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 $\lambda$ 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 $\lambda$ calculus, \textcolor{black}{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.

翻译：复杂适应系统（如生命）如何从简单的组成部分中涌现？上世纪90年代，Walter Fontana与Leo Buss基于形式化计算模型——$\lambda$演算，提出了一种新颖的建模方法以探讨此问题。该模型展示了嵌入在组合可能性巨大空间中的简单规则如何能够产生复杂且动态稳定的组织结构，令人联想到生物化学反应网络。本文重新审视这一被称为AlChemy的经典模型——该模型在过去三十年间研究不足。我们复现了原始结果，并利用当今更强大的计算资源研究了这些结果的鲁棒性。分析揭示了该系统中若干未曾预料到的特性，展现出动态鲁棒性与脆弱性的惊人混合。具体而言，我们发现复杂稳定的组织结构出现频率高于先前预期，这些组织能够抵抗坍缩至平凡不动点，但难以组合为更高阶实体。我们还研究了模型中随机生成器的作用，刻画了两种随机表达式生成器所产生的对象初始分布及其对结果的影响。最后，我们通过构造性证明表明：基于类型化$\lambda$演算的模型扩展\textcolor{black}{能够模拟任意化学反应网络中所有可能状态间的转换，从而揭示了AlChemy与化学反应网络之间的具体联系}。文章最后讨论了AlChemy在现代编程语言自组织及生命起源定量研究中的潜在应用。