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Self-Organization in Computation & Chemistry: Return to AlChemy

Cole Mathis, Devansh Patel, Westley Weimer, Stephanie Forrest

TL;DR

Fontana & Buss's AlChemy investigates how complex, self-organized structures can arise from random λ-expression interactions, treating computation as a constructive chemistry. The paper revisits the original model on modern hardware, reproduces key results, and uncovers robustness and fragility in emergent organizations, while revealing a formal link to chemical reaction networks via typed $\lambda$ calculus. It shows that stable, diverse organizations can persist despite perturbations, yet higher-order mergers of organizations are rare, and that the outcome is highly sensitive to how random expressions are generated and standardized. A constructive proof connects CRNs and typed $\lambda$ expressions, arguing that AlChemy can simulate any CRN state transition, with broad implications for origin-of-life studies, artificial life, and future programming language design.

Abstract

How do complex adaptive systems, such as life, emerge from simple constituent parts? In the 1990s Walter Fontana and Leo Buss proposed a novel modeling approach to this question, based on a formal model of computation known as $λ$ calculus. The model demonstrated how simple rules, embedded in a combinatorially large space of possibilities, could yield complex, dynamically stable organizations, reminiscent of biochemical reaction networks. Here, we revisit this classic model, called AlChemy, which has been understudied over the past thirty years. We reproduce the original results and study the robustness of those results using the greater computing resources available today. Our analysis reveals several unanticipated features of the system, demonstrating a surprising mix of dynamical robustness and fragility. Specifically, we find that complex, stable organizations emerge more frequently than previously expected, that these organizations are robust against collapse into trivial fixed-points, but that these stable organizations cannot be easily combined into higher order entities. We also study the role played by the random generators used in the model, characterizing the initial distribution of objects produced by two random expression generators, and their consequences on the results. Finally, we provide a constructive proof that shows how an extension of the model, based on typed $λ$ calculus, could simulate transitions between arbitrary states in any possible chemical reaction network, thus indicating a concrete connection between AlChemy and chemical reaction networks. We conclude with a discussion of possible applications of AlChemy to self-organization in modern programming languages and quantitative approaches to the origin of life.

Self-Organization in Computation & Chemistry: Return to AlChemy

TL;DR

Fontana & Buss's AlChemy investigates how complex, self-organized structures can arise from random λ-expression interactions, treating computation as a constructive chemistry. The paper revisits the original model on modern hardware, reproduces key results, and uncovers robustness and fragility in emergent organizations, while revealing a formal link to chemical reaction networks via typed calculus. It shows that stable, diverse organizations can persist despite perturbations, yet higher-order mergers of organizations are rare, and that the outcome is highly sensitive to how random expressions are generated and standardized. A constructive proof connects CRNs and typed expressions, arguing that AlChemy can simulate any CRN state transition, with broad implications for origin-of-life studies, artificial life, and future programming language design.

Abstract

How do complex adaptive systems, such as life, emerge from simple constituent parts? In the 1990s Walter Fontana and Leo Buss proposed a novel modeling approach to this question, based on a formal model of computation known as calculus. The model demonstrated how simple rules, embedded in a combinatorially large space of possibilities, could yield complex, dynamically stable organizations, reminiscent of biochemical reaction networks. Here, we revisit this classic model, called AlChemy, which has been understudied over the past thirty years. We reproduce the original results and study the robustness of those results using the greater computing resources available today. Our analysis reveals several unanticipated features of the system, demonstrating a surprising mix of dynamical robustness and fragility. Specifically, we find that complex, stable organizations emerge more frequently than previously expected, that these organizations are robust against collapse into trivial fixed-points, but that these stable organizations cannot be easily combined into higher order entities. We also study the role played by the random generators used in the model, characterizing the initial distribution of objects produced by two random expression generators, and their consequences on the results. Finally, we provide a constructive proof that shows how an extension of the model, based on typed calculus, could simulate transitions between arbitrary states in any possible chemical reaction network, thus indicating a concrete connection between AlChemy and chemical reaction networks. We conclude with a discussion of possible applications of AlChemy to self-organization in modern programming languages and quantitative approaches to the origin of life.
Paper Structure (19 sections, 1 theorem, 3 equations, 6 figures)

This paper contains 19 sections, 1 theorem, 3 equations, 6 figures.

Table of Contents

  1. Introduction
  2. AlChemy
  3. The $\lambda$ calculus
  4. $\lambda$ expressions in AlChemy
  5. Prior Work
  6. Results from the original AlChemy papers
  7. Experimental Results
  8. L0 Organizations
  9. L1 Organizations
  10. L2 Organizations
  11. Generating Random $\lambda$ Expressions
  12. Standardization
  13. Dynamical Consequences
  14. $\lambda$ Expressions Simulate Chemical Reaction Networks (CRNs)
  15. Discussion
  16. ...and 4 more sections

Key Result

Theorem 1

For all CRNs $C = (R, S)$, there exists a mapping $\mathsf{A}: C \to \rho$ such that if $\sigma(t) \to_R^\star \sigma(t+k)$, then there exists some sequence of collisions $\beta^\star$ such that $\rho_1 = \mathsf{A}(\sigma(t)) \to_\beta^\star \mathsf{A}(\sigma(t+k)) = \rho_2$.

Figures (6)

  • Figure 1: Example AlChemy simulation and the organization it produced. (A) The four $\lambda$ expressions remaining at the end of a single simulation run, which constitute the L1 organization. (B) Timeseries of the simulated run where the vertical axis represents the fraction of the population occupied by any given expression. Each expression is assigned a grey-scale value, except the four winners shown in (A). After a few hundred collisions all four expressions in the final organization had emerged, and by $\approx$ 35k collisions they were the only expressions remaining in the simulation. (C) The reaction rules that correspond to the final organization, showing that it is closed under interaction between any two expressions. (D) Network representation of the reaction rules from (C).
  • Figure 2: L0 Simulations. (A) Time series from two different representative simulation runs (red and blue), showing the number of unique expressions through the runs. In both cases the maximum number of objects was set to 1000 and a total of $6\times10^6$ collisions were performed. Every $10^6$ collisions (grey dashed lines) we introduced a perturbation by adding 100 random expressions to the simulation. (B) The distribution of unique expressions across 1000 different simulations after the fifth perturbation (median across the previous $10^6$ timesteps of the simulation), inset shows the same data on log-log scales. The number of expressions at steady state seems to vary across orders of magnitude and is heterogeneously distributed. (C) The organizations stabilize with fixed macroscopic parameters, even though the expressions themselves are continually changing, the average expression length and population entropy (a measure of diversity) are shown for different simulations. Each color is a different simulation with the identical run parameters, each point represents a snapshot of the simulation at different time points. (D) Survival Rate over time of different organizations when they are perturbed by replacing $p$% of the expressions with the identity function. Different colors correspond to different values of $p$.
  • Figure 3: L1 Simulations and Similarity over time. (A) Abundance Distributions illustrating pairs of similar organizations. Each plot shows two different pairs (Blue and Red); each bar corresponds to one expression that is a component of each member of the pair; the ordering of the expressions is determined by the sum of their abundance in both members of the pair. The vertical axis indicates the expression copy number with the red values inverted to facilitate comparison. High symmetry between blue and red indicates similar copy number, low symmetry indicates different copy numbers between the organizations. (B) Stability analysis of four L1 organizations that emerged in simulation. Each organization was initialized with a modification, either "perturb" (10 random expressions added), or "seed" (different random number seed only). The similarity between the modified and original simulations is shown in solid orange lines. The similarity of the original simulation to itself at a previous time-step (500 collisions earlier) is used as a control, and it is shown in the purple dashed lines. Each panel shows a different input organization, selected to highlight the diversity of possible outcomes.
  • Figure 4: L2 Simulations. Composite organizations were formed by combining two different L1 organizations ($L$ and $R$) into the same simulation and running the combination for $10^6$ collisions. (A) time-series of the similarity of the composite to the inputs; color corresponds to different choices of $L$ and $R$. (B) three possible outcomes: (i) Mutual Destruction (the composite retains almost no similarity to either input); (ii) Dominance (one input eventually dominates the other, and the final organization reverts to the dominant input); (iii) Coexistence (the final organization retains significant similarity to both inputs). The relative frequency of these outcomes is summarized in the table.
  • Figure 5: (A) Representative syntax trees for $\lambda$ expressions generated by the two generators. Left (red): Two example trees generated by the permutation method. These trees are uniform in the ratio of abstractions to applications, and they are parameterized only by the size of the tree. Right (blue): Five trees generated using Alchemy's original generator. These trees tend to have long chains of abstractions, seen here as long branches. Thus, their corresponding expressions vary greatly in complexity, and simple expressions (such as one or two-abstraction expressions) are common. (B) Statistical properties of the binary tree representation of the $\lambda$ expressions. (C) The dynamical consequences of these different generators are dramatic; in the permutation method (left) the system collapses to an inert, trivial fixed point, while the original generator (right) produces a diversity of complex organizations.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1: Simulation
  • proof