Primordial Evolution in the Finitary Process Soup

A general and basic model of primordial evolution–a soup of reacting finitary and discrete processes–is employed to identify and analyze fundamental mechanisms that generate and maintain complex structures in prebiotic systems. The processes–$\epsilon$-machines as defined in computational mechanics–and their interaction networks both provide well defined notions of structure. This enables us to quantitatively demonstrate hierarchical self-organization in the soup in terms of complexity. We found that replicating processes evolve the strategy of successively building higher levels of organization by autocatalysis. Moreover, this is facilitated by local components that have low structural complexity, but high generality. In effect, the finitary process soup spontaneously evolves a selection pressure that favors such components. In light of the finitary process soup’s generality, these results suggest a fundamental law of hierarchical systems: global complexity requires local simplicity.

💡 Research Summary

The paper introduces a highly abstract yet mathematically rigorous model of pre‑biotic evolution called the “finitary process soup.” The elementary entities of the soup are ε‑machines, the canonical minimal predictive models defined in computational mechanics. An ε‑machine is a finite‑state stochastic transducer that maps input symbols to output symbols while updating an internal state; its structure is quantified by two well‑established complexity measures: statistical complexity (Cμ), which counts the number of causal states, and topological or predictive complexity (Cν), which reflects the richness of the input‑output mapping.

The authors construct a population of ε‑machines and let them interact according to a predefined set of binary reaction rules. A reaction takes two parent ε‑machines, combines their transition functions, and yields a child ε‑machine that may be a new type or a transformed version of one of the parents. This process mimics catalytic chemistry: low‑complexity machines act as “catalysts” that enable the formation of higher‑complexity machines without being consumed.

Through extensive Monte‑Carlo simulations the authors observe a robust emergence of hierarchical self‑organization. Initially the soup is dominated by low‑Cμ machines that possess high “generality” – they can combine with many different partners because their transition tables are sparse and thus flexible. These simple machines repeatedly generate intermediate‑complexity offspring, which in turn serve as substrates for further reactions. A characteristic pattern is an autocatalytic cascade: machine A + B → C, then C + A → D, and so on, producing ever higher levels of organization. Network‑theoretic diagnostics (modularity, clustering coefficient, average path length, and hierarchical depth) all increase sharply as the simulation proceeds, indicating the formation of distinct modules that are nested within larger modules.

A key insight is that the global complexity of the soup—measured as the sum of Cμ over all distinct ε‑machines—does not arise from the presence of intrinsically complex components. Instead, it is driven by the proliferation of simple, highly general components that act as reusable building blocks. The system therefore creates its own selection pressure: ε‑machines that are easy to replicate and that can catalyze many reactions are preferentially retained, while more complex, specialized machines are only maintained if they contribute to higher‑level autocatalytic loops. This dynamic yields a “law of hierarchical systems”: achieving high overall complexity requires low‑complexity local elements.

The authors also test the robustness of the phenomenon by varying the initial ε‑machine distribution, the density of reaction rules, and the size of the alphabet. Across all parameter regimes the same hierarchical autocatalytic behavior emerges, suggesting that the model captures a universal principle rather than an artifact of a particular setup.

In the discussion, the authors argue that the finitary process soup provides a unifying framework for studying a wide range of pre‑biotic scenarios, from RNA‑world chemistry to mineral surface catalysis, because ε‑machines abstract away chemical details while preserving essential information‑processing capabilities. Moreover, the identified law—global complexity built on local simplicity—offers a potential explanatory lens for other complex systems, such as biological regulatory networks, social organization, and engineered modular architectures.

Overall, the paper makes three major contributions: (1) it formalizes a minimal, computable model of primordial evolution using ε‑machines; (2) it demonstrates, through quantitative network analysis, that hierarchical self‑organization and autocatalysis naturally arise from simple interaction rules; and (3) it proposes a general principle that high‑level complexity is contingent on the presence of low‑complexity, highly general components. These results deepen our theoretical understanding of how life‑like organization could have emerged from non‑living matter and open new avenues for exploring self‑assembly in both natural and synthetic systems.