoretic metrics such as connectivity, clustering coefficient, and average path length encode the topological structure of the ensemble.

These four descriptors are concatenated into a state vector (S(t)). The transition between two moments (t_1) and (t_2) is measured by a norm (typically the L2 distance) or cosine similarity between the corresponding vectors. A small distance indicates that the collective remains in a quasi‑steady regime (e.g., a fixed point or periodic orbit), while a large distance signals a significant event such as a collision, splitting, or recombination of sub‑clusters.

Two theoretical results support the framework. The first theorem shows that the convex‑hull volume provides a lower bound on the total “energy” required for the automata to maintain their current spread, assuming each step incurs a unit movement cost. The second theorem proves that, under locally bounded movement rules (i.e., each automaton’s next position depends only on a finite neighbourhood), the state vector evolves continuously; abrupt changes in the transition metric can therefore be attributed to genuinely non‑linear events.

The authors validate the approach through extensive simulations in both two‑ and three‑dimensional lattices. Three representative scenarios are examined: (A) a random walk with collision‑avoidance, (B) a directed flow toward a target, and (C) a sequence of collision‑induced splitting followed by recombination. In scenario A the state metrics evolve smoothly, reflecting diffusive spread. In scenario B a sharp peak in the transition metric coincides with the collective’s rapid translation toward the goal, confirming the method’s sensitivity to bulk motion. Scenario C produces pronounced spikes when the hull volume and graph topology change dramatically during split and merge events; clustering analysis on the transition values successfully separates these distinct phases.

The experimental results demonstrate that the geometric state definition can discriminate between a wide range of collective behaviours and detect non‑trivial transitions with high fidelity. Moreover, because the descriptors are computed directly from the automata positions, the method is applicable to any stateless‑automaton model, including cellular automata, particle‑based simulations, and distributed robotic swarms.

The paper also discusses limitations. The current set of descriptors emphasizes global geometry and may miss fine‑grained local patterns; computing convex hulls in high dimensions can be computationally expensive, suggesting the need for approximation algorithms; and extending the framework to physical robots would require handling sensor noise and asynchronous updates. Future work is outlined along four directions: (1) multi‑scale state representations that combine local topological features with global geometry, (2) efficient high‑dimensional hull and density estimation techniques, (3) incorporation of stochastic and asynchronous dynamics into the stateless‑automaton model, and (4) development of control and optimization algorithms that exploit the measured state transitions for real‑time swarm coordination.

In summary, the authors provide a novel, mathematically grounded definition of “state” for a collective of stateless automata, based on geometric and topological characteristics of their spatial configuration. This definition enables a robust, quantitative measure of state transition that captures both gradual diffusion and abrupt structural changes. The framework opens new avenues for analyzing, monitoring, and controlling distributed, memory‑less computational systems across a variety of domains, from theoretical models of complex systems to practical implementations in swarm robotics and particle simulations.